FitzHugh-Nagumo Type Research Articles

Synchronization in complete networks of ordinary differential equations of Fitzhugh – Nagumo type with nonlinear coupling

Synchronization is a ubiquitous feature in many natural systems and nonlinear science. This paper studies the synchronization in a complete network consisting of n nodes. Each node is connected to all other nodes by nonlinear coupling and represented by an ordinary differential system of FitzHugh-Nagumo type (FHN) which can be obtained by simplifying the famous Hodgkin-Huxley model. From this complete network, a sufficient condition on the coupling strength is identified to achieve the synchronization. The result shows that the networks with bigger in-degrees of the nodes synchronize more easily. The paper also shows this theoretical result numerically and see that there is a compromise.

Synchronization in complete networks of reaction-diffusion equations of FitzHugh-Nagumo type with nonlinear coupling

The synchronization in complete network consisting of nodes is studied in this paper. Each node is connected to all other ones by nonlinear coupling and is represented by a reaction-diffusion system of FitzHugh-Nagumo type which can be obtained by simplifying the famous Hodgkin-Huxley model. From this complete network, the sufficient condition on the coupling strength to achieve the synchronization is found. The result shows that the networks with bigger in-degrees of nodes synchronize more easily. The paper also presents the numerical simulations for theoretical result and shows a compromise between the theoretical and numerical results.

Travelling wave solutions for fully discrete FitzHugh-Nagumo type equations with infinite-range interactions

We investigate the impact of spatial-temporal discretisation schemes on the dynamics of a class of reaction-diffusion equations that includes the FitzHugh-Nagumo system. For the temporal discretisation we consider the family of six backward differential formula (BDF) methods, which includes the well-known backward-Euler scheme. The spatial discretisations can feature infinite-range interactions, allowing us to consider neural field models. We construct travelling wave solutions to these fully discrete systems in the small time-step limit by viewing them as singular perturbations of the corresponding spatially discrete system. In particular, we refine the previous approach by Hupkes and Van Vleck for scalar fully discretised systems, which is based on a spectral convergence technique that was developed by Bates, Chen and Chmaj.

Mathematical modeling of spatiotemporal patterns formed at a traveling reaction front.

In some chemical systems, the reaction proceeds in the form of a propagating wave. An example is the propagation of a combustion wave. At the front of such a wave, different oscillatory regimes and the appearance of spatiotemporal structures can be observed. We propose a qualitative mechanism for the formation of patterns at the front of the reaction. It is assumed that the reason is the interaction of two subsystems, one corresponding to the propagating front and the other describing the emerging patterns. The appropriate mathematical model contains two blocks: for the travelling front, we use a model of the Fisher-Kolmogorov-Petrovsky-Piskunov type, while patterns at the front are described by the FitzHugh-Nagumo type model. Earlier, we applied this approach to explain the occurrence of autowaves-target waves and spirals-at the front of the reaction. In the present paper, we demonstrate in numerical simulations that this approach also works effectively to explain stationary relative to the front patterns, the so-called Turing or cellular structures, that are observed experimentally, in particular, at the front of a combustion wave. We also investigate the dependence of these patterns on the thickness of the front and its speed, as well as on the degree of diffusion instability achieved within the front layer.

Indentical synchronization in complete networks of reaction-diffusion equations of FitzHugh-Nagumo

Synchronization is a ubiquitous feature in many natural systems and nonlinear science. This paper studies the synchronization in complete network consisting of n nodes. Each node is connected to all other nodes by linear coupling and represented by a reaction-diffusion system of FitzHugh-Nagumo type which can be obtained by simplifying the famous Hodgkin-Huxley model. From this complete network, the author seeks a sufficient condition on the coupling strength to achieve synchronization. The result shows that the more easily the nodes synchronize, the bigger the degrees of the networks. Based on this consequence, the author will test the theoretical result numerically to see if there is a compromise.

Predicting critical ignition in slow-fast excitable models.

Linearization around unstable traveling waves in excitable systems can be used to approximate strength-extent curves in the problem of initiation of excitation waves for a family of spatially confined perturbations to the rest state. This theory relies on the knowledge of the unstable traveling wave solution as well as the leading left and right eigenfunctions of its linearization. We investigate the asymptotics of these ingredients, and utility of the resulting approximations of the strength-extent curves, in the slow-fast limit in two-component excitable systems of FitzHugh-Nagumo type and test those on four illustrative models. Of these, two are with degenerate dependence of the fast kinetic on the slow variable, a feature which is motivated by a particular model found in the literature. In both cases, the unstable traveling wave solution converges to a stationary "critical nucleus" of the corresponding one-component fast subsystem. We observe that in the full system, the asymptotics of the left and right eigenspaces are distinct. In particular, the slow component of the left eigenfunction corresponding to the translational symmetry does not become negligible in the asymptotic limit. This has a significant detrimental effect on the critical curve predictions. The theory as formulated previously uses an heuristic to address a difficulty related to the translational invariance. We describe two alternatives to that heuristic, which do not use the misbehaving eigenfunction component. These new heuristics show much better predictive properties, including in the asymptotic limit, in all four examples.

