# Catastrophe and Hysteresis by the Emerging of Soliton-Like Solutions in a Nerve Model

Authors: Fernando Ongay Larios; Nikolay P. Tretyakov; Maximo A. Agüero
Journal: Journal of Nonlinear Dynamics
 (2014)

---

## Abstract

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.


---

## Body

## 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 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 [8] report the experimental observation of the generation of dark solitons inherently 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 spectra, such as comb compression and chaos. In the work of Rosanov et al. [9] the hysteresis of the velocity of switching waves (fronts) in the bistable regimes has been revealed and discrete dissipative solitons have been found.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 [10]. Recently Heinburg and coworkers [11–13] considerably improved the model and better answered the question considering the wave nerve as a density pulse along the axons. From the scientific medical point of view Barz et al. [14] also supported the idea of thermomechanical responsibility of the nerve pulse propagation and appearance.The aim of this study is to take into consideration the Heinburg model [11] and the results of our recent works [15] directed towards the description of the dark and soliton on background like traveling wave solutions, in such a way to show the emergence of catastrophe and hysteresis as fundamental properties of these solitons.This paper is organized as follows. Section2 discusses the setting of the current problem concerning the mechanical approach along with weak formulation in order to obtain appropriated traveling solutions. In Section 3 we discuss the algebraic analytic conditions for the potential solving the differential equation that governs the evolution of nontopological bubble and pedestal solitons. Section 4 is devoted to fundamental qualitative numerical and analytical studio related to appearance of catastrophe and hysteresis in the model. The qualitative nature of equation solutions depends on equation parameters. Section 5 presents a brief summary and discussion.


## 2. Setting the Problem

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 [11, 16]. An important justification for the assumption of an electromechanical process is the experimental observation of reversible heat changes in phase with the action potential and a zero net heat release during the action potential. Subsequently, this model was further studied by taking into consideration the trivial and condensate boundary conditions [15, 17]. Thus, the equation of motion of density waves along the axon can be represented by
(1)∂2u∂t2=∂∂x{(c02+pu+qu2)∂u∂x}-h∂4u∂x4.Withu(x,t)=ρA-ρ0A representing the change of density with respect to the background densityρ0Aat physiological condition slightly above melting transition.In this paper we will consider the nontrivial boundary condition; that is, far from the excited zone along the axon, the difference densityUremains constant; that is, it is not completely equal to zero.


### 2.1. Weak Formulation

Before the application of the boundary condition we slightly modify (1) bearing in mind the traveling wave solution with the independent variablez=x-vt. By integrating (1) it can be transformed to the following one:
(2)hd2udz2=(c02-v2)u+12pU2+13qU3+C1.HereC1is a constant of the integration. After subsequent integration one can obtain the next equation
(3)h(dudz)2=Cu+(c02-v2)u2+αu3+βu4+V0
withα=p/3,  β=q/6,  C=2C1 and p and q being the parameters that appear in (1). The parametersC,V0are the constants of integration for the first and second integration correspondingly. For (3) to make sense in terms of distribution it is enough that u∈Hloc⁡1(R).Now, let us study (3) by keeping in mind first the nontrivial or the condensate boundary condition; that means, at long distances from the main excited zone of the axon, the perturbation pulse does not vanish while its first derivative tends to zero. Thus, the unknown functionu(z) in the distributional sense satisfies
(4)ifz⟶∞thenu⟶U0dudz⟶0.By applying this restriction, for the constants of integrationU0andV0we obtained
(5)V0=U0[(v2-c02)U0-αU02-βU03-C].As it can be easily seen this constant of integration depends on the background value of the difference densityU0which far from the excited zone will remain unperturbed.In second order equation (3) we will assumeβ>0and we reparametrize the trajectories by usings=β/hzin order to obtain the dimensionless equation
(6)(u˙)2-(Cu+δu2+αu3+u4)=V0
withδ=(c02-v2)/β. Equation (3) can be represented by a canonical structure for conservative systems
(7)u˙1=∂H∂u2,u˙2=-∂H∂u1
withu˙=∂u/∂sand the Hamiltonian
(8)H(u1,u2)=u22-(Cu1+δu12+αu13+u14)
that corresponds to the energy level
(9)H(u1,u2)=V0
beingV0the constant of integration. Then, we rewrite boundary conditions (4) in the following manner:
(10)lim⁡s→±∞(u1(s),u2(s))=(U0,0).
One possible solution of this system is the equilibrium solution(U0,0), that is, the fixed point as a critical point ofH. Conditions (10) indicate the existence of a homoclinic trajectory that joins(U0,0)with itself. In this case this critical point will be a saddle point, and the phase trajectory will surround a local minimum of the potential.Let us assume that(U0,0)is a fixed point of the system; henceU0must be a critical point of the potential function
(11)V(u1)=-(Cu1+δu12+αu13+u14),
with V′(U0)=0. If (U0,0) is a saddle point with trajectories that surround a local minimum, then the potentialVhas a local minimum and reaches a local maximum inU0.


