FitzHugh-Nagumo Model Research Articles

Inference for the stochastic FitzHugh-Nagumo model from real action potential data via approximate Bayesian computation

Inference for the stochastic FitzHugh-Nagumo model from real action potential data via approximate Bayesian computation

Inverse stochastic resonance in adaptive small-world neural networks.

Inverse stochastic resonance (ISR) is a counterintuitive phenomenon where noise reduces the oscillation frequency of an oscillator to a minimum occurring at an intermediate noise intensity, and sometimes even to the complete absence of oscillations. In neuroscience, ISR was first experimentally verified with cerebellar Purkinje neurons [Buchin et al., PLOS Comput. Biol. 12, e1005000 (2016)]. These experiments showed that ISR enables a locally optimal information transfer between the input and output spike train of neurons. Subsequent studies have further demonstrated the efficiency of information processing and transfer in neural networks with small-world network topology. We have conducted a numerical investigation into the impact of adaptivity on ISR in a small-world network of noisy FitzHugh-Nagumo (FHN) neurons, operating in a bi-metastable regime consisting of a metastable fixed point and a metastable limit cycle. Our results show that the degree of ISR is highly dependent on the value of the FHN model's timescale separation parameter ε. The network structure undergoes dynamic adaptation via mechanisms of either spike-time-dependent plasticity (STDP) with potentiation-/depression-domination parameter P or homeostatic structural plasticity (HSP) with rewiring frequency F. We demonstrate that both STDP and HSP amplify the effect of ISR when ε lies within the bi-stability region of FHN neurons. Specifically, at larger values of ε within the bi-stability regime, higher rewiring frequencies F are observed to enhance ISR at intermediate (weak) synaptic noise intensities, while values of P consistent with depression-domination (potentiation-domination) consistently enhance (deteriorate) ISR. Moreover, although STDP and HSP control parameters may jointly enhance ISR, P has a greater impact on improving ISR compared to F. Our findings inform future ISR enhancement strategies in noisy artificial neural circuits, aiming to optimize local information transfer between input and output spike trains in neuromorphic systems and prompt venues for experiments in neural networks.

Vibration-resonance chimeras in coupled excitable systems with heterogeneous aperiodic excitations

In this work, we present a novel chimera-like state known as the vibration-resonance chimera (VRC) state, which is induced by heterogeneous aperiodic (HA) excitations and emerges within the optimal range of amplitude for the HA excitations. We investigate the dynamic behaviors of the FitzHugh-Nagumo model in relation to HA excitations and coupling strength in parameter spaces, identifying specific regions where VRC states exist and analyzing the transition mechanisms between different dynamic behaviors. Using basin stability measures, we find that the mean period of HA excitations affects the multi-stable states of neural systems. Specifically, the model excited by HA excitations with a larger mean period is more likely to exhibit multi-stable states. Additionally, we explore the effects of two levels of heterogeneity in HA excitations, namely amplitude heterogeneity and period heterogeneity, on VRC states. Amplitude heterogeneity can suppress the occurrence of VRC states, whereas period heterogeneity can promote their generation. This work provides valuable insights into complex system dynamics, enhancing our understanding of chimera states in neuronal models and their potential applications in neural networks.

Stability of high-order nonlinear Takagi–Sugeno fuzzy impulsive delayed coupled systems

In this study, we consider the stability of high-order nonlinear Takagi–Sugeno fuzzy impulsive delayed coupled systems (TSFICSs). It should be noted that the linear growth condition is removed in the investigation of high-order nonlinear TSFICSs, which weakens the conditions compared with previous studies of impulsive systems. In addition, the existing research methods do not work for high-order nonlinear TSFICSs, so novel impulsive differential inequalities with high-order terms are established to investigate the stabilization of high-order nonlinear TSFICSs and develop the classic impulsive differential inequalities for high-order nonlinear situations. Next, sufficient conditions for the stability are obtained based on the new impulsive differential inequalities together with graph theory and the Lyapunov method. Finally, the theoretical results are applied to modified impulsive delayed coupled Fitzhugh–Nagumo models, and numerical simulations are provided to illustrate the practical value of the derived results.

