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# Acknowledgements iii Acknowledgements

A Critical Reexamination of Some Assumptions and Implications of Cable Theory in Neurobiology
Thesis by
Gary R. Holt

In Partial Ful llment of the Requirements for the Degree of Doctor of Philosophy

California Institute of Technology Pasadena, California 1998 (Submitted 12 December, 1997)

ii

Acknowledgements

iii

Acknowledgements
I thank God for helping me through the last n + 1 years, and for giving hope through his precious promises. I also want to thank those whom God has brought into my life: | Christof Koch, my advisor, for support and advice on research and academic life; | Rodney Douglas, for warmly inviting me to his lab, and for contributing lots of ideas; | Bashir Ahmed, Neil Berman, and Kevan Martin for some of the intracellular cortical data presented in various plots in chapter 5 and appendix B; | The Sloan Foundation and the O ce of Naval Research for nancial support; | David Kewley for helpful discussions about the extracellular elds of cortical neurons; | Yong-Nam Jun for a few references when I began the analysis of ephaptic interaction; | Mike Holst for donating his Poisson equation solver to me, and for taking time to explain it; | Various people on the net who devoted huge amounts of time to making free software (perl, gcc, and lots of other utilities) work reliably and explaining it; | My parents and grandparents, for their prayers and support in a variety of other ways; | My friends in the Caltech Christian Fellowship for prayers and encouragement.

iv

Abstract

Abstract
Linear cable theory lies at the core of our understanding of how an individual neuron works. Cable theory usually assumes that neurons do not interact signi cantly except at speci c, anatomically specialized locations (synapses and gap junctions). An analysis of the extracellular electrical elds shows that spikes in one neuron could cause a depolarization of several mV in a dendrite or axon passing by its initial segment. This is somewhat larger than typical chemical synapses; such ephaptic interactions could possibly play a role in controlling action potential failure at branch points. Applying conclusions of linear cable theory to nonlinear spiking neurons has led to incorrect ideas about neural function. For example, in linear cable theory changing the membrane conductance can be used to scale the amplitude of EPSPs. Shunting inhibition has therefore been repeatedly proposed as a mechanism for division or normalization. This mechanism does not work if the neuron is spiking, i.e., when the output is ring rate rather than EPSP amplitude. When a neuron spikes, its timeaveraged voltage does not increase much even if the ring rate goes up; therefore current through a shunt resistance is independent of ring rate, and shunting inhibition acts subtractively rather than divisively. Cable theory also predicts that EPSPs are low-pass ltered by the membrane resistance and capacitance, and investigators have therefore assumed that the membrane time constant determines how fast a neuron can respond. Agin, because of the spiking mechanism, the membrane potential never reaches steady state, so the time constant is not obviously relevant. The dynamics of ring rates may be better described by currents than voltages. Applying this principle to the dynamics of simple feedback networks shows that a key factor in the response time of a network is the adaptation current. Without adaptation, the network time constant can be long because it is the gain of the network multiplied by the synaptic time constants. Adaptation can cancel out the long tails of synaptic current, signi cantly speeding up response times. Recurrent inhibition has a similar e ect. Another key factor determining input current is synaptic depression and facilitation. Recurrent networks are especially sensitive to synaptic depression because of the feedback; within a very short period of time the network behaves like a feedforward network because the recurrent synapses have been depressed away. However, facilitation and depression can act together to provide a log-exponential transform, allowing subtractive inhibition at one stage to have a divisive e ect at another.

CONTENTS

v

Contents
Acknowledgements Abstract iii iv

I Foundations of Cable Theory
1 Introduction
1.1 1.2 1.3 1.4 What is a neuron supposed to do? . . . . . . . . . Assumptions and derivation of the cable equation . Implications of cable theory . . . . . . . . . . . . . Small circuits of neurons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
2 3 8 11

2

2 Ephaptic interactions

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Extracellular potential produced by spiking activity . . 2.2.1 Axon in a volume conductor . . . . . . . . . . . 2.2.2 Axon in a sheath: the core conductor model . . . 2.2.3 Cells with dendritic trees: theory and past work 2.2.4 Cells with dendritic trees: model results . . . . . 2.3 E ect of extracellular potential on neural elements . . . 2.3.1 In nite straight cables . . . . . . . . . . . . . . . 2.3.2 Finite or bent axons, cells, and dendrites . . . . 2.4 Where does ephaptic interaction occur? . . . . . . . . . 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Introduction . . . . . . . . . . . . . . . . . . . . Nernst potential of Potassium . . . . . . . . . . Magnitude of potassium ux across membrane Di usion model . . . . . . . . . . . . . . . . . . Analytic solution for large axons . . . . . . . . Numerical solution . . . . . . . . . . . . . . . . Signi cance of changes in EK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
13 14 14 15 20 22 27 27 33 35

3 Extracellular potassium and other di usible signals

38
38 40 40 41 42 43 46

II Implications of Cable Theory
4 Shunting Inhibition and Firing Rates
4.1 4.2 4.3 4.4 4.5 Introduction . . . . . Model description . . Proximal inhibition . Distal inhibition . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48
49
49 50 51 54 56

vi

CONTENTS 5.1 Firing rate dynamics in single compartment neurons . . . . . . . . . . . . . . . . . 5.1.1 Response time of non-spiking or ring-rate neurons . . . . . . . . . . . . . . 5.1.2 Response time of spiking neurons . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Response time of an ensemble of neurons . . . . . . . . . . . . . . . . . . . 5.2 Dynamics of passive spatially extended neurons . . . . . . . . . . . . . . . . . . . . 5.3 Temporal dynamics are primarily dictated by the time course of synaptic currents 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 The Membrane Time Constant and Firing Rate Dynamics

58
58 58 60 62 65 65 68

III Applications to Networks
6.1 6.2 6.3 6.4 Introduction . . . . . . . . . . . . . . . . . . . . . Continuous time systems . . . . . . . . . . . . . . How adaptation helps|heuristic arguments . . . Simpli ed linear model . . . . . . . . . . . . . . . 6.4.1 Dynamics: analysis by Laplace transform 6.4.2 Network time constant . . . . . . . . . . . Detailed linear model|analysis of poles . . . . . 6.5.1 Approximations for net . . . . . . . . . . Reducing the time constant further . . . . . . . . Simpli ed network dynamics in the time domain Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69
70
70 71 72 74 75 75 76 78 80 81 82

6.5 6.6 6.7 6.8

7 Steady state of circuits with dynamic synapses

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Synaptic depression model . . . . . . . . . . . . . . . . . . . . 7.3 E ect on simple recurrent networks . . . . . . . . . . . . . . . 7.3.1 Recurrence greatly ampli es deviations from linearity 7.4 A possible role for layer 6 . . . . . . . . . . . . . . . . . . . . 7.4.1 A simpli ed example . . . . . . . . . . . . . . . . . . . 7.4.2 More realistic synapses . . . . . . . . . . . . . . . . . . 7.4.3 What is the arithmetic operation? . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83
83 84 84 87 88 88 89 90 91

IV Appendices
A The homogenization approximation for parallel cables
A.1 Introduction . . . . . . . . . . . . . . . . . . . A.2 Hexagonal axon array . . . . . . . . . . . . . A.2.1 Parameter values for hexaongal array A.3 Homogenized extracellular space . . . . . . . A.4 Comparison of simulations . . . . . . . . . . . Integrate{and{ re . . . . . Hodgkin-Huxley equations . Cortical cells in slice . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93
94
94 94 96 98 98

B Time average voltage
B.1 B.2 B.3 B.4

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100
100 102 102 103

CONTENTS

vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C Numerical methods

C.1 Software . . . . . . . . . . . . . . . . . . . . . C.2 Methods for chapter 2: Ephaptic interactions C.2.1 Action potential in an axon . . . . . . C.2.2 Potential around an axon . . . . . . . C.2.3 Potential around a cell . . . . . . . . .

105
105 105 105 105 107

Bibliography

112

viii

LIST OF FIGURES

List of Figures
1.1 1.2 1.3 1.4 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 3.1 3.2 3.3 3.4 3.5 4.1 4.2 4.3 4.4 4.5 4.6 5.1 5.2 5.3 5.4 5.5 5.6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 The core-conductor model equivalent circuit . . The slow potential theory . . . . . . . . . . . . Generator potential vs. real spiking mechanism A common generator-potential based model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 9 9 10 16 17 19 25 25 26 28 29 31 32 34 40 44 44 45 46 49 50 51 53 55 56 59 60 61 63 64 66 70 70 73 76 77 78 79 81

Current ow in a propagating action potential . . . . . . . . Extracellular potential around an axon . . . . . . . . . . . . Comparison of line-source and core-conductor models . . . . Field potentials around simulated layer V pyramidal cell . . Closeup of eld potentials near soma . . . . . . . . . . . . . Relative timing of current and eld potential . . . . . . . . Circuit for computing the e ect of extracellular potentials. . Current ow in interacting axons . . . . . . . . . . . . . . . Membrane potential of cable near a cell body . . . . . . . . E ect of cable parameter variations on induced Vm . . . . . Calculation of ephaptic current at a branch point. . . . . .

Potassium reversal potential as a function of the potassium concentration change Change in potassium concentration as a function of axon radius . . . . . . . . . . Change in potassium concentration as a function of distance . . . . . . . . . . . . Spatial distribution of K+ concentration changes . . . . . . . . . . . . . . . . . . . Time course of K+ concentration changes at the membrane . . . . . . . . . . . . . Simpli ed neuron at steady state . . . . . . . . . . . . . . Divisive and subtractive inhibition . . . . . . . . . . . . . E ect of gleak on ring rates . . . . . . . . . . . . . . . . . Shunting inhibition in spiking neurons . . . . . . . . . . . Distal inhibition . . . . . . . . . . . . . . . . . . . . . . . Shunting inhibition when gE is not small compared to gI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Simpli ed circuit models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample spike rasters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response time of ring rate and spiking neurons . . . . . . . . . . . . . . . . . Increasing makes spiking models faster . . . . . . . . . . . . . . . . . . . . . . An ensemble of neurons can respond arbitrarily fast . . . . . . . . . . . . . . . Somatic current in extended neurons is a better predictor of the time course of rate changes than somatic voltage . . . . . . . . . . . . . . . . . . . . . . . . . Simpli ed cortical ampli er . . . . . . . Discrete system impulse response . . . . How adaptation speeds up the response Pole locations of simpli ed linear model Pole locations of detailed linear model . E ect of varying A2 . . . . . . . . . . . E ect of varying A1 . . . . . . . . . . . Step response of simpli ed model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

... ... ... ... ... ring ... . . . . . . . . . . . . . . . . . . . . . . . .

LIST OF FIGURES 7.1 7.2 7.3 7.4 7.5 A.1 A.2 A.3 A.4 A.5 A.6 A.7 B.1 B.2 B.3 B.4 Synaptic current for a depressing synapse . . . . . . . . . E ect of synaptic depression on the cortical ampli er . . . E ect of depression on the gain . . . . . . . . . . . . . . . Inhibitory input to layer 6 has a divisive e ect on layer 4 Gain as a function of inhibitory current to layer 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix 85 86 86 90 91 95 95 96 97 97 99 99 101 102 103 104

Cross section perpendicular to axons . . . . . . . . . . . . . . . . . Element geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . Scheme used for calculation of volume fraction. . . . . . . . . . . . Unit cell of hexagonal array for currents in left{right direction . . . Unit cell of hexaongal array for currents in vertical direction . . . . Test of homogenization approximation: as a function of distance Test of homogenization approximation: adjacent to membrane . Time averaged membrane potential of integrate{and{ re cells hVm i for Hodgkin-Huxley equations . . . . . . . . . . . . . . . Variations in baseline in slice data . . . . . . . . . . . . . . . hVm i of cells in slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C.1 The box method and discretization scheme. . . . . . . . . . . . . . . . . . . . . . . . 106 C.2 Calculating from a line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 C.3 Green's function and Fourier methods of computing extracellular potential around a cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

x

LIST OF TABLES

List of Tables
4.1 Parameters for compartmental models . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Bulk conductivity of hexagonal array of axons . . . . . . . . . . . . . . . . . . . . . . 51 98

1

Part I

Foundations of Cable Theory

The wide world is all about you: you can fence yourselves in, but you cannot for ever fence it out. | J. R. R. Tolkien

2

CHAPTER 1. INTRODUCTION

Chapter 1 Introduction
1.1 What is a neuron supposed to do?
After a century of work, our knowledge of the phenomenology of neural function is staggering. Hundreds of di erent brain areas have been mapped out in various species. Neurons in these regions have been classi ed, subclassi ed, and reclassi ed based on anatomical details, connectivity, response properties, and the channels, neuropeptides, and other markers they express. Hundreds of channels have been quantitatively characterized, and the regulation and gating mechanisms are beginning to be understood. Every issue of neuroscience journals reports identi cation of several new proteins related to vital functions such as synaptic transmission, neurotransmitter synthesis, or various metabolic functions of neurons. Despite this wealth of descriptive data, we still do not have a grasp on how exactly these thousands of proteins are supposed to accomplish computation. For example, how is it that cells in primary visual cortex respond only to stimuli of a certain orientation? Primary visual cortex is probably the most intensively studied piece of the brain, and orientation-selective responses are the most obvious characteristic of the cells. Yet despite thirty years of intensive study, and even a Nobel prize in the eld, the mechanism of orientation selectivity is still hotly debated. We cannot answer with certainty even the most basic questions: How important is inhibition in determining orientation selectivity? Is recurrence important? Are dendritic nonlinearities important? In short, what is it about the neurons or the network that actually gives it orientation selectivity? At least part of the reason for such confusion is an inadequate understanding of the basic biophysics. Contradictory simpli ed models of neurons have been used to understand the same functions. As an extreme example, neurons are sometimes regarded as high-pass lters (coincidence detectors; McCulloch and Pitts, 1943; Abeles, 1982; Softky, 1995), sometimes as low-pass lters (Hop eld, 1984; Carandini et al., 1996b), and sometimes as more complicated kinds of lters. Of course the results are di erent. In some cases, as I will show in chapters 4 and 5, untested assumptions and simpli cations have led to incorrect theories of the operation of cortical neurons. Di cult experiments to test these theories have been performed, experiments which never needed to be done if the fundamentals had been more carefully thought through. To address this problem, this thesis reanalyzes a few of the premises and conclusions from biophysical analyses of neurons. First, I examine in detail the assumptions behind cable theory and compartmental modeling, especially the assumption that neurons communicate only through anatomical specializations like gap junctions or synapses. Models usually assume that if synaptic inputs are completely speci ed, then the output of the neuron is also determined. The goal of biophysical modeling has been to calculate and to understand how this output depends on the synaptic input. The primary mathematical framework, one-dimensional cable theory, assumes that ion concentrations and extracellular potentials do not change appreciably because of the activity of neighboring cells. Is the extracellular environment su ciently constant that we can make this approximation? Only a small fraction (usually about 20%) of the space in the brain is actually extracellular (Nicholson, 1995; Sykova, 1997). At the narrowest, there is only about 20 nm between one cell membrane and the membrane of its neighbor (Van Harreveld, 1966, 1972). Because of the small size of the extracellular space, there has been speculation over the years that some of the assumptions of cable theory might break down. In fact, I show in chapter 2 that they may break down near cell bodies of cortical neurons. A single spike from a cortical neuron is accompanied by an extracellular potential of 3 mV or more near the cell body. This eld potential can have a substantial e ect on nearby neural elements, in some cases much stronger than the e ect of a typical synapse. It is possible that

CHAPTER 1. INTRODUCTION

3

such e ects could play at least a small role in some kinds of computation. The second part of this thesis also analyzes the e ect of a spike from a neuron, but this time on the neuron itself. Surprisingly, although action potentials are the most obvious behavior of neurons, they have been overlooked in many simpli ed neural models. There have been a number of studies which assume that the output of the neuron is the membrane voltage, rather than the ring rate. Over the last thirty years, many people have calculated what factors in uence the membrane voltage|the shape of the dendritic tree, the various active conductances in the dendrites, the conductance of the cell|while ignoring the largest e ect on the membrane voltage, the action potential and subsequent repolarization. Not only crude non-spiking models of neurons, such as variants on the Hop eld point neuron model, are a ected by this oversight. Even sophisticated compartmental models and analytic work often assumes that the output of the neuron is directly proportional to what the voltage at the soma would be if there were no spiking mechanism. In a non-spiking neuron, shunting inhibition (inhibition with a reversal potential near the \resting potential" of the neuron) has a divisive e ect on the membrane potential of the neuron, so it has been postulated as a mechanism for division or normalization. In a spiking neuron, however, I found that it has a subtractive e ect (chapter 4). In a non-spiking neuron, the rate of membrane potential change is governed by the time constant of the cell; the time constant is a measure of how quickly the cell approaches steady state, and so it has been used as a measure of the rate at which a neuron can respond. But a spiking cell has no steady state, and in fact the membrane time constant is not a useful measure of response time; instead, other factors such as synaptic time constants and adaptation conductances determine the dynamics of ring (chapter 5). The third part of this thesis builds on the observation that synaptic time constants and adaptation control the ring rate of an individual cell, and extends this to recurrent networks. Cortical feedback is thought by some to be essential in the response of cortical cells (Douglas and Martin, 1991). However, recurrence greatly ampli es the e ects of any delays in feedback, and recurrent systems can have unreasonably slow response times. In chapter 6, I analyze how spike frequency adaptation can substantially speed up the response of a recurrent system so that it is little slower than a feedforward system. Recurrence also greatly ampli es the e ect of nonlinearities in the feedback. Synaptic depression and facilitation can therefore have a profound impact on the operation of these circuits (chapter 7). What is a neuron supposed to do? We are still a long way from answering this question. This thesis rules out some incorrect answers, and begins to examine other factors such as synaptic depression and ephaptic transmission which may possibly play a larger role than we expected.

1.2 Assumptions and derivation of the cable equation
Most models of neurons are based on the idea that the neural membrane is a leaky capacitor. For \neurons" which do not have any spatial extent, this takes the form of a simple RC circuit (e.g., Hop eld, 1984; Carandini et al., 1996b; see section 1.3 and gure 1.4), possibly with a spiking mechanism (the integrate-and- re neuron; Lapicque, 1907). Point neurons are simple because there is only one voltage to account for, and therefore are widely used in modeling. For some properties of neurons, however, variations of voltage with location can be important. In this case, neurons must be described by the cable equation, a partial di erential equation that takes into account the capacitance and leakiness of the neural membrane and the nite resistance of the intracellular space. Since several parts of this thesis deal with violations of various assumptions of cable theory, it is helpful to set them down explicitly in some detail. Cable theory in neurobiology has a long history, having rst been applied to neurons in 1863 by Matteucci and a decade later by Hermann, and subsequently elaborated by many investigators (see Rall, 1977 for a historical overview). In its most general form, the \core conductor" model, a long thin electrically conducting core is surrounded by a membrane. The core cross section is su ciently small that it can be treated as one-dimensional. Usually, the membrane is surrounded by extracellular space which is assumed to be isopotential

4

CHAPTER 1. INTRODUCTION

(grounded). However, an e ectively one-dimensional extracellular space surrounding the cable is sometimes considered by making a minor modi cation to the equation, and this is the form we will derive here. The assumptions in the derivation are: 1. Magnetic elds are negligible. The magnetic eld makes a di erence of about one part in 109 (Rosenfalck, 1969) because the currents involved are not large. 2. Ion concentrations and uxes may be treated as continuous variables, ignoring stochastic e ects. This is generally a good assumption. For example, although extracellular clefts can be extremely narrow (e.g., 20 nm at the narrowest), there are still 36,000 potassium ions per square micron in such a cleft. All other relevant ions have higher concentrations (except intracellular calcium), so e ects due to a nite number of ions are not likely to be important. A more important source of stochastic variations in ion ux is random channel openings. There may be only 3{4 channels per square micron of membrane (e.g., see Colbert and Johnston, 1996). Random variations in channel openings might even cause spontaneous action potentials in very small structures such as dendritic spines (Strassberg and DeFelice, 1993; Chow and White, 1996). This may explain why very small axons are not myelinated (Franciolini, 1987). However, most structures of interest are not so small. When we assume that ion concentrations and uxes are continuous variables, then ionic ow in the extracellular uid and through membranes is well modeled by the Nernst{Planck equation for electrodi usion (see Hille, 1977):

where Ji is the ux of ions of species i, Di is the di usion coe cient, ni is the concentration, is the potential, zi is the valence, and RT=F = kT=e 27 mV measures how electrical potential trades o against thermal energy. This equation must be coupled with several others that describe how Ji in uences and ni . First, the total amount of any species remains constant, so

Ji = ?Dini Fzi r + rni ; RT ni

(1.1)

@ni = ?r J : i @t
Second, the ow of ions alters by changing the local charge density :

(1.2)

@ = ?F X z @ni : i @t @t i
The charge density a ects the potential through Gauss's law,

(1.3)

r2 = ?
where is the permittivity.

