FitzHugh-Nagumo Model Research Articles

Novel Patterns in Fractional-in-Space Nonlinear Coupled FitzHugh–Nagumo Models with Riesz Fractional Derivative

In this paper, the Fourier spectral method is used to solve the fractional-in-space nonlinear coupled FitzHugh–Nagumo model.Numerical simulation is carried out to elucidate the diffusion behavior of patterns for the fractional 2D and 3D FitzHugh–Nagumo model. The results of numerical experiments are consistent with the theoretical results of other scholars, which verifies the accuracy of the method. We show that stable spatio-temporal patterns can be sustained for a long time; these patterns are different from any previously obtained in numerical studies. Here, we show that behavior patterns can be described well by the fractional FitzHugh–Nagumo and Gray–Scott models, which have unique properties that integer models do not have. Results show that the Fourier spectral method has strong competitiveness, reliability, and solving ability for solving 2D and 3D fractional-in-space nonlinear reaction-diffusion models.

Approximate Analytical Solution for Non-Linear Fitzhugh–Nagumo Equation of Time Fractional Order Through Fractional Reduced Differential Transform Method

Approximate Analytical Solution for Non-Linear Fitzhugh–Nagumo Equation of Time Fractional Order Through Fractional Reduced Differential Transform Method

Bifurcation analysis of mixed-mode oscillations and Farey trees in an extended Bonhoeffer–van der Pol oscillator

The two-variable Bonhoeffer–van der Pol oscillator, which is equivalent to the FitzHugh–Nagumo model, can be represented by a natural circuit, i.e., a circuit consisting only of simple two-terminal elements. An extended Bonhoeffer–van der Pol (BVP) oscillator is a circuit extended to a three-variable system from a two-variable BVP oscillator; this is obtained by adding an inductor–resistor branch to the original BVP oscillator. The extended BVP oscillator is known to generate mixed-mode oscillations. In this study, we classify the bifurcations of the simple sequences and their daughters, which constitute asymmetric Farey trees. Because the extended BVP oscillator is an extremely simple circuit, the parents–daughter processes can be quite precisely explained. We confirm that simple sequences 1s (s≥0) are born via saddle–node bifurcations and can be basic parents, which correspond to each stable fixed point of a Poincaré return map, where 1s represents one large excursion followed by a number s of small peaks. The two basic parents 1s and 1s+1 generate daughters [1s+1,1s×n] sequentially for the successive values of n via mixed-mode oscillation-incrementing bifurcations. The terms [1s+1,1s×n] indicate that 1s+1 is followed by 1sn-times. The daughters are born in a similar fashion to period-adding bifurcations generated by the circle map, and the parents–daughter processes satisfy Farey arithmetic and are terminated by a saddle–node bifurcation through which 1s appears. We create multiple two-parameter bifurcation diagrams and confirm that these phenomena are stably observed in these diagrams over broad ranges, i.e., complex bifurcations, such as codimension-two bifurcations or cusps, do not appear in the diagrams. Furthermore, the theoretical results are confirmed using laboratory measurements and experiments.

Noninvasive inference methods for interaction and noise intensities of coupled oscillators using only spike time data

Measurements of interaction intensity are generally achieved by observing responses to perturbations. In biological and chemical systems, external stimuli tend to deteriorate their inherent nature, and thus, it is necessary to develop noninvasive inference methods. In this paper, we propose theoretical methods to infer coupling strength and noise intensity simultaneously in two well-synchronized noisy oscillators through observations of spontaneously fluctuating events such as neural spikes. A phase oscillator model is applied to derive formulae relating each of the parameters to spike time statistics. Using these formulae, each parameter is inferred from a specific set of statistics. We verify these methods using the FitzHugh-Nagumo model as well as the phase model. Our methods do not require external perturbations and thus can be applied to various experimental systems.

Solitary pulses and periodic wave trains in a bistable FitzHugh-Nagumo model with cross diffusion and cross advection.

