FitzHugh-Nagumo Model Research Articles

Exponential stabilization of semilinear parabolic systems

This work studies the local exponential stabilization of semilinear parabolic systems using multiplicative controls. We build an exponential feedback control that drives the solution to equilibrium. Using the stabilization result, we design a bilinear controller to drive the system solution to the desired state. Applications and simulations are given for the Fitzhugh–Nagumo equation and the heat equation.

Flow-driven control of pulse width in excitable media.

Models of pulse formation in nerve conduction have provided manifold insight not only into neuronal dynamics but also the nonlinear dynamics of pulse formation in general. Recent observation of neuronal electrochemical pulses also driving mechanical deformation of the tubular neuronal wall, and thereby generating ensuing cytoplasmic flow, now question the impact of flow on the electrochemical dynamics of pulse formation. Here, we theoretically investigate the classical Fitzhugh-Nagumo model, now accounting for advective coupling between the pulse propagator typically describing membrane potential and triggering mechanical deformations, and thus governing flow magnitude, and the pulse controller, a chemical species advected with the ensuing fluid flow. Employing analytical calculations and numerical simulations, we find that advective coupling allows for a linear control of pulse width while leaving pulse velocity unchanged. We therefore uncover an independent control of pulse width by fluid flow coupling.

Delay-induced self-oscillation excitation in the Fitzhugh–Nagumo model: Regular and chaotic dynamics

The stochastic FitzHugh–Nagumo model with time delayed-feedback is often studied in excitable regime to demonstrate the time-delayed control of coherence resonance. Here, we show that the impact of time-delayed feedback in the FitzHugh–Nagumo neuron is not limited by control of noise-induced oscillation regularity (coherence), but also results in excitation of the regular and chaotic self-oscillatory dynamics in the deterministic model. We demonstrate this numerically by means of simulations, linear stability analysis, the study of Lyapunov exponents and basins of attraction for both positive and negative delayed-feedback strengths. It has been established that one can implement a route to chaos in the explored model, where the intrinsic peculiarities of the Feigenbaum scenario are exhibited. For large time delay, we complement the study of temporal evolution by the interpretation of the dynamics as patterns in virtual space.

Existence of traveling wave solutions to reaction-diffusion-ODE systems with hysteresis

This paper establishes the existence of traveling wave solutions to a reaction-diffusion equation coupled with a singularly perturbed first order ordinary differential equation with a small parameter ϵ>0. The system is a toy model for biological pattern formation. Traveling wave solutions correspond to heteroclinic orbits of a fast-slow system. Under some conditions, the reduced problem (with ϵ=0) has a heteroclinic orbit with jump discontinuity, while the layer problem (i.e., the fast subsystem obtained as another limit of ϵ→0) has an orbit filling the gap. We thus construct a singular orbit by piecing together these two orbits. The traveling wave solution is obtained in the neighborhood of the singular orbit. However, unlike the classical FitzHugh-Nagumo equations, the singular orbit contains a fold point where the normal hyperbolicity breaks down and the standard Fenichel theory is not applicable. To circumvent this difficulty we employ the directional blowup method for geometric desingularization around the fold point.

Chimera patterns with spatial random swings between periodic attractors in a network of FitzHugh-Nagumo oscillators.

For the study of symmetry-breaking phenomena in neuronal networks, simplified versions of the FitzHugh-Nagumo model are widely used. In this paper, these phenomena are investigated in a network of FitzHugh-Nagumo oscillators taken in the form of the original model and it is found that it exhibits diverse partial synchronization patterns that are unobserved in the networks with simplified models. Apart from the classical chimera, we report a new type of chimera pattern whose incoherent clusters are characterized by spatial random swings among a few fixed periodic attractors. Another peculiar hybrid state is found that combines the features of this chimera state and a solitary state such that the main coherent cluster is interspersed with some nodes with identical solitary dynamics. In addition, oscillation death including chimera death emerges in this network. A reduced model of the network is derived to study oscillation death, which helps explaining the transition from spatial chaos to oscillation death via the chimera state with a solitary state. This study deepens our understanding of chimera patterns in neuronal networks.