## 2.1. Weak Formulation

Before the application of the boundary condition we slightly modify (1) bearing in mind the traveling wave solution with the independent variablez=x-vt. By integrating (1) it can be transformed to the following one:
(2)hd2udz2=(c02-v2)u+12pU2+13qU3+C1.HereC1is a constant of the integration. After subsequent integration one can obtain the next equation
(3)h(dudz)2=Cu+(c02-v2)u2+αu3+βu4+V0
withα=p/3,  β=q/6,  C=2C1 and p and q being the parameters that appear in (1). The parametersC,V0are the constants of integration for the first and second integration correspondingly. For (3) to make sense in terms of distribution it is enough that u∈Hloc⁡1(R).Now, let us study (3) by keeping in mind first the nontrivial or the condensate boundary condition; that means, at long distances from the main excited zone of the axon, the perturbation pulse does not vanish while its first derivative tends to zero. Thus, the unknown functionu(z) in the distributional sense satisfies
(4)ifz⟶∞thenu⟶U0dudz⟶0.By applying this restriction, for the constants of integrationU0andV0we obtained
(5)V0=U0[(v2-c02)U0-αU02-βU03-C].As it can be easily seen this constant of integration depends on the background value of the difference densityU0which far from the excited zone will remain unperturbed.In second order equation (3) we will assumeβ>0and we reparametrize the trajectories by usings=β/hzin order to obtain the dimensionless equation
(6)(u˙)2-(Cu+δu2+αu3+u4)=V0
withδ=(c02-v2)/β. Equation (3) can be represented by a canonical structure for conservative systems
(7)u˙1=∂H∂u2,u˙2=-∂H∂u1
withu˙=∂u/∂sand the Hamiltonian
(8)H(u1,u2)=u22-(Cu1+δu12+αu13+u14)
that corresponds to the energy level
(9)H(u1,u2)=V0
beingV0the constant of integration. Then, we rewrite boundary conditions (4) in the following manner:
(10)lim⁡s→±∞(u1(s),u2(s))=(U0,0).
One possible solution of this system is the equilibrium solution(U0,0), that is, the fixed point as a critical point ofH. Conditions (10) indicate the existence of a homoclinic trajectory that joins(U0,0)with itself. In this case this critical point will be a saddle point, and the phase trajectory will surround a local minimum of the potential.Let us assume that(U0,0)is a fixed point of the system; henceU0must be a critical point of the potential function
(11)V(u1)=-(Cu1+δu12+αu13+u14),
with V′(U0)=0. If (U0,0) is a saddle point with trajectories that surround a local minimum, then the potentialVhas a local minimum and reaches a local maximum inU0.


