^{1}

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We analytically derived the complex Ginzburg-Landau equation from the Liénard form of the discrete FitzHugh Nagumo model by employing the multiple scale expansions in the semidiscrete approximation. The complex Ginzburg-Landau equation now governs the dynamics of a pulse propagation along a myelinated nerve fiber where the wave dispersion relation is used to explain the famous phenomena of propagation failure and saltatory conduction. Stability analysis of the pulse soliton solution that mimics the action potential fulfills the Benjamin-Feir criteria for plane wave solutions. Finally, results from our numerical simulations show that as the dissipation along the myelinated axon increases, the nerve impulse broadens and finally degenerates to front solutions.

Wave propagation and pattern formation in a variety of excitable media can be effectively described by reaction-diffusion equations. The FitzHugh Nagumo (FHN) model [

Myelinated axons modelled by discrete FHN equations incorporate a richer dynamics than its continuous counterparts. Also, the mathematical study of spatially discrete models is challenging because of special and poorly understood phenomena occurring in them that are absent if the continuum limit of these models is taken. For example, in the discrete FHN model, there exists a coupling threshold for its propagation while the con- tinuous system sustains propagation of all coupling strengths [

The effects of dissipation is prominent in most physical and biological systems [

The control of pulse propagation in dissipative media can also be achieved by subjecting the system to high frequency periodic perturbations. For a discrete system like myelinated axons, this method has the advantage that the external frequency can either suppress or enhance the pulse propagation [

The rest of the work is organized as follows; in Section 2, after a brief introduction of the FHN model, we derived the model CGL equation in a weak dissipative medium using the multiple scale expansion in the semi-discrete approach. The analytic solution of the CGL equation is obtained in Section 3 following the method highlighted in [

We consider a one-dimensional chain of FitzHugh-Nagumo equations [

Here

Differentiating Equation (1a) with respect to time and substituting the value of

where

System (2) is the Liénard form of the FHN model of a myelinated nerve fiber which may be considered as the modified form of the van der Pol equation. For

Equation (2) also mimics a one dimensional chain of atoms with unit mass, harmonically coupled to their nearest neighbors, characterized by dissipation and subjected to nonlinear on-site potential. A lot of difficulties is encountered to analytically solve this equation, however we will use a perturbation technique to minimize the effects of dissipation and also attempts to find appropriate solutions. In this light, all the dissipative coefficients

There are always frequency limits within which normal propagation of nerve impulse signals are observed across the FHN myelinated axon. In order to define the appropriate frequency range, we employ a perturbative technique where a suitable solution of the FHN model containing

with

In the semidiscrete approximation,

Collecting terms at order

which is plotted in

the cut-off frequency

The variation of the diffuseness of the plasma membrane through the dispersion coefficient

accounts for the alteration of ions movement across pumps and ion channels at the Ranvier nodes. Clearly for normal propagation of action potential across the myelinated axon, there exists a frequency range

where

this frequency range, the voltage-gated sodium ions channels are highly concentrated in the Ranvier nodes which are myelin free. When an action potential is generated at the axon hillock, the influx of sodium ions causes the adjacent Ranvier nodes to depolarize, resulting in an action potential at the node. This also triggers depolarization of the next Ranvier node and the eventual initiation of an action potential. Action potentials are successively generated at neighboring Ranvier nodes; therefore, the action potential in a myelinated axon appears to jump from one node to the next, a process called saltatory conduction. For

Terms of order

where

Finally, we collect terms proportional to

By considering the reference mobile frame

the wave, we obtain

where P, is the real dispersion coefficient,

Equation (8) is the CGL equation and generally speaking, it represents one of the most-studied nonlinear equations in the physics world today. This is because it gives a qualitative and quantitative description of a myriad of physical activities [

Since we are dealing with a CGL equation, it is incumbent on us to look for a propagating wave solution of the form:

which upon substitution into Equation (8) to obtain

Let us assume that

where d is the chirp parameter and

We now have two second order ordinary differential equations (ODE) relative to the same dependent variable,

The following procedure is employed in order to find the compatibility conditions of the two equations; initially we eliminate the first derivatives from Equation (16) to have

which upon integration and setting the integration constant to zero yields

On the other hand, we eliminate the second derivative from Equation (16), obtaining

Equations (18) and (19) must coincide, consequently leading to the following conditions:

In order for us to obtain pulse solutions with real amplitudes, we assume that

with the amplitude of the solution plotted in

We now obtain the exact analytic solution of the nerve impulse propagating along the myelinated axon by substituting (21) into the FHN model solution (4) to have;

where

Also, when the parameter

Stability is a very important property of a wave profile in a neural network, since it determines whether such patterns can be observed experimentally, or utilized for diagnostic purposes. Recall that the phenomenon of modulational instability results when a steady-state solution is subjected to a weak perturbation, which eventually leads to the exponential growth of its amplitude along the line of propagation due to the interplay between the nonlinearity and dispersive effects of the medium. Initially, we consider the stability of the trivial homogeneous solution

where q is a spatial wavenumber of the perturbation and

and when all the eigenvalues

We now study the existence and stability of plane wave solutions

Due to the symmetry property of the CGL Equation (8) i.e.

We now perturb the plane wave solutions in order to have

where

is a small perturbation term with a growth rate

After substituting (28) into (29) and using Equation (25), we obtain an equation involving two linearly independent functions

with

Since we are looking for non-trivial solutions

Solutions

where

component because it determines the nature of roots of the growth rate

For

For

always stable since

Lastly, for

grows exponentially with time resulting to the instability of the plane waves which tend to self-modulate with a wave vector k. The plane wave solutions of the CGL Equation (8) clearly manifest the the classical Benjamin- Feir scenario where plane waves are unstable for positive

The condition

We now perform the numerical simulations of the CGL Equation (8) by using a Runge-Kutta scheme with fixed step size and initial conditions obtained from the analytic solution (21) of the CGL equation to check the long term effects of modulational instability.

In the first case, i.e.

The corresponding contour plot of the spatiotemporal evolution of the nerve impulse in

In this study, we addressed the issue of nerve pulse propagation along a weakly dissipative myelinated axon modelled by the discrete FHN model. The effect of dissipation is always a big nuisance during the transmission of electrical signals across a neural network, consequently crucial information is always lost. We transformed the FHN model to its Liénard form and minimized the effects of dissipation by perturbing the appropriate parameters to higher order. The proposed solution for the Liénard equation carried the

becomes more unstable and degenerates to fronts.

From this theoretical study carried on a dissipative myelinated axon, we believe it will furnish us with the appropriate knowledge of predicting the physiological state of real neurons. For instance, a sick neuron which is usually considered as dissipative can be clearly distinguished from a healthy neuron and consequently lead to

the appropriate therapeutic action.

N. Oma Nfor appreciates the enriching discussion with F. M. Moukam Kakmeni of the LaRAMaNS research group. The authors are very grateful to the referee for useful comments and the Journal of Modern Physics for subsidizing the publication fee.

Nkeh Oma Nfor,Mebu Tatason Mokoli, (2016) Dynamics of Nerve Pulse Propagation in a Weakly Dissipative Myelinated Axon. Journal of Modern Physics,07,1166-1180. doi: 10.4236/jmp.2016.710106