Behaviour of Solutions of a Neuron Model

We consider a discrete-time network of a single neuron as the discrete dynamical system 1xn+1=βxn-g(xn),n=0,1,…, where β> 1 and an internal decay rate, g is a step signal function given by a piecewise constant function which consists of five steps in the form 2. The considered model is quite simple as a mathematical expression, but with complex dynamics of its solutions. The model is highly sensitive to initial conditions and parameters. Small differences in an initial value and parameters yield widely diverging outcomes for the model, giving a great amount of different periodic orbits. Periodic orbits have been discussed according to the different rage of β. We can find some values of parameters such our considered model has the chaotic behaviour.