FitzHugh-Nagumo Model Research Articles

Mixed-mode oscillations from a constrained extended Bonhoeffer-van der Pol oscillator with a diode.

An extended Bonhoeffer-van der Pol (BVP) oscillator is a circuit that is naturally extended to a three-variable system from a two-variable BVP oscillator. A BVP oscillator is known to exhibit a canard explosion, and the extended BVP oscillator generates mixed-mode oscillations (MMOs). In this work, we considered a case study where the nonlinear conductor in the extended BVP oscillator includes an idealized diode. The idealized case corresponds to a degenerate case where one of the parameters tends to infinity, and circuit dynamics are represented using a constrained equation, and at the expense of the model's naturalness, i.e., in a case in which the solutions of the dynamics are defined only forward in time, the Poincaré return maps are constructed as one-dimensional (1D). Using these 1D return maps, we explain various phenomena, such as simple MMOs and MMO-incrementing bifurcations. In this oscillator, there exists a small amplitude oscillation, which emerges as a consequence of supercritical Hopf bifurcation, and there exists large relaxation oscillation which appears via canard explosion by changing the bifurcation parameter. Between these small and large amplitude oscillations, the MMO bifurcations exhibit asymmetric Farey trees. Furthermore, these theoretical results were verified using laboratory measurements and experiments.

Bifurcation Structures of Nested Mixed-Mode Oscillations

In recently published work [Inaba & Kousaka, 2020a; Inaba & Tsubone, 2020b], we discovered significant mixed-mode oscillation (MMO) bifurcation structures in which MMOs are nested. Simple mixed-mode oscillation-incrementing bifurcations (MMOIBs) are known to generate [Formula: see text] oscillations for successive [Formula: see text] between regions of [Formula: see text]- and [Formula: see text]-oscillations, where [Formula: see text] and [Formula: see text] are adjacent simple MMOs, e.g. [Formula: see text] and [Formula: see text], where [Formula: see text] is an integer. MMOIBs are universal phenomena of evidently strong order and have been studied extensively in chemistry, physics, and engineering. Nested MMOIBs are phenomena that are more complex, but have an even stronger order, generating chaotic MMO windows that include sequences [Formula: see text] for successive [Formula: see text], where [Formula: see text] and [Formula: see text] are adjacent MMOIB-generated MMOs, i.e. [Formula: see text] and [Formula: see text] for integer [Formula: see text]. Herein, we investigate the bifurcation structures of nested MMOIB-generated MMOs exhibited by a classical forced Bonhoeffer–van der Pol oscillator. We use numerical methods to prepare two- and one-parameter bifurcation diagrams of the system with [Formula: see text], and 3 for successive [Formula: see text] for the case [Formula: see text]. Our analysis suggests that nested MMOs could be widely observed and are clearly ordered phenomena. We then define the first return maps for nested MMOs, which elucidate the appearance of successively nested MMOIBs.

Numerical Methods for Monitoring Rare Events in Nonlinear Stochastic Systems

In this article, we consider the development of numerical methods of large deviations analysis for rare events in nonlinear stochastic systems. The large deviations of the controlled process from a certain stable state are the basis for predicting the occurrenceof a critical situation (a rare event). The rare event forecasting problem is reduced to the Lagrange-Pontryagin optimal control problem.The presented approach for solving the Lagrange-Pontryagin problem differs from the approach used earlier for linear systems in that it uses feedback control. In the nonlinear case, approximate methods based on the representation of the system model in the state-space form with state-dependent coefficients (SDC) matrixes are used: the state-dependent Riccati equation (SDRE) and the asymptotic sequence of Riccati equations (ASRE). The considered optimal control problem allow us to obtain a numerical-analytical solutionthat is convenient for real-time implementation. Based on the developed methods of large deviations analysis, algorithms for estimating the probability of occurrence of a rare event in a dynamical systemare presented. The numerical applicability of the developed methods is shown by the example of the FitzHugh-Nagumo model for the analysis of switching between excitable modes. The simulation results revealed an additional problem related to the so-called parameterization problem of the SDC matrices. Since the use of different representations for SDC matrices gives different results in terms of the system trajectory, the choice of matrices is proposed to be carried out at each algorithm iteration so as to provide conditions for the solvability of the Lagrange-Pontryagin problem.

Data-driven prediction of multistable systems from sparse measurements.

