A Critical Reexamination of Some Assumptions and Implications of Cable Theory in Neurobiology

Ph. D. Thesis, Gary R. Holt, California Institute of Technology, Computation and Neural Systems Program, 1998.

You can pick up the whole thing as gzipped postscript (820 kb) or PDF (2.4 Mb), or you can just pick up the relevant chapter(s). Note that if you just pick up the chapters, you won't have the reference list unless you pick that up also.

Abstract
Table of contents Gzipped postscript (40 kb) PDF (90 kb)
Chapter 1: Introduction Gzipped postscript (96 kb) PDF (230 kb)
Chapter 2: Ephaptic interactions Gzipped postscript (280 kb) PDF (450 kb)

Ephaptic interactions (interactions through extracellular electrical fields) are usually thought to be negligible but they have not to my knowledge been carefully analyzed near cell bodies. Large extracellular fields, on the order of a few mV, occur during action potentials (see movie). I find that these fields are sufficiently large to cause an effect on neighboring cells which is considerably larger than a typical synapse.

Chapter 3: Extracellular potassium and other diffusable signals Gzipped postscript (106 kb) PDF (203 kb)

Another way for adjacent neurons to interact non-synaptically is through changes extracellular ionic concentrations. In particular, the extracellular potassium concentration is low and is therefore easier to change by neural activity. I evaluated the magnitude of these changes around axons on very short time scales and found the effects to be negligible.

Chapter 4: Shunting inhibition does not have a divisive effect on firing rates. Gzipped postscript (122 kb) PDF (212 kb)

Shunting inhibition--a conductance change with a reversal potential close to the resting potential of the cell---has been shown to have a divisive effect on subthreshold EPSP amplitudes. It has therefore been assumed to have the same divisive effect on firing rates. However, shunting inhibition actually has a subtractive effect on the firing rate in most circumstances, for three main reasons:

  1. Averaged over several interspike intervals, the spiking mechanism effectively clamps the somatic membrane potential. As a result, the average current through the shunting conductance is approximately independent of the firing rate. This leads to a subtractive rather than a divisive effect for synapses which are electrotonically close to the soma.

  2. Shunting inhibition can only have a divisive effect if the reversal potential of the inhibitory conductance is at the "resting potential" of the cell. This is usually thought to be around -70 mV. However, when the cell is firing, the somatic voltage is clamped to about -50 mV, which means that there is a 20 mV driving force behind inhibition. There is therefore a subtractive effect of distal shunting inhibition which turns out to be much larger than any divisive effect.

  3. Even if the reversal potential of shunting inhibition is set properly for a divisive interaction, the firing rate of the cell is a saturating function of the excitatory input and as a result of this nonlinearity the inhibition acts subtractively when the excitatory synaptic conductance is not small compared to the inhibitory conductance.

Chapter 5: The membrane time constant and firing rate dynamics Gzipped postscript (115 kb) PDF (372 kb)

The subthreshold membrane time constant governs how quickly the membrane potential approaches equilibrium, and has been used to estimate how quickly a neuron can respond to its inputs. However, spiking neurons do not have an equilibrium voltage; subthreshold dynamics do not apply to their firing rates. In fact, a spiking neuron can respond much faster than tau. For current step inputs, a non-adapting spiking neuron reaches its final firing after a single interspike interval, and an ensemble of such neurons can respond arbitrarily fast. Firing rate dynamics are controlled by postsynaptic conductances, adaptation, and other processes in the cell, rather than passive properties of the membrane. The spiking mechanism can speed up neuronal responses, so that information can be passed to successive stages of a feedforward network in considerably less time than tau.

Using these considerations, we develop a formalism which will be used to examine the dynamics of the response of a network of integrate--and--fire units.

Chapter 6: Adaptation and recurrent circuits Gzipped postscript (105 kb) PDF (227 kb)

Adaptation is thought to be useful for emphasizing changes in input rather than the absolute level. In a recurrent network, however, adaptation has a qualitatively different effect. It can speed up the dynamics of the cortical amplifier circuit significantly by cancelling out the long tails of the recurrent EPSPs. This effect is still significant even if parameters are set so that the tails do not exactly cancel.

Chapter 7: Effect of synaptic depression and facilitation on steady state properties Gzipped postscript (82 kb) PDF (181 kb)

Synaptic depression has a profound effect on recurrent circuits because any deviation from linearity is magnified by the recurrent connections. The steady state gain of a "cortical amplifier" falls off dramatically at higher firing rates, even if the synapses are only weakly depressing.

Depression can be used in a cortical amplifier circuit if it is coupled with facilitation to yield a circuit which is linear overall. Synapses from layer 6 cells are known to facilitate and could perhaps undo the effect of depression in layer 4 to layer 6 synapses. The advantage of this operation is that subtractive inhibition in layer 6 will have a divisive effect on layer 4.

Appendix A. Comparison of homogenized and explicitly modeled extracellular space. Gzipped postscript (59 kb) PDF (140 kb)

An evaluation of the homognenous extracellular space approximation for an array of parallel axons.

Appendix B. Time averaged voltage in spiking cells. Gzipped postscript (64 kb) PDF (191 kb)

Koch et al. (1995) suggested that the spiking mechanism can be thought of as a sort of voltage clamp which keeps the time-averaged somatic voltage approximately constant regardless of the firing rate. We test this hypothesis for integrate-and-fire cells, the Hodgkin-Huxley equations, and cortical cells in slice. In each case, to a good approximation, the time average voltage changes very little even though the firing rate changes significantly.

Appendix C. Numerical methods. Gzipped postscript (88 kb) PDF (187 kb)
Bibliography Gzipped postscript (63 kb) PDF (137 kb)

This page last updated 10 Nov 1999.
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