Diverse electrical responses in a network of fractional-order conductance-based excitable Morris-Lecar systems.

The diverse excitabilities of cells often produce various spiking-bursting oscillations that are found in the neural system. We establish the ability of a fractional-order excitable neuron model with Caputo's fractional derivative to analyze the effects of its dynamics on the spike train features observed in our results. The significance of this generalization relies on a theoretical framework of the model in which memory and hereditary properties are considered.

Multistability and multiperiodicity in impulsive hybrid quaternion-valued neural networks with mixed delays.

The existence of multiple exponentially stable equilibrium states and periodic solutions is investigated for Hopfield-type quaternion-valued neural networks (QVNNs) with impulsive effects and both time-dependent and distributed delays. Employing Brouwer's and Leray-Schauder's fixed point theorems, suitable Lyapunov functionals and impulsive control theory, sufficient conditions are given for the existence of 16 attractors, showing a substantial improvement in storage capacity, compared to real-valued or complex-valued neural networks. The obtained criteria are formulated in terms of many adjustable parameters and are easily verifiable, providing flexibility for the analysis and design of impulsive delayed QVNNs.

Dynamics of complex-valued fractional-order neural networks.

The dynamics of complex-valued fractional-order neuronal networks are investigated, focusing on stability, instability and Hopf bifurcations. Sufficient conditions for the asymptotic stability and instability of a steady state of the network are derived, based on the complex system parameters and the fractional order of the system, considering simplified neuronal connectivity structures (hub and ring). In some specific cases, it is possible to identify the critical values of the fractional order for which Hopf bifurcations may occur.

Stability and Hopf bifurcation analysis for the hypothalamic-pituitary-adrenal axis model with memory.

This article generalizes the existing minimal model of the hypothalamic-pituitary-adrenal (HPA) axis in a realistic way, by including memory terms: distributed time delays, on one hand and fractional-order derivatives, on the other hand. The existence of a unique equilibrium point of the mathematical models is proved and a local stability analysis is undertaken for the system with general distributed delays. A thorough bifurcation analysis for the distributed delay model with several types of delay kernels is provided.

Nonlinear dynamics and chaos in fractional-order neural networks.

Several topics related to the dynamics of fractional-order neural networks of Hopfield type are investigated, such as stability and multi-stability (coexistence of several different stable states), bifurcations and chaos. The stability domain of a steady state is completely characterized with respect to some characteristic parameters of the system, in the case of a neural network with ring or hub structure. These simplified connectivity structures play an important role in characterizing the network's dynamical behavior, allowing us to gain insight into the mechanisms underlying the behavior of recurrent networks.

Impulsive hybrid discrete-time Hopfield neural networks with delays and multistability analysis.

In this paper we investigate multistability of discrete-time Hopfield-type neural networks with distributed delays and impulses, by using Lyapunov functionals, stability theory and control by impulses. Example and simulation results are given to illustrate the effectiveness of the results.

Complex and chaotic dynamics in a discrete-time-delayed Hopfield neural network with ring architecture.

This paper is devoted to the analysis of a discrete-time-delayed Hopfield-type neural network of p neurons with ring architecture. The stability domain of the null solution is found, the values of the characteristic parameter for which bifurcations occur at the origin are identified and the existence of Fold/Cusp, Neimark-Sacker and Flip bifurcations is proved. These bifurcations are analyzed by applying the center manifold theorem and the normal form theory.