FitzHugh-Nagumo Neuron Model Research Articles

A fractional-order improved FitzHugh-Nagumo neuron model

Abstract In this paper, we propose a fractional-order improved FitzHugh-Nagumo (FHN) neuron model in terms of a generalised Caputo fractional derivative. Following the existence of a unique solution for the proposed model, we derive the numerical solution using a recently proposed L1 predictor-corrector method. The given method is based on the L1-type discretization algorithm and the spline interpolation scheme. We perform the error and stability analyses for the given method. We perform graphical simulations demonstrating that the proposed FHN neuron model generates rich electrical activities of periodic spiking patterns, chaotic patterns, and quasi-periodic patterns. The motivation behind proposing a fractional-order improved FHN neuron model is that such a system can provide a more nuanced description of the process with better understanding and simulation of the neuronal responses by incorporating memory effects and non-local dynamics, which are inherent to many biological systems.

Neural behaviors and energy properties of Memcapacitor FitzHugh–Nagumo neuron model with Miller effect

Neural behaviors and energy properties of Memcapacitor FitzHugh–Nagumo neuron model with Miller effect

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.

Frequency chimera state induced by time delays in FitzHugh-Nagumo neural networks

The delay of information transmission is an inherent factor of the nervous systems, which has great influences on their collective dynamic behaviors. In this paper, we constructed a ring neural network using FitzHugh-Nagumo (FHN) neuron model as the network node and memristive synapses as the connection mode. Our primary focus was on investigating the effects of time delay and coupling strength on the firing frequency of neurons. Simulation results revealed that the frequency chimera state could be induced in the neural network with appropriate time delay, which is a new type of chimera state characterized by firing frequency rather than traditional membrane potential. By adjusting the time delay properly, the neural network can also display multi-cluster frequency chimera states that coexisted with various incoherent regions and coherent regions. Meanwhile, we exhibit that initial value and coupling strength could have great influences on the effects of time delay on inducing frequency chimera state of the nervous systems.

Influence of sinusoidal forcing on the master FitzHugh–Nagumo neuron model and global dynamics of a unidirectionally coupled two-neuron system

In this paper, we investigate a seven-parameter, five-dimensional dynamical system, specifically a unidirectional coupling of two FitzHugh–Nagumo neuron models, with one neuron being sinusoidally driven. This master–slave configuration features neuron N1 as the master, subjected to an external sinusoidal electrical current, and neuron N2 as the slave, interacting with N1 through an electrical force. We report numerical results for three distinct scenarios where N1 operates in (i) periodic, (ii) quasiperiodic, and (iii) chaotic regimes. The primary objective is to explore how the dynamics of the master neuron N1 influence the coupled system’s behavior. To achieve this, we generated cross sections of the seven-dimensional parameter space, known as parameter planes. Our findings reveal that in the periodic regime of N1, the coupled system exhibits period-adding sequences of Arnold tongue-like structures in the parameter planes. Furthermore, regions of multistability can also be identified in these parameter planes of the coupled system. In the quasiperiodic regime, regions of periodic motion are absent, with only regions of quasiperiodic and chaotic dynamics present. In the chaotic regime of N1, the parameter planes display regions of chaos, hyperchaos, and transient hyperchaos.

High-Matching and Low-Cost Realization of the FHN Neuron Model on Reconfigurable FPGA Board.

The main objectives of neuromorphic engineering are the research, modeling, and implementation of neural functioning in the human brain. We provide a hardware solution that can replicate such a nature-inspired system by merging multiple scientific domains and is based on neural cell processes. This work provides a modified version of the original Fitz-Hugh Nagumo (FHN) neuron using a simple 2V term called Hybrid Piece-Wised Base-2 Model (HPWBM), which accurately reproduces numerous patterns of the original neuron model. With reduced terms, we suggest modifying the original nonlinear term to achieve high matching accuracy and little computing error. Time domain and phase portraits are used to validate the proposed model, which shows that it can reproduce all of the FHN model's properties with high accuracy and little mistake. We provide an effective digital hardware approach for large-scale neuron implementations based on resource-sharing and pipelining strategies. The Hardware Description Language (HDL) is used to construct the hardware on an FPGA as a proof of concept. The recommended model hardly uses 0.48 percent of the resources on a Virtex 4 FPGA board, according to the results of the hardware implementation. The circuit can run at a maximum frequency of 448.236 MHz, according to the static timing study.