## 3. Algebraic-Analytical Conditions for the PotentialV

Assumingαandv(or equivalentlyδ) fixed, the conditions pointed out at the end of the last section could be satisfied for a suitable interval of the parameter valueCas we will see in shortly. In the following paragraphs we will make an algebraic-analytical survey of the function
(12)f(x)=x4+αx3+δx2+Cx,
suggested by (11), in order to determine the desired conditions of the potential
(13)V(u)=-f(u).Sincef(x)is a degree 4-polynomial, the desired conditions imply that it will show three critical points being two relative minima and one relative maximum. Hence the derivativef′has three roots. The derivative is the vertical translation of a degree 3-polynomial,
(14)f′(x)=g(x)+C,
with
(15)g(x)=4x3+3αx2+2δx=x(4x2+3αx+2δ).
Obviouslyf′should have only three roots if its graph is N shaped or that is the same if the graph ofgis N shaped. By derivingg, one found that the necessary and sufficient condition for getting this is
(16)3α2>8δ.
If this condition is satisfied, the functiongreaches a local maximum at
(17)x1=-α4-9α2-24δ12
and a local minimum at
(18)x2=-α4+9α2-24δ12.
It follows thatf′has three real roots if and only if
(19)Cmin⁡(α,δ)=-g(x1)<C<-g(x2)=Cmax⁡(α,δ).
Under these conditions, the critical points offare formed by three real solutions of the next cubic equation
(20)C=-g(x)=-x(4x2+3αx+2δ)
being those a minimum, a maximum, and a minimum.Finally, the graph offhas a vertical axis of symmetry; that is, the minima are at the same level if
(21)C=-g(x1+x22)=-g(-α4),
that is, when the parameters of the motion equation satisfy the following relation:
(22)C=C*(α,δ)=α2(δ-α24).


### 3.1. Bubble and Pedestal Solutions

In this part we will briefly recall the explicit nontopological (bubble and pedestal) solutions of (3) that were reported early in [15, 17]. Let us consider that the parameterCof (21) is delimited by-g(x1)<C<-g(x2)and letV0be chosen in such a manner that the lower maximum of
(23)U~(u)=-f(u)-V0
is tangent to the horizontal axis (axis u) at the pointU0withU0being a solution of (20) andV0=-f(U0). Then, for some constantsmandnthe value ofU~(u)can be obtained from the next equation
(24)U~(u)=-(u-U0)2((u-U0)2+m(u-U0)+n),
with the trait that the polynomial
(25)h(u)=(u-U0)2+m(u-U0)+n
has two distinct real roots ofU~, both different toU0 (see the last figure). Then necessarily it is needed thatm2-4n>0. SinceU0is not situated between the two roots ofh, we have
(26)n=h(U0)>0
(between those rootsh(u)<0, outside of those rootsh(u)>0). With these restrictions in mind, we can rewrite (3) as
(27)(u˙)2=(u-U0)2((u-u0)2+m(u-U0)+n).Insertingr=u-U0into the last equation we obtain
(28)r˙=rr2+mr+n
or, equivalently,
(29)s-s0=∫r0r‍drrr2+mr+n
with2r0<-m-m2-4nor2r0>-m+m2-4n, which can be integrated by standard method as
(30)∫‍drrr2+mr+n=-1ncosh⁡-1(2n+mr±m2-4nr).
From here, the proper solution is obtained:
(31)u(s)=2n±m2-4ncosh⁡(A-ns)-m+U0
with A being some constant of integration.The solution (31) embraces both two types of solitons. Indeed, by taking appropriated values and considering the singularities in the parametric space, one can retrieve the bubble or the pedestal nontopological solitons; see Figures 1 and 2.Figure 1
Pedestal or soliton on background.Figure 2
Bubble or dark like soliton.


