Periodic orbits of single neuron models with internal decay rate 0 < β ≤ 1

Abstract

In this paper we consider a discrete dynamical system

x _n+1=βx _n – g(x _n ), n=0,1,...,

arising as a discrete-time network of a single neuron, where 0 < β ≤ 1 is an internal decay rate, g is a signal function. A great deal of work has been done when the signal function is a sigmoid function. However, a signal function of McCulloch-Pitts nonlinearity described with a piecewise constant function is also useful in the modelling of neural networks. We investigate a more complicated step signal function (function that is similar to the sigmoid function) and we will prove some results about the periodicity of solutions of the considered difference equation. These results show the complexity of neurons behaviour.

Keyword : dynamical system, fixed point, iterative process, nonlinear problem, stability