Oscillatory multipulsons: Dissipative soliton trains in bistable reaction-diffusion systems with cross diffusion of attractive-repulsive type.

One-dimensional localized sequences of bound (coupled) traveling pulses, wave trains with a finite number of pulses, are described in a piecewise-linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross-diffusion terms of opposite signs. The simplest case of two bound pulses, the paired-pulse waves (pulse pairs), is solved analytically. The solutions contain oscillatory tails in the wave profiles so that the pulse pairs consist of a double-peak core and wavy edges. Several pulse pairs with different profile shapes and propagation speeds can appear for the same parameter values of the model when the cross diffusion is dominant. The more general case of many bound pulses, multipulse waves, is studied numerically. It is shown that, dependent on the values of the cross-diffusion coefficients, the multipulse waves upon collision can pass through one another with unchanged size and shape, exhibiting soliton behavior. Moreover, multipulse collisions with the system boundaries can generate a rich variety of wave transformations: the transition from the multipulse waves to pulse-front waves and further to simple fronts or to annihilation as well the transition to solitary pulses or to multipulse waves with lower numbers of pulses. Analytical and numerical results for the pulse pairs agree well with each other.

Destroying synchrony in an array of the FitzHugh–Nagumo oscillators by external DC voltage source

A control method for desynchronizing an array of mean-field coupled FitzHugh–Nagumo-type oscillators is described. The technique is based on applying an adjustable DC voltage source to the coupling node. Both, numerical solution of corresponding nonlinear differential equations and hardware experiments with a nonlinear electrical circuit have been performed.

On the Bidomain equations driven by stochastic forces

The bidomain equations driven by stochastic forces and subject to nonlinearities of FitzHugh-Nagumo or Allen-Cahn type are considered for the first time. It is shown that this set of equations admits a global weak solution as well as a stationary solution, which generates a uniquely determined invariant measure.

Patterns of interaction of coupled reaction–diffusion systems of the FitzHugh–Nagumo type

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the FitzHugh–Nagumo slow–fast system with diffusion and coupling. In the case of diffusion, the system provides a canonical example of Turing–Hopf bifurcation. By analyzing the linear stability of the local equilibrium, the occurrence of Turing–Hopf bifurcation, Turing–Turing bifurcation and coupled Turing–Hopf bifurcation are obtained. The normal form associated with the Turing–Hopf bifurcation is obtained by using the procedure of Song for calculating the normal form of PDEs. Further, in the case of two coupled FitzHugh–Nagumo reaction–diffusion, the Turing–Hopf–Turing bifurcation occurs, and we also find the case about the spatial resonance of Turing–Turing bifurcation arising, and two kinds spatially steady-state solutions are found which are synchronous or anti-phased. Finally, sample numerical results are reported.

Steady states of FitzHugh-Nagumo system with non-diffusive activator and diffusive inhibitor

In this paper, we consider a diffusion equation coupled to an ordinary differential equation with FitzHugh-Nagumo type nonlinearity. We construct continuous spatially heterogeneous steady states near, as well as far from, constant steady states and show that they are all unstable. In addition, we construct various types of steady states with jump discontinuities and prove that they are stable in a weak sense defined by Weinberger.The results are quite different from those for classical reaction-diffusion systems where all species diffuse.

Global attractors and exponential stability of partly dissipative reaction diffusion systems with exponential growth nonlinearity

We study the long-time behavior of the solutions of the partly dissipative reaction diffusion systems of the FitzHugh–Nagumo type with exponential growth nonlinearity. More precisely, we prove the existence of weak solutions, the regularity of the global attractor and the exponential stability of stationary solutions of the systems.

Investigation of the mechanism of emergence of autowave structures at the reaction front.

A qualitative mechanism for autowave pattern formation at the reaction front, observed in certain chemical systems including combustion, is suggested. It is assumed that patterns are formed as a result of interaction of two subsystems, one of which is responsible for the reaction front propagation while the other determines the formation of waves at the front. A corresponding phenomenological model is constructed in which reaction front propagation is described by a submodel of the Fisher-Kolmogorov-Petrovskii-Piskunov type and waves on the front are described by a submodel of the FitzHugh-Nagumo type. In the three-dimensional numerical analysis, it is demonstrated that the model is able to qualitatively explain the emergence of wave patterns of both spiral and target types, which are experimentally observed at the reaction front. The dependence of these patterns on the velocity and thickness of the front is examined.

