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

Marat Akhmet a, $^{∗}$, Mehmet Onur Fen $^{b}$, Mokhtar Kiranec 

$^{a}$Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey 

$^{b}$Neuroscience Institute, Georgia State University, Atlanta, Georgia 30303, USA 

$^{c}$Laboratoire de Mathématiques, Image et Applications, Pôle Sciences et Technologies, Université de La Rochelle, Avenue Michel Crépeau, 17042 La Rochelle, France 

## Abstract 

We consider a new model for shunting inhibitory cellular neural networks, retarded functional dif- ferential equations with piecewise constant argument. The existence and exponential stability of almost periodic solutions are investigated. An illustrative example is provided. 

Keywords: Shunting inhibitory cellular neural networks; Retarded functional differential equations; Alternate constancy of argument; Bohr almost periodic solutions; Exponential stability 

## 1 Introduction 

Cellular neural networks ($_{CNNs}$) have been paid much attention in the past two decades [1]-[10]. Ex- ceptional role in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing is played by shunting inhibitory cellular neural networks ($_{SICNNs}$), which was in- troduced by Bouzerdoum and Pinter [11]. One of the most attractive subjects for this type of neural networks is the existence of almost periodic solutions. This problem has been investigated for models with different types of activation functions [12]-[20]. In the present study, we investigate a new model of $_{SICNNs}$by considering deviated as well as piecewise constant time arguments, and prove the existence of exponentially stable almost periodic solutions. All the results are discussed for the general type of activation functions, but they can be easily specified for applications. 

Extended information about differential equations with generalized piecewise constant argument [21] can be found in the book [22]. As a subclass, they contain differential equations with piecewise constant argument ($_{EPCA}$) [23]-[30], where the piecewise constant argument is assumed to be a multiple of the greatest integer function. 

Differential equations with piecewise constant argument are very useful as models for neural networks. This was shown in the studies [31]-[33], where the authors utilized EPCA. We propose to involve a 

