Exponential transient length generated by a neuronal recurrence equation
We study the sequences generated by neuronal recurrence equations of the form x(n) = 1[ Sigma(j=1)(k) a(j)x(n - j) - theta], where k is the size of memory (k represents the number of previous states x(n - 1),x(n -2),...,x(n - k) which intervene in the calculation of x(n)). We are interested in the number of steps (transient length) from an initial configuration to the cycle, where the length of the cycle represents the period. We show that under certain hypotheses it is possible to build a neuronal recurrence equation of memory size (s + 1)6m, whose dynamics contains an evolution of transient length (s+ 1)(3m + 1 + lcm(p(0), p(1),..., p(s-1), 3m - 1)) and a cycle of length (s + 1) lcm(p(0), p(1),..., p(s-1)), where lcm denotes the least common multiple and p(0), p(1),..., p(s-1) are prime numbers lying between 2m and 3m. (C) 2003 Elsevier B.V. All rights reserved.
transient length neuronal recurrence equation cycle length