A cellular automaton model for collective neural dynamics

Abstract

A stochastic epidemic model for the collective behaviour of a large set of Boolean automata placed upon the sites of a complete graph is revisited. In this paper we study the generalisation of the model to take into account inhibitory neurons. The resulting stochastic cellular automata are completely defined by five parameters: the number of excitatory neurons, N , the number of inhibitory neurons, M , the probabilities of excitation, α , and inhibition, γ , among neurons and the spontaneous transition rate from the firing to the quiescent state, β . We propose that the background of the electroencephalographic signals could be mimicked by the fluctuations in the total number of firing neurons in the excitatory subnetwork. These fluctuations are Gaussian and the mean-square displacement from an initial state displays a strongly subdiffusive behaviour approximately given by σ 2 ( t ) = A ( 1 − e − t / τ ) , where N A = β / ( β + M γ ) , τ = 2 ( N α − β ) . Comparison with real EEG records exhibits good agreement with these predictions.