A set of five generalised memristive synapses for the hidden nonlinear dynamics in three coupled neurons

This article, reports a new generalised set of five memristive synapses for three sets of coupled neurons (i) 2D Fitzhugh-Nagumo (FN) and 3D Hundmarsh-Rose (HR) neuron model, (ii) 2D Fitzhugh-Nagumo and 2D Fitzhugh-Nagumo neuron model and (iii) 3D Hundmarsh-Rose neuron and 3D Hundmarsh-Rose neuron model. The proposed synapses with a coupled 2D3D neurons, 3D3D neurons or 2D2D neurons leads to various dynamical behaviours. The hidden-attracted dynamics of the coupled neurons with the proposed synapses is discussed after finding the equilibrium point of the models. Using the numerical tools, the proposed model depicts periodic, quasiperiodic, stable and chaotic bursts. Furthermore, a free control technique is applied on the states of a model by using the control of its amplitude, frequency and offset.

An image encryption scheme based on an improved memristive neuron chaotic system

With fast-developing Internet and communication, the security transmission of image in network has become a research highlight. So far, researchers have designed a lot of image encryption methods based on chaotic models, some of them are not secure enough. To enhance security of image transmission on the network, in this paper, an image encryption method is developed from a 3D memeristive FitzHugh-Nagumo (FHN) neuron. First, a 3D memeristive FHN model is obtained by connecting a memeristor into a 2D FHN model, and dynamics for 3D FHN model are estimated by applying phase diagrams, bifurcation and Lyapunov exponent. Then, an image encryption algorithm is proposed by using this 3D model. Finally, security of encryption algorithm is estimated. Simulation results confirm the effectiveness of encryption scheme.

Stability of the Ionic Parameters of a Nonlocal FitzHugh-Nagumo Model of Cardiac Electrophysiology

Stability of the Ionic Parameters of a Nonlocal FitzHugh-Nagumo Model of Cardiac Electrophysiology

New convolved Fibonacci collocation procedure for the Fitzhugh–Nagumo non-linear equation

Abstract This article is dedicated to propose a spectral solution for the non-linear Fitzhugh–Nagumo equation. The proposed solution is expressed as a double sum of basis functions that are chosen to be the convolved Fibonacci polynomials that generalize the well-known Fibonacci polynomials. In order to be able to apply the proposed collocation method, the operational matrices of derivatives of the convolved Fibonacci polynomials are introduced. The convergence and error analysis of the double expansion are carefully investigated in detail. Some new identities and inequalities regarding the convolved Fibonacci polynomials are introduced for such a study. Some numerical results, along with some comparisons, are provided. The presented results show that our proposed algorithm is efficient and accurate.

Numerical analysis of coupled dynamical biological networks: Modeling electrical information exchange among nerve cells using finite volume method

An innovative approach to modeling the conduction of electrical impulses via intricate neuronal structures is introduced in this paper, which offers a theoretical and computational examination of parameter estimation in a coupled FitzHugh–Nagumo model. With this goal in mind, we present a finite volume approach to solving the FitzHugh–Nagumo model and check the numerical method’s accuracy against previous findings. To further assess and contrast the efficacy and precision of the model’s outputs, a finite difference formulation is incorporated. To clarify the basic qualitative properties of the inhibitor–activator mechanism intrinsic to the coupled FitzHugh–Nagumo model, the analysis uses dynamical system approaches and linear stability analysis. The results show that the suggested schemes are very accurate, with conditional stability, reaching fourth-order spatial and second-order temporal precision. The results are given in both tabular and graphical forms. According to numerical results, the suggested finite volume method outperforms the finite difference method in accurately and efficiently solving the nonlinear coupled FitzHugh–Nagumo model. Neuronal activity and electrical communication are complex biological systems with a lot of investigated nonlinear differential equations; this research helps us understand more about these topics.

Cell-to-cell signaling in cell populations with large cell size variability.

