Stochastic dynamics of time-summating binary neural networks

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Abstract
An analysis is given of the stochastic dynamics of time-summating binary neural networks. Such networks have a memory or trace of their previous output activity reaching back to some initial time. Particular attention is given to a class of networks based on a discrete time version of leaky-integrator shunting networks. The stochastic dynamics is formulated as a linear Markov process describing the evolution of densities on the infinite-dimensional space of neuronal activation states. Using certain results from the theory of linear Markov operators, due to Lasota and Mackey [Probabilistic Properties of Deterministic Systems (Oxford University, Oxford, 1986); Physica D 28, 143 (1987)], conditions are derived for asymptotic stability in which the network converges to a unique limiting density. Moreover, the limiting density is shown to be a differentiable function of the parameters of the network such as the weights and decay factors. Finally, dynamical mean-field equations are derived that have periodic and chaotic solutions, implying a breaking of asymptotic stability in the thermodynamic limit.

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