(1.4)

3. The extracellular and intracellular space can be treated as if it were homogeneous, rather than modeling all membranes explicitly. Intracellular space in most neurons has numerous membranes, including various vesicles and the endoplasmic reticulum. Extracellular space is densely packed with dendrites, axons, and glia. It would be di cult to make any general statements unless it is possible to ignore the complex geometry of the space. Fortunately, the distance scale of electrical potential variations is usually much larger than such inhomogeneities. It is therefore a good approximation to replace the potential by its average over some small volume, and adjust the dielectric constant appropriately. See appendix A for an explicit test of this idea for an array of parallel axons.

CHAPTER 1. INTRODUCTION

5

As a result, only the plasma membrane of the cell being studied needs to be explicitly modeled, and the extracellular and intracellular space can be replaced by homogeneous resistive uids. 4. Extracellular and intracellular ion concentrations do not change appreciably. This assumption is examined critically for extracellular potassium in chapter 3. This assumption has been examined for the intracellular space by Qian and Sejnowski (1989, 1990) and found to hold everywhere except in very small structures like dendritic spines. If ion concentrations are roughly constant, the Nernst potentials for the various ions are wellde ned, and ow through ion channels can be expressed as Ii = gi (Ei ? Vm ), where g is possibly voltage-dependent. Also, the di usion term in equation 1.1 can be neglected. Combining the ux of all ions into a total ux of charge J = F zi Ji , equations 1.1{1.3 become

where

J=? r @ = ?r J @t
F = RT
2 X

(1.5) (1.6) (1.7)

i

zi Di ni

5. The intra- and extra-cellular uids are electrically neutral. Equation 1.4 can be combined with equations 1.5{1.6 to obtain which indicates that the system relaxes to electroneutrality with a time constant of = e . for extracellular space is about 0.3 S/ m (300 -cm; see, e.g., Ranck, 1963 for empirical measurements; this can also be calculated directly from equation 1.7 using di usion coe cients adjusted for tortuosity, as in section 3.4 on page 41). Taking into account the dielectric constant of water, and the e ect of all the membranes in the extracellular space, the permittivity of tissue is about 105 times the permittivity of empty space in the relevant frequency range (Rosenfalck, 1969), so the time constant will be about 3 s. To a good approximation, then, the charge density is always 0 on the time scales we are interested in.1 This is an important simpli cation; it means that the capacitive e ects of inactive membranes can be ignored because very little current crosses them. Both the extracellular and intracellular space are almost purely resistive, and therefore the potential is an instantaneous function of the transmembrane current. In other words, the only time dependence is in the membrane of the cells producing the extracellular potential (in the activation and inactivation of conductances and in the capacitance); other membranes play a negligible role. When we set = 0 and @ =@t = 0, equations 1.5 and 1.6 turn into Laplace's equation,

is the conductivity.

@ =? @t

(1.8)

r ( r )=0
with the boundary conditions

(1.9) (1.10)

1 The same conclusion can also be reached from the phase of complex impedance measurements at relevant frequencies (e.g., Plonsey, 1969).

r n = Jm ;

6 PSfrag replacements Ve (z ? dz )

CHAPTER 1. INTRODUCTION

Ve (z )

Ve (z + dz )
Extracellular

re (z ? dz=2) dz

re (z + dz=2) dz im dz gm dz Em

cm dz

Vm = V i ? V e

ri (z ? dz=2) dz Vi (z ? dz ) Vi (z )

im dz ri (z + dz=2) dz Vi (z + dz )

Intracellular

Figure 1.1: Schematic for the core-conductor model including a one dimensional extracellular space. where n is the normal to the membrane and Jm is the transmembrane current per area (Rall, 1969). 6. The intracellular space can be treated as one-dimensional. In other words, the radial voltage drop is negligible compared to the longitudinal voltage drop on the relevant distance scale; essentially all the current ow is axial. A mathematical analysis is too lengthy to present here, but this assumption has been studied in detail (Clark and Plonsey, 1966; Rall, 1969) and found to be true in almost any practical case. This reduces equation 1.9 to a one dimensional PDE. However, it is often more intuitive to think in terms of equivalent circuits; a one dimensional intracellular space is properly described by the bottom half of the circuit in gure 1.1. Summing the currents into the bottom node gives Vi (z ? dz ) ? Vi (z ) + i dz + Vi (z + dz ) ? Vi (z ) = 0; (1.11) where Vi is the intracellular potential, im is the current through the membrane into the cell per unit length, ri = a2 = i is the resistance per unit length of cytoplasm, a is the radius of the axon, and i is the intracellular resistivity. After simplifying,

ri dz

m

ri dz

@ 2 Vi = ?r i : im @z 2

(1.12)

7. No other neurons inject current into the extracellular space. This assumption will be analyzed in detail in chapter 2. This implies that neurons' electrical activity are entirely independent of each other: there is no cross talk. The extracellular potential is either 0 or due solely to the neuron being considered. 8. The extracellular space can be treated as either one-dimensional or isopotential (grounded). The validity of a one-dimensional extracellular space model will be analyzed in detail in section 2.2.2. It is equivalent to assuming that most of the extracellular current is

CHAPTER 1. INTRODUCTION

7

owing radially rather than axially. Physically, this means that the cable is surrounded by a non-conducting sheath with a radius not too much larger than the cable itself. This is the case, for example, if an axon is removed from the animal and placed into oil with only a thin layer of water around it, as was commonly done in the early days of neurophysiology. It also may be a good approximation for axons in peripheral nerve because of the connective tissue surrounding the bers. Summing the currents into the top node of gure 1.1, where re = ( b2 ? a2 )= e is the resistance per unit length of the extracellular space, b is the radius of the sheath surrounding the neuron, and e is the extracellular resistivity. The large volumes in the central nervous system cannot be modeled as one-dimensional, because the extracellular current ow is not mostly axial (see section 2.2.1). In this case, Laplace's equation (equation 1.9 must be solved. However, eld potentials are usually small in comparison with the transmembrane potentials (on the order of tens of microvolts). For this reason, the extracellular potential is usually set to 0 (i.e., re = 0). 9. The membrane can be modeled by a capacitor in parallel with a conductance as P shown in gure 1.1. If there is more than one conductance in the membrane, then gm = gi P P and Em = ( gi Ei )= gi , where Ei is the Nernst potential for each species of ion. As long as the gi have a suitable voltage dependence, this will always be the case; however, analytic results from cable theory are only useful if gi does not depend strongly on voltage or time. This is often approximately true in neurons when the membrane potential is not close to threshold. When neurons are ring then gi cannot be considered approximately constant, and the predictions of linear cable theory break down. While it may seem obvious, neglect of this fact has led to some important misconceptions (chapters 4 and 5). From gure 1.1,

@ 2 Ve = r i @z 2 e m

(1.13)

im = gm (Vm ? Em ) + cm @Vm : @t

(1.14)

The neuron may be accurately modeled by cable theory if and only if these assumptions hold. Adding equations 1.12 and 1.13 gives

Vm (re + ri )im = @ Vi ? @@zV2e = @@z 2 : @z 2 Substituting this into equation 1.14 gives the full cable equation:
1 @ 2 Vm = g (V ? E ) + c @Vm m m m m @t re + ri @z 2 or
2 2 @ Vm @z 2

2

2

2

(1.15) (1.16)

where 2 = 1=gm(re + ri ) and = cm =gm. The e ect of the extracellular space, with these assumptions, is merely to increase the e ective intracellular resistance, i.e., to change ri into ri + re . This is because the axial intracellular currents are equal and opposite to the axial extracellular currents.

= Vm ? Em + @Vm @t

(1.17)

8

CHAPTER 1. INTRODUCTION

Near a cell body, extracellular elds may be several mV. When considering axons and dendrites passing near the cell body of another cell, assumptions 7 and 8 are violated, and the extracellular elds must be explicitly taken into account. I show that axons and dendrites passing near the cell body feel a depolarization approximately equal to the amplitude of the extracellular potential (chapter 2).

1.3 Implications of cable theory
Some early researchers such as Lorente de No (1947b) argued on the basis of extracellular records that active propagation of impulses in dendrites was important for function. Intracellular work in motoneurons, however, showed that if dendrites are active, their threshold for impulse generation is certainly much higher than the axon and soma (Fuortes et al., 1957; Coombs et al., 1957a, 1957b), and many properties of the dendrites could be well accounted for by passive cable theory (Rall, 1977). The classic conception of a neuron, as developed by Eccles, Rall, and others, is that dendrites sum up and smooth synaptic inputs, and deliver the results as an EPSP to the soma. The soma, it is assumed, converts the EPSP into a ring rate. A great deal of experimental and theoretical work has gone into measuring how various manipulations a ect the amplitude of the somatic EPSP. How does the soma convert EPSPs into ring rates? This is a much more subtle question than it appears at rst sight. Before the origins of the action potential were well understood, it was not appreciated how signi cantly the action potential itself a ects the membrane potential. For example, when studying the stretch receptor in a muscle, Katz (1950) noted that in response to a steady stimulus there was \a local preliminary depolarization which re-develops after each discharge and which varies with the strength of the stimulus." Today, after years of compartmental models and theoretical analysis of the cable equation, it seems that the best way to understand this depolarization is as an e ect of a transmembrane current (see below). But since the current could not be measured directly at the time, the earlier physiologists tended to regard the observable voltage change as the fundamental quantity, which they sometimes called a generator potential (Granit, 1947, 1955; Granit et al., 1963). In this example, Katz (1950) disabled spiking pharmacologically in muscle stretch receptors and measured the amplitude of the generator potential. He found that it was proportional to the ring rate for the same stimulus without local anesthesia, and concluded that \the local spindle potential appears to be an essential link between the input and output of the sense organ...." The same idea is carried somewhat further by the classic textbook of Stevens (1966). The slow potential theory ( gure 1.2) is the idea that \an above-threshold depolarization whose magnitude changes relatively slowly" is faithfully encoded in the ring frequency of the cell. Temporal summation in the dendrites of a postsynaptic cell will yield a depolarization proportional to the frequency. As a result, \nerve impulse frequency appears to be translated back into a depolarization similar to the one which originally generated the axonal nerve impulses." Regarding the depolarization as the cause of the spikes, as Katz and other early investigators did, is certainly correct as far as it goes, since clearly if there is no depolarization there will be no spikes. However, treating the generator potential as if it is a real potential can be misleading. When spikes are disabled, the generator potential is a long slow depolarization, as shown schematically in gure 1.3A. When spikes are not disabled, the observed potential is not a set of spikes riding on top of the unaltered generator potential (shown in dotted lines), as would be expected from a model which rst computes the generator potential and then computes a ring rate. Instead, at the spike initiation zone the depolarization is chopped o by each spike and begins anew after the spike repolarizes ( gure 1.3B). That was why the physiologists had to disable spikes to see the generator potential. There is no place in the neuron where the membrane potential is equal to the generator potential2 .
2 The case where the spike mechanism is electrotonically distant from the site of input will be dealt with later, in the chapters devoted to shunting inhibition (chapter 4) and time constants (chapter 5). The spiking mechanism can

CHAPTER 1. INTRODUCTION

9

This gure has been censored for copyright reasons.

Figure 1.2: The slow potential theory. Taken from Stevens (1966), gure 4-1. Used completely without permission.

A. Generator potential

B. With spiking mechanism

Figure 1.3: A schematic of the membrane voltage with spikes disabled (the generator potential; A) and with spikes enabled (B). When spikes are enabled, the membrane potential is not the generator potential with spikes riding on top of it (dotted lines in A); the potential is truncated at threshold (grey lines) and begins anew after each spike.

10 PSfrag replacements

CHAPTER 1. INTRODUCTION
Firing rate

C

gleak Vleak

ge Ve

gi Vi

gshunt Vshunt

Figure 1.4: The model of Carandini and Heeger (1994) and Carandini et al. (1996b, 1997). The key feature of this model is that an above-threshold voltage is computed using the usual RC model of a neuron. This above-threshold voltage is turned into a ring rate, but the ring mechanism itself has no e ect on the voltage. It may seem uncharitable to quibble with the best physiologists in the world about one of their ideas that after a half century of intensive research looks a tri e naive. The reason for examining it in some depth is that although we do not often use the term \generator potential" any more, many modern modeling and experimental e orts are based on exactly the same idea. This idea leads to some important misconceptions about neural function. Consider, for example, the recent model of Carandini and Heeger (1994) and Carandini et al. (1996b, 1997) in gure 1.4. This model has been quite popular for explaining some features of normalization in cortex, and in fact has set experimentalists looking to con rm its biophysical prediction of a large conductance change. It explicitly computes an above-threshold membrane potential Vm (e ectively the generator potential, though these authors did not call it so) using a conductance-based model of a neuron. It then converts this membrane potential into a ring rate but the spiking mechanism itself has no in uence on the membrane potential. This model is singled out only because it has been quite popular lately. Many other pointneuron models are based on the same idea (e.g., Sejnowski, 1977; Hop eld, 1984; Nelson, 1994) and the results from this thesis apply equally to them. The same is also true of some compartmental models which examine closely the magnitude of the somatic EPSP, and treated the time course and amplitude of this EPSP as indicative of how the cell's ring rate will be a ected, without actually modeling the ring mechanism (e.g., Koch et al., 1983; parts of Bernander et al., 1991; Rapp et al., 1992). But since there is no place in the spiking neuron where the membrane voltage is equal to the generator potential, di erent physics apply to the membrane voltage when the neuron is ring and when it is not. Models which use passive cable properties to compute a generator potential, and then convert the above-threshold potential into a ring rate, are not properly based on the underlying biophysics. One consequence of this mistake is that shunting inhibition was thought to have a divisive e ect on ring rates (chapter 4). A second consequence is that the membrane time constant was thought to be critical in determining the temporal dynamics of ring rate (chapter 5). When the generator potential is a long slow depolarization as shown in gure 1.3, it is often better to think of the current injected into the neuron as fundamental rather than the voltage. The generator potential amplitude is proportional to the injected current, so the same information is present in both numbers. But the current is a real physical current, and to a rst approximation its time-average value is una ected by the existence of spikes (chapter 4). For this reason, it is a better predictor of the ring rate than an above-threshold membrane potential.
still in uence the average voltage at electrotonically distant sites and still should not be neglected.

CHAPTER 1. INTRODUCTION

11

When the generator potential is not above threshold most of the time, but instead spends most of the time below threshold and has only short excursions above, then it may more appropriate to think in terms of voltage rather than current to predict the spiking pattern. Since the voltage is only occasionally above threshold, spiking will not have as drastic an in uence. This is a fundamentally di erent mode of operation: the spiking output is not determined by the sum of large numbers of small, independent inputs, but by the timing of uctuations (Shadlen and Newsome, 1994; Softky, 1995). If the uctuations arise from simultaneous synaptic inputs, then the neuron can be thought of as a coincidence detector (Abeles, 1982). Which mode do neurons actually operate in? The classical conception of a neuron is based primarily on motoneurons, and it seems clear that their primary function is to sum up inputs and reporting the result as an analog value. It is still a matter of debate to what extent other neurons operate in di erent modes. In some cases, neurons obviously act as coincidence detectors, such as in the sound localization pathway, and they have clear anatomical specializations to enable them to do so. In neocortex, it is not yet clear what the best way to think of a neuron is. Most experimentalists and modelers assume the summing mode. This is true of the normalization models such as the model of Carandini and Heeger (1994) discussed above. It is also true of the so-called \canonical microcircuit" or cortical ampli er models that inspired the network analysis in chapters 6 and 7 (Douglas and Martin, 1991; Maex and Orban, 1992, 1996; Douglas et al., 1995; Somers et al., 1995; Ben-Yishai et al., 1995; Suarez et al., 1995; Holt et al., 1996); in fact, attempts to explain the behavior of these models led to the considerations discussed in this section. For this reason, most of this thesis pertains to the summing mode.

1.4 Small circuits of neurons
The classic Hubel{Wiesel model postulates that orientation selectivity in simple cells in cat visual cortex arises primarily from convergent input of thalamic cells chosen so that the response is orientation selective. In the thirty years since their pioneering work, the mechanism has been hotly debated, especially the importance of inhibition in sharpening the response. It appears that the thalamic input onto cortical simple cells is orientation tuned but only weakly so (Pei et al., 1994; Vidyasagar et al., 1996; but see Reid and Alonso, 1995), although it is clear that the total synaptic input onto cortical cells is sharply orientation tuned (Ferster, 1986, 1987). Experiments with iontophoresed bicuculline appear to indicate that inhibition is important in generating direction and orientation selectivity (Sillito, 1975, 1977); however, blocking inhibition intracellularly in a single cell has only a small e ect on the sharpness of orientation tuning (Nelson et al., 1994). IPSPs are strongest for stimuli of the preferred orientation rather than for other orientations, and in fact seem to have the same orientation tuning as EPSPs (Ferster, 1986). Searches for hidden IPSPs (shunting inhibition) have uncovered no signi cant conductance changes (Douglas et al., 1991; Dehay et al., 1991; Berman et al., 1991, 1992; Douglas and Martin, 1991; Ferster and Jagadeesh, 1992; but see Borg-Graham et al., 1996). Models from our lab and other labs (Douglas and Martin, 1991; Maex and Orban, 1992, 1996; Douglas et al., 1995; Somers et al., 1995; Ben-Yishai et al., 1995; Suarez et al., 1995; Holt et al., 1996) have been developed to explain these puzzling observations. Previously, all models had assumed that only feed-forward connections were important in generating the response. Yet over 95% of the excitatory synapses onto a layer IV cortical cell come from other cortical cells (Peters and Payne, 1993; Peters et al., 1994). It is clear from a number of anatomical studies that cortical cells usually make excitatory connections only to other cortical cells in the same orientation column, or with cells in other columns that have similar receptive elds (Kisvarday et al., 1997).Thus most of the excitatory input a cell receives comes from other cells with almost the same receptive eld. This observation suggests that the cortical circuitry could be described as an ampli er. A small amount of thalamic current causes cortical cells to spike. Activation of recurrent synapses causes the cells to spike more, and so on; the resulting ring rate is much larger than if there were only feedforward

12

CHAPTER 1. INTRODUCTION

input. Inhibition may be necessary for sharpening the orientation tuning, but only a small amount of inhibition is necessary to counter the small amount of geniculate input which triggers the ampli cation cascade; it is not surprising that it might not have been seen in intracellular recordings. The inhibition needs to be only slightly more broadly tuned than excitation, as found recently in anatomical work (Dalva et al., 1997; Kisvarday et al., 1997), to have a signi cant sharpening e ect (Douglas et al., 1995; Somers et al., 1995); it is not surprising that experiments nd that inhibition has approximately the same tuning as excitation. Blocking inhibition in a single cell has only a small e ect, because most of the excitatory input into that cell is already orientation-tuned since it comes from other cortical cells. How long does the recurrent circuitry take to act? A thalamic input triggers a few spikes which cause more recurrent input, leading to more spikes, leading to more recurrent input; but the whole process takes a certain amount of time. In positive feedback loops in electrical engineering, the time constant of an ampli er system is usually approximately the gain of the ampli er times the time constant involved in the feedback. Our models have assumed a gain of around 5, and the feedback will have a time constant of on the order of 80 ms if NMDA receptors are involved. As a result, the network time constant ought to be around 400 ms. This is rediculously long. Celebrini et al. (1993) argued that since orientation tuning seems fully developed in the rst spikes, a recurrent mechanism is ruled out. However, recurrent models of orientation tuning show the same property (Somers et al., 1995), and in general cortical ampli er models do not have such a long time constant. However, no detailed analysis of the factors that in uence the dynamics of the response has been performed. In chapter 6, I use the considerations and formalism developed in chapter 5 to provide a simple explanation for why network models do not show such long time constants. A second issue with recurrent models is the sensitivity to nonlinearities in the feedback. One kind of nonlinearity which is known to be present in some synapses is depression: if the presynaptic cell res a train of action potentials, later EPSPs in the postsynaptic cell are smaller than the rst EPSP. Although this phenomenon is well known in other systems (see Zucker, 1989 for a review), the extent of synaptic depression in neocortex was not appreciated until recently (Thomson and West, 1993; Thomson et al., 1993b; Thomson and Deuchars, 1994; Stratford et al., 1996; Markram and Tsodyks, 1996; Tsodyks and Markram, 1997; Abbott et al., 1997). Synaptic depression turns out to have a profound e ect on properties of such circuits, so much so that there is practically no steady state ampli cation, unless it is countered by facilitation at other synapses (chapter 7).

