Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model
Ferenc A. Bartha Ábel Garab
Journal of Computational Dynamics 2014, 1(2): 213-232 doi: 10.3934/jcd.2014.1.213
We consider the global asymptotic stability of the trivial fixed point of the difference equation $x_{n+1}=m x_n-\alpha \varphi(x_{n-1})$, where $(\alpha,m) \in \mathbb{R}^2$ and $\varphi$ is a real function satisfying the discrete Yorke condition: $\min\{0,x\} \leq \varphi(x) \leq \max\{0,x\}$ for all $x\in \mathbb{R}$. If $\varphi$ is bounded then $(\alpha,m) \in [|m|-1,1] \times [-1,1]$, $(\alpha,m) \neq (0,-1), (0,1)$ is necessary for the global stability of $0$. We prove that if $\varphi(x) \equiv \tanh(x)$, then this condition is sufficient as well.
keywords: neural networks. Global stability Neimark–Sacker bifurcation rigorous numerics strong resonance graph representations interval analysis
Unique periodic orbits of a delay differential equation with piecewise linear feedback function
Ábel Garab
Discrete & Continuous Dynamical Systems - A 2013, 33(6): 2369-2387 doi: 10.3934/dcds.2013.33.2369
In this paper we study the scalar delay differential equation \linebreak $\dot{x}(t)=-ax(t) + bf(x(t-\tau))$ with feedback function $f(\xi)=\frac{1}{2}(|\xi+1|-|\xi-1|)$ and with real parameters $a>0,\ \tau>0$ and $b\neq 0$, which can model a single neuron or a group of synchronized neurons. We give necessary and sufficient conditions for existence and uniqueness of periodic orbits with prescribed oscillation frequencies. We also investigate the period of the slowly oscillating periodic solution as a function of the delay. Based on the obtained results we state an analogous theorem concerning existence and uniqueness of periodic orbits of a certain type of system of delay differential equations. The proofs are based among others on theory of monotone systems and discrete Lyapunov functionals.
keywords: period function delayed cellular neural networks periodic orbit. Delay differential equation discrete Lyapunov functional neural networks

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