Discrete delays and piecewise constant argument in neuronics

Sufficient conditions are established for the convergence of solutions of difference equation xn+1 = a tanh[xn - bn-1 - c], n = 0,, 1, 2, 3, ., discritized from the mother version of continuous case dx(t)/dt = -x(t) + a tanh[x(t) - bx(t - τ) - c], t > 0, with a, b, c and τ are all positive, that models the dynamics of a single effective neuron with dynamical threshold with delay. A similar approach can also be applied to establish the stability conditions for a differential equation with piecewise constant arguments as in the following dx(t)/dt = -ax(t) + tanh[bx(t) + cx([t])], t > 0, t ≠ 0, 1, 2, .... The stability analysis of the generalized time delay model dx(t)/dt = -ax(t) + tanh{bx(t) +∑j=0k cjx([t - j])}, t > 0, t ≠ 0, 1, 2, ..., is also discussed using the lemma due to Cooke and Huang 1991. Copyright © 2011 Dynamic Publishers, Inc.