^{1}

^{2}

^{1}

^{1}

^{2}

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.

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 [

After this successful beginning several other works concerning specifically soliton-like structures in the HH model have been done. Indeed, for example, Katz in [

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. This 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 process.

On the other hand, there are relevant works which report unusual behavior of soliton emergency due to hysteric processes. For example, Wu and coworkers [

The principal argument taken further is considering the neural activity as related to mechanical and thermodynamical properties in the axon. However, the idea of the neural signal as a mechanical wave was proposed early in 1912 by Wilke, University of Heidelberg, Germany [

The aim of this study is to take into consideration the Heinburg model [

This paper is organized as follows. Section

A mathematical model representing the propagation of nerve waves along an axon by considering the neural signals as propagating density pulses has been formulated. For more details on this hypothesis see [

With

In this paper we will consider the nontrivial boundary condition; that is, far from the excited zone along the axon, the difference density

Before the application of the boundary condition we slightly modify (

Here

Now, let us study (

By applying this restriction, for the constants of integration

As it can be easily seen this constant of integration depends on the background value of the difference density

In second order equation (

Let us assume that

Assuming

Since

Finally, the graph of

In this part we will briefly recall the explicit nontopological (bubble and pedestal) solutions of (

Inserting

The solution (

Pedestal or soliton on background.

Bubble or dark like soliton.

In this section, we will perform a qualitative analysis of parametric family of the following system of equations:

Equation (

The inequalities (

Figure

A slide of the admissible region with

Let us consider that the parameter

Bubble phase trajectory,

If we restrict ourselves to the parametric region defined by (

In the example illustrated by Figures

Bifurcation: the homoclinic trajectory of the previous figure becomes in two heteroclinic trajectories joining the saddle points.

A pedestal type solution emerges from different saddle like equilibrium point. In this case we use the condensate boundary condition value

Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analyzing how the qualitative nature of equation solutions depends on the parameters that appear in the equation. This may lead to sudden and dramatic changes, for example, the unpredictable timing and magnitude of a landslide. This is the subject matter of catastrophe theory, a branch of bifurcation theory, which was originated with the work of the French mathematician René Thom in the 1960s. Catastrophe theory represents the propensity of structural stable systems to display discontinuity behavior. Small changes in certain parameter values of a nonlinear system could cause equilibrium to appear or disappear, or it could change the system state from attracting to repelling one and vice versa. The large and sudden changes in the system behavior could eventually promote the generation of divergences and hysteresis. Hysteresis as is well known is the tendency of certain materials to conserve one of its properties, in absence of the stimulus that generated it. For extension it is applied to phenomena that not only depend on the actual circumstances but also depend on how these properties have been acquired by the material.

Examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures. In our case, the part of the cubic surface (

Homoclinic path for a bubble type solution with condensate boundary condition

Homoclinic path for a bubble solution with condensate boundary condition

Catastrophe switching suddenly from a bubble to a pedestal type of solution. Phase space for the parameter values

Homoclinic trajectory for a pedestal type solution with condensate boundary condition

Homoclinic trajectory for a pedestal type solution with condensate boundary condition

The catastrophic situation takes place in accordance with a hysteresis-like behavior of the family: let us suppose that we begin with traveling waves of certain initial velocity (say

In other words, we start from a bubble type solution and we return to pedestal type solution and vice versa. This behavior in the parameter space is illustrated in Figure

The hysteresis in this model can be observed in the parametric space determined by two parameters: the boundary condition

Hysteresis of condensate boundary condition

We have discussed in this work the hysteresis and catastrophe phenomena that appear in the parameter space when nontopological type of solitons, that is, bubble and pedestal solitons, emerges from background along the axon. These finding were done on the basis of the pioneering model developed by Heinburg and Jackson [

Both solutions, the bubble and the soliton (pedestal), on the background obtained as particular soliton-like solutions for specific values of parameters, could be used by the nerve for enhancing confidentiality in communication tasks. For instance, as the bubble soliton amplitude vanishes or minimizes during propagation along the nerve, this wave could be used to perform communication transmission for security, whereas the required information can be retrieved by the dark/bright soliton conversion on the background. Apparently, these solutions could conform some informational code structures for preserving and transmitting valid information along the nerve. In this case we can observe that the hysteresis could be considered as a natural filter mechanism for securely transmitting signals along the nerve system.

The authors declare that there is no conflict of interests regarding the publication of this paper.

Maximo A. Agüero is indebted to Professor V. G. Makhankov for constant support and discussions. This work was supported in part by the Secretary of Education of Mexico under the Project PROMEP 103.5/13/9347 for developing research scientific groups.