Localized nonlinear excitations in diffusive memristor-based neuronal networks

Abstract

We extend the existing ordinary differential equations modeling neural electrical activity to include the memory effect of electromagnetic induction through magnetic flux, used to describe time varying electromagnetic field. Through the multi-scale expansion in the semi-discrete approximation, we show that the neural network dynamical equations can be governed by the complex Ginzburg-Landau equation. The analytical and numerical envelop soliton of this equation are reported. The results obtained suggest the possibility of collective information processing and sharing in the nervous system, operating in both the spatial and temporal domains in the form of localized modulated waves. The effects of memristive synaptic electromagnetic induction coupling and perturbation on the modulated action potential dynamics examined. Large electromagnetic induction coupling strength may contribute to signal block as the amplitude of modulated waves are observed to decrease. This could help in the development of a chemical brain anaesthesia for some brain pathologies.

The generation of diverse traveling pulses and its solution scheme in an excitable slow-fast dynamics

“…One of these states give rise to traveling pulses. Here we derive analytically the emergence of traveling pulses using the semi-discrete approach [30,29]. In particular, we transform system (1) in the wave form by reducing its first and second equations into the second order differential equation…”

“…The evolution of modulated waves in the network is described by a modified CGLE. To find the traveling wave profiles, we need to reduce the system into a CGLE and find its solution using the semi-discrete approximation [30,29]. We consider the new variables m k and n k as…”

“…In the following, we will use the semi-discrete approximation [30,29] to derive analytically the localized traveling pulses that propagate in a diffusively connected excitable network of FHR neurons (1). In order to investigate the resulting interactions, the system is considered in such a way that each neuronal node follows the same coupling topology.…”

“…Generally, the wave number is real and the propagation frequency complex. Substituting the solutions ( 29) and (30) in Eqs. ( 27) and ( 28), we derive the following linear homogeneous equations for a 10 and a 20…”

“…Our study is motivated by theoretical and experimental works that assume the existence of localized excitations in certain neuronal populations [27,28,29,30,1]. Many research articles have investigated the existence of envelope solitons of the nerve impulse, not only in diffusively connected networks but also in memristive neuron models [29,30]. In [1], we have shown the existence of solitary wave profiles in the HR model using the tanh method [31,32].…”