We develop a data-driven method, based on semi-supervised classification, to predict the asymptotic state of multistable systems when only sparse spatial measurements of the system are feasible. Our method predicts the asymptotic behavior of an observed state by quantifying its proximity to the states in a precomputed library of data. To quantify this proximity, we introduce a sparsity-promoting metric-learning (SPML) optimization, which learns a metric directly from the precomputed data. The optimization problem is designed so that the resulting optimal metric satisfies two important properties: (i) it is compatible with the precomputed library and (ii) it is computable from sparse measurements. We prove that the proposed SPML optimization is convex, its minimizer is non-degenerate, and it is equivariant with respect to the scaling of the constraints. We demonstrate the application of this method on two multistable systems: a reaction-diffusion equation, arising in pattern formation, which has four asymptotically stable steady states, and a FitzHugh-Nagumo model with two asymptotically stable steady states. Classifications of the multistable reaction-diffusion equation based on SPML predict the asymptotic behavior of initial conditions based on two-point measurements with 95% accuracy when a moderate number of labeled data are used. For the FitzHugh-Nagumo, SPML predicts the asymptotic behavior of initial conditions from one-point measurements with 90% accuracy. The learned optimal metric also determines where the measurements need to be made to ensure accurate predictions.

A popular reaction-diffusion model fractional Fitzhugh-Nagumo equation: analytical and numerical treatment

A popular reaction-diffusion model fractional Fitzhugh-Nagumo equation: analytical and numerical treatment

A Nearly Exact Discretization Scheme for the FitzHugh–Nagumo Model

A Nearly Exact Discretization Scheme for the FitzHugh–Nagumo Model

Hubs, diversity, and synchronization in FitzHugh-Nagumo oscillator networks: Resonance effects and biophysical implications

Using the FitzHugh-Nagumo equations to represent the oscillatory electrical behavior of β-cells, we develop a coupled oscillator network model with cubic lattice topology, showing that the emergence of pacemakers or hubs in the system can be viewed as a natural consequence of oscillator population diversity. The optimal hub to nonhub ratio is determined by the position of the diversity-induced resonance maximum for a given set of FitzHugh-Nagumo equation parameters and is predicted by the model to be in a range that is fully consistent with experimental observations. The model also suggests that hubs in a β-cell network should have the ability to "switch on" and "off" their pacemaker function. As a consequence, their relative amount in the population can vary in order to ensure an optimal oscillatory performance of the network in response to environmental changes, such as variations of an external stimulus.

Synchronization mode transition induced by bounded noise in multiple time-delays coupled FitzHugh–Nagumo model

ABSTRACT Noise and time-delays are ubiquitous in physical and biological systems. In this paper, the multiple time-delays coupled FitzHugh-Nagumo (FHN) models is employed to investigate the synchronization mode transition. The orbital projection method is used to study the difference of membrane potential between two FHN neurons in the phase plane, and a measure of anti-phase is defined to characterize the synchronization state of neural system. It is shown that the synchronization mode of coupled neurons is different while changing the parameters of the system. In the absence of noise, as the coupling strength increases, the firing mode of two coupled neurons undergoes a succession of transitions (i.e., from the asynchronous state, to the completely synchronized state, then the anti-phase state, and finally to the completely synchronized state again). In the presence of noise, the synchronization mode of neurons becomes more diversified with the increasing of noise intensity. Moreover, by changing the time-delay and coupling strength, the sensitivity of two-neuron to noise can be changed, thereby the synchronization mode transition can be adjusted.

Transient dynamics and multistability in two electrically interacting FitzHugh-Nagumo neurons.

We analyze the existence of chaotic and regular dynamics, transient chaos phenomenon, and multistability in the parameter space of two electrically interacting FitzHugh-Nagumo (FHN) neurons. By using extensive numerical experiments to investigate the particular organization between periodic and chaotic domains in the parameter space, we obtained three important findings: (i) there are self-organized generic stable periodic structures along specific directions immersed in a chaotic portion of the parameter space; (ii) the existence of transient chaos phenomenon is responsible for long chaotic temporal evolution preceding the asymptotic (periodic) dynamics for particular parametric combinations in the parameter space; and (iii) the existence of various multistable domains in the parameter space with an arbitrary number of attractors. Additionally, we also prove through numerical simulations that chaos, transient chaos, and multistability prevail even for different coupling strengths between identical FHN neurons. It is possible to find multistable attractors in the phase and parameter spaces and to steer them apart by increasing the asymmetry in the coupling force between neurons. Such a strategy can be essential to experimental matters, as setting the right parameter ranges. As the FHN model shares the crucial properties presented by the more realistic Hodgkin-Huxley-like neurons, our results can be extended to high-dimensional coupled neuron models.