## 3.1. Bubble and Pedestal Solutions

In this part we will briefly recall the explicit nontopological (bubble and pedestal) solutions of (3) that were reported early in [15, 17]. Let us consider that the parameterCof (21) is delimited by-g(x1)<C<-g(x2)and letV0be chosen in such a manner that the lower maximum of
(23)U~(u)=-f(u)-V0
is tangent to the horizontal axis (axis u) at the pointU0withU0being a solution of (20) andV0=-f(U0). Then, for some constantsmandnthe value ofU~(u)can be obtained from the next equation
(24)U~(u)=-(u-U0)2((u-U0)2+m(u-U0)+n),
with the trait that the polynomial
(25)h(u)=(u-U0)2+m(u-U0)+n
has two distinct real roots ofU~, both different toU0 (see the last figure). Then necessarily it is needed thatm2-4n>0. SinceU0is not situated between the two roots ofh, we have
(26)n=h(U0)>0
(between those rootsh(u)<0, outside of those rootsh(u)>0). With these restrictions in mind, we can rewrite (3) as
(27)(u˙)2=(u-U0)2((u-u0)2+m(u-U0)+n).Insertingr=u-U0into the last equation we obtain
(28)r˙=rr2+mr+n
or, equivalently,
(29)s-s0=∫r0r‍drrr2+mr+n
with2r0<-m-m2-4nor2r0>-m+m2-4n, which can be integrated by standard method as
(30)∫‍drrr2+mr+n=-1ncosh⁡-1(2n+mr±m2-4nr).
From here, the proper solution is obtained:
(31)u(s)=2n±m2-4ncosh⁡(A-ns)-m+U0
with A being some constant of integration.The solution (31) embraces both two types of solitons. Indeed, by taking appropriated values and considering the singularities in the parametric space, one can retrieve the bubble or the pedestal nontopological solitons; see Figures 1 and 2.Figure 1
Pedestal or soliton on background.Figure 2
Bubble or dark like soliton.


## 4. Qualitative Analysis

In this section, we will perform a qualitative analysis of parametric family of the following system of equations:(32)u˙1=∂H∂u2(u1,u2,α,δ,C)=u2,u˙2=-∂H∂u1(u1,u2,α,δ,C),
with H being the Hamiltonian defined by (8). This system is equivalent to second order nonlinear equation (2). The potential piece of this Hamiltonian takes the form
(33)V(u,α,δ,C)=-f(u,α,δ,C).Equation (2) or the equivalent conservative system (32) depends essentially on the parametersα,δ, andC; the parameterδdepends on the velocityvof the traveling wave, which is a solution of the original differential equation of the model. The valueδ=0corresponds to traveling waves with sound velocityc0, whileδ<0corresponds to supersonic solutions and δ>0 corresponds to subsonic solutions. There is an upper bound forδwhenv=0:δ≤c02/β; theoretically there is also a lower bound for the velocityvimposed by the light velocity, but since this value is much greater thanc0we can assume thatδis not lower-bounded. The parameterCis a constant of integration determined by initial conditions or by some other conditions.The inequalities (19) define a region in the parameter space(α,δ,C)that is contained in one of the two regions determined by the parabolic cylinder3α2=8δin accordance with the inequality (16). We will call the region (19) as the region of admissible parameters for the existence of desired solutions (cf. the second section). Inside the admissible region, system (32) has three equilibria (except for a measure zero set) along the horizontal axisu1: a center between two saddle points.Figure3 is a slide of the admissible region with α being constant. The vertical straight line on the right is the intersection of the parabolic cylinder (16) with the planeα=cte.Figure 3
A slide of the admissible region withα=-2. The straight line which divides the dashed regions is the bifurcation set (22). Below this line pedestal like solutions “live” while above this line bubble like solutions exist. The dotted points represent the hysteresis path; see Figures 7 to 11 and the hysteresis curve is represented in Figure 12.


