Period-halving Bifurcation of a Neuronal Recurrence Equation

Abstract

We study the sequences generated by neuronal recurrence equations of the form $x(n)=1[{\sum }_{j=1}^h{a}_{j}x(n-j)-\theta ]$. From a neuronal recurrence equation of memory size $h$ which describes a cycle of length $\rho (m)\times lcm(p_0,p_1,...,p_{-1+\rho (m)})$, we construct a set of $\rho (m)$ neuronal recurrence equations whose dynamics describe respectively the transient of length $O(\rho (m)\times lcm(p_0,...,p_d))$ and the cycle of length $O(\rho (m)\times lcm(p_{d+1},...,p_{-1+\rho (m)}))$ if $0\le d\le -2+\rho (m)$ and 1 if $d=\rho (m)-1$. This result shows the exponential time of the convergence of neuronal recurrence equation to fixed points and the existence of the period-halving bifurcation.

50 pages. This paper was submitted to Complex in July 2010. This paper is the full version of the paper to appear in Volume 20 Issue 4 of Complex Systems