Averaged neural networks

Cohen and Grossberg (1983) studied an almost gradient system of ordinary differential equations with application to neural networks and used a Liapunov function, together with an invariance principle, to show that some equilibrium points attract solutions. Independently, Hopfield (1984) modeled a neural network by means of a system of ordinary differential equations which turn out to be a special case of the Cohen-Grossberg system, as pointed out by Cohen (1990). In the Hopfield model it is clear that the functions involved in the equations are averages and that current will flow through a synapse only if a certain threshold is reached; however, none of the models take into account an averaging technique. Investigators have been interested in sustained oscillations in neural networks and have produced them in computer simulations when there is a pointwise delay. The linearized systems with a delay have also exhibited sustained oscillations. But our conjecture is that oscillations are not caused by a delay. This paper is intended to put substance to that conjecture by examining models with both pointwise and distributed delays. None of the models have solutions with sustained oscillations.