Dynamics of coupled mode solitons in bursting neural networks

Periodic soliton trains and informational code structures in an improved soliton model for biomembranes and nerves

Many experiments have shown that the action potential propagating in a nerve fiber is an electromechanical density pulse. A mathematical model proposed by Heimburg and Jackson is an important step in explaining the propagation of electromechanical pulses in nerves. In this work, we consider the dynamics of modulated waves in an improved soliton model for nerve pulses. Application of the reductive perturbation method on the resulting generalized Boussinesq equation in the low-amplitude and weak damping limit yields a damped nonlinear Schrödinger equation that is shown to admit soliton trains. This solution contains an undershoot beneath the baseline ("hyperpolarization") and a "refractory period," i.e., a minimum distance between pulses, and therefore it represents typical nerve profiles. Likewise, the linear stability of wave trains is analyzed. It is shown that the amplitude of the fourth-order mixed dispersive term introduced here can be used to control the amount of information transmitted along the nerve fiber. The results from the linear stability analysis show that, in addition to the main periodic wave trains observed in most nerve experiments, five other localized background modes can copropagate along the nerve. These modes could eventually be responsible for various fundamental processes in the nerve, such as phase transitions and electrical and mechanical changes. Furthermore, analytical and numerical analyses show that increasing the fourth-order mixed dispersion coefficient improves the stability of the nerve signal.