Cyclic Evolution of Neuronal Automata with Memory When All the Weighting Coefficients are Strictly Positive

In this paper we study the sequences generated by a neuronal equation of the form xn " 1[!ki"1 aixn#i # Θ], where k is the size of the memory. We show that in the case where all the parameters (ai)1%i%k are strictly positive reals and k is a multiple of 6 (i.e., k " 6m), there exists a subset B " &w1, w2, . . . , wΑ( of &1, 2, . . . , m # 1( and for which there exists a neuronal equation of memory lengthΑk (i.e., k " 6m) with strictly positive weighting coefficients that generates a sequence of period Α ! lcm(3m # w1, . . . , 3m # wΑ). As an application; first, for all integer m greater than or equal to 3, we exhibit a neuronal recurrence equation of memory length 2k with weighting coefficients that are all strictly positive and which generates a sequence of period 2(3m # 1)(3m # 2); and second, for all odd integer m greater than or equal to 5, we exhibit a neuronal equation of memory length 3k with strictly positive weighting coefficients that generates a sequence of period 3(3m # 1)(3m # 2)(3m # 4).