MODELING THE CUMULATIVE DISTRIBUTION FUNCTION OF SPIKES IN NEURAL NETWORKS

Abstract

Important components of neural networks are input synapses, action potential generators and output synapses. Rather than modeling a whole neuron in terms of a few ionic channels or as having Hodgkin–Huxley, Morris–Lecar or FitzHugh–Nagumo dynamics, we describe a neuron's action potential generator (APG). An APG may be at the hillock region at the base of an axon or another specific region of a cell. We model it using bifurcation theory based on observations by A. F. Hodgkin about membrane excitability. The result is a simplified model that leads us to view a neural network as comprising input and output synapses (electrical or chemical) that network APGs. These centers of activity are coupled by transfer functions from input synapses to an APG and from an APG to output synapses. The transfer functions account for time delays and signal attenuation that result from input and output structures. While this falls far short of a complete biophysical model of specific neurons in a network, it is consistent with empirical data, it is easily formulated, it is analytically tractable, and computer simulations based on it are straightforward. One outcome is a precise description of the cumulative distribution function (CDF) of action potentials. Since records of cell firing amount to collections of CDFs, the model is for a variable that is accessible to experimental observation. This methodology is applied here to describe bursting neural circuits and embedded loop networks similar to those occurring in basal ganglia.