Bifurcation analysis of a modified FitzHugh-Nagumo neuron with electric field

This paper is devoted to investigating the analytical solutions of the modified FitzHugh-Nagumo (FHN) neuron model with external electric fields through by the discrete mapping method, which can provide a new perspective to the study of complex motions of the neuron. The bifurcation trees of period-1 to period-8 motions and period-3 to period-12 motions of FHN neuron are presented through the implicit discrete mapping method, and the stable and unstable orbits, which cannot study through the traditional numerical method, are calculated. The bifurcations and stability of the periodic orbits are developed via eigenvalues. The spiking firing of neurons can be observed through the discrete nodes in phase portraits and time histories of membrane potential. Moreover, the antimonotonicity of improved FHN model is reported with varying the controlled parameters. The two-parameter diagrams are employed to intuitively explain the generation and evolution of periodic motions. Moreover, the modified FHN system is implemented via hardware circuit, the experiments of which validate the semi-analytical method. The investigation results are useful for the development of artificial intelligence and medicine.

Determinants of collective failure in excitable networks.

We study collective failures in biologically realistic networks that consist of coupled excitable units. The networks have broad-scale degree distribution, high modularity, and small-world properties, while the excitable dynamics is determined by the paradigmatic FitzHugh-Nagumo model. We consider different coupling strengths, bifurcation distances, and various aging scenarios as potential culprits of collective failure. We find that for intermediate coupling strengths, the network remains globally active the longest if the high-degree nodes are first targets for inactivation. This agrees well with previously published results, which showed that oscillatory networks can be highly fragile to the targeted inactivation of low-degree nodes, especially under weak coupling. However, we also show that the most efficient strategy to enact collective failure does not only non-monotonically depend on the coupling strength, but it also depends on the distance from the bifurcation point to the oscillatory behavior of individual excitable units. Altogether, we provide a comprehensive account of determinants of collective failure in excitable networks, and we hope this will prove useful for better understanding breakdowns in systems that are subject to such dynamics.

Turing instability mechanism of short-memory formation in multilayer FitzHugh-Nagumo network.

The study of brain function has been favored by scientists, but the mechanism of short-term memory formation has yet to be precise. Since the formation of short-term memories depends on neuronal activity, we try to explain the mechanism from the neuron level in this paper. Due to the modular structures of the brain, we analyze the pattern properties of the FitzHugh-Nagumo model (FHN) on a multilayer network (coupled by a random network). The conditions of short-term memory formation in the multilayer FHN model are obtained. Then the time delay is introduced to more closely match patterns of brain activity. The properties of periodic solutions are obtained by the central manifold theorem. When the diffusion coeffcient, noise intensity np, and network connection probability p reach a specific range, the brain forms a relatively vague memory. It is found that network and time delay can induce complex cluster dynamics. And the synchrony increases with the increase of p. That is, short-term memory becomes clearer.

Responsive fungal insoles for pressure detection

Mycelium bound composites are promising materials for a diverse range of applications including wearables and building elements. Their functionality surpasses some of the capabilities of traditionally passive materials, such as synthetic fibres, reconstituted cellulose fibres and natural fibres. Thereby, creating novel propositions including augmented functionality (sensory) and aesthetic (personal fashion). Biomaterials can offer multiple modal sensing capability such as mechanical loading (compressive and tensile) and moisture content. To assess the sensing potential of fungal insoles we undertook laboratory experiments on electrical response of bespoke insoles made from capillary matting colonised with oyster fungi Pleurotus ostreatus to compressive stress which mimics human loading when standing and walking. We have shown changes in electrical activity with compressive loading. The results advance the development of intelligent sensing insoles which are a building block towards more generic reactive fungal wearables. Using FitzHugh-Nagumo model we numerically illustrated how excitation wave-fronts behave in a mycelium network colonising an insole and shown that it may be possible to discern pressure points from the mycelium electrical activity.

On the “traveling pulses” of the limit of the FitzHugh–Nagumo equation when [formula omitted

A solution (u(s),v(s)) of the differential system u′=v,v′=−cv−u(u−a)(1−u)+w,w′=−(ɛ/c)(u−γw).with a,c,ɛ∈R such that (u(s),v(s))→(0,0) when s→±∞ is a traveling pulse of the FitzHugh–Nagumo equation. The limit of this differential system when ɛ→0 gives rise to the polynomial differential system u′=v,v′=−cv−u(u−a)(1−u)+w,where now a,c,w∈R. We give the complete description of its phase portraits in the Poincaré disc (i.e. in the compactification of R2 adding the circle S1 of the infinity) modulo topological equivalence.

Bifurcations to transient and oscillatory excitations in inhomogeneous excitable media: Insights into arrhythmogenesis in long QT syndrome.