Chimera states in FitzHugh–Nagumo networks with reflecting connectivity

We investigate the effects of reflecting connectivity in a network composed of FitzHugh–Nagumo (FHN) elements linked in a ring topology. Reflecting connectivity is inspired by the linking between the opposite hemispheres in the mammalian brain. To study synchronization phenomena and coexistence of synchronous and asynchronous domains (chimera states) under the influence of this connectivity, we use two versions of the FHN model: Version I where membrane and recovery potentials are interlinked via a rotational matrix (Omelchenko et al. PRL 110:224101, 2013), and version II where only the membrane potentials are linked and not the recovery ones (Shepelev et al., Phys. Lett. A 381:1398, 2017). In both realizations, the reflecting connectivity forces the activity to organise in two connected semirings. Our numerical results give evidence that, for FHN-I and positive (negative) coupling strength, coherent (incoherent) oscillatory regions develop and reside in the junctions between the two semirings. For the FHN-II model which supports multistability, our simulations indicate that for appropriate values of the parameters the two semirings are separated via oscillating elements. The center parts of the two semirings are composed either by elements which are frozen at the FHN fixed points (coexistence of fixed points and oscillatory states), or by coherent oscillatory elements (coexistence of coherent and incoherent nodes). Notable, here, is the case of oscillation death, which has also been reported for other oscillators coupled via one of their variables. These results can be useful for interpreting inhomogeneous and localised coordinated activity observed in the mammalian brain.

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.

Near-Pulse Solutions of the FitzHugh–Nagumo Equations on Cylindrical Surfaces

We introduce a geometrical extension of the FitzHugh-Nagumo equations describing propagation of electrical impulses in nerve axons. In this extension, the axon is modeled as a warped cylinder, rather than a straight line, as is usually done, while pulses propagate on its surface, as is the case with real axons. We prove the stability of electrical impulses for a standard cylinder and existence and stability of pulse-like solutions for warped cylinders whose radii are small and vary slowly along their lengths.

The kinetic Fokker–Planck equation with general force

We consider the kinetic Fokker–Planck equation with a class of general force. We prove the existence and uniqueness of a positive normalized equilibrium (in the case of a general force) and establish some exponential rate of convergence to the equilibrium (and the rate can be explicitly computed). Our results improve results about classical force to general force case. Our result also improves the rate of convergence for the Fitzhugh–Nagumo equation from non-quantitative to quantitative explicit rate.

Design of Suboptimal Local Piecewise Fuzzy Controller With Multiple Constraints for Quasi-Linear Spatiotemporal Dynamic Systems.

This paper discusses the problem of suboptimal local piecewise H ∞ fuzzy control of quasi-linear spatiotemporal dynamic systems with control magnitude constraints. A Takagi-Sugeno fuzzy partial differential equation (PDE) model with space-varying coefficient matrices is first assumed to be derived for exactly describing nonlinear system dynamics. In the light of the fuzzy model, a local piecewise fuzzy feedback controller is then constructed to guarantee the exponential stability with a prescribed H ∞ disturbance attenuation level for the resulting closed-loop system, while the control constraints are also ensured. A sufficient condition on the existence of such fuzzy controller is developed by the Lyapunov direct method and an integral inequality and presented in terms of space algebraic linear matrix inequalities (LMIs) coupled with LMIs. By virtue of extreme value theorem, a suboptimal-constrained local piecewise H ∞ fuzzy control design in the sense of minimizing the disturbance attenuation level is formulated as a minimization optimization problem with LMI constraints. Finally, the proposed method is applied to solve the feedback control of a quasi-linear FitzHugh-Nagumo equation with space-varying coefficients, and simulation results show its effectiveness and merit.

