Stability of periodic travelling wave solutions of a nerve conduction equation

Summary Nagumo's nerve conduction equation has travelling wave solutions of pulse type and periodic wave type. We consider the stability of the latter ones. We denote byL(c) the minimum spatial period of a periodic travelling wave solution whose propagation speed isc. It is shown that this travelling wave solution is unstable ifL′(c)<0.     

Existence and stability of periodic travelling wave solutions to Nagumo's nerve equation

Summary Nagumo's nerve conduction equation has a one-parameter family of spatially periodic travelling wave solutions. First, we prove the existence of these solutions by using a topological method. (There are some exceptional cases in which this method cannot be applied in showing the existence.) A periodic travelling wave solution corresponds to a closed orbit of a third-order dynamical system. The Poincaré index of the closed orbit is determined as a direct consequence of the proof of the existence. Second, we prove that the periodic travelling wave solution is unstable if the Poincaré index of the corresponding closed orbit is + 1. By using this result, together with the result of the author's previous paper, it is concluded that the slow periodic travelling wave solutions are always unstable. Third, we consider the stability of the fast periodic travelling wave solutions. We denote by L(c) the spatial period of the travelling wave solution with the propagation speed c. It is shown that the fast solution is unstable if its period is close to L_min, the minimum of L(c).     

Temporal stability of solitary impulse solutions of a nerve equation

We study a differential equation that models nerve impulse transmission. The nonlinearity is simplified to be piecewise linear in order to allow explicit solution. In general, two solitary impulse solutions are exhibited. The temporal stability of these solutions is analyzed by a technique that identifies the number of unstable modes. These results extend the results of Rinzel and Keller (1973, Biophys. J. 13:1313) by showing that the slower unstable solution has only one unstable mode, and that the fast solution, as conjectured, has no unstable modes and is therefore stable.     

Existence of traveling wave solutions in a diffusive predator-prey model

 We establish the existence of traveling front solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator-prey model with Holling type-II functional response. The traveling front solutions are equivalent to heteroclinic orbits in R ⁴ and the small amplitude traveling wave train solutions are equivalent to small amplitude periodic orbits in R ⁴ . The methods used to prove the results are the shooting argument and the Hopf bifurcation theorem. Received: 25 May 2001 / Revised version: 5 August 2002 / Published online: 19 November 2002 RID="*" ID="*" Research was supported by the National Natural Science Foundations (NNSF) of China. RID="*" ID="*" Research was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. On leave from the Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada. Mathematics Subject Classification (2000): 34C35, 35K57 Key words or phrases: Traveling wave solution – Wazewski set – Shooting argument – Hopf bifurcation Acknowledgements. We would like to thank the two referees for their careful reading and helpful comments.     

Existence of periodic solutions of a nerve equation

A recently developed continuous neuronal model is transformed into a system which is closely related to the theory of cellular control processes. In a broad region of parameter values periodic solutions of the equations describing the neuron do exist.     

Numerical solution of a nonlinear advance-delay-differential equation from nerve conduction theory

A functional differential equation which is nonlinear and involves forward and backward deviating arguments is solved numerically. The equation models conduction in a myelinated nerve axon in which the myelin completely insulates the membrane, so that the potential change jumps from node to node. The equation is of first order with boundary values given at t=±. The problem is approximated via a difference scheme which solves the problem on a finite interval by utilizing an asymptotic representation at the endpoints, cubic interpolation and iterative techniques to approximate the delays, and a continuation method to start the procedure. The procedure is tested on a class of problems which are solvable analytically to access the scheme's accuracy and stability, then applied to the problem that models propagation in a myelinated axon. The solution's dependence on various model parameters of physical interest is studied. This is the first numerical study of myelinated nerve conduction in which the advance and delay terms are treated explicitly.Supported in part by NSF Grant MCS8301724 and by a Biomedical Research Support Grant 2SO7RR0706618 from NIH     

Explicit solutions of Fisher's equation for a special wave speed

The travelling waves for Fisher's equation are shown to be of a simple nature for the special wave speeds . In this case the equation is shown to be of Painlevé type, i.e. solutions admit only poles as movable singularities. The general solution for this wave speed is found and a method is presented that can be applied to the solution of other nonlinear equations of biological and physical interest.     

Periodic and traveling wave solutions to Volterra-Lotka equations with diffusion

Analytic and numerical solutions to two coupled nonlinear diffusion equations are studied. They are the modified equations of Volterra and Lotka for the spatially stratified predatorprey population model. In a bounded domain with the reflecting boundary, equilibrium, stability, and transition to time-periodic solutions are analyzed. For a wide class of initial states, the solutions to the initial boundary-value problem evolve into their corresponding stable, space-homogeneous, periodic oscillations. In an unbounded domain, a family of traveling wave solutions is found for certain exponential, initial distributions in the limit as the diffusion coefficientv ₁ of the prey tends to zero. In the presence of both diffusions, the results of a numerical simulation to an initial-value problem showed the rapid formation of the Pursuit-Evasion Waves whose speed of propagation and amplitudes increase with the diffusion coefficientv ₁. Presented at the 1974 SIAM Fall Meeting.     

