Influence of magnetic flow on novel nonlinear periodic wave patterns in Hindmarsh-Rose neural networks

We employ multiple-scale expansion to show that the dynamics of the Hindmarsh-Rose neural network under magnetic flow effect is governed by a nonlinear differential-difference equation. This discrete equation is eventually transformed to a complex cubic Ginzburg-Landua amplitude equation via the continuum limit approximation. The modulational instability analysis establishes the Benjamin-Feir instability criterion, with the electric coupling and magnetic flux strengths greatly influencing the modulational instability gain. Exact analytical solution of the amplitude equation are obtained as Jacobi elliptic dn-function, by exploring appropriate decoupling ansatz technique. By keeping all other network parameters fixed and varying the elliptic modulus, electric and magnetic field strengths; various spatial nonlinear periodic patterns and small-amplitude localized modes of the membrane potential and associated ion currents are obtained. Results of numerical simulations confirm the analytical predictions and strongly suggest that a decrease in the magnetic flux strength induces chaotic regimes in the neural networks. Control spatiotemporal chaos can interfere and disrupt the regular synchronous patterns triggered by neuronal disorders. This work thus establishes a strong theoretical framework of magnetic flow effects on neuro-pathological abnormalities, such as epileptic seizures, Parkinson disease and loss of brain flexibility.