Funnel control for the monodomain equations with the FitzHugh-Nagumo model

We consider a nonlinear reaction diffusion system of parabolic type known as the monodomain equations, which model the interaction of the electric current in a cell. Together with the FitzHugh-Nagumo model for the nonlinearity they represent defibrillation processes of the human heart. We study a fairly general type with co-located inputs and outputs describing both boundary and distributed control and observation. The control objective is output trajectory tracking with prescribed performance. To achieve this we employ the funnel controller, which is model-free and of low complexity. The controller introduces a nonlinear and time-varying term in the closed-loop system, for which we prove existence and uniqueness of solutions. Additionally, exploiting the parabolic nature of the problem, we obtain Hölder continuity of the state, inputs and outputs. We illustrate our results by a simulation of a standard test example for the termination of reentry waves.

The probability density function of interspike intervals in an FHN model with α-stable noise

In this paper, we expand the paradigmatic Fitzhugh–Nagumo (FHN) model by including α-stable noise and study the firing rate and coherence resonance via probability density function (PDF) of the interspike intervals (ISIs). In the previous work, different indicators and measurements are given to illustrate the firing rate and coherence resonance. However, we find that they can be described simply by the PDF of ISIs, which contains the underlying information of spiking neuron signals. For example, the width of the peak of the PDF of ISIs can characterize the phenomenon of coherence resonance. The narrower width of peak stands for the better coherence resonance.

Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: Fronts, pulses, and wave trains.

We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [Zemskov et al., Phys. Rev. E 77, 036219 (2008) and Phys. Rev. E 95, 012203 (2017) for fronts and for pulses and wave trains, respectively]. In addition, these constructions allow us to describe novel waves that are specific to the tristable system. Most interesting is the pulse solution with a zigzag-shaped profile, the bright-dark pulse, in analogy with optical solitons of similar shapes. Numerical simulations indicate that this wave can be stable in the system with asymmetric thresholds; there are no stable bright-dark pulses when the thresholds are symmetric. In the latter case, the pulse splits up into a tristable front and a bistable one that propagate with different speeds. This phenomenon is related to a specific feature of the wave behavior in the tristable system, the multiwave regime of propagation, i.e., the coexistence of several waves with different profile shapes and propagation speeds at the same values of the model parameters.

Control of coherence resonance in multiplex neural networks

We study the dynamics of two neuronal populations weakly and mutually coupled in a multiplexed ring configuration. We simulate the neuronal activity with the stochastic FitzHugh-Nagumo (FHN) model. The two neuronal populations perceive different levels of noise: one population exhibits spiking activity induced by supra-threshold noise (layer 1), while the other population is silent in the absence of inter-layer coupling because its own level of noise is sub-threshold (layer 2). We find that, for appropriate levels of noise in layer 1, weak inter-layer coupling can induce coherence resonance (CR), anti-coherence resonance (ACR) and inverse stochastic resonance (ISR) in layer 2. We also find that a small number of randomly distributed inter-layer links are sufficient to induce these phenomena in layer 2. Our results hold for small and large neuronal populations.

Mathematical Model of the Fitzhugh–Nagumo Hereditary Oscillator

The paper proposes a new mathematical FitzHugh—Nagumo model with memory, which describes the propagation of a nerve impulse in a membrane. The model consists of an integro-differential equation with initial conditions (the Cauchy problem). The difference kernel (memory function) of the model equation was chosen as a power function so that it can be rewritten in terms of the fractional derivative. For the Cauchy problem, an explicit finite-difference scheme is constructed, and an investigation of its stability and convergence was performed in computer experiments. The finite-difference scheme was implemented in the Maple computer program, with the help of which the simulation results were visualized, and oscillograms and phase trajectories were obtained.

Bifurcation and chaotic behavior in the discrete BVP oscillator

In this work, a new class of discrete Bonhoeffer–van der Pol (BVP) system with an odd function is proposed and investigated. At first, the necessary and sufficient conditions on the existence and stability of the fixed points for this system are given. We then show the system passes through various bifurcations of codimension one, including pitchfork bifurcation, saddle–node bifurcation, flip bifurcation and Neimark–Sacker bifurcation under some certain parameter conditions. The center manifold theorem and bifurcation theory are the main tools in the analysis of the local bifurcations. Furthermore, we prove rigorously there exists Marotto’s chaos in this discrete BVP system, which means the fixed point eventually evolves into a snap-back repeller. Finally, numerical simulation evidences are provided not only to further demonstrate our theoretical analysis, but also to exhibit the complex dynamical phenomena, such as the period-9, -17, -18 orbits, attracting invariant cycles, quasi-periodic orbits, ten-coexisting chaotic attractors, etc. These phenomena illustrate relatively rich dynamical behaviors of the discrete BVP oscillator.