CHAPTER 2. EPHAPTIC INTERACTIONS

13

Chapter 2 Ephaptic interactions
2.1 Introduction
Studies on squid giant axons (Arvanitaki, 1942), crab motoneurons (Katz and Schmitt, 1940, 1942), and even algal strands (Tabata, 1990) showed that when two axons were placed in a medium with reduced extracellular conductivity, activity in one axon could depolarize the other. Such interactions are called ephaptic (Greek \touching onto", rather than synaptic, \touching together"; Arvanitaki, 1942). The early studies were done before the chemical nature of synaptic transmission in the CNS was understood, and were thought to be evidence that transmission was purely electrical (see Eccles, 1964; Faber and Korn, 1989). In extreme cases an action potential can be induced in an inactive axon by a nearby one. In fact, ephaptic transmission may underly pathological activity in motor neurons some kinds of facial spasms, or in crushed nerves or in nerves damaged by multiple sclerosis (see Faber and Korn, 1989; Je erys, 1995). What determines the magnitude of the potential around a neuron? Suppose, for example, we have an isotropic spherical cell with no dendrite. Because of conservation of charge and electroneutrality of the cell, the net current into the extracellular space is 0, and the extracellular potential will always be 0 no matter what electrical activity happens at the cell's membrane.1 The same holds for a space-clamped axon. In fact, the total current through the membrane of any neuron must always equal 0, no matter what the neuron does, because of conservation of charge. However, current may enter at one point and exit at another. In this case, the current loop must be completed through the extracellular space. Current ow in the extracellular space causes potential di erences which can be measured. Currents in the extracellular space come from two distinguishable sources: synaptic currents and action potential currents. Synaptic currents are small in comparison with the action potential currents, but they last for several ms and in laminar structures the synaptic currents from thousands of neurons can sum up to make eld potentials on the order of several mV. On the other hand, currents from action potentials are quite large, but they are usually very brief and diphasic, unlike the synaptic currents. As a result, it is rare for action potentials from adjacent cells to be aligned precisely enough to sum up; if they are misaligned by a fraction of a ms, they will tend to cancel out. It is fairly clear that most slow eld potentials (with time courses longer than a ms) and also potentials recorded over large areas (e.g., the EEG) are due primarily to synaptic currents rather than action potential currents (Creutzfeld and Houchin, 1974). In fact, a number of models of extracellular eld potentials do not even include action potentials in the calculation (Rall and Shepherd, 1968; Klee and Rall, 1977; Wilson and Bower, 1989). Current source-density (CSD) analysis has been used extensively in laminar structures like neocortex and olfactory cortex to understand the sequence of events in response to electrical shock stimulation (see Mitzdorf, 1985), so a great deal is known about the extracellular potentials from synaptic currents. There is some evidence that the electrical elds due to summed synaptic input from thousands of neurons are strong enough to in uence ring signi cantly. Even elds as small as 2.5 mV/mm can signi cantly modulate population responses, and such elds certainly occur during normal operation (see Je erys, 1995). These elds could have a small general excitatory or inhibitory e ect over a large area of cortex, but the interactions are unlikely to be speci c. I have chosen to concentrate rather on interaction based on the elds from single action potentials because their elds are less
1 This can also be seen from circuits. The equivalent circuit is the same as in gure 1.1 on page 6 with the intracellular resistive grid removed; the intracellular node (Vi ) is oating and not connected to anything. It is clear from the circuit that there can be no transmembrane current, because the capacitive current must always equal the ionic current.

14

CHAPTER 2. EPHAPTIC INTERACTIONS

well studied and could have more speci c e ects. See Rosenfalck (1969) for a thorough review of the earlier literature pertaining to axons, and Faber and Korn (1989) for a thorough review of ephaptic interaction in general. Plonsey (1988) discusses solutions and approximations for unmyelinated axons. A relatively readable review on the general theory of volume conductors is Malmivuo and Plonsey (1995), ch. 8. A general but old review on potentials from cells is Hubbard et al. (1969), ch. 7.

2.2.1 Axon in a volume conductor

2.2 Extracellular potential produced by spiking activity

The elds around axons have been extensively studied, largely because the geometry is fairly regular and the pulse propagates down the axon without change of shape. Since there are no barriers to conduction over large volumes of the central nervous system (assuming the ventricles and pial surface and other boundaries are far away), the potential around axons in the brain can be modeled by Laplace's equation (equation 1.9 on page 5),

r2 = 0
with the boundary condition that = 0 at the point at in nity and
er

(2.1)

n = Jm ; (2.2) where Jm is the transmembrane current density and n is the normal to the membrane, at membranes
which are active. These equations are identical to the equation for potential due to charges located at the same position as the current sources; e is dual to the permittivity , and Jm is dual to the charge density on the sheet. As a result, solutions from electrostatics can be immediately applied to the problem of potential in a volume conductor. For example, if we approximate the axon as a line (neglecting its thickness), then we can use the solution for potential from a line of charge. The potential from a single point2 a distance R away is = I=4 R e , so the potential from a whole line of sources a distance r away is (r; z ) = ?
Z

im(z 0 ) dz 0 ?1 4 e r2 + (z ? z 0 )2
p

1

(2.3)

where im (z ) = 2 aJm is the transmembrane current per length. This approximation, rst made by Lorente de No (1947a), leads to relatively understandable expressions for the potential and shows qualitatively what factors in uence it3 . Furthermore, in an unbounded volume conductor, this approximation is very good even at the axon membrane (see section C.2.3). The transmembrane currents from a simulated axon are shown in gure 2.1, and the corresponding eld in gure 2.2. For axons, the extracellular potential at the neuron membrane is roughly
2 The nite volume of the axon has only a negligible e ect on the potential from a point source, so it is su ciently accurate to use the Green's function for a point source in free space. 3 It is often inconvenient to specify the transmembrane current Jm , since it is di cult to measure. The extracellular potential e can also be computed directly from the intracellular potential (Geselowitz, 1966), since Jm can be computed from the spatial variation of the intracellular potential Vi (equation 1.12 on page 6) rather than from the membrane conductance and capacitance (equation 1.14). The resulting expression is Z1 i @ 2 Vi dz : (2.4) = ?1 e @z 2 4 2 a2 r

CHAPTER 2. EPHAPTIC INTERACTIONS

15

proportional to the transmembrane current there, because the action potential is spread out over a large spatial distance and only nearby points contribute to the sum in equation 2.3. The transmembrane current in an action potential is roughly a dipole, or two back-to-back dipoles (an axial quadrupole, sometimes in this con guration called a tripole): current ows out of the axon ahead of the action potential and behind it, and into the axon during the action potential ( gure 2.1). Analytic expressions can be written for the potential from dipoles in an in nite volume conductor (put delta functions into equation 2.3). These considerations suggest that far away from the neuron, potentials should decline as 1=r2 as they do for a dipole (Rosenfalck, 1969) or 1=r3 as for a quadrupole (Plonsey, 1977). In mammals, unmyelinated axons with diameters much larger than 1 m are rare; one would not expect extracellular potentials to be orders of magnitude greater than shown in gure 2.2. Measured potentials from single axon activity are in general quite small, usually on the order of tens of microvolts. Potentials do not fall o drastically near the ber, so there is no particular reason to think that the extracellular potentials are not accurately measured by the electrode because of tissue destruction or nite electrode size. It is unlikely that a unmyelinated mammalian axon in a volume conductor will have an extracellular potential larger than a few tens of microvolts. This is not enough for signi cant interaction (see below).

2.2.2 Axon in a sheath: the core conductor model

In the nerves of the peripheral nervous system, the extracellular space around axons is not well modeled as an in nite volume conductor. Peripheral nerves, even those which are unmyelinated, are surrounded by Schwann cells and a sheet of collagenous tissue (the endoneurium ). Many axons join together into a fascicle which is surrounded by a second, thicker sheet of connective tissue, the perineurium. Fascicles are joined together to form the nerve, which is surrounded by a third, thick layer of connective tissue called the epineurium. One function of these sheaths, particularly the perineurium, appears to be analogous to the blood-brain barrier: to isolate the axons from changes in the extracellular environment (see Low, 1976; Peters et al., 1991, ch. XII). The perineurium is often modeled by a single cylindrical resistive or non-conducting sheath surrounding the axon at some distance from it. (The same model applies to experimental preparations where a nerve is lifted out of the tissue into air or oil; Stein and Pearson, 1971.) In this case, if the sheath is close enough to the axon, the current ow is essentially radial, and the core-conductor model holds. If there is a sheath that surrounds the neuron and the diameter b of the sheath is small enough that the core-conductor approximation is valid, then the amplitude of the potential can be easily computed (Clark and Plonsey, 1968; Rosenfalck, 1969; Stein and Pearson, 1971; Plonsey, 1977):

2 2 re (2.5) = ? r + r Vm = ? a2 + i ab2 ? a2 ) Vm = ? i a2 Vi i e i e( eb Note that the core conductor model predicts that the potential has the same shape as the intracellular action potential. This is di erent from the case of an unlimited volume conductor, where for a monophasic action potential there is a triphasic extracellular potential (Rosenfalck, 1969). If there is no sheath or if the sheath is too far away to use the core-conductor approximation, then the solution is more complicated and there is no useful simple expression (see Plonsey, 1977 for a review). Obviously the solution must collapse to the core-conductor model when the sheath radius b is not too large compared to the radius of the ber, but it has been di cult to nd a simple criterion for when this occurs. Rosenfalck (1969) argued that the core-conductor model breaks down for b > 2a. Trayanova et al. (1990) after a detailed analysis found empirically that it was valid up for b < 5a but this will depend on the value of a (25 m in their case) and shape of the action potential as well so this result is not generally useful. Stein and Pearson (1971) suggest without proof that the core-conductor model is valid as long as radial voltage variations are not signi cant, i.e., when

16

CHAPTER 2. EPHAPTIC INTERACTIONS

40 0 mV ?40 ?80 0.0004

Vm

S m2 0.0002
0 0.002

gNa gK

nA m2 PSfrag replacements

0 Total current Capacitive current Ionic current 0 1000 2000 Distance (?m) 3000

?0.002

?0.004

Figure 2.1: Current ow across the membrane in a propagating action potential propagating to the left at an instant in time. Top, transmembrane voltage; Middle, sodium and potassium conductances; Bottom current density as a function of position along the axon. The total current is rst positive, then negative, and then there is a smaller positivity. Numbers are for an axon with a diameter of 1 m and the conductance values from Hodgkin and Huxley (1952). The action potential propagates with a speed of 440 m/ms. Only a short segment of the axon is shown. The capacitive (displacement) current is of course the derivative of Vm , The ionic current is close to ?dVm =dt except that it is shifted. See Jack et al. (1983), ch. 9, for a discussion of the currents.

CHAPTER 2. EPHAPTIC INTERACTIONS

17

10

5

0 ¦Õ (?V)

?5

?10

?15 3 2.5 2 1.5 2000 1 1500 0.5 0 0 z (?m) 1000 500 2500 3000

PSfrag replacements

r log10 a

Figure 2.2: The extracellular potential around the axon of gure 2.1. See section C.2.2 for how the potential was computed.

18

CHAPTER 2. EPHAPTIC INTERACTIONS

the conductance between the neuron and the sheath is small compared to the conductance along the axon. This works out to (b2 ? a2 ) log b=a 2l2 where l is the length of the rising phase of the action potential.4 Another way of deriving limits on the validity of the core-conductor model is to compare it to another approximation which is known to be valid when b is large. The line-source model discussed above is known to be within 5% of the true solution when b > 5a (Trayanova and Henriquez, 1991). Unfortunately, there is no simple expression for the potential from a line source when a sheath is present. Instead, it is necessary to resort to analysis in the spatial frequency domain. We rst separate variables for a non-zero a, and later we take the limit as a ! 0 later; otherwise singularities appear in the solution too early. We assume = R(r)Z (z ). Laplace's equation then turns into:

Z 00 + k2 Z = 0 rR00 + R0 ? k2 rR = 0

(2.6) (2.7)

to which the solution is Z = eikz and R = AI0 (jkjr) + BK0 (jkjr) where A and B are constants determined by the boundary conditions. At the sheath boundary r = b, dR=dr = 0; also, R is normalized so that R(a) = 1. After some manipulation,

K ( k b)I ( k r + I ( k b)K ( k r R = K 1(jjkjjb)I 0(jjkjja) + I1 (jjkjjb)K0 (jjkjja) ) )
1

(k; r) =

Z

0

1

?1

eikz R(k; r) ^(k; a) dk

1

0

(2.8) (2.9)

where ^(k; a) is the Fourier transform of the potential at r = a. Since R(k; a) = 1, ^(k; r) = ^(k; a)R(k; r).] We now apply the boundary condition at the membrane surface:

J =@ @r e Z
= implying that

eikz @R r=a ^(k; a) dk @r ?1

1

?1 ^ ^(k; a) = J @R e @r r=a ^ K ( k b)I ( k r = Jkj K 1(jjkjjb)I 1(jjkjja) ? I1 (jjkjjb)K1 (jjkjjr) ) + I1 ( k b)K0 ( k a) ej 1 0 To obtain a line-source model, we now let the axon radius a ! 0 while keeping the total current ^ ^ constant. De ning J = J0 a0 =a, after some algebra ^ k (2.10) lim ^(k; r) = J0 a0 K1((jjkjjbb)) I0 (jkjr) + K0 (jkjr) : a!0 I1 e This should be a good approximation as long as ka0 1 and a0 b. The core-conductor model is valid if the radial variation of potential is not signi cant, i.e., if the potential at r = b is about the same as the potential at the surface of the axon. Using the identity
potential instead of a transmembrane potential. Since the surface extracellular potential varies strongly with the radius of the sheath, this is not a useful way to understand what a ects extracellular potentials.
4 Stein and Oguztoreli (1978) also attempted to solve for extracellular potential, but they assumed a given surface

CHAPTER 2. EPHAPTIC INTERACTIONS

19

I0 (z )K1(z ) + I1 (z )K0 (z ) = 1=z , the potential at r = b is
^ ^(k; b) = J0 a0 1 kbI (kb)
e
1

(2.11)

By comparison, the extracellular potential from the core-conductor model is given by:

@ 2 = 2 aJ @z 2 (b2 ? a2 ) e ^ ^core (k) = ? 22Ja 2 2 2Ja02 e (b ? a )k e k2 b
Therefore ^(k; b) kb ^core (k) = 2I1 (kb) :

(2.12) (2.13)

(2.14)

This is shown in gure 2.3. The core-conductor prediction is within 10% of the line-source prediction for kb < 1. This corresponds to l > 2 b, where l is the shortest signi cant spatial scale of the action potential.
1 0.8

0.6 0.4 0.2 0 0

PSfrag replacements

Figure 2.3: A comparison of the predictions of the core-conductor model with the line-source model as a function of spatial frequency normalized by sheath radius. The core-conductor prediction is quite good up until kb = 1, i.e., for spatial wavelengths in the action potential longer than 2 times the distance to the sheath.
0.5 1 1.5

^= ^core

kb

2

2.5

3

If the action potential has a rise time of say 0.1 ms and a speed of 0.5 mm/ms, then the characteristic length l will be around 0.1 ms 0.5 mm/ms = 50 m. Hence the core-conductor model could be valid for sheath radii up to 300 m. In general, the actual potential will be larger than what is given by the core-conductor approximation5 . Therefore the core-conductor approximation can be used to decide when extracellular potentials will be signi cant. It is reasonable to suppose that a 1 mV potential might have a signi cant e ect (see section 2.3), while something smaller would probably be negligible. Equation 2.5 predicts that if Vm = 100 mV, then i a2 = e b2 > 0:01 for a 1 mV potential. i = e is at most 5, so one would predict signi cant extracellular potentials only if a=b > 0:04. For an axon of radius a = 1 m, this implies b < 25 m. Mammalian perineurial sheaths usually have a radius greater than 50 m (Low, 1976; Peters et al., 1991), so interaction is unlikely. There is a substantial body of work predicting the extracellular elds of myelinated axons (e.g., Marks and Loeb, 1976; Ganapathy and Clark, 1987; Stephanova et al., 1989; Struijk, 1997). Since myelin decreases the capacitance of the membrane, much less current ows and the potentials are 5 ^core ^ for small b, and for large b, and ^core ! 0, while ^ does not approach 0.

20

CHAPTER 2. EPHAPTIC INTERACTIONS

smaller. In keeping with this, it is much more di cult to record any neural activity in white matter (David Kewley, personal communication).

2.2.3 Cells with dendritic trees: theory and past work

Extracellular potentials from action potentials around cell bodies and dendritic arbors are di erent from the potentials around axons. Most obviously, they can be much larger, sometimes 5 mV or more6 (Freygang, 1958; Freygang and Frank, 1959; Terzuolo and Araki, 1961; Rosenthal, 1972; Towe, 1973). A much larger area (the proximal dendritic tree) is simultaneously depolarized by currents at or near the soma, so the current that ows must be larger. Also, the shapes can be di erent because the action potential does not propagate in the same way. Analysis is complicated by the irregular geometry of dendrites and cell body, and in general the elds are not well understood. One approximation which is sometimes made is based on the line source model, equation 2.3. Since potential at any point is a sum of contributions of all points in the neuron weighted by 1=R, where R is the distance to the point, only nearby points will have a large e ect on potential unless there are very large current sources far away. Therefore, the potential at the membrane ought to be approximately proportional to the transmembrane current at that point. This is quite accurate for axons (compare gure 2.1 with gure 2.2). Although was once used to understand eld potentials around cells, it does not accurate in this case (Rall, 1962) because the e ect of sources and sinks near the soma or axon initial segment can be large compared to the e ect of current through the dendrites at the point being considered. Nevertheless, determining where a particular peak in the extracellular eld is largest can be a clue to its origin. This generalization also breaks down if a large population of spatially distributed neurons is active exactly simultaneously (Klee and Rall, 1977), such as in response to an electrical shock, or when the eld is due primarily to slower events such as synaptic input. A second tool for intuition is to formulate the problem in terms of volume sources. Because dendritic trees are complicated and irregular, the boundary conditions for solution of Laplace's equation (equation 1.9 on page 5) are prohibitively complicated. One can average the tissue over a small volume, changing current ux across membranes into current source density. From conservation of charge,

r J=I r r = ?I:

(2.15)

where I (the current source density) is the sum of all the transmembrane currents in some small volume. Since J = ? r (equation 1.5 on page 5), (2.16) If is a scalar (conductivity is roughly isotropic) then equation 2.16 has exactly the same form as Gauss's law in electrostatics (equation 1.4 on page 4). Current source density in volume conductors is dual to charge density in electrostatics. This means that the mathematical techniques which have been useful in electrostatics apply equally well to the elds around cells. This was used by Bishop and O'Leary (1942) and Lorente de No (1947b) for a population of neurons with a regular geometry to try to explain the di erent shapes of extracellular action potentials along the microelectrode track. For example, suppose that the dendritic eld of a given cell7 is approximately spherically symmetric. Then the eld from an action potential originating at or near the soma will also have spherical symmetry. Taking integrals of both sides of equation 2.16 over a spherical volume of radius r and
6 So called \giant potentials" of 20 mV or more can be recorded when the electrode is pushed up against the membrane. These are caused by a resistive seal between the membrane and the electrode and are therefore an artifact of the electrode's presence. See Hubbard et al. (1969), pp. 282-283. 7 Or population of simultaneously activated cells in a nucleus, as Lorente de No (1947b) originally analyzed it. Approximations to this situation are not uncommon in the various nuclei in the brain.

CHAPTER 2. EPHAPTIC INTERACTIONS applying the divergence theorem,
ZZ

21
ZZZ

E dS =
Z

I (r) dV

(2.17)

where E = ?r is the electric eld and I (r) is the total current inside a sphere of radius r. Using spherical symmetry, 4

r2 E (r) =

r

0

I (r0 ) 4 r0 2 dr0

(2.18)

8 A population with cell bodies at the periphery and dendrites in the center, such as in the superior olive, also is a closed eld; its eld potentials are opposite in sign from the closed eld discussed above.