Dynamical analysis of the FitzHugh–Nagumo model with memristive synapse

Understanding the transition mechanism between different activity patterns in realistic neuron models brings forward fundamental challenges for the dynamical systems theory. The knowledge about the mechanisms can present precious predictions and intuitions for determining fundamental principles of neuron functioning. This paper aims to present a dynamic analysis of the FitzHugh–Nagumo neuron model with a memristive synapse. In this article, the effects of memristive synapse's conductance gain, applied current, and the speed of electrical signal propagation along axons and dendrites on the system's behavior and the neuron's membrane potential are investigated. Analyzes are performed by investigating bifurcation and Lyapunov exponent diagrams versus various model parameters. The physiological role of some parameters and the effect of parameter changes on the model behaviors are discussed. The results demonstrate that when no current is applied to the system, the maximum value of membrane potential is larger, and also, a lower memristive synapse conductance gain leads to a larger maximum value of membrane potential. So, it can be concluded that the neuron in an amplitude death state may be triggered and start firing by decreasing the conductance gain or not applying current.

A novel memristive neuron model and its energy characteristics.

The functional neurons are basic building blocks of the nervous system and are responsible for transmitting information between different parts of the body. However, it is less known about the interaction between the neuron and the field. In this work, we propose a novel functional neuron by introducing a flux-controlled memristor into the FitzHugh-Nagumo neuron model, and the field effect is estimated by the memristor. We investigate the dynamics and energy characteristics of the neuron, and the stochastic resonance is also considered by applying the additive Gaussian noise. The intrinsic energy of the neuron is enlarged after introducing the memristor. Moreover, the energy of the periodic oscillation is larger than that of the adjacent chaotic oscillation with the changing of memristor-related parameters, and same results is obtained by varying stimuli-related parameters. In addition, the energy is proved to be another effective method to estimate stochastic resonance and inverse stochastic resonance. Furthermore, the analog implementation is achieved for the physical realization of the neuron. These results shed lights on the understanding of the firing mechanism for neurons detecting electromagnetic field.

The rotation-transition procedure of the Fitzhugh-Nagumo neuron model and its hardware verification

The rotation-transition procedure of the Fitzhugh-Nagumo neuron model and its hardware verification

Forecasting coherence resonance in a stochastic Fitzhugh–Nagumo neuron model using reservoir computing

We delve into the intriguing realm of reservoir computing to predict the intricate dynamics of a stochastic FitzHugh–Nagumo neuron model subjected to external noise. Through innovative reservoir design and training, we unveil the remarkable capacity of a reservoir computer to forecast the behavior of this stochastic system across a wide spectrum of noise intensity variations. Notably, our reservoir computer astutely replicates the intricate phenomenon of coherence resonance in the stochastic FitzHugh–Nagumo neuron, signifying the superior modeling capabilities of this approach. A detailed examination of the microscopic dynamics within the reservoir’s hidden layer reveals the emergence of distinct neuronal clusters, each displaying unique behaviors. Certain neurons within the reservoir are adept at faithfully reproducing the dynamical traits of the neuron, particularly the spike generation mechanism. In contrast, the remaining neurons within the reservoir seem to emulate stochastic influences with remarkable precision, accurately capturing the moments of spike generation in the neuron under the sway of noise. This innovative reservoir design proves to be highly effective across a diverse range of noise control parameters, faithfully replicating the essential characteristics of the original stochastic FitzHugh–Nagumo neuron. These findings illuminate the potential of reservoir computing to model and predict the dynamics of complex stochastic systems, showcasing its adaptability and versatility in understanding and simulating natural phenomena.

