Dynamics of Shunting Inhibitory Cellular Neural Networks with Variable Two-Component Passive Decay Rates and Poisson Stable Inputs

Abstract

Shunting inhibitory cellular neural networks with continuous time-varying rates and inputs are the focus of this research. A new model is considered with compartmental passive decay rates which consist of periodic and Poisson stable components. The first component guarantees the Poisson stability of the dynamics, and the second one causes irregular oscillations. The inputs are Poisson stable to take into account the more sophisticated environment of the networks. The rates and inputs are synchronized to obtain Poisson stable outputs. A new efficient technique for checking the recurrence, the method of included intervals, is applied. Sufficient conditions for the existence of a Poisson stable solution and its asymptotic stability were obtained. Numerical simulations of Poisson stable outputs as well as inputs are provided. Examples of the model with Poisson stable rates, inputs and outputs confirm the feasibility of theoretical results. Discussions were undertaken to provide additional light on the relation of the obtained results with practical and theoretical potentials of neuroscience. Quantitative characteristics are suggested, which can be useful for the future applications of the results. In particular, the center of antisymmetry for the degree of periodicity is determined.

Unpredictable and Poisson Stable Oscillations of Inertial Neural Networks with Generalized Piecewise Constant Argument

“…That is, we need a special kappa property [ 35 ], which establishes a correspondence between periodicity and the unpredictability. The existence of such factors should be due to the higher possibility of the selection of the triple , when they satisfy property [ 34 ], and in addition, the kappa property must be fulfilled [ 35 , 36 , 37 , 38 ]. In this paper, we utilize a stronger state when this triple is a Poisson triple, which is more comfortable in applications.…”

Symmetrical Impulsive Inertial Neural Networks with Unpredictable and Poisson-Stable Oscillations

“…That is, we have to check that there exists a sequence t n , t n → ∞ such that for each Φω(t) ∈ Ξ, Φω(t + t n ) → Φω(t) uniformly on each closed and bounded interval of the real axis. We will use the method of intervals considered in [26] and other our papers. Fix a section [a, b], where a, b ∈ R with a < b and a positive real number ε.…”

“…The proof of Poisson stability is based on the method of included intervals, which was considered in [24,26] and appears to be an efficient instrument for verifying convergence. In this paper, the method is used to show the existence and uniqueness of unpredictable and Poisson-stable oscillations for impulsive inertial neural networks.…”

Inertial Neural Networks with Unpredictable Oscillations

“…To simplify this task, in papers [14,15], we have proposed a relation between intervals of convergence for inputs and outputs of models. It is called the method of included intervals, and has been successfully applied to the study of Poisson stable motions in neural networks [16,17].…”

“…In papers [19,20,21], the synchronization of Poincare chaos in semiconductor gas discharge models was considered. Currently, Poisson stable and unpredictable oscillations of Hopfield type neural networks [22,23], shunting inhibitory cellular neural networks [16,24], and inertial neural networks [17], have been investigated.…”