Modelling of neural systems in continuous time

Abstract

The activity of a network of N interconnected neurons is modelled in terms of a coupled system of N nonlinear and N linear first-order ordinary differential equations. The nonlinear equations govern the time development of the neuronal firing rates X i and take account of the sigmoid response to input with finite warm-up; while the linear equations trace the time course of fatigue variables y i which describe accommodation of individual neuronal response to maintained input. The input to neuron i from neuron j is assumed to be proportional to the firing rate of j, with a coefficient C ij which reflects the synaptic efficacy. In general, the assembly so modelled may display one or more stable fixed points or periodic attractors, or deterministic chaos. Two examples are considered in some detail: (a) the basic circuit of a simple cortical structure, the mammalian olfactory bulb; and (b) randomly connected networks. For the former, results are obtained which may cast light on the microsopic origin of the observed EEG activity; for the latter, instances of chaotic activity are established, attesting to the rich behavior accessible to this class of neurobiological models.