A practical synthesis and analysis of the fractional-order FitzHugh-Nagumo neuronal model

A practical synthesis and analysis of the fractional-order FitzHugh-Nagumo neuronal model

Impact of time varying interaction: Formation and annihilation of extreme events in dynamical systems.

This study investigates the emergence of extreme events in two different coupled systems: the FitzHugh-Nagumo neuron model and the forced Liénard system, both based on time-varying interactions. The time-varying coupling function between the systems determines the duration and frequency of their interaction. Extreme events in the coupled system arise as a result of the influence of time-varying interactions within various parameter regions. We specifically focus on elucidating how the transition point between extreme events and regular events shifts in response to the duration of interaction time between the systems. By selecting the appropriate interaction time, we can effectively mitigate extreme events, which is highly advantageous for controlling undesired fluctuations in engineering applications. Furthermore, we extend our investigation to networks of oscillators, where the interactions among network elements are also time dependent. The proposed approach for coupled systems holds wide applicability to oscillator networks.

Uniform ultimate boundedness and approximate synchronization of coupled reaction-diffusion systems with parameter mismatches

We present a framework for studying the uniform ultimate boundedness and approximate synchronization of coupled systems. The individual subsystems of the coupled systems are described by general reaction-diffusion (RD) systems, and we account for parameter mismatches between individual subsystems while considering a general coupling scheme. We derive several criteria for the uniform ultimate boundedness of coupled RD systems under different assumptions regarding the properties of the reaction terms and coupling schemes. Using M-matrix theory and iterative methods, we establish an approximate synchronization criterion for the coupled RD systems. This criterion allows the coupled systems to synchronize with synchronization errors arising from parameter mismatches, and perfect synchronization is achieved when there are no parameter mismatches. Moreover, the criterion provides an estimated bound for the synchronization error, which decreases as the degree of inhomogeneity caused by parameter mismatches reduces. To demonstrate the effectiveness of our approaches, we apply our theories to the FitzHugh-Nagumo (FHN) neuron model.

Analysis of two spectral Galerkin approximation schemes for solving the perturbed FitzHugh-Nagumo neuron model

In this paper, we mainly study two spectral Galerkin approximation schemes of the FitzHugh-Nagumo (FHN) neuron model for the nerve impulse propagation, which contains a small parameter perturbation and strong nonlinearity. To this end, we consider two kinds of temporal discretization by a semi-implicit Euler method and a Crank-Nicolson method, and the spatial discretization by the spectral Galerkin method for the perturbed FHN neuron model. In particular, in order to obtain the second-order approximation in time for the numerical scheme, we modify the nonlinear term f(un) in the Crank-Nicolson spectral Galerkin (CNSG) scheme. We derive the error estimations with optimal convergence rates of the numerical solutions by the first-order semi-implicit spectral Galerkin (SISG) approximation scheme and the second-order modified CNSG (MCNSG) approximation scheme for the perturbed FHN neuron model. Finally, some numerical examples of one-dimensional and two-dimensional nonlinear FHN models with periodic boundary conditions are carried out to verify that the approximate solutions of the proposed SISG and MCNSG schemes satisfy the proved theoretical results.

Conformable fractional-order derivative based adaptive FitzHugh-Nagumo neuron model

Abstract Various neuron models have been proposed and are extensively examined in the scientific literature. The FitzHugh-Nagumo neuron model is one of the most well-known and studied models. The FitzHugh-Nagumo model is not biologically consistent but operationally simple. A fractional-order derivative is described as a derivative with a non-integer order. Caputo, Grünwald-Letnikov, and Riemann-Liouville are some of the well-known fractional order derivatives. However, a simple fractional-order derivative called the conformable fractional-order derivative has been proposed in the literature and it is much simpler to use. In literature, there are already neuron models with fractional-order derivatives. In this study, a FitzHugh-Nagumo model circuit with a conformable fractional derivative capacitor and conformable fractional derivative inductor is proposed. The proposed circuit is modelled, and its simulation results are given. The simulation results reveal that the model circuit shows both slow and fast adaptation in firing frequency under sustained current stimulation.