We describe analytically, and simulate numerically, traveling waves with oscillatory tails in a bistable, piecewise-linear reaction-diffusion-advection system of the FitzHugh-Nagumo type with linear cross-diffusion and cross-advection terms of opposite signs. We explore the dynamics of two wave types, namely, solitary pulses and their infinite sequences, i.e., periodic wave trains. The effects of cross diffusion and cross advection on wave profiles and speed of propagation are analyzed. For pulses, in the speed diagram splitting of a curve into several branches occurs, corresponding to different waves (wave branching). For wave trains, in the dispersion relation diagram there are oscillatory curves and the discontinuous curve of an isola with two branches. The corresponding wave trains have symmetric or asymmetric profiles. Numerical simulations show that for large values of the period there exist two wave trains, which come closer and closer together and are subject to fusion into one when the value of the period is decreasing. Other types of waves are also briefly discussed.

Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model

<p style='text-indent:20px;'>We investigate the formation of stationary patterns in the FitzHugh-Nagumo reaction-diffusion system with linear cross-diffusion terms. We focus our analysis on the effects of cross-diffusion on the Turing mechanism. Linear stability analysis indicates that positive values of the inhibitor cross-diffusion enlarge the region in the parameter space where a Turing instability is excited. A sufficiently large cross-diffusion coefficient of the inhibitor removes the requirement imposed by the classical Turing mechanism that the inhibitor must diffuse faster than the activator. In an extended region of the parameter space a new phenomenon occurs, namely the existence of a double bifurcation threshold of the inhibitor/activator diffusivity ratio for the onset of patterning instabilities: for large values of inhibitor/activator diffusivity ratio, classical Turing patterns emerge where the two species are in-phase, while, for small values of the diffusion ratio, the analysis predicts the formation of out-of-phase spatial structures (named <i>cross-Turing patterns</i>). In addition, for increasingly large values of the inhibitor cross-diffusion, the upper and lower bifurcation thresholds merge, so that the instability develops independently on the value of the diffusion ratio, whose magnitude selects Turing or cross-Turing patterns. Finally, the pattern selection problem is addressed through a weakly nonlinear analysis.</p>

Understanding Harmonic Structures Through Instantaneous Frequency.

The analysis of harmonics and non-sinusoidal waveform shape in time-series data is growing in importance. However, a precise definition of what constitutes a harmonic is lacking. In this paper, we propose a rigorous definition of when to consider signals to be in a harmonic relationship based on an integer frequency ratio, constant phase, and a well-defined joint instantaneous frequency. We show this definition is linked to extrema counting and Empirical Mode Decomposition (EMD). We explore the mathematics of our definition and link it to results from analytic number theory. This naturally leads to us to define two classes of harmonic structures, termed strong and weak, with different extrema behaviour. We validate our framework using both simulations and real data. Specifically, we look at the harmonic structures in shallow water waves, the FitzHugh-Nagumo neuronal model, and the non-sinusoidal theta oscillation in rat hippocampus local field potential data. We further discuss how our definition helps to address mode splitting in nonlinear time-series decomposition methods. A clear understanding of when harmonics are present in signals will enable a deeper understanding of the functional roles of non-sinusoidal oscillations.

Wiggly canards: Growth of traveling wave trains through a family of fast-subsystem foci

<p style='text-indent:20px;'>A class of two-fast, one-slow multiple timescale dynamical systems is considered that contains the system of ordinary differential equations obtained from seeking travelling-wave solutions to the FitzHugh-Nagumo equations in one space dimension. The question addressed is the mechanism by which a small-amplitude periodic orbit, created in a Hopf bifurcation, undergoes rapid amplitude growth in a small parameter interval, akin to a <i>canard explosion</i>. The presence of a saddle-focus structure around the slow manifold implies that a single periodic orbit undergoes a sequence of folds as the amplitude grows. An analysis is performed under some general hypotheses using a combination ideas from the theory of canard explosion and Shilnikov analysis. An asymptotic formula is obtained for the dependence of the parameter location of the folds on the singular parameter and parameters that control the saddle focus eigenvalues. The analysis is shown to agree with numerical results both for a synthetic normal-form example and the FitzHugh-Nagumo system.</p>

Design of the monodomain model by artificial neural networks

<p style='text-indent:20px;'>We propose an optimal control approach in order to identify the nonlinearity in the monodomain model, from given data. This data-driven approach gives an answer to the problem of selecting the model when studying phenomena related to cardiac electrophysiology. Instead of determining coefficients of a prescribed model (like the FitzHugh-Nagumo model for instance) from empirical observations, we design the model itself, in the form of an artificial neural network. The relevance of this approach relies on the approximation capacities of neural networks. We formulate this inverse problem as an optimal control problem, and provide mathematical analysis and derivation of optimality conditions. One of the difficulties comes from the lack of smoothness of activation functions which are classically used for training neural networks. Numerical simulations demonstrate the feasibility of the strategy proposed in this work.</p>