For r larger than the radius of the dendritic eld, the right hand side is 0 since all current that ows out through the soma must ow back in through the dendrites somewhere. Therefore no potential at all will be visible beyond a certain radius, no matter what activity there is in the dendrites; hence this is called a \closed eld" (Lorente de No, 1947b).8 For r within the radius of the dendrites, there will be a measurable potential. When the action potential is in the rising phase at the soma, I at the soma will be negative, but in the dendrites I will be positive. When the action potential begins to repolarize, I will be positive at the soma and negative in the dendrites. The potential from this cell is therefore diphasic, in contrast to the triphasic potential from an axon. The shape of the potential will always be approximately the same (negative then positive) wherever it is recorded. This is di erent from the potential from an \open eld" where the potential can be triphasic or be rst positive then negative. Rall (1962) used the closed eld approximation to estimate the magnitude of the extracellular potential for stellate cells. Many kinds of cells do not have spherically symmetric dendrites, but part of their tree can be regarded as spherical. For example, in the case of a pyramidal cell the potential from the whole cell can be regarded as the sum of a closed eld (the soma and basal dendrites) and a dipole source (the soma and apical dendrites). The extracellular potential is large and initially negative near the soma, and declines rapidly with distance. Probably most of the current from the somatic spike ows out through the basal dendrites, but a small fraction of it ows up into the apical dendrite. As a result, potentials around the apical dendrites are small and initially positive. Of course, this approximation breaks down since action potentials propagate actively up the dendrites, but it turns out that the potential due to dendritic spikes is much smaller than the potential due to the somatic spike. It has been di cult to make general statements about extracellularly recorded action potentials beyond these qualitative considerations (Lorente de No, 1947b, 1953). The shapes and magnitudes of the electric potentials depend sensitively on the membrane currents and when the action potential reaches di erent parts of the neuron. In early work the extracellular potentials at di erent positions were used to answer questions such as: Is synaptic transmission electrical or chemical? (For example, see the lively discussions by Eccles and others at a symposium on the spinal cord, Malcolm and Gray, 1953) Where does the spike initiate? The correct answer, in the axon, was determined by a number of extracellular studies (thalamic cells: Freygang, 1958; motoneurons: Freygang and Frank, 1959; Terzuolo and Araki, 1961) and incorrectly by others (Fatt, 1957). Do dendrites carry spikes? Extracellular recordings were interpreted to support dendritic spikes in some cell types for many years (cortical pyramids: Clare and Bishop, 1955; spinal motoneurons: Fatt, 1957; Terzuolo and Araki, 1961; hippocampal pyramids: Sperti et al., 1967; Buzsaki et al., 1996; Purkinje cells: Eccles et al., 1966; Nicholson and Llinas, 1971), although there were some disagreements (Freygang, 1958; Freygang and Frank, 1959; Nelson and Frank, 1964; Rosenthal, 1972). Lorente de No (1947b) even showed that the action potential does

22

CHAPTER 2. EPHAPTIC INTERACTIONS

not propagate as far into the dendrite of motoneurons in later spikes of a burst, evidently due to branch point failure. This result caused considerable excitement when it was rediscovered fty years later with di erent techniques (Spruston et al., 1995; Ho man et al., 1997). Analysis of extracellular action potentials has been largely neglected in recent years (an exception is Buzsaki et al., 1996), partly because extracellular potentials are di cult to interpret, and partly because dendrites were considered passive for theoretical convenience. More emphasis has been placed on the origin of eld potentials due to synaptic input in a population of neurons (Mitzdorf, 1985; Bullock, 1997).

2.2.4 Cells with dendritic trees: model results

9 In EM views of axon initial segments, there is a dense undercoating below the plasma membrane similar to the dense undercoating at the nodes of Ranvier, and in fact uorescent neurotoxin probes speci c for the sodium channel stain the hillock of cultured motoneurons and cortical cells more brightly than the cell body (Angelides et al., 1988). However, physiological measurements in hippocampal pyramidal cells by Colbert and Johnston (1996) indicated only a low density of sodium channels in the hillock/initial segment. It is still unclear exactly where in the axon the action potential begins. 10 The sheath of satellite cells is probably more for maintaining a constant concentration of potassium and other ions; see chapter 3.

Since little is known about the elds around cells, I computed the extracellular potential directly from a compartmental model. The purpose of this thesis is not to develop an accurate detailed model of a cell, but instead to understand the electrical elds around it; so I used a previously published model of a neocortical cell (Mainen and Sejnowski, 1996) without modi cation. This model was chosen because its spikes initiate in the axon (Mainen et al., 1995). Proper spike initiation is critical for computing the extracellular potential, because the potentials are much larger near the region of spike initiation than anywhere else (see below). In this model, the action potential initiates in the axon hillock and initial segment of the axon, where there is postulated to be a very large density of sodium channels, as large as at a node of Ranvier.9 A second advantage of the model of Mainen and Sejnowski (1996) is that the dendrites are weakly active, and action potentials propagate actively up into the tree, so it can be used to study how active propagation in the dendrites a ects the elds. The model uses measured sodium channel densities in the dendrites (for young rats; see Stuart and Sakmann, 1994). There is some controversy about these values, however, because they evidently change signi cantly with age. In the hippocampus in older rats, for example, the sodium channel density is much higher, but the sodium currents are balanced out by a high density of potassium A currents so the dendrites are still only weakly active (Ho man et al., 1997). In the peripheral nervous system, cell bodies of neurons are usually surrounded by a sheath of Schwann cells. In the central nervous system, ensheathment of the cell body and axon initial segment is more variable. Muller cells in the retina ensheath ganglion cells (Stone et al., 1995). Purkinje cells have a complete covering of astroglial processes around the cell body. Neocortical pyramidal cells appear not to have any covering at all. In neocortex, one nds dendrites and axons in abundance even directly apposed to the cell body and axon hillock (Peters et al., 1991). Thus in neocortex we are justi ed in ignoring the complicating e ects of a sheath.10 Calculation of the eld potential was based on the line source model, with a homogeneous unbounded extracellular space with a conductivity of 0.3 m/ S (330 -cm; e.g., Ranck, 1963). The potential and eld amplitudes are directly proportional to 1= e so the e ect of a change in e is trivial to calculate. See section C.2.3 for details. Results are shown in gure 2.4 and gure 2.5. It is di cult to account for every detail of the eld potential, but most of the obvious features can be understood. First consider the largest potentials, which occur near the axon hillock and initial segment. This area of the neuron has an extremely high density of sodium channels in the model (maximum conductance if all are open is 30,000 pS/ m 2 , in comparison with 20 pS/ m 2 in the soma and dendrites). The inward currents are large because current through the axon hillock is what depolarizes the soma and proximal dendrites.

CHAPTER 2. EPHAPTIC INTERACTIONS

23

400

300

200

100

0

-100

-200

-150

-100

-50

0

50

100

150

200

3 ms

0.01 mV

0.05 mV

0.25 mV

Figure 2.4: caption on page 25

24

CHAPTER 2. EPHAPTIC INTERACTIONS

60

40

20

0

-20

-40

-60 -40 -30
0.1 mV

-20

-10

0
0.5 mV

10

20

30
2.5 mV

40

Figure 2.5: caption on page 25

CHAPTER 2. EPHAPTIC INTERACTIONS

25

Figure 2.4: Field potentials in a plane around the simulated layer V cortical pyramidal cell. x and y axes are in units of m. Each trace was taken from a point at its center. Shaded areas indicate di erent voltage scales. Closer than 20 m to the axon hillock, eld potentials are much larger than shown here (see gure 2.5 for closeup). The soma is at (0; 0), and the axon descends straight below it. The apical dendritic trunk is slightly to the left of x = 0 and goes up approximately straight, so the larger potentials at the top center are from the apical dendrite. The eld potentials look roughly similar in other slices through the volume, so only this slice is shown. Figure 2.5: Field potentials near the soma of the simulated layer V cortical pyramidal cell. The peak eld potential is slightly over -5 mV and occurs next to the axon initial segment. Note that the peak amplitude on this graph is much higher than in gure 2.4 because the traces are closer to the axon initial segment. The action potential initiates in the distal part of the initial segment (actually in the rst node of Ranvier). It then propagates up the initial segment (not shown), slowing down considerably in the axon hillock because of the increasing diameter and the load of the soma and the dendrites. The initial negativity in the extracellular potential (the \A spike" in the nomenclature of Fuortes et al., 1957 for motoneurons) is due to the action potential in the distal initial segment, since it occurs at the same time as the maximum current from the distal initial segment (see gure 2.6). Because of charge conservation, current owing in through the initial segment must ow out somewhere in the cell; in fact, there is an initial positivity near the apical dendrite ( gure 2.4) because the potential from the local outward current there was larger than the potential from the inward current at the initial segment. This sign reversal in the apical tree is commonly observed in physiology (e.g., Sperti et al., 1967; Rosenthal, 1972). A second negativity (the \B spike") in the eld potential is due to the ring of the axon hillock, especially the proximal part ( gure 2.6). This negativity is larger because the area of the hillock is larger than the initial segment, and also because the axon hillock is driving the depolarization of the soma and proximal dendrites. In fact, the eld potential can be as large as -5 mV within a few microns of the hillock. Again, this reverses in sign far up the apical dendrite because of current out ow there. Is this double-peaked structure observed, or is it an artifact of this model? The amplitude of the rst peak is quite small in most places. With a noise level of 40 V (primarily due to ring of other neurons; David Kewley, personal communication), it is unlikely that it could be resolved except by spike-triggered averaging. A peak with a shoulder might be observable very near the axon initial segment. Often a shoulder on the waveform is seen near the soma of a variety of neurons, including CA1 pyramidal cells (Sperti et al., 1967; Buzsaki et al., 1996), possibly pyramidal tract neurons ( gure 1 of Rosenthal, 1972), and motoneurons (Fatt, 1957; Terzuolo and Araki, 1961; Nelson and Frank, 1964). In some of these examples, two separate peaks rather than just a shoulder can sometimes be discerned (e.g., Fatt, 1957). By direct comparison of intracellular with extracellular voltage, the extracellular A spike has classically been attributed to the axon hillock/initial segment, and the B spike to the soma and possibly proximal dendrites (Terzuolo and Araki, 1961). In the model here, however, the B spike is due not to currents from soma/dendrites but from the axon hillock; this probably could not be discerned experimentally because the potential in the soma very closely follows the potential in the axon hillock, and the transmembrane currents cannot be measured directly. As illustrated by this model, extracellular eld potentials can reveal a good deal about spike initiation and propagation within the cell. As mentioned above, there is still some controversy about exactly where in the cortical cell the action potential initiates, and whether the axon hillock/initial segment has su cient numbers of sodium channels to be the site of initiation. Extracellular elds are a sensitive indicator of the location of currents. In the slice, where it is possible to visualize

26

CHAPTER 2. EPHAPTIC INTERACTIONS

0.02 0 ¦Õ (mV) ?0.02 ?0.04 ?0.06 ?0.08 40 20 Vm (mV) 0 ?20 ?40 0.2 0 Current (nA/?m )
2

(?100, 0) (?50, 150) (?15, ?15) x 0.1

?0.2 ?0.4 ?0.6 ?0.8 35.5

First node Distal initial segment Proximal initial segment Proximal axon hillock Soma (x 10) Apical dendrite (x 10) 36 Time (ms) 36.5 37

Figure 2.6: A comparison of the eld potential and transmembrane currents. The rst negativity in the eld potential corresponds to the maximum current from the distal initial segment. The second negativity corresponds to the maximum current from the proximal axon hillock.

CHAPTER 2. EPHAPTIC INTERACTIONS

27

a single neuron within the tissue, it would be relatively easy to examine the extracellular elds around a cell which can be stimulated by an intracellular electrode. The location where where the earliest negativities are largest is an indicator of where the spike initiates. One could also determine whether the large negativity comes from the axon hillock as predicted from the model of Mainen and Sejnowski (1996), from the soma/proximal dendrites (from the classical motoneuron model), or from further out in the axon (Colbert and Johnston, 1996). In the model, the action potential propagates up the apical tree11 , but dVm =dt is very small because of the much lower sodium channel density. The eld potentials are therefore extremely small (sometimes less than 10 V) and are di cult to observe directly. However, Lorente de No (1947b) was able to examine the elds from thousands of simultaneously activated neurons to conclude that action potentials propagate into the dendrites, and that often propagation fails for successive spikes in a burst. Buzsaki et al. (1996) found the same result for a single hippocampal cell in vivo using spike triggered averaging to estimate the average extracellular waveform more accurately. There are several important features of the extracellular action potential for ephaptic interaction. First, as noted above, it is very large, much larger than the extracellular elds around axons. Second, it is more con ned in space than elds from axons. For an axon in a sheath, for example, / ?Vm (equation 2.5 on page 15), so the eld may be spread out over a mm or more depending on the speed of propagation (see gure 2.1 on page 16). In contrast, the extracellular eld has a large amplitude over only a small region (50 m for this particular model; sometimes over 100 m measured experimentally). The eld has a much larger gradient, and this turns out to be important for interaction.

2.3 E ect of extracellular potential on neural elements
Ephaptic interactions between axons have been observed in a variety of preparations where the extracellular resistance is arti cially increased, including squid giant axon (Arvanitaki, 1942; Ramon and Moore, 1978), crab motoneuron axons (Katz and Schmitt, 1940, 1942), frog sciatic nerve (Kocsis et al., 1982), and even algal strands (Tabata, 1990). This situation has attracted a good deal of theoretical interest because of the simplicity of the geometry. A number of studies have addressed the e ects of electrical elds on axons, particularly myelinated axons, because of their importance in electrical stimulation experiments. Some modeling work in the hippocampus has shown the importance of coupling to the extracellular potential (Traub et al., 1985a, 1985b) in epileptiform bursts. There has been almost no attempt to examine the e ect of the eld potential produced by a single spike on other cells. Rall (1962) guesses that coupling could have an e ect based on preliminary calculations which he does not explain. I am unaware of any subsequent work on the problem of ephaptic interaction of cells rather than axons. Obviously, if the potential of the whole brain is raised relative to a distant ground, there will be no e ect at all on neural activity, since the intracellular potentials will rise by the same amount. It is the gradients of the extracellular potential, rather than the magnitude of the potential itself, determine the e ect.

2.3.1 In nite straight cables
Ve , as shown in gure 2.712 . Summing the currents into the junction gives (Clark and Plonsey, 1971;
11 A movie of this can be seen at http://www.klab.caltech.edu/~holt/thesis/. 12 This gure assumes that Ve does not change signi cantly from one side of the cable to the other, i.e., that there is no dependence in Ve . If Ve does depend noticeably on , Ve and Vm can be replaced with their average over

Consider rst the case of a passive unmyelinated cable with a varying extracellular electric potential

and exactly the same equation results.

28 PSfrag replacements

CHAPTER 2. EPHAPTIC INTERACTIONS

Ve (z )
Extracellular

cm dz ri (z ? dz=2) dz Vi (z ? dz ) Vi (z )

gm dz ri (z + dz=2) dz Vi (z + dz )

Vm = V i ? V e
Intracellular

Figure 2.7: Circuit for computing the e ect of extracellular potentials. Plonsey and Barr, 199513) 1 2 cm @Vm + gm Vm = r @ Vi @t i @z 2 (2.19) 1 @ 2 Vm + @ 2Ve = r @z 2 @z 2 i since Vm = Vi ? Ve . With the usual de nitions = cm =gm and 2 = 1=ri gm, this becomes14 @Vm + V = 2 @ 2 Vm + @ 2 Ve (2.20) m @t @z 2 @z 2 The extracellular potential acts like a distributed current (the ephaptic current ) with a magnitude of ieph = (1=ri )@ 2 Ve =@z 2 per unit length. For an intuitive understanding of what this means, it is helpful to consider the Fourier transform of equation 2.20 in both space (k) and time (!): ^ ^ ^ ^ i! Vm + Vm = ?k2 2 Vm ? k2 2 Ve (2.21) or

?2 2 ^ ^ Vm = 1 + i!k + k2 2 Ve :

(2.22)

^ ^ Clearly the biggest that Vm (k) can ever be is ?Ve (k), and that occurs for large k2 2 . Therefore the largest Vm (z ) can be is ?Ve (z ). This occurs when Vi (z ) is approximately constant, i.e., when the spatial scale of the changes of Ve (z ) is much less than the length constant of the ber at the relevant frequencies.
13 An alternative formulation based only on Vi and Ve instead of Vm and Ve is possible but involves time derivatives of Ve (Rubinstein and Spelman, 1988). Electrodes sense Vi ; however, membrane channels sense Vm , so Vm is more relevant for the biophysics. 14 Note that this value is di erent from in the core-conductor model when the extracellular space was treated as one-dimensional (equation 1.17 on page 7). In that case we assumed that the only contribution to Ve was from the axon under study; here we assume that the contribution to Ve from the axon under study is negligible, and the only contributions come from other neurons.

CHAPTER 2. EPHAPTIC INTERACTIONS

29

Interaction between two axons

This upper bound is not reached in practice for axonal interaction. Instead, for the term i! in the denominator of equation 2.22 dominates, implying that most of the current is capacitive (Clark and Plonsey, 1971). For example, in the simulations of gure 2.1, if the axon is placed in a sheath, Ve / ?Vm , so the dominant spatial frequency in Ve is about k = =1000/ m (based on the half-width of the action potential), and the dominant temporal frequency will be ! = kv = 440 =1000/ms. If there is another axon at rest with the same parameters nearby ( = 220 m and = 1:1 ms), then k2 2 0:5 and ! 1:5; the di erence between these is more extreme for the axon considered by Clark and Plonsey (1971). k2 2 in the denominator can be neglected: ^ ^ i! Vm ?k2 2 Ve : (2.23) Taking the inverse transforms,

If the action potential is propagating with a velocity , then z = t, and equation 2.24 becomes

@Vm @t

2 2 @ Ve @z 2

(2.24) (2.25)

Vm

2 @Ve :

The membrane will therefore be excited with a spatial distribution given by @Ve =@z . Since in a nerve surrounded by a sheath Ve is proportional to the negative of the intracellular voltage of the active ber (equation 2.5 on page 15), an inactive ber's membrane will rst be hyperpolarized, then depolarized. This pattern of depolarization and hyperpolarization also explains the result when both axons conduct action potentials. If the action potentials are almost abreast, their current sources and sinks are lined up ( gure 2.8 top). Each attempts to hyperpolarizes the other, and both will slow down. If one is slightly behind the other ( gure 2.8 bottom), the leading axon will depolarize the lagging axon and speed its action potential up. Staggered action potentials are stable; the outward currents in one axon line up with the inward currents in another. As a result, even if the propagation velocities are di erent when each axon is alone, the two action potentials will still propagate in step. Both action potentials can propagate faster than either would alone (Katz and Schmitt, 1942; Maeda et al., 1980; Tabata, 1990; Barr and Plonsey, 1992) because propagation of an action potential in the second ber changes the e ective extracellular resistance.
Aligned Action Potentials

@z

Staggered Action Potentials

Figure 2.8: Patterns of current ow in two interacting axons. Top, action potentials exactly aligned. In this case, both axons try to hyperpolarize the other at the front of the action potential, so both slow down. Bottom, action potentials staggered, with sources and sinks lined up. The currents from the leading axon depolarize the lagging axon at the front of its action potential, speeding it up.

A battery of theoretical studies has addressed the phase-locking of action potentials in adjacent bers mathematically and with simulations15. However, unless precise spike timing (less than 1
15 Markin, 1970a, 1970b, 1973a, 1973b; Scott, 1977; Scott and Luzader, 1979; Maeda et al., 1980; Eilbeck et al., 1981; Keener, 1989; Barr and Plonsey, 1992; Bose and Jones, 1995

30

CHAPTER 2. EPHAPTIC INTERACTIONS

ms) is important for the information carried by small unmyelinated bers, the synchronization and phase-locking e ects will be unimportant even if the extracellular potentials were large enough to have a noticeable e ect. For this reason, I will not consider ephaptic interaction between two axons further. Myelinated axons are more sensitive to extracellular elds for several reasons. They have larger diameters and lower membrane conductances. Therefore is longer, and the k2 2 terms in equation 2.22 are relatively larger. The capacitance is much lower, lowering the i! term. Therefore Vm will be closer to its limit of ?Ve . The nodes of Ranvier have low thresholds, making it more likely that a transmembrane potential shift will initiate an action potential. Several studies have estimated the response of myelinated axons to various kinds of electrical stimulation by the experimenter (e.g., Ranck, 1975; MacNeal, 1976; Tranchina and Nicholson, 1986; Rattay, 1987; Altman and Plonsey, 1990; Rubinstein, 1991). Although myelinated axons are more sensitive to extracellular elds, they generate much smaller electrical elds, so interaction seems unlikely to be signi cant.

