Research Article Catastrophe and Hysteresis by the Emerging of Soliton-Like Solutions in a Nerve Model Fernando Ongay Larios, 1 Nikolay P. Tretyakov, 2 and Maximo A. Agüero 1 1 Faculty of Sciences, Autonomous University of the State of Mexico, 50000 Toluca, Mexico 2 Department of Applied Mathematics, Russian State Social University, Moscow 129226, Russia Correspondence should be addressed to Fernando Ongay Larios; fernando ongay@yahoo.com.mx Received 15 June 2014; Revised 6 September 2014; Accepted 13 September 2014; Published 7 October 2014 Academic Editor: Mitsuhiro Ohta Copyright © 2014 Fernando Ongay Larios et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e nonlinear problem of traveling nerve pulses showing the unexpected process of hysteresis and catastrophe is studied. e 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. e 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. e 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. at means the membrane contains pro- teins 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]. Aſter this successful beginning several other works con- cerning 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 heat releasing during the evolution of electrical nerve signals along the axons. is problem could be discarded if nature could choose the way for transmitting information and energy as evolving processes with or without little amount of heat liberation, that is, like an adiabatic proc- ess. On the other hand, there are relevant works which report unusual behavior of soliton emergency due to hysteric proc- esses. For example, Wu and coworkers [8] report the experi- mental observation of the generation of dark solitons inher- ently hysterical, through self-modulational instability moving the operating point frequency across a selected dipole gap in the dipole-exchange spin wave mode dispersion response. Higher powers lead to further changes in the power frequency Hindawi Publishing Corporation Journal of Nonlinear Dynamics Volume 2014, Article ID 710152, 8 pages http://dx.doi.org/10.1155/2014/710152