Memristor initial-offset boosting and its bifurcation mechanism in a memristive FitzHugh-Nagumo neuron model with hidden dynamics

Electromagnetic induction plays a vital role in impacting the neuron dynamics, since the electromagnetic induction is triggered alongside with the complex interaction of membrane potential and various ions transport. In other words, the electromagnetic induction is reversely subjected to the membrane potential fluctuation and ions distribution. Flux-controlled memristor described by flux and voltage has been employed to restrict the electromagnetic induction effect. This paper presents a flux-controlled memristor with sinusoidal mem-conductance function and hyperbolic tangent function modulated input. The memristor can availably reflect the non-uniform ions distribution inside and outside neuron membrane. Then, an improved memristive FitzHugh-Nagumo (mFHN) neuron model is built to explore the dynamical effect of the memristive electromagnetic induction. Numerical simulations and theoretical analysis are performed, which reveal that the mFHN neuron model possesses no equilibrium point and can generate abundant hidden dynamics. Interestingly, hidden coexisting behavior is triggered by memristor initial-offset boosting, which evokes the emergence of infinitely hidden coexisting firing patterns. Besides, bifurcation mechanism of the memristor initial-offset boosting behavior is theoretically analyzed. Furthermore, PowerSIM-based (PSIM-based) analog circuit simulations and microcontroller unit based (MCU-based) digital experiments are executed, the captured results perfectly verify the correctness of numerical results and the feasibility of analog/digital hardware experiments.

Identification of Two FitzHugh-Nagumo Neuron Models Based on the Speed-Gradient and Filtering

The paper is devoted to the problem of parameter identification of two FitzHugh-Nagumo neuron models. The FitzHugh-Nagumo model is a simplification of the Hodgkin-Huxley model and it is very valuable for using on practice thanks to its simplicity. However, within an experiment only one variable of the FitzHugh-Nagumo model, the membrane potential, is measured, while another variable of cumulative effects of all slow ion currents responsible for restoring the resting potential of the membranes and both variables’ derivatives cannot be measured. This circumstance brings additional difficulties to the parameters estimation problem and, therefore, this case needs special attention. Firstly, the model was transformed to more simple form without unmeasured variables. Variables obtained from applying second-order real filter-differentiator were used instead of unmeasured derivatives in model’s equations. As a result, a linear equation was gotten and for this equation the identification goal, which guarantees correct parameters’ adjustment, was formulated and an adaptive system, parameters of which are estimations of original system’s parameters and an output of which estimates the output of the linear equation, was constructed. Then, the integral objective function was defined and the algorithm for the original model parameters identification was designed with the speed-gradient method. The results of computer simulation in the Simulink environment are presented. These results demonstrate that estimates of the model’s state and parameters converge to their true values rather fast. Unlike existing solutions of the FitzHugh-Nagumo identification problem, we propose a much easier deterministic algorithm. Moreover, the parameters are estimated for a system collected from two FitzHugh-Nagumo models, which opens perspectives for using the proposed method in modeling neuron population activity.

Complete dynamic analysis of homeostatic model: a feedback signal from extracellular matrix to FitzHugh–Nagumo neuron model

Complete dynamic analysis of homeostatic model: a feedback signal from extracellular matrix to FitzHugh–Nagumo neuron model

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.

Self-tuned criticality: Controlling a neuron near its bifurcation point via temporal correlations.

Previous work showed that the collective activity of large neuronal networks can be tamed to remain near its critical point by a feedback control that maximizes the temporal correlations of the mean-field fluctuations. Since such correlations behave similarly near instabilities across nonlinear dynamical systems, it is expected that the principle should control also low-dimensional dynamical systems exhibiting continuous or discontinuous bifurcations from fixed points to limit cycles. Here we present numerical evidence that the dynamics of a single neuron can be controlled in the vicinity of its bifurcation point. The approach is tested in two models: a two-dimensional generic excitable map and the paradigmatic FitzHugh-Nagumo neuron model. The results show that in both cases, the system can be self-tuned to its bifurcation point by modifying the control parameter according to the first coefficient of the autocorrelation function.