Localized nonlinear waves in a myelinated nerve fiber with self-excitable membrane

We systematically study the evolution of modulated nerve impulses in a myelinated nerve fiber, where both the ionic current and membrane capacitance provide the necessary nonlinear feedbacks. This is achieved by using a perturbation technique, in which the Liénard form of the modified discrete Fitzhugh–Nagumo equation is reduced to the complex Ginzburg–Landau amplitude equation. Three distinct values of the capacitive feedback parameter are considered. At the critical value of the capacitive feedback parameter, it is shown that the dynamics of the system is governed by the dissipative nonlinear Schr?dinger equation. Linear stability analysis of the system depicts the instability of plane waves,which is manifested as burst of modulated nerve impulses that fulfills the Benjamin–Feir criteria. Variations of the capacitive feedback parameter generally influences the plane wave stability and hence the type of wave profile identified in the neural network. Results of numerical simulations mainly confirm the propagation, collision, and annihilation of nerve impulses in the myelinated axon.

Investigation of bright and dark solitons in α,β-Fermi Pasta Ulam lattice

We consider the Hamiltonian ofα,β-Fermi Pasta Ulam lattice and explore the Hamilton-Jacobi formalism to obtain the discrete equation of motion.By using the continuum limit approximations and incorporating some normalized parameters,the extended Korteweg-de Vries equation is obtained,with solutions that elucidate on the Fermi Pasta Ulam paradox.We further derive the nonlinear Schrodinger amplitude equation from the extended Korteweg-de Vries equation,by exploring the reductive perturbative technique.The dispersion and nonlinear coefficients of this amplitude equation are functions of theαandβparameters,with theβparameter playing a crucial role in the modulational instability analysis of the system.Forβgreater than or equal to zero,no modulational instability is observed and only dark solitons are identified in the lattice.However forβless than zero,bright solitons are traced in the lattice for some large values of the wavenumber.Results of numerical simulations of both the Korteweg-de Vries and nonlinear Schr¨odinger amplitude equations with periodic boundary conditions clearly show that the bright solitons conserve their amplitude and shape after collisions.