Interaction of cell bodies and in nite cables

Axons and dendrites in cortex also pass next to cell bodies, where the extracellular elds are much larger. The elds around cells are much more con ned than around axons, and therefore k is much bigger. From gure 2.6, the dominant spatial frequency in the action potential is around k = =50/ m and the dominant temporal frequency is ! = =0:15/ms. For the axon of gure 2.1 passing by this cell body, k2 2 190 and ! 20. Unlike axonal interaction, then, the capacitive current is much smaller than the axial currents. As a result, Vm ?Ve and has the same time course as Ve . Despite the crudity of this analysis, it is not a bad good predictor of the transmembrane potential. The actual transmembrane potential is shown in gure 2.9. In fact, Vm is almost equal to ?Ve , especially near the cell body where Ve is changing rapidly with position. Although ephapses can be characterized by localized current injection (section 2.3.1), just as synapses can, this example shows that they have somewhat di erent properties. Because the ephaptic current depends on the second derivative of the extracellular potential (equation 2.20), a peak in the extracellular potential produces an ephaptic current which is depolarizing at the peak and hyperpolarizing anking the peak (see the currents in gure 2.9). In fact, the total ephaptic current into an cable is given by

Itot =

Z

1 = r zlim @Ve ? z!?1 @Ve lim @z i !1 @z = 0: As a result, the transmembrane potential is more localized and does not spread in the same way as a point current source injection would. The Fourier analysis also predicts what the e ect of parameter variations is on induced potential. Since k2 2 ! , changing = cm =gm by changing the capacitance within reasonable limits p should have little e ect on the result ( gure 2.10A). Also changing = i a=2gm by changing the intracellular resistivity ( gure 2.10B) or the cable diameter ( gure 2.10C) within physiological limits has little e ect because k2 2 appears in both the numerator and the denominator. Changing the membrane conductance gm has the same e ect on both k2 2 and ! and has little e ect on the result as long as ! + k2 2 1 ( gure 2.10D). In general, for cables passing near cell bodies, parameter variations within reasonable physiological limits have only small e ects on the peak amplitude or the time course of the induced transmembrane voltage. The only parameter that has a large e ect on the magnitude of ephaptic voltage transients is the location of the cable. The magnitude of the extracellular electrical eld

1 1 @ 2 Ve dz ?1 ri @z 2

CHAPTER 2. EPHAPTIC INTERACTIONS

31

37 36.9 36.8 36.7 36.6 36.5 36.4 Time (ms) 36.3 36.2 36.1 36 35.9 35.8 35.7 35.6 35.5 ?100 ?50 0 50 Position along axon (?m) ?Ve Vm ieph 1.5 mV 0.05 nA/?m 100

Figure 2.9: The membrane potential of a long straight passvive cable located near a cell body. Extracellular potential, induced transmembrane potential, and the ephaptic current are shown as a function of position for several di erent times during the action potential. This axon was perpendicular to the plane of gure 2.4 and intersected it at (5; ?20) (near the axon hillock).

32

CHAPTER 2. EPHAPTIC INTERACTIONS

Membrane Capacitance 3 2 1 0 ?1 Potential (mV) 3 2 1 0 ?1 35 36 37 35 36 Radius 3 37 38 3

Intracellular Resistivity 2 1 0 ?1 35 36 37 38

A.

?Ve 2 0 ?F/cm 0.8 1.6

B.

?Ve 35 ? cm 70 300

Membrane Leak ?Ve 1 ?m 0.5 0.2 0.05 2 1 0 38 ?1 35 36 37 38

C.

D.

?Ve 1.4 2 7 ?S/?m 35

Time (ms)

Figure 2.10: E ect of varying cable parameters on the transmembrane voltage induced by a spike in a nearby cell. Only the voltage at the center of the cable (the peak voltage) is shown. A: Variations in capacitance cm , a ecting only . B: Variations in intracellular resistivity, a ecting only . C: Variations in the radius of the axon, which a ects only . D: Variations in the membrane conductivity, which a ects both 2 and proportionately. The parameter con guration used for gure 2.9 is shown with a solid black line in all four panels. In all cases, there is little e ect within physiological parameter regimes.

CHAPTER 2. EPHAPTIC INTERACTIONS

33

decreases sharply with distance; more than 10 m away from the axon hillock, the peak extracellular potential amplitude has dropped to less than 1 mV, and induced transmembrane voltages also drop by the same amount.

2.3.2 Finite or bent axons, cells, and dendrites
Neural elements are straight for long distances only in nerve tracts, and perturbations of action potential timing are unlikely to be of great signi cance in such cases. Potentially more interesting interactions occur in neuropil, where it is much more important to consider neural elements with sharp bends, terminations, and branches. The e ect of the extracellular potential is mediated by the derivative of its gradient in the direction of the axon or dendrite, and the gradient changes abruptly when the direction of the axon or dendrite changes. Therefore the largest e ects can be seen at bends in neural processes (Markin, 1973b; Tranchina and Nicholson, 1986). In neuropil, the modi ed cable equation 2.19 on page 28 must be rewritten in terms of an arc length parameter s instead of distance z , where the cable is described parametrically by x = x(s), y = y(s), z = z (s). With this modi cation, summing the currents into the node in gure 2.7 gives

@ cm @Vm + gm Vm = @s @t @ = @s

ri @s 1 @Vm + @ 1 @Ve ri @s @s ri @s

1 @Vi

(2.26)

As discussed previously, this is exactly the standard cable equation except that Vm replaces Vi and there is a distributed current (the ephaptic current) injected into the cell of magnitude

@ 1 1 2V @ 1 ieph = @s r @Ve = @s r @Ve + r @@s2e i @s i @s i

(2.27)

per unit length. The derivatives of the extracellular potential at the surface of the cable Ve depends on the extracellular potential in the tissue volume in a somewhat complicated way. We assume that on one side of the cable is approximately the same as on the other side of the cable, so we can treat the cable as one dimensional and use Ve = at a point instead of de ning Ve as the average of over the cable's circumference.

@Ve = r T = ?E T @s
where E is the electric eld and T is the normalized tangent vector,

(2.28)

T = dx ; dy ; dz : ds ds ds
The e ective current is proportional to the second derivative,

(2.29)

@ 2 Ve = ?T @ (Ex ; Ey ; Ez ) T ? E dT @s2 @ (x; y; z ) s

(2.30)

where @ (Ex ; Ey ; Ez )=@ (x; y; z ) is the Jacobian of E (the Hessian of ? ). Axons and dendrites in neuropil tend to have kinks rather than smooth bends, so T is discontinuous and dT=ds is a sum of functions. As a result, the current source consists of a distributed current (the rst term in equation 2.30) and a series of point current sources at each bend (the

34

CHAPTER 2. EPHAPTIC INTERACTIONS

second term). The magnitude of each point current source is

Ieph = ?E

T(s+) ? T(s? )
ri (s+ ) ri (s? )

(2.31)

where T(s+ ) and T(s? ) are the tangent vectors on each side of the kink. A similar situation occurs at a branch point. Summing the currents into the node in gure 2.11 gives 1 @Via = 1 @Vib + 1 @Vic ; (2.32) or 1

@Vma + @Vea = ria @sa @sa 1 @Vmb + @Veb + 1 @Vmc + @Vec : (2.33) rib @sb @sb ric @sc @sc

Ta

If we consider the transmembrane potential instead of in- Figure 2.11: Calculation of ephaptracellular potential, it looks like there is a current of magni- tic current at a branch point. tude Ieph = ? r1 @Vea + r1 @Veb + r1 @Vec a @sa b @sb c @sc (2.34) 1E T ? 1E T ? 1E T =r a r b r c
a b c

injected at the node. There is also an e ective current injected at the ends of axons or dendrites. At the end of the cable, no intracellular axial current ows (sealed end condition): 1 @Vm @Ve 1 @Vi ri @s = 0 = ri @s ? @s : Once again, in terms of Vm , it is as if there is a point current source of magnitude I = 1 @Ve = ? 1 E T
eph

(2.35)

ri @s

ri

(2.36)

located at the end. These special cases are all subsumed by a general rule: at any point along the cable, the ephaptic current is (2.37) I =?1E T
eph

ri

summed over all of the cable segments that join at that point. This rule is also applicable for discontinuous changes in ri . How much in uence do these ephaptic currents have? Extracellular potentials produced by cell spiking, at least from this particular pyramidal cell model, change very rapidly in space, and the induced transmembrane potential is already almost equal to the extracellular potential (section 2.3.1); Vm cannot grow any larger. For elds which do not change so rapidly in space, such discontinuities focus the e ects of the electrical eld and are likely to be important. For example, if electrical stimulation causes an action potential, it is much more likely to initiate at a discontinuity (Tranchina and

Tc
r ib ds b

ria @sa

rib @sb

ric @sc

PSfrag replacements

ria dsa

ric

ds

c

Tb

CHAPTER 2. EPHAPTIC INTERACTIONS

35

Nicholson, 1986). With regard to interaction between cells, if the model for generation of extracellular potential is inaccurate and elds do not change as rapidly as it predicts, then kinks, terminals, and bifurcations will be the only places where Vm ?Ve .

2.4 Where does ephaptic interaction occur?
Ephaptic interaction has observed experimentally in a number of non-physiological situations, such as axons placed next to each other in a highly resistive medium (Katz and Schmitt, 1940, 1942; Arvanitaki, 1942; Ramon and Moore, 1978; Tabata, 1990) and in simultaneous electrical stimulation of many axons (Kocsis et al., 1982) or cells (Nelson, 1966; Magherini et al., 1976; Dalkara et al., 1986; Turner and Richardson, 1991). It is clearly important in a number of pathological situations (reviewed in Faber and Korn, 1989; Je erys, 1995). So far, there have been only two clear demonstrations of its e ect in normal operations. In the case of the Mauthner cell (see Korn and Faber, 1980; Faber and Korn, 1989) the extracellular resistivity in the surrounding space is much larger than in most other systems by a factor of ve or so, and as a result extracellular eld potentials from ring of the Mauthner cell can be as large as 50 mV. In this case, inhibition has been observed bidirectionally, from the Mauthner cell to its inhibitory a erents and vice versa. In the mammalian cerebellum, basket cells form a cap around Purkinje cell bodies. Because of glial cells and tight junctions, there is a strong barrier to current ow. The spike does not propagate actively into the synapses of the basket cell; as a result, there is a large current out ow from the synapse rather than an in ow, and the extracellular potential from the synapse is positive rather than negative. The Purkinje cell thus experiences a transient electrical hyperpolarization, followed by inhibition from the chemical transmitter (Korn and Axelrad, 1980). The purpose for this electrical inhibition is unclear, since there is no obvious reason why a very fast inhibition is necessary. In fact, the purpose of the system may have nothing to do with ephaptic inhibition. For example, suppose that the tight junctions are present for some other reason (e.g., to prevent di usion of neurotransmitter or some other chemical) and the ephaptic interaction is a side e ect. There is also a reciprocal ephaptic interaction: when the Purkinje cell res, the basket cell's synaptic terminals will be strongly depolarized. If active conductances were present in the terminals, it might be possible for a spike to be initiated accidently ephaptically in the basket cell terminals. Perhaps for this reason the terminals are not active, and an unintended side-e ect of inactive terminals is ephaptic inhibition. So far there has been no clear evidence for ephaptic interactions in systems without such unusual geometries. For a long time there has been a suspicion that ephaptic e ects could be important in the hippocampus, where the extracellular resistivity is somewhat higher than elsewhere in the brain. Several studies have shown signi cant e ects of eld potentials in response to electrical stimulation (Dalkara et al., 1986; Turner and Richardson, 1991) but so far no interactions without electrical stimulation are known except in epilepsy (Snow and Dudek, 1984; Traub et al., 1985a, 1985b). Ephaptic e ects are probably not interesting if all they do is accelerate or retard an action potential, as for a nerve bundle (section 2.3.1). Much more interesting are the cases where an action potential which would not otherwise be present occurs because of an action potential in a neighboring cell, or where an action potential that would have been present disappears. Ephaptic e ects are not likely to be very large, since extracellular elds are almost never over a few mV, and probably never cause a spike on their own except in some pathological cases which are only of medical interest. They can a ect action potential generation only in structures close to threshold: near the spike initiation zone of a neuron, or at points with low safety factor for propagation. It is still unclear exactly where the spike initiates (Regehr and Armstrong, 1994). Early studies of motoneurons indicated clearly that the spike initiated somewhere in the axon, presumably the initial segment (Fuortes et al., 1957; Coombs et al., 1957a, 1957b; Dodge and Cooley, 1973). More recent studies reached the same conclusion for neocortical pyramidal cells (Stuart and Sakmann, 1994). In hippocampus, it was formerly thought that the fast prepotentials observed in somatic

36

CHAPTER 2. EPHAPTIC INTERACTIONS

recordings were dendritic spikes, but now it seems more likely that these are impulses of adjacent neurons conducted through gap junctions (MacVicar and Dudek, 1981). In these cells too, action potentials appear to initiate in the axon, probably at the rst node of Ranvier rather than the axon initial segment, and propagate into the dendrites (Colbert and Johnston, 1996). In all of these cases, the site of action potential initiation is somewhat removed electrotonically from the synaptic input and has a very high density of sodium channels in order for it to have a lower threshold than the soma. For several reasons one might expect ephaptic e ects to be noticeable occasionally at the sight of initiation. First, it is close to several di erent geometrical inhomogeneities. There is a change in diameter from the soma and initial segment. Furthermore, bends and bifurcations near the soma are not uncommon. Second, cell bodies are lined up in layers in cortex and especially hippocampus; the sites of initiation are where the electrical elds are the largest are thus close to each other. Once the action potential initiates, it propagates down the axon and up into the dendrites. For many years there has been speculation that under some circumstances propagation can fail at branch points in the axonal arborization (Swadlow et al., 1980). For example, action potentials from cerebellar granule cells might fail at the T-junctions with the parallel bers (James Bower, personal communication). More recently, there is solid experimental evidence that failure occurs at dendritic branch points (Ho man et al., 1997) and also presumably at axonal branch points (Debanne et al., 1997). In each of these cases, the place where the action potential is most likely to fail is also the place where ephaptic e ects are largest: at a bifurcation. Interaction could occur if the action potential causing the large extracellular eld is almost exactly coincident with the action potential in the dendrite or axon. Alternatively, if an A current is present (as suggested by recent results of Ho man et al., 1997 in dendrites and Debanne et al., 1997 in axons), and it can be somewhat inactivated by ephaptic depolarization, ephaptic interaction might reduce branch point failure even if the spikes are not exactly coincident. Dendritic bundles have been another candidate for ephaptic interaction (e.g., see Schmitt et al., 1976; Roney et al., 1979; Faber and Korn, 1989). Dendrites of many kinds of cells in many regions of the brain and spinal cord, including apical dendrites of neocortical pyramidal cells, come together in bundles (see Roney et al., 1979 for a review) and the dendrites are often weakly active. Sometimes the dendrites are coupled through gap junctions16 ; sometimes gap junctions are absent and it is possible that ephaptic interactions mediate coupling (Matthews et al., 1971; Zupanc, 1991). However, I nd that potentials surrounding dendrites are likely to be very small, except possibly very near the soma, unless all the cells in the bundle re together. If the purpose of dendritic bundling is to facilitate interaction, then the interaction is probably chemical rather than merely electrical. Ephapses around cell bodies can be considerably stronger than typical excitatory chemical synapses between pyramidal cells. Unlike chemical synapses, however, they are di cult to modify once a neurite is grown because the magnitude of the depolarization is roughly independent of the cable properties of the post-ephaptic membrane (section 2.3.1). If ephaptic e ects computationally useful, then it would be reasonable to expect that growth cones are directed by the electric elds set up by cell spiking activity. Electric elds are known to have an in uence on some growth cones. Depending on the type of cell and the experimenter, the magnitudes of the e ect range from no e ect even at elds as high as 200 mV/mm, to a noticeable e ect at elds as weak as 2.5 mV/mm (see McCaig, 1988). Neurites are usually attracted to the cathode but in some cases they grow toward the anode. Axons and dendrites of the same cell may be a ected di erently; for example, dendritic growth cones on hippocampal cells in culture turned toward a cathodal microelectrode but axonal growth cones were una ected (Davenport and McCaig, 1993). Substantial DC voltage gradients exist in the developing embryo, and electric elds may be one of the cues that growth cones use for guidance (McCaig and Zhao, 1997). The vast majority of work on galvanotropism is concerned only with DC electrical elds; as far as I know, only one paper examines the e ect of elds of the kind that might be produced by
16 See Dermietzel and Spray (1993). In neocortex in development: Peinado et al. (1993); in adult hippocampus: MacVicar and Dudek (1981); in motoneurons: Magherini et al. (1976).

CHAPTER 2. EPHAPTIC INTERACTIONS

37

neural spiking activity (Patel and Poo, 1984). These experimenters found that Xenopus neural tube neurons grew asymmetrically when pulsed elds as low as 50 mV/mm (the lowest value tested) were applied; in general, the amount of asymmetry was the same as if the neurons were exposed to a DC eld with the same time averaged value. Most neurites tended to grow toward the cathode, both in DC and pulsed elds. The same experimenters also performed time-lapse photography studies which showed that individual growth cones turned toward nearby microelectrodes when the eld strength at the growth cone was as low as 6 mV/mm as long as the pulse frequency was su ciently high. Fields of this magnitude certainly exist around spiking neurons, though they are very brief. If elds from action potentials do have an e ect, it seems that they would guide the growing neurites nearer to cell bodies of neurons which are active in development. Can the brain use short range ephaptic interaction to do anything useful, or is it just unwanted cross talk? Ephaptic interactions which a ect branch point failure in axons could be used as a kind of switch: if cells in one region have just been active, then action potentials are more successfully propagated into that region. The purpose of action potential propagation in dendritic trees is not yet as clear; if it has to do with Hebb-based learning, ephaptic interactions could a ect the signal that conveys the cell's own ring rate to its distal tips, so learning would occur only if the cell and neighbors located at strategic positions have both been active. In both of these cases, however, ephaptic interactions would be much smaller than the e ect of repetitive activity in the axon or dendrite itself. Ephaptic interaction is not trivial to observe experimentally using microelectrodes. Some investigators have used two electrodes, or a double-barelled pipette electrode and compared the extracellular and intracellular potentials directly. In hippocampal epileptic bursts, Snow and Dudek (1984) found that often the intracellular potential brie y decreased immediately before an action potential, raising the question of what triggered the spike. However, the transmembrane potential showed a clear depolarization leading up to the spike. This nding, that the actual change in Vm is opposite in sign to Ve , is typical of ephaptic interactions; Ve and Vi move in the same direction, but Vi moves less than Ve . A two-electrode experiment to test for ephaptic interaction on the distance scales examined here would be di cult because the two electrodes have to be so close together; one must be inside a small structure like an axon or dendrite, and the other must be within a few microns of the membrane. Another way to look for speci c ephaptic interaction would be to work in slices where synaptic transmission has been pharmacologically disabled. Because of new microscopy techniques, it is now possible to record from the soma and the distal apical dendrites of the same pyramidal cell. Propagation of action potentials up the apical tree, and failure at branch points, have been observed in such preparations (Spruston et al., 1995; Ho man et al., 1997). Action potentials could be evoked in the soma and recorded in the dendrites, while at the same time activating nearby cells either by glutamate iontophoresis, a minimal stimulation protocol, or even intracellular stimulation. Since normal chemical synaptic transmission is disabled, any e ects on propagation failure must be electrical. Gap junctions can be ruled out by closing them pharmacologically, or possibly by examining the intracellular records for depolarization in one cell when the other spikes. Of course, propagation of action potentials up the apical tree of a neuron in vivo will be strongly in uenced by the extensive ongoing synaptic activity, so this experiment can only indicate whether the e ects are large enough to be signi cant. This technique is more di cult to apply to axonal branch point failure unless one can reliably trace an axon and measure intracellularly at di erent points in the arbor. Ephaptic interactions with magnitudes of several mV are just on the border of being signi cant. The magnitude of these interactions is approximately proportional to the extracellular resistivity; if this were much higher, ephaptic e ects would be much more widespread in the central nervous system. Some cross-talk may occasionally be useful for computation, but widespread crosstalk would probably be damaging. Extracellular resistivity is controlled primarily by the size and tortuosity of the extracellular medium. Since ephaptic e ects are just on the border of being widespread, it is possible that the fundamental limits on average spacing between neural elements are set by the constraint to minimize ephaptic interactions.