Ventricular arrhythmias are the leading cause of sudden cardiac death. Understanding the mechanisms of arrhythmia initiation is important for developing effective therapeutics for prevention. Arrhythmias can be induced via premature external stimuli or occur spontaneously via dynamical instabilities. Computer simulations have shown that a large repolarization gradient due to regional prolongation of the action potential duration can result in instabilities leading to premature excitations and arrhythmias, but the bifurcation remains to be elucidated. In this study we carry out numerical simulations and linear stability analyses using a one-dimensional heterogeneous cable consisting of the FitzHugh-Nagumo model. We show that a Hopf bifurcation leads to local oscillations, which, once their amplitudes are large enough, lead to spontaneous propagating excitations. Depending on the degree of heterogeneities, these excitations can range from one to many and to be sustained oscillations, manifesting as premature ventricular contractions (PVCs) and sustained arrhythmias. The dynamics depends on the repolarization gradient and the length of the cable. Complex dynamics is also induced by the repolarization gradient. The mechanistic insights from the simple model may help in the understanding of the genesis of PVCs and arrhythmias in long QT syndrome.

Efficient numerical method for the Fitzhugh-Nagumo equations with Neumann boundary conditions

Efficient numerical method for the Fitzhugh-Nagumo equations with Neumann boundary conditions

On the nonlinear characteristics of two-phase flow system as modified Fitzhugh-Nagumo model

Two-phase flow systems are used in many industries for high heat transfer because these systems utilize latent heat of vaporization. However, these systems are susceptible to two-phase flow instabilities, which have been studied for the last several decades, mainly using numerical simulations and, to some extent, nonlinear stability analysis. These instabilities lead to flow excursions or flow oscillations and have recently been related to various nonlinear phenomena obtained by numerical continuation software. Further research of the instabilities shows that such systems’ behavior is like the FitzHugh-Nagumo (FHN) model. Therefore, the existing two-phase flow model (with some reasonable approximations) is modified into a form identical to the FHN model. It is demonstrated that the modified model reasonably captures the physical attributes of the existing model. Moreover, it is shown that the phenomena described in the FHN model are the same as predicted by earlier models. Since FHN models and their various attributes are relatively well established, they can control the instabilities or operate the two-phase system at desired conditions. The characteristics of the system dynamics with various types of disturbance (instantaneous and periodic) have been investigated. The system provides an unattainable periodicity for periodic perturbation. In comparison, it responds with a single large amplitude or spike dynamics for instantaneous perturbation.

Experimental and numerical study of the synchronization of canards in identical coupled canard-generating Bonhoeffer-van der Pol oscillators

This Letter discusses the complete and in-phase synchronization of canards generated in identical and nearly identical coupled canard-generating Bonhoeffer-van der Pol oscillators. In particular, motivated by the fact that canards are extremely sensitive to variations in the parameters describing the system, we investigate the manner in which synchronization occurs. We show numerically that the complete and in-phase synchronization of canards occurs in systems in which the coupling parameter is relatively small. Furthermore, we show that the complete synchronization of canards can be experimentally observed. Experimentally, it is found, however, that the complete synchronization of canards occurs in systems with stronger coupling. We explain why a strong coupling is necessary to experimentally observe complete synchronization.

Concentration Phenomena in Fitzhugh–Nagumo Equations: A Mesoscopic Approach

We consider a spatially extended mesoscopic FitzHugh–Nagumo model with strong local interactions and prove that its asymptotic limit converges towards the classical nonlocal reaction-diffusion FitzHugh–Nagumo system. As the local interactions strongly dominate, the weak solution to the mesoscopic equation under consideration converges to the local equilibrium, which has the form of a Dirac distribution concentrated to an averaged membrane potential.

Dynamics and synchronization control of fractional conformable neuron system.