Sizes of multiple cells vary when they communicate with each other. Differences in cell size result in variations in the cell surface area and volume, as well as the number of enzymes and receptors involved in signal transduction. Although heterogeneity in cell size may inhibit uniformity in signaling, cell-to-cell signaling is still possible. The outcome when cell size changes to an extreme degree remains unclear. Hence, we inhibited cell division in Dictyostelium cells, a model organism for signal transduction, to gain insights into the consequences of extreme cell size variations. Measurements of cell signals in this population using fluorescence microscopy indicated that the giant cells can communicate with normal-sized cells by suppressing the signal level. Simulations of signal transduction based on the FitzHugh-Nagumo model also suggested similar results.Our findings suggest that signaling mechanism homogenizes cell-to-cell signaling in response to cell size.

Adaptive delayed feedback control for stabilizing unstable steady states.

Delayed feedback control is a commonly used control method for stabilizing unstable periodic orbits and unstable steady states. The present paper proposes an adaptive tuning delay time rule for delayed feedback control focused on stabilizing unstable steady states. The rule is designed to slowly vary the delay time, increasing the difference between the past and current states of dynamical systems, which induces the delay time to automatically fall into the stability region. We numerically confirm that the tuning rule works well for the Stuart-Landau oscillator, FitzHugh-Nagumo model, and Lorenz system.

Analysis of the nonlinear Fitzhugh–Nagumo equation and its derivative based on the Rabotnov fractional exponential function

The Fitzhugh–Nagumo equation is an essential governing equation that explains the process of impulse transmission from the nerves. This work suggests a novel generalized arbitrary-order Fitzhugh–Nagumo equation that is based on the recently developed nonsingular fractional operator. The modified Fitzhugh–Nagumo equation has been generalized using fractional Yang–Abdel–Cattani operator to provide a more accurate representation of the equation with memory effects and non-local behavior. Further, we discussed some interesting and new properties of the proposed fractional operator and the Rabotnov exponential fractional kernel with a modified Sumudu integral transform. In order to solve the proposed nonlinear differential equation, the homotopy perturbation method coupled with the modified Sumudu integral transform technique is implemented to examine the result of a newly proposed Yang–Abdel–Cattani fractional Fitzhugh–Nagumo equation, which converges to the exact solution. The nonlinear terms of this equation are handled in terms of He’s polynomials. The existence and uniqueness of the solution have been demonstrated using the Banach space fixed point theorem. The convergence and error analysis of the suggested technique are also discussed. The findings are expressed in terms of generalized Mittag-Leffler functions. The obtained results are plotted with the help of MATLAB R2016a mathematical software. The results in this work are more accurate, and it is proposed that the new Yang–Abdel–Cattani fractional derivative operator is an effective tool for finding the results of any other nonlinear problems arising in science and engineering. All three problems have been computed for different orders to confirm the significance of the fractionalizing concept in the proposed equation. The increase of the fractional order value ℘ to 1 confirms that our proposed time-fractional Fitzhugh–Nagumo equation in the Yang–Abdel–Cattani sense gets close to the analytic solution.

Modal Analysis of Cerebrovascular Effects for Digital Health Integration of Neurostimulation Therapies-A Review of Technology Concepts.

Transcranial electrical stimulation (tES) is increasingly recognized for its potential to modulate cerebral blood flow (CBF) and evoke cerebrovascular reactivity (CVR), which are crucial in conditions like mild cognitive impairment (MCI) and dementia. This study explores the impact of tES on the neurovascular unit (NVU), employing a physiological modeling approach to simulate the vascular response to electric fields generated by tES. Utilizing the FitzHugh-Nagumo model for neuroelectrical activity, we demonstrate how tES can initiate vascular responses such as vasoconstriction followed by delayed vasodilation in cerebral arterioles, potentially modulated by a combination of local metabolic demands and autonomic regulation (pivotal locus coeruleus). Here, four distinct pathways within the NVU were modeled to reflect the complex interplay between synaptic activity, astrocytic influences, perivascular potassium dynamics, and smooth muscle cell responses. Modal analysis revealed characteristic dynamics of these pathways, suggesting that oscillatory tES may finely tune the vascular tone by modulating the stiffness and elasticity of blood vessel walls, possibly by also impacting endothelial glycocalyx function. The findings underscore the therapeutic potential vis-à-vis blood-brain barrier safety of tES in modulating neurovascular coupling and cognitive function needing the precise modulation of NVU dynamics. This technology review supports the human-in-the-loop integration of tES leveraging digital health technologies for the personalized management of cerebral blood flow, offering new avenues for treating vascular cognitive disorders. Future studies should aim to optimize tES parameters using computational modeling and validate these models in clinical settings, enhancing the understanding of tES in neurovascular health.