38

CHAPTER 3. EXTRACELLULAR POTASSIUM AND OTHER DIFFUSIBLE SIGNALS

Chapter 3 Extracellular potassium and other di usible signals
3.1 Introduction
Besides the ephaptic e ects considered in chapter 2, there are a number of other possible non-synaptic interactactions between neurons. Neurons clearly use some means of communicating without morphological specializations, e.g., gaseous messengers such as NO or CO. A number of other chemical interactions are possible, mostly based on byproducts of normal operation. For example, there has been considerable interest in di usion of neurotransmitter away from the synapse, possibly activating adjacent neural elements. Often glia are present around synapses to prevent such di usion (see Peters et al., 1991, pp. 288-290), but in some cases there is evidence of a signi cant e ect (see Fuxe and Agnati, 1991). Extracellular pH changes in neural activity because of neural metabolism, and extracellular pH has a profound in uence on neuronal excitability. Rapid pH decreases of 0.1{0.2 pH units results in a net decrease in excitability (Sykova, 1991). However, even during intense stimulation the pH rarely changes by more than 0.1 units except in pathological cases, so pH is unlikely to be a means of communication. Other kinds of non-synaptic interaction have been suggested, mostly based on glia. For example, it is clear that NMDA receptors require glycine or D-serine in the extracellular medium to function at all. It is unclear where the required amounts come from. Interestingly, astrocytes do contain both glycine and D-serine, and they release it into the extracellular space non-synaptically via an antiporter mechanism in response to depolarization. Glial cells depolarized in response to ambient glutamate or other neurotransmitters or extracellular potassium increases (Attwell et al., 1993; Shell et al., 1995). Extracellular concentrations of other kinds of neurotransmitter are also controlled by glia (Cull-Candy, 1995; Pfrieger and Barres, 1996). It is possible, then, that glia modulate synapses in computationally interesting ways through controlling transmitter concentrations in the extracellular space. This area is not well enough explored experimentally to make any conclusions yet. Probably the most important possible kind of chemical non-synaptic interaction is due to changes in extracellular ion concentrations. Concentration di erences of ions across neural membranes give rise to most electrical activity in neurons, and therefore changes in these concentrations are potentially of great signi cance for the computations neurons perform (see reviews in Somjen, 1979; Erulkar and Weight, 1979; Sykova, 1983, 1991, 1997; Nicholson and Rice, 1991; Erecinska and Silver, 1994; Je erys, 1995). Outside the neuron, the concentration of potassium is low and the concentration of sodium is high. For this reason, the extracellular sodium concentration does not change signi cantly, while the extracellular potassium ion concentration can sometimes double or triple in periods of intense activity. Intracellularly, the situation is reversed, so potentially sodium accumulation could be important. However, the intracellular volume is usually much larger than the e ective extracellular volume so concentration changes play little role (except in dendritic spines; Qian and Sejnowski, 1989, 1990). In the peripheral nervous system, extracellular potassium seems to be cleared primarily through di usion and reuptake by neurons. As a result, potassium transients can last up to 50{100 s (Hoppe et al., 1991). In the central nervous system, potassium homeostasis seems to be one of the primary functions of astrocytes. At rest, their membrane has a large potassium permeability, enabling them to bu er it rapidly. They can also pump potassium faster than neurons. They are coupled through gap junctions, enabling potassium and other molecules to be moved more rapidly over long

CHAPTER 3. EXTRACELLULAR POTASSIUM AND OTHER DIFFUSIBLE SIGNALS

39

distances (Gardner-Medwin and Nicholson, 1983; Gardner-Medwin, 1983a, 1983b). There has been some speculation that the astrocytic network could regulate neuronal activity in computationally important ways by distributing extracellular potassium according to some blueprint, but at the present this seems unlikely (see Somjen, 1979). Changes in extracellular potassium occur over large regions in epilepsy or when a nerve is repeatedly shocked electrically, because many cells are simultaneously active and the potassium homeostasis mechanisms are overloaded. In such cases, K+ ] can rise from 3 mM to 10 mM or more (20{50 mM in the case of spreading depression). Extracellular potassium changes are therefore undoubtably important for various kinds of epilepsy (see Sykova, 1983, 1997; Je erys, 1995). It is less clear whether they are important in normal operation. Small increases in extracellular potassium concentrations do occur in response to sensory stimuli. Large changes of up to 3 mM ( EK 20 mV) can be measured in spinal cord 5{10 minutes after injecting formalin or turpentine into a rat's paw, apparently due to abnormal self-sustained neuronal ring in response to such painful stimuli (see Sykova, 1991). More normal sensory stimuli cause less signi cant but still noticeable changes. For example, light touches or pinches cause a change of 0.1 to 0.5 mM ( EK = 0:8 to 4.2 mV) in spinal cord (Sykova, 1991). When light is turned on a toad eyecup, there is a slow decrease in K+ ] of up to 0.5 mM between the photoreceptors and the pigment epithelial cells, and an increase of 0.4 mM ( EK = 3:4 mV) primarily in the inner plexiform layer (Karwoski and Proenza, 1987). In visual cortex, passing a bar across a cell's receptive eld caused a change of 1{2 mV in EK (Singer and Lux, 1975; Lux, 1976) or possibly higher ( EK = 7 mV, calculated from depolarization of glial cells). It is hard to see how such increases in extracellular potassium could be computationally useful, since they occur over a large area and they have a time course of hundreds of milliseconds; all neurons in the area would be a ected, and the a ect would last long after the stimulus is gone. More interesting from a computational viewpoint is that extracellular potassium ions can act like a speci c neurotransmitter over short distances. The action potential in the presynaptic neuron causes K+] in the synaptic cleft to rise, depolarizing the postsynaptic cell. In some specialized systems, this e ect is known to be important. For example, the calyx synapse from the type I hair cell completely surrounds the cell body of the hair cell. Goldberg (1996a, 1996b) has shown that because of this geometry, K+ ] rises signi cantly in the cleft and augments the response to the traditional neurotransmitter (ACh). In some cases, K+ is the only neurotransmitter. The giant interneurons of the cockroach come together in close apposition, and there is clearly an excitatory interaction between them, but there is no chemical synapse or gap junction. Instead, potassium released by an action potential in one is su cient to depolarize the other (Yarom and Spira, 1982; Spira et al., 1984). There have only been a few attempts to record potassium concentration changes from single spikes. A change of 0.02 mM ( EK = 0:2 mV) was recorded for single spikes from mesencephalic reticular formation neurons. Bursts of activity caused larger responses (0.2 mM, EK = 1:7 mV; Sykova, 1991). For technical reasons, however, the actual change near the neuron is certainly much higher. First, ion-selective microelectrodes have a response time of at least several milliseconds due to the ion exchanger; simulations show that the largest local increases have a time course shorter than this (see below). Second, recording electrodes necessarily destroy the extracellular space. This is not a problem when recording eld potentials because the voltage does not change much over distances comparable to the size of the electrode, but it is a serious problem when measuring concentrations, because the peak concentration change is expected to be localized to only the cleft immediately adjacent to the active neuron (see below). Hence the actual K+ ] transients due to single action potentials cannot be measured experimentally with current technology except at considerable distances from the cell. The transmembrane current caused by changes in extracellular potassium is proportional to the area of the membrane over which the change occurs (see section 3.7. As a result, the e ect of changes in extracellular potassium will not be signi cant if it only occurs over a tiny area. Since I am not attempting to study increases in potassium over large volumes caused by the activity of

40

CHAPTER 3. EXTRACELLULAR POTASSIUM AND OTHER DIFFUSIBLE SIGNALS

many neurons, the only remaining kinds of possible interaction are cases where neural elements are apposed to each other for large areas, e.g., dendritic or axonal bundles. The changes in potassium due to a spike at a cell body is likely to be larger because the potassium currents are larger, but it is also much more localized and so will have much less e ect. In this chapter, I have not attempted to compute the changes in extracellular potassium very precisely; instead, I have used simple approximations to estimate whether the e ect is appreciable and worth modeling in some detail. These approximations indicate that potassium from single spikes is not likely to have a signi cant e ect anywhere, so it is not worth simulating in detail on a ne spatial scale. A number of other theoretical studies have examined extracellular potassium on a much coarser spatial scale and longer time scale (Vern et al., 1977; Cordingley and Somjen, 1978; Green and Tri et, 1985).

3.2 Nernst potential of Potassium
If the baseline extracellular potassium concentration is nK;o before = 3 mM (1:8 106 K+ ions/ m3; see Erecinska and Silver, 1994 for dozens of references) and it changes to nK;o after, the potassium reversal potential changes by

EK = kT log nK;o after ? kT log nK;o before e nK;i e nK;i kT log nK;o after = e nK;o before EK is approximately a linear function of nK over the possible parameter range ( gure 3.1) with a slope of kt=enK;o = 1:4 10?5 mV- m3 /ion.
12 10 8 E (mV) 6 4 2 0 0 2 4 63 ?n (ions/?m )
K

Figure 3.1: Potassium reversal potential as a function of the change in concentration of potassium. Over the possible relevant parameter regime, this is approximately linear.

K

8

10 5 x 10

3.3 Magnitude of potassium ux across membrane
If the membrane capacitance is 0.8 F/cm2 , it is easy to show that about 5000 ions/ m2 must ow across the membrane to change the voltage by 100 mV (approximately the height of an action potential). The number of potassium ions that actually ow across the membrane is considerably larger than this, however, because the sodium channel has not entirely shut o when the potassium channels open; there is overlap in the conductances. In measurements with radioactive tracers in a variety of unmyelinated invertebrate systems, the actual number of ions is usually about a factor of 3

CHAPTER 3. EXTRACELLULAR POTASSIUM AND OTHER DIFFUSIBLE SIGNALS

41

larger than this minimum estimate (Cohen and De Weer, 1977). As a result, about 15,000 potassium ions/ m2 cross the membrane. (I could calculate exactly how much K+ crosses the membrane given a particular model of the action potential, but I do not have a model I am su ciently con dent in to make the prediction better than the empirical estimate.) In the discussion that follows, I will assume that the ux of potassium, JK , is constant for the time of repolarization and then suddenly drops to 0. It would be possible to calculate exactly the detailed time course of the extracellular potassium given a particular model, but since model parameters are not known, there is no advantage in using a smoothly varying time-dependence for JK . Since the action potential repolarization phase lasts about 0.5 ms, this gives us JK 30; 000 ions/ m2 /ms. This provides an upper limit on K+ ] changes since it is likely that a signi cant fraction of the potassium crosses the membrane more than 0.5 ms later than the repolarization of the action potential.

3.4 Di usion model
On the time scales we are interested in, potassium mobility is governed entirely by di usion (the Nernst{Planck equation, equation 1.1 on page 4) and passive transport across membranes; active pumping is much too slow. The extracellular space is full of barriers to di usion; there are membranes of axons, dendrites, and glia. To obtain an upper bound, I assume that these membranes are relatively impermeable to potassium on. Glial membranes are highly permeable to potassium, and this undoubtably a major means of potassium transport over long distances (Gardner-Medwin and Nicholson, 1983; Gardner-Medwin, 1983a, 1983b). However, it turns out that peaks in potassium concentration are localized to areas so small that there might not be any glia. The Nernst{Planck equation can be used in a tortuous medium such as the extracellular space simply by modifying the di usion constant:

DK = DK 2

(3.1)

where is an empirical constant called the tortuosity of the medium (Nicholson, 1980; Nicholson and Phillips, 1981; see reviews in Nicholson and Rice, 1986, 1991; Nicholson, 1995). Tortuosities have been measured for a variety of tissues and are usually around 1.6 (Nicholson, 1995; Sykova, 1997). If DK = 2:5 m2 /ms (Gardner-Medwin, 1983b; Qian and Sejnowski, 1989), then DK = 0:75 m2 . This may be an overestimate; measurements suggest a value about 5{6 times lower than the coe cient in physiological saline (0.39 m2 /ms; see references in Vern et al., 1977; Cordingley and Somjen, 1978). The other necessary modi cation to the equation is that only a small fraction (usually about 20%, but about 30% in the molecular layer of the cerebellum; Nicholson, 1995; Sykova, 1997) of the volume of the tissue is actually extracellular. This means that the concentrations of ions change by a factor of ve more than they would in a liquid with the same DK . The Nernst-Planck equation describes electrodi usion in three dimensions. However, for axons, we can take advantage of cylindrical symmetry and discard the angular dependence of and nK . Furthermore, action potentials are spread out over a signi cant length of axon; if the propagation velocity is 0.1 mm/ms (for the slowest of the parallel bers in the mammalian cerebellum), and the repolarization lasts 0.5 ms, potassium is being exuded over a length of 50 m, which is much longer than the relevant di usion lengths for short times (see below). In other axons, the action potential will be spread out over much larger distances so axial di usion will be even less important. Therefore, to estimate the short time concentration changes, we also discard the dependence on location along the axon. The problem is reduced to nding the radial dependence of concentration changes around an in nite cylinder. The most signi cant out ow of potassium ions occurs during the repolarization phase of the action potential, where the extracellular eld is positive and the gradient is directed away from

42

CHAPTER 3. EXTRACELLULAR POTASSIUM AND OTHER DIFFUSIBLE SIGNALS

the axon. This electric eld will encourage positive charges to migrate away from the axon, so the concentration changes near the axon will be smaller than they would be without the electric eld. Mathematically, this can be seen from equation 1.1 on page 4 by noting that for xed JK , a larger jr j means a smaller jrnK j. JK will of course not be exactly xed, but unless concentration changes or electric potentials are very large, it is approximately independent of nK and . Therefore if we want to estimate the largest possible magnitude of the concentration changes, it is a reasonable simpli cation to ignore the electric eld. This is in fact not a bad approximation, since the extracellular potential is usually very small, on the order of microvolts except in situations where current ows through a restricted volume (section 2.2.2). Furthermore, the extracellular potentials decay less steeply than the concentration gradients1. So if extracellular potassium ion concentrations change signi cantly, then jrnK j=nK will be signi cantly greater than jr j= K . After all these simpli cations, the problem is reduced to a one-dimensional equation:

@nK = D @ 2 nK + 1 @nK (3.2) K @r2 @t r @r with the boundary condition that JK = DK @nK =@r at the membrane (r = r0 ) is a given value. nK =
Z

The Green's function solution to equation 3.2 can be obtained using an integral transform (Ozisik, 1993, p. 111, or Carslaw and Jaeger, 1959, pp. 341-345). The full solution is
t
Z 1 e?D 2 t R (r)R (r0 ) d G(r; tjr0 ; t0 ) = N( ) 0 R (r) = J0 ( r)Y1 ( r0 ) ? Y0 ( r)J1 ( r0 ) 2 N ( ) = J1 ( r0 ) + Y12 ( r0 );

0

r0 G(r; tjr0 ; t0 ) JK dt0

(3.3) (3.4) (3.5) (3.6)

where nK is the change in potassium concentration. (The absolute concentration nK is unimportant if we neglect the electric elds.) We are only interested in sources located at r0 = r0 (at the membrane). Using the identity that J1 (z )Y0 (z ) ? J0 (z )Y1 (z ) = 2= z , the Green's function G(r; tjr0 ; t0 ) simpli es to

G(r; r0 ; t) = 2 r

Z

1

0 0

Y1 ( r ) ? Y e?D 2 t J0 ( r)J 2 ( r 0) + Y 02(( r)J1 ( r0 ) d 1 0 1 r0 )

(3.7)

I believe this integral cannot be solved in closed form in terms of common special functions (see, e.g., Jaeger, 1942, and the asymptotic expansions in Zonneveld and Berghuis, 1955)2. I approximated this under some limiting conditions and analyzed it numerically for the remainder.3

3.5 Analytic solution for large axons
It is possible to treat the axon as a thine line instead of a cylinder. The solution for this special case can be expressed in terms of exponential integrals, a well-understood special function. Unfortunately, although the line source approximation worked well for the elds around axons (section 2.2.1 on
1 The same equation governs extracellular current ow as extracellular ionic di usion, except that for practical purposes extracellular current ow is always in equilibrium whereas time is very important for di usion. Therefore the the gradients will be the same as for di usion which has reached steady state, and these gradients are smaller than for time-dependent di usion. 2 Some variations on this problem can be solved easily, such as a Gaussian cylinder (Vern et al., 1977) or a point source (Nicholson and Phillips, 1981). 3 A similar unpublished analysis (a Ph. D. thesis) is brie y described in Cordingley and Somjen (1978) but only for much longer time scales.

CHAPTER 3. EXTRACELLULAR POTASSIUM AND OTHER DIFFUSIBLE SIGNALS

43

page 14), it does not work well for di usion of potassium. Potassium falls o very steeply with distance on short time scales (see below), and the fallo over the radius of the axon as predicted by the line source model is signi cant. Instead of treating the radius of the axon as very small, as in the line source model, it is better to treat it as extremely large. Qualitatively, we expect larger changes in potassium ion concentration around large axons than small axons, simply because there is more space within a given distance around a small axon than a large axon.4 Thus to estimate the maximum e ects we should look at large axons. If we let r0 ! 1, the geometry changes from a cylinder to an in nite sheet. The Green's function for this geometry is well known to be5 (3.8) G(r; r0 ; t) = pQ e?(r?r0)2 =4D t Dt where r is the distance from the sheet. If the only source is at r = 0 and di usion is only allowed in the positive r direction, then the concentration is: Z t n = G(0; r; t0 ) JK dt0
K

J = pK

0

Z

D

t e?r2 =4D t0

0

p0
t

dt0

This is still an intractable integral6 but if we look at the place where the concentration is the highest (r = 0), it is easy:

nK;max

= 2JK

r

t D

(3.9)

Using the values discussed above, we nd that after a 0.5 ms repolarization, nK = 1:4 105 ions/ m3 and EK = 2 mV. While this only gives the upper limit on a concentration change, it roughly shows how the change should depend on the various parameters. For example, if the same amount of potassium is exuded over a 1 ms time course p instead of a 0.5 ms time course, then JK is halved and t is doubled, decreasing nK by a factor of 2.

3.6 Numerical solution
4 Crudely speaking, suppose ions can di use a distance r in the relevant time. Then the volume they occupy will 2 be (r0 + r)2 ? r0 = (2r0 r + r2 ). Into this volume, 2 r0 JK t ions/ m are discharged. The change in concentration will be 2 r0 JK t= (2r0 r + r2 ). This function initially increases linearly with r0 but then saturates; the more exact solution has the same qualitative behavior. 5 This can also be derived as a limiting case of equation 3.7. 6 Actually, it can be written in closed form in terms of degenerate hypergeometric functions (Whittaker's functions). However, I didn't nd this form enlightening. 7 For r = r0 , we can use the identity J0 ( r0 )Y1 ( r0 ) ? J1 ( r0 )Y0 ( r0 ) = ?2= r0 again to simplify the computation: 2 JK Z 1 1 ? e?D 2 t=r0 d (3.10) nK = 42 D r0 2 0 3 J1 ( ) + Y12 ( )] 2 2 where = r0 . For large , J1 ( ) + Y12 ( ) 2= , and e?D 2 =r0 0, so the integral from some point onward can be evaluated analytically. The remainder was evaluated numerically using Matlab's 4'th order adaptive Runge{ Kutta{Fehlberg method.

This upper limit is attained for realistic values of r0 . I evaluated equation 3.3 numerically7 and found that nK has basically reached its maximum value of about 2 mV for r0 = 2 m. Note that nK is proportional to JK , so if the ion ux is di erent, the y axis in gure 3.2 can be rescaled proportionately.

44

CHAPTER 3. EXTRACELLULAR POTASSIUM AND OTHER DIFFUSIBLE SIGNALS
0.25 2 0.2

?nK (mM)

0.15 0.1 0.05 0 0.1 1 r0 (?m) 10 1 0.5 0 100

?EK (mV) ?EK (mV)

1.5

Figure 3.2: Change in concentration of potassium ions at the axon membrane as a function of axon radius at the end of 0.5 ms for JK = 30; 000 ions/ m 2 .

1 0.1 ?nK (mM) 0.01

r0 = 0.1 r0 = 0.2 r0 = 1 r0 = 2 r0 = 5

1 0.1 0.01 0.001 0.0001

0.001 0.0001 0 0.5 1 r-r0 (?m) 1.5 2

Figure 3.3: Change in potassium ion concentration as a function of distance from the axon after all the ions have been exuded (t = 0:5 ms). This is the maximum concentration at r = r0 ; maximum concentration changes occur slightly later for r > r0 (see gure 3.4). The concentration falls o roughly exponentially with distance.