### 4.1. Bifurcations

Let us consider that the parameterCsatisfies the next inequality
(34)Cmin⁡(α,δ)<C<C*(α,δ),
with C* being defined by (22). Then, the first minimum of the function  f is greater than the second one. Analogously for the inverse case we have
(35)C*(α,δ)<C<Cmax⁡(α,δ).
When the parametric triplet satisfies (34), system (32) supports a homoclinic trajectory that joins the left saddle point with itself. In this case we will call its corresponding solution aspedestal like solution. On the other hand, when (35) is satisfied, the system supports now a homoclinic trajectory that joins the right saddle point with itself and we will call its corresponding solutions asbubble like solution. These subregions of admissible regions are shown in Figure 3 separated by a straight line; the upper one corresponds to bubble type solutions. The cubic surface (22) is the set of bifurcations between “pedestals” and “bubbles” which is a heteroclinic bifurcation of the system (32). We illustrate these bifurcations by showing in Figures 4 to 6 the “evolution” of phase trajectories while the relevant parameters change.Figure 4
Bubble phase trajectory,δ=0.8,  C=0.4: the solution is of the bubble type with condensate boundary condition,  U0=1, located at the saddle on the right.If we restrict ourselves to the parametric region defined by (19), then we observe that the condensate boundary condition valueU0is one of the three solutions of the cubic equation
(36)g(x)=-C.
Outside the admissible region, this equation has only one real solution (in the border of the admissible region it has two real solutions). If (10) is accomplished, then (36), for eachU0, represents a plane in the parametric space (αδC) with equation
(37)C=-2U0δ-4U03-3αU02;
additionally, it is subjected to the restriction
(38)Cmin⁡(α,0)≤-4U03-3αU02≤Cmax⁡(α,0)
(e.g., ifα<0,Cmin⁡(α,0)=0andCmax⁡(α,0)=-α3/4).In the example illustrated by Figures4, 5, and 6 we chose the value forα=-2and the boundary conditionU0=1. The other parameter values are taken from the straight line
(39)C=-2δ+2
that are inside of the admissible region. Bifurcation occurs forδ=1andC=0as can be seen in Figure 5 and the trio(α,δ,C)lies in the bifurcation set (22).Figure 5
Bifurcation: the homoclinic trajectory of the previous figure becomes in two heteroclinic trajectories joining the saddle points.δ=1,C=0.Figure 6
A pedestal type solution emerges from different saddle like equilibrium point. In this case we use the condensate boundary condition valueU0=0.12andδ=1.1,C=-0.2.


### 4.2.  Catastrophe and Hysteresis

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 (22) contained in the admissible region is like a catastrophe surface for the parametric family (32) in the following sense: if a parametric trajectory crosses the surface (22), then the system suddenly switches from a bubble type solution to pedestal type one or vice versa (see Figures 7, 8, 9, 10, and 11).Figure 7
Homoclinic path for a bubble type solution with condensate boundary conditionU0=1.2; this will be keeping up with the parametric path until bifurcation occurs,(δ,C)=(0,1.728).Figure 8
Homoclinic path for a bubble solution with condensate boundary conditionU0=1.2. (δ,C)=(0.26,1.104).Figure 9
Catastrophe switching suddenly from a bubble to a pedestal type of solution. Phase space for the parameter values(δ,C)=(0.52,0.48)showing heteroclinic trajectories joining the actual condensate boundary condition U0=1.2with an emerging boundary condition U0=-0.2.Figure 10
Homoclinic trajectory for a pedestal type solution with condensate boundary conditionU0=-0.2.  (δ,C)=(0.26,0.376).Figure 11
Homoclinic trajectory for a pedestal type solution with condensate boundary conditionU0=-0.2,  (δ,C)=(0,0.272).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 (sayv=c0see Figure 7) specifically with a bubble type of solution and the boundary condition U0 that will be maintained fixed as the velocity changes (we will preserve the value of the parameter α unchanged). Thus, by modifying the velocity and keeping the boundary condition U0 fixed, the parametersδandCchange according to the lineal relationship (37). Following this trajectory by decreasing the velocityv, the surface (22) will be crossed, while the first border condition changes abruptly to a new border conditionU0¯. At this point the pedestal soliton-like solution suddenly appears. Maintaining now this new border condition and the same solution fixed, we try to return back to the initial state with soliton velocity (v=c0). But unexpectedly performing these changes the system does not return to the initial dynamical behavior; instead the system shows a pedestal type of behavior.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 Figure3 with the aid of Figures 7 to 11, where it is shown that the broken dotted straight line that corresponds to hysteresis and the point at the edge represents the catastrophe.The hysteresis in this model can be observed in the parametric space determined by two parameters: the boundary conditionU0and the soliton velocityvas it is depicted in Figure 12. One starts with velocityv=c0and moves keeping upU0=1.2 and decreasingvuntil the critical value ofc1=c02-δ0βis achieved while the condensate boundary condition switches suddenly toU0=-0.2. Next, one maintains this boundary condition fixed and the velocityvreturns to its initial valuec0. The parameterδ0is the critical value when the system has two heteroclinical trajectories (see Figure 9). We can observe that the velocity of the nonlinear structures is restricted by the boundary conditions and the constant of integrationsCthat also depends on the other parameters of equation of motion (2). Eventually the switching between bubble and pedestal states of the system occurs without releasing or absorbing energy in absence of other types of forces. Thus, the energy necessary for appearing bubble excitations should be equal to the energy of pedestal excitations. This conjecture subsequently will provide us with the new restriction for determining the relationship between the soliton velocities in dependence on the main parameters of the motion equation. As we can observe, it is supposed that in the gel or “solid” state the pedestal excitations could emerge, while in the liquid sate bubble perturbations should propagate along the nerve. This behavior could tell us that inside the membrane the second type of phase transition occurs, when the boundary conditions for the soliton-like solutions in the model equation are of condensate typeU0. As usual we can consider second order phase transition only those of the transitions in which below and above the critical point only one phase lives in each sector.Figure 12
Hysteresis of condensate boundary conditionU0as function of the traveling wave velocityv. Starting with v=c0, keeping upu=U0=1.2, and decreasingvuntil the critical valuec1=c02-δ0β is achieved, the condensate boundary condition switches suddenly toU0=-0.2maintaining it fixed whilevreturns to the initial valuec0. Other parameter values were chosen asα=-2,c0=2, β=0.173, and δ0=0.52.


