Periodic orbits of single neuron models with internal decay rate \(0 < \beta \leq 1\) (Q2846818)

This paper considers the following difference equation, NEWLINE\[NEWLINE x_{n+1}=\beta x_n-g(x_n), \qquad n=0,1,\ldots, NEWLINE\]NEWLINE which describes the dynamics of a single neuron. Here \(0<\beta\leq 1\) is the internal decay rate and the signal function \(g\) is a more complicated step function defined below, NEWLINE\[NEWLINE g(x)=\begin{cases} -b, &x\leq -\alpha, \\ -a, &-\alpha <x<0, \\ 0, &x=0, \\ a, &0<x<\alpha, \\ b, &\alpha \leq x, \end{cases} NEWLINE\]NEWLINE where \(b>a>0\) and \(\alpha>0\). Some results on the periodicity of solutions and stability of periodic solutions are established. These results generalize some existing ones.