The potassium ion concentration changes fall o roughly exponentially with distance over the relevant range ( gure 3.3). Given an exponential decay, unit analysis predicts that the form of the p p function must be Ae?Br= D t where A and B are unitless constants, since pD t is the only way to combine the relevant parameters to give units of length. I found that B = 2 provides close ts to the slope. Figure 3.4 shows how the the potassium ion concentration changes vary with both distance and time for various values of r0 . It is clear that the spatial scale of potassium di usion is very limited on these time scales; noticeable concentration changes will only be seen by the cell's immediate neighbor, and in fact only at the places where the membranes are apposed, not on the other side. In fact, the limited spatial scale suggests that a continuum di usion model is not appropriate; it might be more useful to make a model with discrete elements. I do not expect changes of orders of magnitude in these numbers with such a model, however. In fact, a model by Lebovitz (1996) explicitly includes only the immediately adjacent 20 nm of extracellular space and models the rest of the tissue by a simple permeability to a constant reservoir. The time scale for changes in extracellular potassium is very short. Concentration falls precipitously after JK drops to 0. This can be seen more clearly in gure 3.5 for several di erent r0 values. Modulations in extracellular potassium are important only during the time that the potassium is actually crossing the membrane; as soon as potassium ceases to cross the membrane, extracellular levels fall very rapidly. If there is short range interaction between neurons based on extracellular potassium, it has a very short time course.

CHAPTER 3. EXTRACELLULAR POTASSIUM AND OTHER DIFFUSIBLE SIGNALS
r0 = 0.1 ?m 2 Distance from axon (?m) Distance from axon (?m) 2 r0 = 0.5 ?m

45

1.5

1.5

1

1

0.2

0.2

0.5

0.1 0.2
0.3

0.

1

0.5

0.2

0.4

0.4

0.6
Time (ms) r0 = 1 ?m 1 1.5 0 0 0.5

0 0

0.5

Time (ms) r0 = 2 ?m

1

1.5

2 Distance from axon (?m) Distance from axon (?m)

2

1.5

1.5

0.2
0.2

0.2
1
2 0.
0.4

0.2

1
0. 2

0.4

0.4 0.6

0.4

0.5

0.6
0.8

0.5

0.4 0.6 0.8

0.6
1

0.8

0 0

0.5

Time (ms) r0 = 5 ?m

1

1.5

0 0

0.5

Time (ms)

1

1.5

2
0.2

Distance from axon (?m)

1.5

0.2

1

0.

2

0.4
0.6

0.4

0.5

4 0.

0.6 0.8
1

0.8

0.6

0 0

0.5

Time (ms)

1

1.5

Figure 3.4: Contour plots of the changes in K+ concentration around an axon for various sizes of axons. Numbers on the contours are changes in EK (mV).

46

CHAPTER 3. EXTRACELLULAR POTASSIUM AND OTHER DIFFUSIBLE SIGNALS
0.25 0.2 r0 = 0.1 r0 = 0.5 r0 = 1 r0 = 2 r0 = 5 2 ?EK (mV) 1.5 1 0.5 0 0 0.2 0.4 0.6 0.8 1 Time (ms) 1.2 1.4

0.15 0.1 0.05 0

Figure 3.5: The time course of a change in extracellular K+ at the membrane of the axon. Extracellular K+ concentrations drop rapidly from their peaks as soon as the current is shut o (at t = 0:5 ms). The di erent curves correspond to axons of di erent sizes.

3.7 Signi cance of changes in EK
Are these changes in extracellular K+ signi cant? No matter what happens, they cannot result in a larger membrane potential change than 2 mV, since EK changes by at most 2 mV. However, an EPSP of 2 mV would be large in cortex and could have a signi cant e ect. A better way to estimate the signi cance is to calculate the extra current due to the increased extracellular potassium concentration:

?nK (mM)

I = g K EK

(3.11)

In dendrites, a typical time constant is = 20 ms and the membrane capacitance is 8 10?6 nF/ m2 (0.8 F/cm2 ), so the membrane conductance is 4 10?7 S/ m2 , most of which is due to potassium channels. Roughly, then, if EK = 2 mV, there is a peak current of 8 10?7 nA/ m2 . It seems unreasonable to suppose that membranes could be in apposition for more than 100 m2 , so the maximum current into the cell would be 0.08 pA. By contrast, the peak current from a typical cortical synapse with a peak conductance of 0.5 nS and a 60 mV driving force is 30 pA, and this current may last several ms. In dendrites, one also nds that action potentials take much longer than 0.5 ms to repolarize, so the actual ux of potassium may be much lower. Brief potassium transients from single action potentials are therefore unlikely ever to be signi cant in interaction between dendrites in bundles. Sustained potassium increases of this magnitude have been shown to cause small increases in conduction velocity of cerebellar parallel bers, presumably because the axon is about 1 or 2 mV more depolarized everywhere (Kocsis et al., 1983). Axons typically have shorter time constants (about 1 ms or less), implying that they could come close to their steady state value during the time course of an action potential. As a result, it is possible that in a bundle of unmyelinated axons, very transient extracellular potassium increases could speed up conduction slightly. However, as discussed in section 2.3.1 on page 29, a small change in latency is unlikely to be critical for most computations. Lebovitz (1996) in his simulations found signi cant e ects of extracellular K+ changes only when he assumed a very narrow extracellular cleft and a large distance (20 m) to the reservoir where K+ ] was held constant. His model is di cult to compare directly because he modeled the extracellular space in terms of a monolayer of tiny rectangular cells surrounding one very large cell. It appears, however, that in the cases where he found signi cant interaction via K+ , his extracellular volume fraction was much too small (5% or even much less). He also adjusted his di usion coe cient for tortuosity even though he explicitly models all of the membranes; however, this only changes the di usion constant by a factor of 1.5, which does not explain the three orders of magnitude di erence. There are some other details of his calculations which are unexplained in the paper.

CHAPTER 3. EXTRACELLULAR POTASSIUM AND OTHER DIFFUSIBLE SIGNALS

47

With the numbers he gives, the peak EK is about 7 mV, and yet for that case he calculates that the transmembrane voltage change is also about 7 mV. This is di cult to understand since the shift in EK is very short-lived by my calculations. Lebovitz does not show the time course of nK but he ts it to an exponential; in my calculations it seems to fall much faster than exponentially from its peak. The e ect of the slow evoked potassium transients that are measurable with ion-selective electrodes (e.g., Singer and Lux, 1975; Karwoski and Proenza, 1987) will be much larger than the transients from single spikes. First, the relevant area is larger, since K+ ] is elevated over the entire surface of the neuron, which is from 104 m2 to 5 104 m2 . This raises the above estimate of current by two or three orders of magnitude. Second, such large scale extracellular potassium transients are very slow, so the result will be a sustained current with an amplitude comparable to the peak of a synaptic current. It is not surprising, then, that such potassium transients cause a noticeable change in neural excitability. Increases in extracellular potassium also a ect synaptic transmission because they depolarize presynaptic terminals. Initially, this results in an increase in release probability. Beyond a certain point, however, it results in a decrease in transmitter release, evidently because the spike fails to propagate into the terminal because sodium conductances are inactivated (Somjen, 1979; Sykova, 1983, 1987).

48

Part II

Implications of Cable Theory

In the space of one hundred and seventy-six years the Mississippi has shortened itself two hundred and forty-two miles. Therefore... in the Old Silurian Period the Mississippi River was upward of one million three hundred thousand miles long.... Seven hundred and forty-two years from now the Mississippi will be only a mile and three-quarters long.... There is something fascinating about science. One gets such wholesome returns of conjecture out of such a tri ing investment of fact. | Mark Twain

CHAPTER 4. SHUNTING INHIBITION AND FIRING RATES

49

Chapter 4 Shunting Inhibition and Firing Rates
4.1 Introduction
Many neuronal models treat the output of a neuron as an analog value coded by the ring rate of a neuron. Often the analog value is thought of as what the somatic voltage would be if spikes are pharmacologically disabled (sometimes called a generator potential ; see section 1.3). The neuron is modeled as if the spiking conductances were not present, as shown in black in gure 4.1. PSfrag replacements V Figure 4.1: A simpli ed neuron at steady state. Synaptic current can be discharged either through the leak or the spiking mechanism. If the cell does not spike, then the current through the resistor will be equal to Isyn ; changing G will change V proportionately. If the cell is spiking, then some of the synaptic current will be discharged through the spike mechanism (grey) so changing G will not have the same e ect.

Isyn

G

Ileak

Ispike

If there are no spikes, all the synaptic current Isyn must ow through the the leak ( gure 4.1), so the voltage will be proportional to the synaptic current: where Isyn is the synaptic current and G is the input conductance. A ring rate is then computed directly from this above-threshold membrane potential:

V = Isyn G

f = g(V ) where g is some monotonic function. For example, fout / V 2 in Carandini and Heeger (1994) or fout = tanh(V ) in Hop eld (1984). Varying G, for instance via activation of inhibitory input with a reversal potential close or equal
to the cell's resting potential (also known as \silent" or \shunting" inhibition), will directly a ect the generator potential V in a divisive manner. A recent and quite popular model (Carandini and Heeger, 1994; Nelson, 1994) has suggested that changing G by shunting inhibition would be a useful way to control the gain of a cell: when the inhibitory input rate increases, the slope of the input{output relationship decreases ( gure 4.2A) but the threshold does not change much. On the other hand, inhibition which is not of the shunting variety should have a subtractive e ect on the input{output relationship. If the reversal potential of the inhibition is far from the spiking threshold, then the inhibitory synapse will act more like a current source; the cell's conductance is not changed much, but a hyperpolarizing current is injected. This current simply shifts the input{output relationship by changing Isyn to Isyn ? Iinh ( gure 4.2B) where Iinh is the inhibitory current. Simpli ed models based on a generator potential ignore the e ect of the spiking mechanism ( gure 4.1 and assume that the behavior of the neuron above threshold is adequately described by the subthreshold equations. But when the cell is spiking, not all the current ows through the conductance G. The spiking mechanism itself removes charge, primarily through the potassium
Most of the contents of this chapter can be found in Holt and Koch (1997).

50
150 Firing Rate (Hz) 100 50 0

CHAPTER 4. SHUNTING INHIBITION AND FIRING RATES

A. Divisive

B. Subtractive

0

0.5

1

1.5

0

0.5

1

1.5

2

Isyn (nA)
Figure 4.2: A comparison of divisive and subtractive inhibition. A. Divisive inhibition changes the slope of the input{output relationship. In this case, f = g(V ) was a linear function of V and G was varied from 10 to 70 nS in equal steps. B. Subtractive inhibition shifts the curves by subtracting a current. Here Iinh varies from 0.08 to 0.058 nA in equal steps. conductances that are responsible for repolarizing the spike (Koch et al., 1995). Because of the spiking mechanism, we nd that changing the membrane leak conductance by shunting inhibition does not have a divisive e ect on ring rate, casting doubt on the hypothesis that such a mechanism serves to normalize a cell's response. A similar conclusion has been reached independently by Payne and Nelson for certain classes of neuron models (personal communication; Payne and Nelson, 1996).

4.2 Model description
Compartmental simulations were done using the model described by Bernander et al. (1991, 1994) and Bernander (1993) and Koch et al. (1995). The geometry for the compartmental models were derived from a large layer 5 pyramidal cell and a much smaller layer 4 spiny stellate cell stained during in vivo experiments in cat (Douglas et al., 1991) and reconstructed. Geometries of both cells are shown as insets in gure 4.3B. Each model has the same eight active conductances at the soma, including an A current and adaptation currents (see Koch et al., 1995 for details). The somatic conductance values were di erent for each cell, but the same conductance per unit area was used for each type of channel. Dendrites were passive. Simulations were performed with the program Neuron (Hines, 1989, 1993a). To study the e ect of GABAA synapses and shunting inhibition, we did not explicitly model each synapse; we set the membrane leak conductance and reversal potential at each location in the dendritic tree to be the time-averaged values expected from excitatory and inhibitory synaptic bombardment at presynaptic input ring rates fE and fI (as described in Bernander et al., 1991). To compute the time averaged values, synapses were treated as alpha functions with a given time constant and maximum conductance (g(t) = gmax te?t= e?1 = ). The area density of synapses at a given location on a dendrite was a function of the length of dendrite that separated the area from the soma (see table 4.1 on the next page). Two di erent sets of densities (\near" and \far") were used, depending on whether inhibitory synapses were near the soma or far from the soma. The \near" con guration is identical to the distribution used by Bernander et al. (1991) and re ects the anatomical observation that inhibitory synapses are mostly located near the soma in cortical pyramidal cells. The \far" con guration is not intended to be anatomically realistic. For simplicity, we used the same number of synapses for both the spiny stellate cell and the layer V pyramidal cell models.

CHAPTER 4. SHUNTING INHIBITION AND FIRING RATES Type AMPA Reversal Number gmax potential 0 mV 4000 1 nS 500 500 Area density Near Far x ? 40 1 + tanh x ? 100 0.5 ms 1 + tanh 22:73 22:73 x ? 100 ?x=50 1 nS 5 ms e 1 + tanh 22:73 0.1 nS 40 ms xe?x=50

51

GABAA ?70 mV GABAB ?95 mV

Table 4.1: Parameters of synapses in the compartmental models. x is the length of dendrite in m that separates this compartment from the soma. The \near" area density was used for gure 4.3; the \far" area density was used for gure 4.5 on page 55. The normalization for the area density is not included in the expression because it depends on the geometry; di erent neurons have di erent fractions of their membrane at a given distance from the soma.

A. 100
80 Firing rate (Hz) 60 40 20 0 0 0.5 1 Current (nA) 1.5

B.100
80 Firing rate (Hz) 60 40 20 0 0 2 4 6 8 10 0 2 4 Excitatory synaptic input rate (Hz) 6 8

Figure 4.3: Changing gleak has a subtractive rather than a divisive e ect on ring rates. A. Current discharge curves for the integrate{and{ re model, with gleak varying from 10 to 70 nS (from left to right) in steps of 10 nS. B. Fully adapted ring rates of the two cells as a function of excitatory input rate for di erent inhibitory input rates. From left to right, the curves correspond to a GABAA inhibitory rate of 0.5, 2, 4, and 6 Hz. Note that in all of these cases, the curve shifts rather than changes slope. In this case, inhibitory synapses were near the soma (\near" con guration in table 4.1), as found in cortical cells. Our integrate{and{ re model is described by where gleak is the input conductance and C is the capacitance and Isyn is the synaptic input. When the voltage V exceeds a threshold Vth the cell emits a spike and resets its voltage back to 0. We used C = 1 nF, gleak = 16 nS, Vth = 16:4 mV, which matches the adapted current{discharge curve of the layer 5 pyramidal cell model quite well (not shown). Results are not changed if a refractory period or an adaptation conductance is added to the integrate{and{ re model (also not shown).

C dV = ?V gleak + Isyn dt

for V < Vth

(4.1)

4.3 Proximal inhibition
Changing gleak does not change the slope of the current{discharge curve for integrate{and{ re cells ( gure 4.3A); it primarily shifts the curves. It therefore has a subtractive rather than a divisive e ect. The compartmental models behave very similar to the integrate{and{ re unit. For two di erent

52

CHAPTER 4. SHUNTING INHIBITION AND FIRING RATES

geometries (a layer V pyramid and a layer IV spiny stellate cell), we computed the fully adapted ring rate as a function of the excitatory synaptic input rate for various di erent rates of inhibitory input to synapses with GABAA receptors ( gure 4.3B). The slope of the input{output relationship does not change when the GABAA input amplitude is changed; the entire curve shifts. The same e ect can be observed when considering the current{discharge curves (not shown). This e ect is most easily understood in the integrate{and{ re model. In the absence of any spiking threshold, V would rise until V = Isyn =gleak ( gure 4.4 on the facing page). Under these conditions, the steady state leak current is proportional to the input current. However, if there is a spiking threshold, V never rises above Vth . Therefore no matter how large the input current is, the leak current can never be larger than Vth gleak . We can replace the leak conductance by a current whose value is equal to the time-average value of the current through the leak conductance (hIleak i = gleak hV i), and simplify the leaky integrate{and{ re unit to a perfect integrator. (Now, however, hIleak i will be a function of Isyn .) If theRcurrent is suprathreshold, the cell will still re at exactly the same rate because the same charge 0T (Isyn ? Ileak ) dt is deposited on the capacitor during one interspike interval T , although for the leaky integrator the deposition rate is not constant. For constant just supra-threshold inputs, hV i will be close to Vth and hIleak i will be large. For larger synaptic input currents, the time-averaged membrane potential becomes less and less (since V has to charge up from the reset point) and, therefore, the time-averaged leak current decreases for increasing inputs (compare gure 4.4A and B). It can be shown that

hIleak i = :V g 1 Isyn + th leak V g log(1 ? Vth gleak =Isyn ) th leak

8 <

Isyn

if Isyn < Vth gleak otherwise.

(4.2)

For large Isyn , and even for quite moderate levels of Isyn just above Vth gleak , the lower expression is approximately equal to gleak Vth =2, independent of Isyn ( gure 4.4C). Therefore it is a good approximation to replace the leak conductance by a constant o set current. The current{discharge curve for the resulting perfect integrate{and{ re neuron is simply1

? Isyn leak f (I ) = Isyn CVhIleak i = CV ? g2C : (4.3) th th Shunting inhibition (varying gleak ) above threshold acts like a constant, hyperpolarizing current

source, quite distinct from its subthreshold behavior. For currents just above the threshold, the initial slope is larger2 for larger gleak values ( gure 4.3A), which is the opposite of divisive normalization. One way of understanding this is to observe that for larger gleak there is a larger range of synaptic currents where Ileak is decreasing noticeably since gleak is a scale parameter in equation 4.2. This e ect is not visible in the compartmental models because they include adaptation currents which tend to linearize the current{discharge curve. The magnitude of this e ect is also strongly reduced by noise (not shown). A similar mechanism explains the result for the compartmental models ( gure 4.3B). In these, the voltage does rise above the ring threshold. However, spiking conductances are so large that during a spike any proximal synaptic conductances will be ignored. Furthermore, the spiking mechanism acts as a kind of voltage clamp on a long time scale (Koch et al., 1995) so that the time averaged voltage including the spike remains approximately constant (see appendix B and especially gure B.4 on page 104).
1 This expression can also be derived from the Laurent expansion of the current{discharge curve for a leaky integrator, f (Isyn ) = ?gleak =C log(1 ? Vth gleak =Isyn ), in terms of 1=Isyn around Isyn = 1 (Stein, 1967). 2 Actually, the slope is in nite for the leaky integrator at Isyn = Ith ; however, for Isyn slightly greater than Ith it decreases more slowly to the constant value of 1=CVth when gleak is higher.

CHAPTER 4. SHUNTING INHIBITION AND FIRING RATES

53

Ileak (t) = V(t) g leak

1

1

A.
0.5 0 Isyn=0.27 nA 0.5

B.
Isyn=1 nA V thg leak 0 300 0 Time (ms)

0

100

200

100

200

300

0.4 <I leak>

C.
0.2 0 Spikeless Integrate-and-fire 0 0.5 Isyn (nA) 1 1.5

V thg leak V thg leak/2

Figure 4.4: Why shunting inhibition has a subtractive rather than a divisive e ect on an integrate{ and{ re unit. A. The time-dependent current across the leak conductance Ileak (in nA) in response to a constant 0.5 nA current into a leaky integrate{and{ re unit with (solid line) and without (dashed line) a voltage threshold, Vth . The sharp drops in Ileak occur when the cell res, since the voltage is reset. B. Same for a 1 nA current. Note that Ileak with a voltage threshold has a maximum value which is well below Ileak without a voltage threshold. C. Time-averaged leak current (hIleak i) in nA as a function of input current, computed from the analytic expression. Below threshold, the ring rate model and the integrate{and{ re models have the same Ileak , but above threshold hIleak i drops for the integrate{and{ re model because of the voltage threshold. For Isyn just greater than threshold, the cell spends most of its time with V Vth , so hIleak i is high (panel A; at threshold, Ileak = Vth gleak ). For high Isyn , the voltage increases approximately linearly with time, so V has a sawtooth waveform as shown in panel B. This means that hIleak i = (max Ileak )=2 = Vth gleak =2.

