Almost periodic solutions of retarded SICNNs with functional response on piecewise constant argument

Abstract

We consider a new model for shunting inhibitory cellular neural networks, retarded functional differential equations with piecewise constant argument. The existence and exponential stability of almost periodic solutions are investigated. An illustrative example is provided. new type of systems, retarded functional differential equations with piecewise constant argument of generalized type, in the modeling. It will help to investigate a larger class of neural networks.In paper [21], differential equations with piecewise constant argument of generalized type (EP CAG) were introduced. We not only maximally generalized the argument functions, but also proposed to reduce investigation of EP CAG to integral equations. Due to that innovation, it is now possible to analyze essentially non-linear systems, that is, systems non-linear with respect to values of solutions at discrete moments of time, where the argument changes its constancy. Previously, the main and unique method for EP CA was reduction to discrete equations and, hence, only equations in which values of solutions at the discrete moments appear linearly [23]- [30] have been considered.The crucial novelty of the present paper is that the piecewise constant argument in the functional differential equations is of alternate (advanced-delayed) type. In the literature, biological reasons for the argument to be delayed were discussed [34,35]. However, the role of advanced arguments has not been analyzed properly yet. Nevertheless, the importance of anticipation for biology was mentioned by some authors. For example, in the paper [36], it is supposed that synchronization of biological oscillators may request anticipation of counterparts behavior. Consequently, one can assume that equations for neural networks may also need anticipation, which is usually reflected in models by advanced argument.Therefore, the systems taken into account in the present study can be useful in future analyses of SICN N s. Furthermore, the idea of involving both advanced and delayed arguments in neural networks can be explained by the existence of retarded and advanced actions in a model of classical electrodynamics [37]. Moreover, mixed type deviation of the argument may depend on traveling waves emergence in CN N s [7]. Understanding the structure of such traveling waves is important due to their potential applications including image processing (see, for example, [1]-[7]). More detailed analysis of deviated arguments in neural networks can be found in [38]-[40].Shunting inhibition is a phenomenon in which the cell is "clamped" to its resting potential when the reversal potential of Cl − channels are close to the membrane resting potential of the cell [11,41]. It occurs through the opposition of an inward current, which would otherwise depolarize the membrane potential to threshold, by an inward flow of Cl − ions [41]. From the biological point of view, shunting inhibition has an important role in the dynamics of neurons [42]-[44]. According to the results of Vida et al.[42] networks with...