Existence and stability of travelling wave solutions of a nonlinear integral operator

In this paper, we establish the existence and stability property of travelling wave solutions of a nonlinear integral operator in the inferior case.     

A quantitative approximation scheme for the traveling wave solutions in the Hodgkin-Huxley model

We introduce an approximation scheme for the Hodgkin-Huxley model of nerve conductance that allows calculation of both the speed and shape of the traveling pulses, in quantitative agreement with the solutions of the model. We demonstrate that the reduced problem for the front of the traveling pulse admits a unique solution. We obtain an explicit analytical expression for the speed of the pulses that is valid with good accuracy in a wide range of the parameters.     

Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics

We consider a nonlinear diffusion equation proposed by Shigesada and Okubo which describes phytoplankton growth dynamics with a selfs-hading effect.We show that the following alternative holds: Either (i) the trivial stationary solution which vanishes everywhere is a unique stationary solution and is globally stable, or (ii) the trivial solution is unstable and there exists a unique positive stationary solution which is globally stable. A criterion for the existence of positive stationary solutions is stated in terms of three parameters included in the equation.     

Model of traveling waves in a coral nerve network

Coral polyps contract when electrically stimulated and a wave of contraction travels from the site of stimulation at a constant speed. Models of coral nerve networks were optimized to match one of three different experimentally observed behaviors. To search for model parameters that reproduce the experimental observations, we applied genetic algorithms to increasingly more complex models of a coral nerve net. In a first stage of optimization, individual neurons responded with spikes to multiple, but not single pulses of activation. In a second stage, we used these neurons as the starting point for the optimization of a two-dimensional nerve net. This strategy yielded a network with parameters that reproduced the experimentally observed spread of excitation.     

The effect of inhibition on the existence of traveling wave solutions for a neural field model of human seizure termination

In this paper we study the influence of inhibition on an activity-based neural field model consisting of an excitatory population with a linear adaptation term that directly regulates the activity of the excitatory population. Such a model has been used to replicate traveling wave data as observed in high density local field potential recordings (González-Ramírez et al. PLoS Computational Biology, 11(2), e1004065, 2015). In this work, we show that by adding an inhibitory population to this model we can still replicate wave properties as observed in human clinical data preceding seizure termination, but the parameter range over which such waves exist becomes more restricted. This restriction depends on the strength of the inhibition and the timescale at which the inhibition acts. In particular, if inhibition acts on a slower timescale relative to excitation then it is possible to still replicate traveling wave patterns as observed in the clinical data even with a relatively strong effect of inhibition. However, if inhibition acts on the same timescale as the excitation, or faster, then traveling wave patterns with the desired characteristics cease to exist when the inhibition becomes sufficiently strong.     

An asymptotic solution for traveling waves of a nonlinear-diffusion Fisher's equation

We examine traveling-wave solutions for a generalized nonlinear-diffusion Fisher equation studied by Hayes [J. Math. Biol. 29, 531–537 (1991)]. The density-dependent diffusion coefficient used is motivated by certain polymer diffusion and population dispersal problems. Approximate solutions are constructed using asymptotic expansions. We find that the solution will have a corner layer (a shock in the derivative) as the diffusion coefficient approaches a step function. The corner layer at z = 0 is matched to an outer solution for z < 0 and a boundary layer for z > 0 to produce a complete solution. We show that this model also admits a new class of nonphysical solutions and obtain conditions that restrict the set of valid traveling-wave solutions. Supported by a National Science Foundation graduate fellowship. This work was performed under National Science Foundation grant DMS-9024963 and Air Force Office of Scientific Research grant AFOSR-F49620-94-1-0044.     

A study of conduction velocity in nonmyelinated nerve fibers.

By treating a nonmyelinated nerve fiber as a continuous cable consisting of three distinct zones (Resting, transitional, and excited), the following mathematical expression was derived: (formula: see text) where v is the conduction velocity, d the diameter of the fiber, R the resistance of the membrane of unit area at the peak of excitation, rho the resistivity of the medium inside the fiber, and C the capacity of membrane per unit area. The validity of this expression was demonstrated by using squid giant nerve fibers intracellularly perfused with dilute salt solutions. The relationship between these results and previous theories and experiments on conduction velocity is discussed.     

Travelling front solutions of a nonlocal Fisher equation

We consider a scalar reaction-diffusion equation containing a nonlocal term (an integral convolution in space) of which Fisher‘s equation is a particular case. We consider travelling wavefront solutions connecting the two uniform states of the equation. We show that if the nonlocality is sufficiently weak in a certain sense then such travelling fronts exist. We also construct expressions for the front and its evolution from initial data, showing that the main difference between our front and that of Fisher‘s equation is that for sufficiently strong nonlocality our front is non-monotone and has a very prominent hump. Received: 8 August 1999 / Revised: 3 March 2000 / Published online: 14 September 2000     

神经传导的新概念:无动作电位的兴奋传导

近年来研究揭示在神经传导中存在着无动作电位的兴奋传导机制,它主要涉及细胞膜中胞膜窖专门化脂筏内的分子转导作用,这对兴奋只能通过电现象沿神经进行传导的传统理念是一个重大的挑战.这一机制可能是神经科学基础领域研究的重要进展.