54

4.4 Distal inhibition

CHAPTER 4. SHUNTING INHIBITION AND FIRING RATES

Shunting inhibition does not act divisively for an \anatomically correct" distribution of inhibition, where synapses are close to the cell body, because the spiking mechanism clamps the somatic voltage. However, distal synapses are not so tightly coupled electrically to the soma, so one might expect that distal GABAA inhibition might act divisively. Since the spiking mechanism can be thought of as a kind of voltage clamp (Koch et al., 1995), one can study the neuron's response by examining the current into the soma when it is clamped at the time-averaged voltage Vs (Abbott, 1991). For analysis, we simplify the dendritic tree into a single nite cable which has an excitatory synapse (conductance gE , reversal potential EE ) and an inhibitory synapse (gI and EI ) located at the other end. The cable has length l, radius r, speci c membrane conductance Gleak , intracellular resistivity p Ri and a length constant = r=(2Ri Gleak ). Using the cable equation, one can show that at steady-state the current owing into the soma from the cable is
?L) V e?L) V ?L Isoma = ?g1Vs + 2g1 e?L gE (EEg? +sg )(1 + gI?(2EI) ? Vs e(1 + +?g21) s e L +g L ( ?e e
E I

1

(4.4)

where L = l= is the electrotonic length of the cable and g1 = r3=2 2Gleak =Ri is the input conductance of the cylinder but with in nite length. (In this equation, all voltages are relative to the leak reversal potential, not to ground.) Despite the simpli cation involved in equation 4.4, it it qualitatively describes the response of the compartmental model. First, in the absence of any cable (L = 0), this equation becomes linear in both gE and gI and inhibition acts to subtract a constant amount from Isoma , as we have shown above. When L 6= 0, some divisive e ect is expected since gI appears in the denominator. However, a subtractive e ect will persist due to the term containing gI in the numerator. The reversal potential of GABAA synapses (increasing a membrane conductance to chloride ions) is in the neighborhood of ?70 mV relative to ground, while the time-averaged voltage when the model neuron is spiking is around ?50 mV (Koch et al., 1995). When a cell is spiking, therefore, a non-zero driving force exists for GABAA inhibition. In the pyramidal cell model, the subtractive e ect turns out to be much more prominent than the divisive e ect even for quite distant inhibition ( gure 4.5 on the next pageA). Both the inhibitory and excitatory synapses have been moved to more than 100 m (which is more than 1 ) away from the soma. To a good approximation, inhibition still subtracts a constant from both the current delivered to the soma and the ring rate of the cell. Equation 4.4 predicts that if the term containing gI is removed from the numerator, then a divisive e ect might be visible. When we changed the reversal potential EI of the GABAA synapses as well as the leak reversal potential Eleak to ?50 mV , we found that there is indeed an observable change in slope at low ring rate ( gure 4.5B). For higher ring rates, however, inhibition still acts approximately subtractively. We demonstrate this in the extreme case of moving both EI as well as the reversal potential associated with the leak conductance to ?50 mV (which is also the value to which the somatic terminal of the cable is clamped). Under these conditions, equation 4.4 simpli es to (4.5) Isoma (gE ; gI ) = constant + A g +ggE + B E I where A and B are independent of gE and gI . Clearly if gE is small, changing gI simply changes the slope. When gE is not small, then it turns out that changing gI has an e ect which is more subtractive than divisive ( gure 4.6). Another way of thinking about this is that Isoma (gE ; gI ) log(gE =gI ) for gE > gI . Thus Isoma (gE ; gI ) = log(gE ) ? log(gI ) and inhibition is subtractive. Since in the \far" model both kinds of synapses have the same distribution, their ring rates are proportional to the conductances. With the synaptic parameters we have used (table 4.1), gE =gI = 0:8fE =fI ; therefore, we expect to see a subtractive e ect when gE > 6 Hz, and this is

p

CHAPTER 4. SHUNTING INHIBITION AND FIRING RATES

55

A. EI = ?70 mV
2 Clamp 1 Current (nA) 0 80 60 Firing Rate 40 (Hz) 20 0 0 5 10

B. EI = ?50 mV

15 0

5

10

15

Excitatory Firing Rate (Hz)

Figure 4.5: The e ect of more distant inhibition on ring rates for the layer 5 pyramidal cell. A: The e ect on the input{output relationship when inhibitory synapses are more distant from the soma (\far" con guration in table 4.1 on page 51). Inhibitory rates are 0.5 Hz (solid) and 8 Hz (dashed). In these simulations, EI , the GABAA reversal potential, had its usual value of ?70 mV. Top : The voltage clamp current when the soma is clamped to ?50 mV, close to the spiking threshold of the cell. Bottom : The adapted ring rate when the soma is not clamped. Very little divisive e ect is visible on the ring rate; there is a slope change for ring rates less than 20 Hz, but this is too small to have a signi cant e ect. B: Same as A, except that EI and the reversal potential of all leak conductances were changed to ?50 mV so there is no driving force behind the GABAA synapses or the membrane passive conductance. A clear change in slope for low ring rates is evident. However, even for this rather unphysiological parameter manipulation, subtraction prevails at moderate and high input rates.

56
Isoma,1

CHAPTER 4. SHUNTING INHIBITION AND FIRING RATES Figure 4.6: When gE is not small compared to gI , then inhibition acts more subtractively than divisively even when the IPSP reversal potential is equal to the somatic voltage. The two upper curves are Isoma as a function of gE for two di erent values of gI such that gI + B changes by a factor of two in equation 4.5. The lower curve is the di erence between those two curves. For gE > gI + B , it is approximately constant over a large range. Units on gE are chosen such that gE =(gI + B ) = 1 for the smaller value of gI .

Isoma,2

Difference 0 0.5 1 1.5 2 gE 2.5 3 3.5 4

approximately true ( gure 4.5B).

4.5 Conclusions
Divisive normalization of ring rates has become a popular idea in visual cortex (Albrecht and Geisler, 1991; Heeger, 1992a; Heeger et al., 1996). It has been suggested that this is accomplished through shunting inhibitory synapses activated by cortical feedback (Carandini and Heeger, 1994; Nelson, 1994). Most discussions of shunting inhibition have assumed that the voltage at the location of the shunt is not constrained and may rise as high as necessary (e.g., Blom eld, 1974; Torre and Poggio, 1978; Koch et al., 1982, 1983). However, when the shunt is located close to the soma, the voltage at the site of the shunt cannot rise above the spike threshold. Therefore the current that can ow into the cell through the shunting synapse is limited, and at moderate rates becomes a constant o set ( gure 4.4C; appendix B). The current through the shunt is approximately independent of the ring rate. For this reason, shunting inhibition under these circumstances implements a linear subtractive operation. Even if the conductance change is not located close to the soma, it may not have a divisive e ect ( gure 4.5A). First, when the cell is spiking, shunting inhibition is not \silent": there is a signi cant driving force behind GABAA inhibition, since the somatic voltage is clamped to approximately ?50 mV by the spiking mechanism and the reversal potential for GABAA inhibition is in the neighborhood of ?70 mV. Second, even if the reversal potential for GABAA and the leak reversal potential are set to ?50 mV, inhibition acts divisively only if the excitatory synaptic conductance is small compared to the inhibitory conductance ( gures 4.5B and 4.6). Large excitatory conductances are expected when the cell receives signi cant input (Bernander et al., 1991; Rapp et al., 1992) so the subtractive e ect of large conductances is relevant physiologically. Current{discharge curves are a ected as predicted by the simple integrate{and{ re model in response to IPSPs and GABA iontophoresis in motoneurons in vivo (Granit et al., 1966; Kernell, 1969) and cortical cells in vitro (Connors et al., 1988; Berman et al., 1992). The input{output curves for di erent amounts of inhibition do not diverge for larger inputs, as would be required for a divisive e ect; in fact, they converge at high rates because of the refractory period (Douglas and Martin, 1990). In recordings from Limulus eccentric cells, current{discharge curves show both a slope change and a shift (Fuortes, 1959) because the site of current injection is distant from the site of spike generation.3 Rose (1977) showed that iontophoresis of GABA onto an in vivo cortical network appeared to act divisively. Because shunting inhibition has a subtractive e ect on single cells, this could possibly be caused by a network e ect (Douglas et al., 1995).
3 In this case, only the current{discharge curve was measured; the considerations in Figs. 4.5 and 4.6 are not relevant. A slope change is expected for current injection but not synaptic input.

CHAPTER 4. SHUNTING INHIBITION AND FIRING RATES

57

For synapses close to the spike generating mechanism, as well as for the integrate{and{ re unit, the subtractive e ect of conductance changes does not depend on the reversal potential of the conductance. Changing the reversal potential is equivalent to adding a constant current source in parallel with the conductance, and in a single compartment model this will obviously merely shift the current{discharge curve. Therefore, like inhibitory input, proximal excitatory input does not change the gain of other superimposed excitatory input. Similarly, our results are not a ected by a \weak reset" (where the voltage is reset to a voltage closer to Vth instead of to 0; Tsodyks and Sejnowski, 1995; Troyer and Miller, 1996); such models are mathematically equivalent to an integrate{and{ re model of the kind we consider here with a lower Vth and a leak conductance with a non-zero reversal potential. Our analysis assumes that synaptic inputs change on a time scale slower than an interspike interval. High temporal frequencies may be present in synaptic input currents for irregularly spiking neurons. Furthermore, our analysis assumes passive dendrites; active dendritic conductances complicate the interaction of synaptic excitation and inhibition. Although we cannot rule out that under some parameter combinations shunting inhibition could act divisively on the ring rates, we have not found such a range for physiological conditions. In combination with our integrate{and{ re and single cable models, we believe that a di erent mechanism is necessary to account for divisive normalization. The compartmental models and associated programs are available from ftp://ftp.klab.caltech. edu/pub/holt/holt_and_koch_1997_normalization.tar.gz.

58

CHAPTER 5. THE MEMBRANE TIME CONSTANT AND FIRING RATE DYNAMICS

Chapter 5 The Membrane Time Constant and Firing Rate Dynamics
5.1 Firing rate dynamics in single compartment neurons
Action potentials report only intermittently the activation of their source neuron. Therefore, the post-synaptic cells must estimate the activation either by averaging single presynaptic action potentials over times longer than the average inter-event interval, or by taking the near instantaneous average over multiple sources of similar presynaptic input. Whatever combination of these strategies is used, the speed of response of the post-synaptic cell is interesting because it bears on the rapidity with which a signal can propagate through a network of neurons. The passive membrane time constant is often used to characterize the time scale of a neuron's response to changes in its input. Injecting a constant current into the soma of a cortical neuron causes the membrane potential to increase gradually with a time course governed by the membrane's passive properties, if active currents are disabled. In a simpli ed one-compartment model, such as the kind considered by Hop eld (1984) and Carandini and Heeger (1994), the passive response to somatic input can be approximated by a simple equivalent electronic circuit consisting of a capacitance and a conductance in parallel (black lines in gure 5.1). The voltage response at the output node of this circuit to a current input is given by where V is the membrane potential, C is the membrane capacitance, R is the membrane resistance, and = RC is the membrane time constant. This model can be simply extended to account for the e ects on the time constant of variable parallel synaptic conductances that could control the gain and temporal integration of neurons (Bernander et al., 1991; Rapp et al., 1992; Carandini and Heeger, 1994). A neuron with a complicated dendritic geometry behaves qualitatively similarly in response to somatic current injection: the rate of somatic voltage change is also governed by the time constant (Rall, 1962, 1969). The di erences between an extended and a point neuron will be considered in section 5.2. The important issue is that the passive time constant is often seen to be crucial to the response, even if the response consists of action potentials. As discussed in section 1.3 on page 8, in some older literature and a number of newer models, voltage is viewed as the primary signal, and action potentials are thought of as merely minor perturbations. For this reason neuronal models that are concerned with average neuronal activation rather than individual spike timing commonly reduce the neuron to a single compartment whose dynamics are given by equation 5.1, and whose ring rate is some monotonic function of the somatic voltage:

5.1.1 Response time of non-spiking or ring-rate neurons

dV = ? V + I dt RC C

(5.1)

f = g(V )

(5.2)

Usually, g is sigmoidal, that is, a monotonic increasing, positive and saturating function (as in the popular choice of f = tanh V in Hop eld (1984)). However, other functions have also been used (e,g,
This work was done in collaboration with Rodney Douglas and Misha Mahowald, to whom I am indebted for some of the text as well.

CHAPTER 5. THE MEMBRANE TIME CONSTANT AND FIRING RATE DYNAMICS

59

V

C

R

Figure 5.1: Simpli ed circuit models which ignore the spatial extent of the neuron and collapse it to a single point are usually based on this circuit. The time constant, = RC , is the time it takes for the membrane voltage V to reach 1 ? 1=e of its nal value in response to a step change in current. Often in ring rate models (black), the voltage is computed using this circuit and then a ring rate is computed as a function of the voltage. The time constant then determines the dynamics of the ring rate because ring rate dynamics are essentially the same as the voltage dynamics. In models which represent spikes explicitly (grey), when the voltage reaches a threshold, it is reset to 0. The presence of the spiking mechanism completely changes the dynamics not only of the voltage but also of the ring rate; the voltage never reaches an equilibrium, so there is no time constant that governs the ring rate.

f / V 2 in Carandini and Heeger (1994)). Such ring-rate models incorporating a low-pass lter

to capture the passive properties of the underlying membrane have been applied widely in abstract neural network analysis (e.g., Hop eld, 1984 and the innumerable papers spawned from it, such as Kleinfeld, 1986) and also in more biological models (e.g., Wilson and Bower, 1989; Worgotter and Holt, 1991; Abbott, 1991; Carandini and Heeger, 1994; Carandini et al., 1996a, 1996b, 1997). In this class of model the ring rate can only change gradually in response to a rapid change in I . For small steps in input current g(V ) is approximately linear. In the linear regime, the ring rate, f , is simply I convolved with a rst order low pass lter with time constant , and is therefore a smoothed version of the input. The response to a step input current provides a useful case for comparisons. Suppose a ring-rate neuron has a constant input current I0 = 0, and then at time t = 0 the current is suddenly changed to I1 . How long will it take the ring rate to re ect the new input value? The remarkable property of linear systems is that they take a single characteristic time to approach their equilibrium state, regardless of how far it is. Doubling the input current doubles the distance to the equilibrium state but the system still only takes a time to reach 1 ? 1=e of equilibrium state. For this reason, the subthreshold time constant has been used as a measure of how long each stage in a feedforward neural network will take.1 It is possible to shorten the response time by introducing a nonlinearity. For example, if g(V ) is a saturating function (Hop eld, 1984) and the steady state ring rate of the neuron is close to its maximum rate, then the ring rate may reach 1 ? 1=e of its nal value much earlier than V reaches 1 ? 1=e of its nal value. Convergence time in a saturating neuron is the time that it takes V to rise high enough to saturate the ring rate, not the time for voltage to reach equilibrium. Increasing the current will allow V to reach that level faster. However, in most parts of the brain, neurons are typically not either silent or ring at their maximum rates. Furthermore, this kind of a saturating nonlinearity has several disadvantages. First, the ring rate of this neuron responds more slowly to decreases in current than a linear neuron for precisely the same reason as it responds more rapidly to increases in current: the ring rate changes slowly as a function of voltage when the voltage is high. Second, the saturating neuron does not transmit analog information; it either discharges near its maximum rate, or does not.
1 Note that the e ective time constant can be increased in the presence of positive feedback; recurrent networks often have much longer time constants than their individual components (chapter 6).

60

CHAPTER 5. THE MEMBRANE TIME CONSTANT AND FIRING RATE DYNAMICS Figure 5.2: Sample spike rasters in response to a step current injection. Arrows mark onset of current. The ring rate in spiking cells does not gradually increase; the e ect of the change in current is fully visible in the rst interspike interval. A. An integrate{and{ re unit, R = 20 M , C = 1 nF, Vth = 16:4 mV, I = 1:6 nA. B. A layer V compartmental model from Bernander et al. (1991), 1.5 nA. This model shows adaptation; the ring rate reaches its maximum after the rst ISI and declines slowly after that. C. A cell in vivo from area 17 of the anesthetized cat responding to a 0.6 nA injection current. Taken from Ahmed et al. (1993). D. For comparison, the ring rate as a function of time from a non-spiking non-adapting model neuron with a time constant of 20 ms. Unlike the spiking neurons, this neuron's ring rate increases gradually.

A.

B.

C. D.
50 ms

5.1.2 Response time of spiking neurons
Another important kind of nonlinearity is the spiking mechanism itself. The integrate{and{ re model is the simplest spiking model, and has been used extensively ever since it was proposed by Lapicque (1907). This model neuron has a membrane voltage which also obeys equation 5.1. However, when the voltage reaches a threshold Vth , a spike is emitted and the membrane voltage is reset ( gure 5.2A). The dynamics of the ring rate of a leaky integrate and re neuron di er fundamentally from the dynamics of the ring rate models governed by equation 5.2. In ring rate models or in a subthreshold integrate{and{ re neuron, measures the time to reach 1?1=e of a steady state membrane potential. However, if the neuron is ring the voltage never reaches equilibrium. Therefore is not a measure of response time. A spiking neuron's steady state is a limit cycle (an oscillation), not an equilibrium value. In its simplest form, the integrate{and{ re unit has only one state variable, its membrane voltage. When the neuron spikes, and the state variable is reset, it loses memory of the previous input current, and begins to respond to the new current by charging toward the threshold. If there is a step change in current, from these considerations it follows that the rst complete interspike interval after the change re ects the new current; everything during the rst interspike interval after the change is exactly the same as during the second interspike interval, so the second interval will not be di erent from the rst. Figure 5.2 shows the step response of an integrate{and{ re unit, a compartmental model of a cortical pyramidal neuron, and an experimental record derived from a neuron in cat visual cortex in vivo. The rst interspike interval already re ects the new ring rate|the convergence occurs on as short a time interval as can be de ned (i.e., the interspike interval). More complicated spiking neurons behave similarly. Compartmental models ( gure 5.2B) and cortical cells in vitro ( gure 5.2C) also reach their maximum ring rate by the rst interspike interval. Thereafter, the ring rate decreases slowly because of adaptation. Unlike the ring rate neuron, the response time for a spiking neuron is not the time to reach an equilibrium voltage which is proportional to the input. In the case of the simplest possible spiking mechanism, the integrate-and- re neuron, the neuron is on its new steady-state limit cycle immediately, as soon as the current changes, so in one sense it can be said that there is no delay at all. If one is looking at the spiking output of an ensemble of neurons, this means that the change

CHAPTER 5. THE MEMBRANE TIME CONSTANT AND FIRING RATE DYNAMICS

61

A
60 50 Time (ms) 40 30 20 10 0 0 0.5

B
Tth
Time (ms) 20 10 5 2 1 0.5 0.2 1 1.5 2 2 5 10 20

PSfrag replacements

Tth R (M )
50 100 200 500

I (nA)

Figure 5.3: Measures of the response times of di erent kinds of neurons. Tth is the maximum time it takes for an integrate{and{ re cell to re one spike; since the ring rate of such a neuron has reached its nal value after the rst interspike-interval after a change in input, this is a measure of the speed of response. is the subthreshold time constant, and is a measure of the response speed of the ring rate models. A. E ect of changing the current I on the response time. Using this measure, responses of a spiking neuron become faster with larger changes. B. E ect of changing the resistance R. Paradoxically, the spiking neuron becomes faster as increases. Parameters: C = 0:207 nF, Vth = 16:4 mV. In A, R = 38:3 M , = RC = 8 ms. In B, I = Vth =Rmin = 4:3 nA. can be detected instantly (see below); however, if one can only examine a single neuron's output, there will be no measurable e ects until that neuron actually spikes. For a single neuron, then, the relevant response time is the time required for V to reach a particular voltage: the threshold, Vth . In the case of the non-spiking neuron, the response time is the time to reach (1 ? 1=e) of its steady state value, which varies with the input; in this case, the response time is the time to reach a voltage which does not vary with the input. As a result, the response time will be shorter for stronger inputs. Unless inhibition is strong, V will always be greater than the reset voltage. The maximum latency is therefore the time the membrane takes to charge up from reset with the current I1 :

V Tth ?RC log 1 ? I th 1R

(5.3)

When the nal ring rate is higher, the latency is shorter. Although the time constant, = RC , is independent of the input current, the latency, Tth , decreases with increasing current ( gure 5.3A). The situation is slightly di erent for step decreases in input current, because it takes more time to measure a low ring rate. In this sense, the response to a decrease in ring rate is slow because to measure the new ring rate a postsynaptic neuron will have to wait for one interspike interval at the new low ring rate. However, after waiting for just one interspike interval at the old rate, a postsynaptic neuron can detect that the ring rate has decreased because the spike does not occur at the expected time. It may have to wait much longer to nd out exactly how much the ring rate has dec

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