Enhancing model identification with SINDy via nullcline reconstruction.

Many dynamical systems exhibit oscillatory behavior that can be modeled with differential equations. Recently, these equations have increasingly been derived through data-driven methods, including the transparent technique known as Sparse Identification of Nonlinear Dynamics (SINDy). This paper illustrates the importance of accurately determining the system's limit cycle position in phase space for identifying sparse and effective models. We introduce a method for identifying the limit cycle position and the system's nullclines by applying SINDy to datasets adjusted with various offsets. This approach is evaluated using three criteria: model complexity, coefficient of determination, and generalization error. We applied this method to several models: the oscillatory FitzHugh-Nagumo model, a more complex model consisting of two coupled cubic differential equations with a single stable state, and a multistable model of glycolytic oscillations. Our results confirm that incorporating detailed information about the limit cycle in phase space enhances the accuracy of model identification in oscillatory systems.

Pulsation waves along the Ciona heart tube reverse by bimodal rhythms expressed by a remote pair of pacemakers.

The heart of ascidians (marine invertebrate chordates) has a tubular structure, and heartbeats propagate from one end to the other. The direction of pulsation waves intermittently reverses in the heart of ascidians and their relatives; however, the underlying mechanisms remain unclear. We herein performed a series of experiments to characterize the pacemaker systems in isolated hearts and their fragments, and applied a mathematical model to examine the conditions leading to heart reversals. The isolated heart of Ciona robusta autonomously generated pulsation waves at ∼20 to 25beatsmin-1 with reversals at ∼1 to 10 min intervals. Experimental bisections of isolated hearts revealed that independent pacemakers resided on each side and also that their beating frequencies periodically changed as they expressed bimodal rhythms, which comprised an ∼1.25 to 5.5min acceleration/deceleration cycle of a beating rate of between 0 and 25beatsmin-1. Only fragments including 5% or shorter terminal regions of the heart tube maintained autonomous pulsation rhythms, whereas other regions did not. Our mathematical model, based on FitzHugh-Nagumo equations applied to a one-dimensional alignment of cells, demonstrated that the difference between frequencies expressed by the two independent terminal pacemakers determined the direction of propagated waves. Changes in the statuses of terminal pacemakers between the excitatory and oscillatory modes as well as in their endogenous oscillation frequencies were sufficient to lead to heart reversals. These results suggest that the directions of pulsation waves in the Ciona heart reverse according to the changing rhythms independently expressed by remotely coupled terminal pacemakers.

A Combined Taylor–Bernstein Approximation for Solving Non-linear Fitz-Hugh–Nagumo Equation

A Combined Taylor–Bernstein Approximation for Solving Non-linear Fitz-Hugh–Nagumo Equation

Soliton solutions of nonlinear stochastic Fitz-Hugh Nagumo equation

This study deals with the stochastic Fitz-Hugh Nagumo (FHN) equation and its multiple soliton solutions. The underlying model has numerous applications in neuroscience that express the pulse behavior of neurons. In general, different kinds of noise affect neurons, e.g., oscillations in the opening and closing of ion stations within cell membranes and the fluctuations of the different conductivities in the system. This fluctuation creates a sequence of stochastic excitation. Various applications are branching Brownian motion process, flame propagation, nuclear reactor theory, autocatalytic chemical reaction, mobility in neurons, population growth in the open environment, and liquid environment. So, need of the hour to consider the FHN equation under the impact of noise. The ϕ6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi ^6$$\\end{document}-model expansion method is used to extract the analytical solutions that give a dynamic attitude of the transmission for the nerve impulses of a nervous system. The different constraint conditions for the existence of these solutions are also discussed. The solutions of this model are represented in hyperbolic, trigonometric, and rational forms. The 2 and 3-dimensional behavior of these solutions are depicted by choosing the different values of parameters. The impact of noise on the physical system is analyzed and its real-world applications are discussed. The spikes in the solutions are controlled through the Borel function. These important results will open a new horizon of research for the young researchers.

