A Mathematical Foundation for Statistical Neurodynamics

Abstract

The brain is a large-scale system composed of an enormous number of neurons. In order to understand its functioning, we need to know the macroscopic behavior of a nerve net as a whole. Statistical neurodynamics treats an ensemble of nets of randomly connected neurons and derives macroscopic equations from the microscopic state transition laws of the nets. There arises, however, a theoretical difficulty in deriving the macroscopic state equations, because of possible correlations among the microscopic states. The situation is similar to that encountered in deriving the Boltzmann equation in statistical mechanics of gases. We first elucidate the stochastic structures of random nerve nets. We then derive macroscopic state equations which apply to a wide range of ensembles of random nets. These equations are shown to hold in a weak sense: we prove that the probability that these equations are valid within an arbitrarily small error and for an arbitrarily long time converges to 1 as the number n of the component elements in a random net tends to infinity. We also derive the macroscopic equations describing the dynamic behavior of the state correlations, and prove that these equations hold in the weak sense. The strong assertion which states the uniform convergence in time, is proved for a special class of random nets.