On the Nonlinear Characteristics of Two-Phase Flow System as Modified Fitzhugh-Nagumo Model

On the Nonlinear Characteristics of Two-Phase Flow System as Modified Fitzhugh-Nagumo Model

Processing and model design of the gamma oscillation activity based on FitzHugh-Nagumo model and its interaction with slow rhythms in the brain

The brain is processing information 24 hours a day. There are millions of processes proceeding in it accompanied by various spectra of rhythms. This paper tests the hypothesis that the slow delta rhythm excites the gamma rhythm oscillations. Unlike other papers, we determine the slow rhythm spectrum not at the hypothesis stage but during the experiment. We design algorithms of filtering, envelope extraction, and correlation coefficient calculation for signal processing. Moreover, we examine the data on all electroencephalogram channels, which allows us to make a more reasonable conclusion. We confirm that a slow delta rhythm excites a fast gamma rhythm with an amplitude-phase type of interaction and calculate a delay between these two signals equal to about half a second.

Modelling the coupling of the M-clock and C-clock in lymphatic muscle cells

Chronic dysfunction of the lymphatic vascular system results in fluid accumulation between cells: lymphoedema. The condition is commonly acquired secondary to diseases such as cancer or the associated therapies. The primary driving force for fluid return through the lymphatic vasculature is provided by contractions of the muscularized lymphatic collecting vessels, driven by electrochemical oscillations. However, there is an incomplete understanding of the molecular and bioelectric mechanisms involved in lymphatic muscle cell excitation, hampering the development and use of pharmacological therapies. Modelling in silico has contributed greatly to understanding the contributions of specific ion channels to the cardiac action potential, but modelling of these processes in lymphatic muscle remains limited. Here, we propose a model of oscillations in the membrane voltage (M-clock) and intracellular calcium concentrations (C-clock) of lymphatic muscle cells. We modify a model by Imtiaz and colleagues to enable the M-clock to drive the C-clock oscillations. This approach differs from typical models of calcium oscillators in lymphatic and related cell types, but is required to fit recent experimental data. We include an additional voltage dependence in the gating variable control for the L-type calcium channel, enabling the M-clock to oscillate independently of the C-clock. We use phase-plane analysis to show that these M-clock oscillations are qualitatively similar to those of a generalised FitzHugh-Nagumo model. We also provide phase plane analysis to understand the interaction of the M-clock and C-clock oscillations. The model and methods have the potential to help determine mechanisms and find targets for pharmacological treatment of lymphoedema.

Numerical scheme and stability analysis of stochastic Fitzhugh–Nagumo model

This article deals with the Fitzhugh–Nagumo equation in the presence of stochastic function. A numerical scheme has been developed for the solution of such equations which preserves the certain structure of the unknown functions, also we have given the stability analysis, consistency of the problem, and explicitly optimal a priori estimates for the existence of solutions of such equations. A unique solution has been guaranteed. The corresponding explicit estimates in the function spaces are formulated in the form of theorems. Lastly, one important feature of the article is the simulation of the proposed numerical scheme in the form of the 2D and 3D plots which shows the efficacy of the stochastic analysis of such nonlinear partial differential equations.

Analysis of a quasi-reversibility method for nonlinear parabolic equations with uncertainty data

In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed—i.e., the solution does not depend continuously on the data. To regularize the instable solution, we develop some new methods to construct some new regularized solution. We also investigate the convergence rate between the regularized solution and the solution of our equations. In particular, we establish results for several equations with constant coefficients and time dependent coefficients. The equations with constant coefficients include the heat equation, extended Fisher–Kolmogorov equation, Swift–Hohenberg equation, and many others. The equations with time dependent coefficients include Fisher-type logistic equations, the Huxley equation, and the Fitzhugh–Nagumo equation. The methods developed in this paper can also be applied to get approximate solutions to several other equations including the 1-D Kuramoto–Sivashinsky equation, 1-D modified Swift–Hohenberg equation, strongly damped wave equation, and 1-D Burgers equation with randomly perturbed operator. Some numerical examples are given which illustrate the effectiveness of our method.