Nonstandard Nearly Exact Analysis of the FitzHugh–Nagumo Model

The FitzHugh–Nagumo model has been used empirically to model certain types of neuronal activities. It is also a non-linear dynamical system applicable to chemical kinetics, population dynamics, epidemiology and pattern formation. In the literature, many approaches have been proposed to study its dynamics. In this paper, initially, we have employed cutting-edge tools from discrete dynamics for discretization and fixed points. It has been proven that an exact discrete scheme exists for this paradigm. This project also considers the phase space and integral surfaces of these evolutionary equations. In addition, it carries out a thorough symmetry analysis of this reaction diffusion system to find equivalent systems. Moreover, steady-state solutions are obtained using ansatzes for traveling wave solutions. The existence of infinite traveling wave solutions has also been proven. Yet again, this investigation establishes the potential of symmetry methods to unravel non-linearity. Finally, singular perturbation theory has been employed to obtain analytical approximations and to study stability in different parameter regimes.

A Reconfigurable Hardware Realization for Real-Time Simulation of Cardiac Excitation and Conduction

Dynamic simulation of complex cardiac excitation and conduction requires high computational time. Thus, the hardware techniques that can run in real-time simulation were introduced. However, according to existing studies, the hardware simulation requires high power consumption and involves a large size of the physical parts. Due to the drawbacks, this research presents the adaptation of nonlinear Ordinary Differential Equation (ODE)-based cardiac excitable models of Luo-Rudy Phase I (LR-I) and FitzHugh-Nagumo (FHN) in a reconfigurable hardware of field-programmable gate array (FPGA). FPGA rapid prototyping using MATLAB Hardware Description Language (HDL) Coder was used to convert a fixed-point MATLAB Simulink blocks design of the cardiac models into a synthesizable VHSIC Hardware Description Language (VHDL) code and verified using the FPGA-In-the Loop (FIL) Co-simulator. The Xilinx FPGA Virtex-6 XC6VLX240T ML605 evaluation board was chosen as the FPGA platform. The cardiac excitation response characteristics using the LR-I model and the simulations of reentrant initiation and annihilation using the FHN model were verified in Virtex-6 FPGA. This means a new real-time simulation-based analysis technique of cardiac electrical excitation and conduction wassuccessfully developed using the reconfigurable hardware.

Dynamic modulation of external excitation enhance synchronization in complex neuronal network

Understanding and controlling neural network synchronization is crucial for neuroscience in revealing brain functions and addressing neurological disorders. This study explores the innovative use of dynamic learning of synchronization (DLS) technology to enhance synchronization within neuronal networks. Using the Hodgkin-Huxley model across various network topologies, including Erdős-Rényi random graphs, small-world, and scale-free networks, it dynamically adjusts external electrical excitation to study its effects on network synchrony. To further demonstrate the universality of DLS technology, this study also validates the main results using larger-scale networks and the Izhikevich and FitzHugh-Nagumo models. The research quantifies the enhancement of synchrony through DLS, using root-mean-square error (RMSE) and synchronization factors as metrics. Findings show that DLS effectively boosts network synchrony by dynamically adjusting external excitation in response to node differences, significantly in both small-world and scale-free networks, irrespective of synaptic connections. Furthermore, DLS demonstrates potential for targeted synchronization enhancement in specific region of network. This paper highlights DLS technology's effectiveness in modulating external excitation to improve complex neural network synchrony, providing new insights into neural synchronization and information transmission.