Preface

Summary

In Volume 1, Linear Cable Theory and Dendritic Structure, the classical theory of membrane potentials, the Lapicque model neuron, and linear cable theory were introduced. The subthreshold responses described were always linear and the mathematical theory relatively simple.

In this volume, Nonlinear and Stochastic Theories, more complicated models are considered with concomitant increases in mathematical complexity. The principal new elements are nonlinear systems of reaction—diffusion systems and probability and stochastic processes. However, the required mathematics is developed in the text. The material also involves numerical methods of solution of partial differential equations and the statistical analysis of point processes. It is expected that this volume would be suitable for a second one-term course in quantitative neurophysiology with Volume 1 as a prerequisite.

Chapter 7 introduces reversal potentials for ionic currents into the cable equation with emphasis on those that arise during synaptic transmission. A derivation of the cable equation with synaptic reversal potentials is given as there did not appear to be one in the literature. The space-clamped problems are dealt with first, for the sake of completeness of presentation, and are used to explore the effects of relative timing of excitation and inhibition. Using the Green's function technique, exact solutions are then obtained for the full cable equation with periodic synaptic excitation.