Periodic and Chaotic Orbits of a Neuron Model

Abstract

In this paper we study a class of difference equations which describes a discrete version of a single neuron model. We consider a generalization of the original McCulloch-Pitts model that has two thresholds. Periodic orbits are investigated accordingly to the different range of parameters. For some parameters sufficient conditions for periodic orbits of arbitrary periods have been obtained. We conclude that there exist values of parameters such that the function in the model has chaotic orbits. Models with chaotic orbits are not predictable in long-term.

Behaviour of Solutions of a Neuron Model

“…In this case is necessary that α > x 3 > 0 > −α ≥ x 2 > x 1 > x 0 . If we take β = 3, a = 4, b = 6 and α = 1, we get a periodic orbit {− 23 8 , − 21 8 , − 15 8 , 3 8 } (see Fig. 6).…”

“…In [2] we have already obtained some results about the periodicity of a neuron model (1) with parameter 0 < β ≤ 1 and a signal function (2). Also in [3] we had analysed model (1) with a different step signal function-a step function with two thresholds. In [4,5] is considered model (1) with periodic internal decay rate.…”

Eventually periodic solutions of single neuron model

“…In [2,3], the authors studied models by applying a different signal function (with more than one threshold). In [14], the authors investigated a discrete neuron model with periodic solutions.…”