Periodic solutions of difference equations ✩

Consider the difference equation xn+1 = βxn − g(xn), n = 0, 1, . . . , (∗) arising as a discrete-time network of single neuron, where β is the internal decay rate, g is a signal function with McCulloch–Pitts nonlinearity. For any positive integers k and m, necessary and sufficient conditions are obtained for (∗) has a periodic solution of period 2k + 1 or 2m. In addition, sufficient conditions are also given for (∗) has a (2k + 1)2m-periodic solution.  2003 Elsevier Inc. All rights reserved