Long-range memory effects in a magnetized Hindmarsh-Rose neural network

•Long-range interactions and electromagnetic induction are combined in the study of neuronal activity.•Modulational instability, chaos and synchronization phenomena are investigated.•Strong local couplings, high electromagnetic induction and strong long-range are believed to support modulated waves within the system.•Neuronal activity is governed by quasi-periodic oscillations, chaotic dynamics, chimera and synchronized states.•At both spiking and bursting regimes, strong long-range interaction induces more Chaotic activity than weak long-range interactions. We consider a model network of diffusively coupled Hindmarsh-Rose neurons to study both analytically and numerically, long-range memory effects on the modulational instability phenomenon, chaotic, synchronous and chimera states within the network. The multiple scale method is used to reduce the generic model into a discrete nonlinear Schrödinger equation. The latter is explored in the linear stability analysis and the instability criterion along with the critical amplitude are derived. The analytical results predict that strong local coupling, high electromagnetic induction and strong long-range interactions may support the formation of highly localized excitations in neural networks. Through numerical simulations, the largest Lyapunov exponents are computed for studying chaos, the synchronization factor and the strong of incoherence are recorded for studying, respectively synchronous and chimera states in the network. We find the appropriate domains of space parameters where these rich activities could be observed. As a result, quasi-periodic synchronous patterns, chaotic chimera and synchronous states, strange chaotic and non-chaotic attractors are found to be the main features of membrane potential coupled with memristive current during long-range memory activities of neural networks. Our results suggest that a combination of long-range activity and memory effects in neural networks may produces a rich variety of membrane potential patterns which are involved in information processing, odors recognition and discrimination and various diseases in the brain.