Analysis of the limiting behavior of a biological neurons system with delay

In this work an analytical and numerical analysis of the limiting behaviors of a system consisting of a pair of biological neurons was carried out. In this case connection between neurons will occur with a delay. As a neuron model, the FitzHugh-Nagumo model was chosen as a model that can reproduce many dynamic behaviors of a real neuron and, at the same time, is not very complex computationally.

Painlevé properties, gauge invariance and solitons of some classes of reaction-diffusion equations

The role of symmetries and conservation laws are well known mechanisms for the reduction of systems of differential equations and, used in conjunction, lead to double reductions of the underlying models. We show here that variational and gauge symmetries have additional applications in the integrability of differential equations. In particular, we present a broad class of diffusion type equations, viz., the Fisher–Kolmorov and Fitzhugh–Nagumo equations, that satisfy Pinlevé properties, of their respective travelling wave forms and solitons, under the existence of gauge symmetries following a variational principle.

Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons

Abstract The paper is focused on the analysis of effect of coupling strength and time delay for a pair of connected neurons on the dynamics of the system. The FitzHugh–Nagumo model is used as a neuron model. The article contains analytical conditions for Hopf bifurcations in the system. A numerical verification of the results is given. Several examples of global bifurcation in the system were analyzed.

Extreme events in a forced BVP oscillator: Experimental and numerical studies

In this work, we report the existence of extreme events in the well known Bonhoeffer-van der Pol (BVP) oscillator under the excitation of a periodically forced voltage. Extreme events refer to the sudden and random increase in the amplitude of one or more of the state variables of the dynamical system and arise because of the incidence of interior crises or the presence of discontinuous boundaries or intermittency. We have chosen this system because of the fact that it has spawned several systems modelling neuronal dynamics such as Hindmarsh-Rose (HR) and Hodgkin-Huxley (HH) models. Our investigations involve both laboratory experiments and numerical simulations. We have obtained time plots, phase portraits, Poincaré maps, bifurcation graphs, Lyapunov exponents and signal to noise ratio (SNR) to study the general dynamics and to confirm the presence of extreme events, we have used statistical measures such as phase slip analysis, distribution functions for both experimental and numerical data. To the best of our knowledge, we believe that it is for the first time that the occurrence of extreme event has been reported using both real time experimental and numerical studies on this forced BVP system.

A Frequency Domain Analysis of the Excitability and Bifurcations of the FitzHugh-Nagumo Neuron Model.

The dynamics of neurons consist of oscillating patterns of a membrane potential that underpin the operation of biological intelligence. The FitzHugh–Nagumo (FHN) model for neuron excitability generates rich dynamical regimes with a simpler mathematical structure than the Hodgkin–Huxley model. Because neurons can be understood in terms of electrical and electrochemical methods, here we apply the analysis of the impedance response to obtain the characteristic spectra and their evolution as a function of applied voltage. We convert the two nonlinear differential equations of FHN into an equivalent circuit model, classify the different impedance spectra, and calculate the corresponding trajectories in the phase plane of the variables. In analogy to the field of electrochemical oscillators, impedance spectroscopy detects the Hopf bifurcations and the spiking regimes. We show that a neuron element needs three essential internal components: capacitor, inductor, and negative differential resistance. The method supports the fabrication of memristor-based artificial neural networks.

Spiral waves in a hybrid discrete excitable media with electromagnetic flux coupling.

Though there are many neuron models based on differential equations, the complexity in realizing them into digital circuits is still a challenge. Hence, many new discrete neuron models have been recently proposed, which can be easily implemented in digital circuits. We consider the well-known FitzHugh-Nagumo model and derive the discrete version of the model considering the sigmoid type of recovery variable and electromagnetic flux coupling. We show the various time series plots confirming the existence of periodic and chaotic bursting as in differential equation type neuron models. Also, we have used the bifurcation plots, Lyapunov exponents, and frequency bifurcations to investigate the dynamics of the proposed discrete neuron model. Different topologies of networks like single, two, and three layers are considered to analyze the wave propagation phenomenon in the network. We introduce the concept of using energy levels of nodes to study the spiral wave existence and compare them with the spatiotemporal snapshots. Interestingly, the energy plots clearly show that when the energy level of nodes is different and distributed, the occurrence of the spiral waves is identified in the network.