## 4.1. Bifurcations

Let us consider that the parameterCsatisfies the next inequality
(34)Cmin⁡(α,δ)<C<C*(α,δ),
with C* being defined by (22). Then, the first minimum of the function  f is greater than the second one. Analogously for the inverse case we have
(35)C*(α,δ)<C<Cmax⁡(α,δ).
When the parametric triplet satisfies (34), system (32) supports a homoclinic trajectory that joins the left saddle point with itself. In this case we will call its corresponding solution aspedestal like solution. On the other hand, when (35) is satisfied, the system supports now a homoclinic trajectory that joins the right saddle point with itself and we will call its corresponding solutions asbubble like solution. These subregions of admissible regions are shown in Figure 3 separated by a straight line; the upper one corresponds to bubble type solutions. The cubic surface (22) is the set of bifurcations between “pedestals” and “bubbles” which is a heteroclinic bifurcation of the system (32). We illustrate these bifurcations by showing in Figures 4 to 6 the “evolution” of phase trajectories while the relevant parameters change.Figure 4
Bubble phase trajectory,δ=0.8,  C=0.4: the solution is of the bubble type with condensate boundary condition,  U0=1, located at the saddle on the right.If we restrict ourselves to the parametric region defined by (19), then we observe that the condensate boundary condition valueU0is one of the three solutions of the cubic equation
(36)g(x)=-C.
Outside the admissible region, this equation has only one real solution (in the border of the admissible region it has two real solutions). If (10) is accomplished, then (36), for eachU0, represents a plane in the parametric space (αδC) with equation
(37)C=-2U0δ-4U03-3αU02;
additionally, it is subjected to the restriction
(38)Cmin⁡(α,0)≤-4U03-3αU02≤Cmax⁡(α,0)
(e.g., ifα<0,Cmin⁡(α,0)=0andCmax⁡(α,0)=-α3/4).In the example illustrated by Figures4, 5, and 6 we chose the value forα=-2and the boundary conditionU0=1. The other parameter values are taken from the straight line
(39)C=-2δ+2
that are inside of the admissible region. Bifurcation occurs forδ=1andC=0as can be seen in Figure 5 and the trio(α,δ,C)lies in the bifurcation set (22).Figure 5
Bifurcation: the homoclinic trajectory of the previous figure becomes in two heteroclinic trajectories joining the saddle points.δ=1,C=0.Figure 6
A pedestal type solution emerges from different saddle like equilibrium point. In this case we use the condensate boundary condition valueU0=0.12andδ=1.1,C=-0.2.


