The nonlinear problem of traveling nerve pulses showing the unexpected process of hysteresis and catastrophe is studied. The analysis was done for the case of one-dimensional nerve pulse propagation. Of particular interest is the distinctive tendency of the pulse nerve model to conserve its behavior in the absence of the stimulus that generated it. The hysteresis and catastrophe appear in certain parametric region determined by the evolution of bubble and pedestal like solitons. By reformulating the governing equations with a standard boundary conditions method, we derive a system of nonlinear algebraic equations for critical points. Our approach provides opportunities to explore the nonlinear features of wave patterns with hysteresis. 1. Introduction As is well known, one of the fundamental problems in physics applied to natural biological processes is to investigate the ways the nature transports information and energy between two or more points of living organism. On nerve fibers, for example, there exist several mathematical descriptions. The remarkable one was done by Hodgkin and Huxley (HH) in the 50s [1]. By using this model it has been established that the dynamic of ionic currents through voltage channels is responsible for the change of the membrane potential in nerve tissues. That means the membrane contains proteins with specific behaviors of selectivity with respect to the conduction of sodium and potassium ions though the membrane. Later, Hodgkin and Huxley system was developed independently by many authors and specifically the work of Fitzhugh and Nagumo [2, 3] suggested, as analogous neuronal, a nonlinear electrical circuit, controlled by an equation system also similar to those of Van Der Pol currents. Further the model was extended by modeling the nerve pulse as collective excitations from the point of view of dynamical systems; see, for example, [4]. After this successful beginning several other works concerning specifically soliton-like structures in the HH model have been done. Indeed, for example, Katz in [5] proposed the existence of traveling soliton-like pulse in this model. Muratov in his paper [6] obtained solitary waves for nerve pulses with velocities that are in certain agreement with experimental results. However, some further investigations lead to observing the dissipation of heat [7] that should have considerable distort at large distance and time the proper behavior of nerve cells. It is suggested to say that, despite those very important achievements on the HH model, there is still a current problem concerning the
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