Dynamic analysis, electrical coupling and synchronization control of the conformable FitzHugh-Nagumo neuronal models have been presented in this work. Firstly, equations of the Adomian-Decomposition-Method and conformable neuron model have been introduced. The Adomian-Decomposition-Method has been employed for the numerical simulation analysis, since it converges fast and provides serial solutions. Fractional order and external current stimulus have been considered as bifurcation parameters and their effects on neuron model dynamics have been examined in detail. Then, the electrical coupling of the two conformable neuronal models without any controller has been revealed and the significance of the coupling strength parameter has been evaluated. To eliminate impact of the coupling strength parameter on synchronization status of neurons, Lyapunov control method has been employed for synchronization control. In the last step, the numerical simulation studies have been experimentally verified using the Texas Instrument Delfino digital signal processor board. Numerical simulation results together with experimental validation have showed that the types of dynamics of the related neuron model are not affected from the change of the fractional order of conformable derivative, but the frequency of the dynamic response of the neuronal model is changed from the alteration of the fractional order. The frequency of response of the neuron model increases with decreasing values of the fractional order. On the other hand, if there is no synchronization control method, the coupled neuron models exhibit response ranging from synchronous to asynchronous depending on the sign and value of the coupling parameter. Additionally, decreasing values of the fractional order may allow the coupled neurons to enter the synchronous state more quickly due to increasing frequency of response of the neuronal model. Finally, the coupled neuron models could exhibit synchronous behavior, that is determined by calculating the standard deviation results, regardless of the value of the coupling parameter by using the Lyapunov control method.

Dynamics and control of spiral waves under feedback derived from a moving measuring point

Feedback control of spiral waves by the signal derived from a moving measuring point is investigated in the two-dimensional FitzHugh–Nagumo model, where the measuring location is changed with the motion of the spiral tip according to two schemes. In the first scheme, the direction of the line joining the spiral tip and the moving measuring point at the same time remains unchanged during the feedback control, and the application of the feedback signal can make the spiral tip gradually drift to the non-flux boundary of the system, which eventually causes the disappearance of the whole spiral pattern. It is interesting that the drift unit can always travel in a straight line after a period of time, which is more effective for eliminating spiral waves. The direction angles of the straight drift and the initial drift depend on the parameters for determining the measuring location. The second scheme is designed to has a fixed included angle between the joining line and the tangent line of the tip path at the instantaneous tip position during the control. In response to the feedback control, the system can show a few types of dynamical behaviors, including the transition from the inward-petal to outward-petal tip paths, the shrinkage of the attractor region for outward-petal tip paths with a larger unexcited core, and the transition from meandering to rigidly rotating spirals.

Non‐instantaneous impulsive stochastic FitzHugh–Nagumo equation with fractional Brownian motion

This article is devoted to examine the exponential behavior of stochastic FitzHugh–Nagumo equation in Hilbert space. Initially, the proposed model is reformulated into an abstract space, and the existence of mild solution is constructed by employing Mönch fixed‐point theorem and Hausdorff measure of non‐compactness. Furthermore, suitable conditions are developed to prove the proposed model's exponential behavior which reflects the stable generation and transmission of electric signal in neurons. At last, an example is presented to verify the established theoretical concepts.

Exploration of new solitons solutions for the Fitzhugh–Nagumo-type equations with conformable derivatives

The Fitzhugh–Nagumo equation is an important nonlinear reaction-diffusion equation used to model the transmission of nerve impulses. This equation is used in biology as population genetics, the Fitzhugh–Nagumo equation is also frequently used in circuit theory. In this study, we gave solutions to the fractional Fitzhugh–Nagumo (FN) equation, the fractional Newell–Whitehead–Segel (NWS) equation, and the fractional Zeldovich equation. We have obtained exact solutions within time fractional conformable derivative for these equations.

Nested mixed-mode oscillations, Part III: Comparison of bifurcation structures between a driven Bonhoeffer–van der Pol oscillator and Nagumo–Sato piecewise-linear discontinuous one-dimensional map

In our previous studies (Inaba and Kousaka, (2020); Inaba and Tsubone, (2020)), we discovered bifurcation structures represented by nested mixed-mode oscillations (MMOs) generated by a driven Bonhoeffer–van der Pol (BVP) oscillator. BVP oscillators are equivalent to FitzHugh–Nagumo models and have been a subject of intense research for the last six decades. In this study, we consider the case in which the diode included in a driven BVP oscillator is assumed to operate as an ideal switch. In this case, Poincaré return maps can be rigorously constructed one-dimensionally, which consist of two downward convex branches. We also consider the Poincaré return map that is approximated as a two-segment piecewise-linear discontinuous one-dimensional map. Such a piecewise-linear map was proposed by Nagumo and Sato and generates nested period-adding bifurcations. We show that un-nested, singly, and doubly nested MMO-incrementing bifurcations generated by the driven BVP oscillator coincide with one of the possible un-nested, singly, and doubly nested period-adding bifurcations, respectively, generated with the Nagumo–Sato map.