Approximation of Neural Network Dynamics by Reaction-Diffusion Equations

The equations of the Hopfield network, without the constraint of symmetry, can have complex behaviours. Cottet borrowed techniques from particle methods to show that a class of such networks with symmetric, translation-invariant connection matrices may be approximated by a reaction–diffusion equation. This idea is extended to a wider class of network connections yielding a slightly more complex reaction–diffusion equation. It is also shown that the approximation holds rigorously only in certain spatial regions (even for Cottet's special case) but the small regions where it fails, namely within transition layers between regions of high and low activity, are not likely to be critical.