Discrete delays and piecewise constant argument in neuronics

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Abstract

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.
Original languageEnglish
Pages (from-to)197-210
JournalNeural, Parallel & Scientific Computations
Volume19
Issue number1-2
Publication statusPublished - Mar 2011

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Piecewise Constant Argument
Discrete Delay
Convergence of Solutions
Stability Condition
Difference equation
Stability Analysis
Lemma
Neuron
Time Delay
Differential equation
Sufficient Conditions
Model

Citation

Leung, I. K. C. (2011). Discrete delays and piecewise constant argument in neuronics. Neural, Parallel & Scientific Computations, 19(1-2), 197-210.