Large time behaviour and synchronization of complex networks of reaction–diffusion systems of FitzHugh–Nagumo type

Large time behaviour and synchronization of complex networks of reaction–diffusion systems of FitzHugh–Nagumo type

Bifurcation control of Fitzhugh-Nagumo models

A theoretical bifurcation control strategy is presented for a single Fitzhugh-Nagumo (FN) type neuron. The bifurcation conditions are tracked for varying parameters of the individual FN neurons. A MATLAB package called as MATCONT is utilized for this purpose and all parameters of the neuron is analyzed one-by-one. Analysis by MATCONT revealed five Hopf (H) and one Limit-Point/Saddle Point (LP) bifurcation. The Hopf type of bifurcations are controlled by a washout filter supported by projective control theory. Washout filters are designed as first and second order. First order washout filter which is also physically applicable appeared to be more advantageous than the second order version. It appeared that, the LP case could not be stabilized by the aid of a washout filter. To solve this issue, a nonlinear controller is proposed. The only drawback associated with that is its inability to keep the original equilibrium point. Simulations are also provided to validate the research done.

Heterogeneity-induced effects for pulse dynamics in FitzHugh–Nagumo-type systems

Particle like structures have been observed in many fields of science. In a homogeneous medium, a stable, standing pulse is a localized wave that may arise when nonlinear and dissipative effects are in balance. In this paper, we investigate certain phenomena associated with the dynamics of pulse solutions for a FitzHugh–Nagumo reaction–diffusion model. When two pulses are located far from one another initially, their weak interaction drives the subsequent slow dynamics. Our comprehension of the standing pulse profiles allows us to quantitatively characterize their interplay; when the diffusivity of the activator is small compared to that of the inhibitor, the two pulses repel. In addition, using a center-manifold reduction to study the presence of heterogeneities in the environment, we demonstrate that the pulses will move so as to maximize the strength of activation or minimize that of inhibition. The pulse motion will also be influenced by the reaction mechanism.

Oscillatory pulse-front waves in a reaction-diffusion system with cross diffusion.

We explore traveling waves with oscillatory tails in a bistable piecewise linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross diffusion. These waves differ fundamentally from the standard simple fronts of the kink type. In contrast to kinks, the waves studied here have a complex shape profile with a front-back-front (a pulse-front) pattern. The characteristic feature of such pulse-front waves is a hybrid type of the speed diagram, which on the one hand reflects the typical dynamical behavior of the fronts in the FitzHugh-Nagumo model, related to the nonequilibrium Ising-Bloch bifurcation, and on the other hand exhibits also the solitary pulse scenario where several waves appear simultaneously with different speeds of propagation. We describe analytically the wave profiles and heteroclinic trajectories in the phase plane and discuss their morphology and transformation. The phenomena of wave formation and propagation are also studied by numerical simulations of the model partial differential equations. These simulations support the view that the pulse-front waves are constructed of fronts and pulses.

On a Variational Problem Arising from the Three-component FitzHugh-Nagumo Type Reaction-Diffusion Systems

We study a variational problem arising from the three-component Fitzhugh-Nagumo type reaction diffusion systems and its shadow systems. In [15], Oshita studied the two-component systems. He revealed that a minimizer of energy corresponding to the problem oscillates under an appropriate condition and also studied its stability. Moreover, he mentioned its energy estimate without a proof. We investigate the behavior of a minimizer corresponding to the three-component problem, its stability and its energy estimate and extend some results of Oshita to the three-component systems and its shadow systems. In particular, we give a necessary and sufficient condition that the minimizer highly oscillates as $ \epsilon \to 0 $. Also, we establish a precise order of the energy estimate of the minimizer as $ \epsilon \to 0 $. In the proof of the energy estimate, we propose a new interpolation inequality.

Upper and Lower Solutions for the FitzHugh– Nagumo Type System of Equations

We consider a moving front solution of a singularly perturbed FitzHugh–Nagumo type system of equations. The solution contains an internal transition layer, that is, a subdomain where a sharp change in the values of the functions describing the solution occurs. In initial-boundary value problems with moving front solutions, there naturally exists a small parameter that is equal to the ratio of the inner transition layer width to the width of the considered region. Taking into account this small parameter leads to the fact that the equations become singularly perturbed, thus the problems are classified as ”hard”, the numerical solution of which meets certain difficulties and does not always give a reliable result. In connection with this, the role of an analytical investigation of the existence of a solution with an internal transition layer increases. For these purposes the use of differential inequalities method is especially effective. The method consists in constructing continuous functions, which are called upper and lower solutions. An important role is played by the so-called ”quasimonotonicity condition” for functions which describe reactive terms. In this paper, we present an algorithm for constructing the upper and the lower solutions of a parabolic system with a single-scale internal transition layer. It should be mentioned that the quasimonotonicity condition in the present paper differs from the analogous condition in previous publications. The above algorithm can be further generalized to more complex systems with two-scale transition layers or to systems with discontinuous reactive terms. The study is of great practical importance for creating mathematically grounded models in biophysics.

Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type

We focus on the long time behavior of complex networks of reaction-diffusion systems. We prove the existence of the global attractor and the $L^{∞}$-bound for networks of $n$ reaction-diffusion systems that belong to a class that generalizes the FitzHugh-Nagumo reaction-diffusion equations.