Approximation of neural network dynamics by reaction-diffusion equations
Date
2010-05-03T20:48:38Z
Authors
Edwards, Roderick
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Abstract
'Attractor' neural network models have useful properties, but biology suggests that lack of symmetry and more varied dynamics may be significant. The equations of the Hopfield network, without the constraint of symmetry, can have complex behaviours which have been little studied. G.-H. Cottet [C.R. Acad. Sci. Paris, 312-I (1991), pp. 217-221] borrowed techniques from particle methods to show that a class of such networks with symmetric, translation-invariant connection matrices may be approximated by reaction-diffusion equations. A single, albeit infinite-dimensional, equation of known type is easier to analyse than an enormous system of equations, keeping track of the activity of every neuron. 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.
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Keywords
neural network, reaction-diffusion, particle method, technical reports (mathematics and statistics)