Generalized Period-halving Bifurcation of a Neuronal Recurrence Equation

We study the sequences generated by neuronal recurrence equations of the form x(n) = 1[ ∑h j=1 ajx(n − j) − θ]. From a neuronal recurrence equation of memory size h which describes a cycle of length ρ(m)× lcm(p0, p1, . . . , p−1+ρ(m)), we construct a set of ρ(m) neuronal recurrence equations whose dynamics describe respectively the transient of length O(ρ(m)×lcm(p0, . . . , pd)) and the cycle of length O(ρ(m)×lcm(pd+1, . . . , p−1+ρ(m))) if 0 ≤ d ≤ −2 + ρ(m) and 1 if d = ρ(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.