## 4.2.  Catastrophe and Hysteresis

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 (22) contained in the admissible region is like a catastrophe surface for the parametric family (32) in the following sense: if a parametric trajectory crosses the surface (22), then the system suddenly switches from a bubble type solution to pedestal type one or vice versa (see Figures 7, 8, 9, 10, and 11).Figure 7
Homoclinic path for a bubble type solution with condensate boundary conditionU0=1.2; this will be keeping up with the parametric path until bifurcation occurs,(δ,C)=(0,1.728).Figure 8
Homoclinic path for a bubble solution with condensate boundary conditionU0=1.2. (δ,C)=(0.26,1.104).Figure 9
Catastrophe switching suddenly from a bubble to a pedestal type of solution. Phase space for the parameter values(δ,C)=(0.52,0.48)showing heteroclinic trajectories joining the actual condensate boundary condition U0=1.2with an emerging boundary condition U0=-0.2.Figure 10
Homoclinic trajectory for a pedestal type solution with condensate boundary conditionU0=-0.2.  (δ,C)=(0.26,0.376).Figure 11
Homoclinic trajectory for a pedestal type solution with condensate boundary conditionU0=-0.2,  (δ,C)=(0,0.272).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 (sayv=c0see Figure 7) specifically with a bubble type of solution and the boundary condition U0 that will be maintained fixed as the velocity changes (we will preserve the value of the parameter α unchanged). Thus, by modifying the velocity and keeping the boundary condition U0 fixed, the parametersδandCchange according to the lineal relationship (37). Following this trajectory by decreasing the velocityv, the surface (22) will be crossed, while the first border condition changes abruptly to a new border conditionU0¯. At this point the pedestal soliton-like solution suddenly appears. Maintaining now this new border condition and the same solution fixed, we try to return back to the initial state with soliton velocity (v=c0). But unexpectedly performing these changes the system does not return to the initial dynamical behavior; instead the system shows a pedestal type of behavior.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 Figure3 with the aid of Figures 7 to 11, where it is shown that the broken dotted straight line that corresponds to hysteresis and the point at the edge represents the catastrophe.The hysteresis in this model can be observed in the parametric space determined by two parameters: the boundary conditionU0and the soliton velocityvas it is depicted in Figure 12. One starts with velocityv=c0and moves keeping upU0=1.2 and decreasingvuntil the critical value ofc1=c02-δ0βis achieved while the condensate boundary condition switches suddenly toU0=-0.2. Next, one maintains this boundary condition fixed and the velocityvreturns to its initial valuec0. The parameterδ0is the critical value when the system has two heteroclinical trajectories (see Figure 9). We can observe that the velocity of the nonlinear structures is restricted by the boundary conditions and the constant of integrationsCthat also depends on the other parameters of equation of motion (2). Eventually the switching between bubble and pedestal states of the system occurs without releasing or absorbing energy in absence of other types of forces. Thus, the energy necessary for appearing bubble excitations should be equal to the energy of pedestal excitations. This conjecture subsequently will provide us with the new restriction for determining the relationship between the soliton velocities in dependence on the main parameters of the motion equation. As we can observe, it is supposed that in the gel or “solid” state the pedestal excitations could emerge, while in the liquid sate bubble perturbations should propagate along the nerve. This behavior could tell us that inside the membrane the second type of phase transition occurs, when the boundary conditions for the soliton-like solutions in the model equation are of condensate typeU0. As usual we can consider second order phase transition only those of the transitions in which below and above the critical point only one phase lives in each sector.Figure 12
Hysteresis of condensate boundary conditionU0as function of the traveling wave velocityv. Starting with v=c0, keeping upu=U0=1.2, and decreasingvuntil the critical valuec1=c02-δ0β is achieved, the condensate boundary condition switches suddenly toU0=-0.2maintaining it fixed whilevreturns to the initial valuec0. Other parameter values were chosen asα=-2,c0=2, β=0.173, and δ0=0.52.


## 5. Conclusions

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 [11]. These solutions propagate over the spatially homogeneous background. It is interesting to note that the appearance of solitons does not manifest as a Markovian process; instead the propagation of pulses apparently occurs with memory. The hysteresis curve in dependence on the parameterv, the traveling wave velocity, is depicted in Figure 12. The appearance of catastrophe and hysteresis suggests for us that if the system initially supports a bubble structure on the condensate, then, after parameter variations, the condensate changes to another value while the bubble solution is transformed into pedestal soliton lying on different background or condensate with slightly minor value.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.



---

*Source: 710152-2014-10-07.xml*