FitzHugh Nagumo 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

Complex dynamic behavioral transitions in auditory neurons induced by chaotic activity

Chaotic sequences are widely used in secure communication due to their high randomness. Chaotic resonance (CR) refers to the resonant response of a system to weak signals induced by chaotic activity, but its practical application remains limited. This study designs a simplified FitzHugh-Nagumo (FHN) auditory neuron model by simulating the physiological activities of auditory neurons and considering the combined stimulation of chaotic activity and sound signals. It is found that the neuron dynamics depend on both external sound stimuli and chaotic current intensity. Chaotic currents induce spikes in the neuron output sequence through CR, and the spikes become more frequent with increasing current intensity, eventually leading to a chaotic state regardless of the initial state. However, the sensitivity of the initial value of this chaotic sequence shifts to the chaotic current excitation system. The injection of chaotic currents can reduce the system's average Hamiltonian energy under certain conditions. By measuring the complexity of the generated sequences, we find that the addition of chaotic currents can enhance the complexity of the original sequences, and the enhancement ability increases with the intensity. This provides a new approach to enhance the complexity of original chaotic sequences. Moreover, different chaotic currents can induce different chaotic sequences with varying abilities to enhance the complexity of the original sequences. We hope our work can contribute to secure communication.

Asymptotic behavior of non-autonomous fractional stochastic lattice FitzHugh–Nagumo system driven by linear mixed white noise

This paper is concerned with the asymptotic behavior of the non-autonomous fractional stochastic lattice FitzHugh–Nagumo system driven by the linear mixed white noise, which simultaneously contains linear additive noise and multiplicative noise. For the sake of the long-term behavior of the system we considered, we need to utilize a different Ornstein–Uhlenbeck transformation than the general one. First, the existence and uniqueness of pullback random attractors are demonstrated. Then, we prove the upper semicontinuity of random attractors when the intensity of noise approaches zero.

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.

Deterministic analysis of stochastic FHN systems based on Gaussian decoupling

This paper systematically explores the deterministic characteristics of FitzHugh-Nagumo (FHN) systems’ response to synaptic noise in statistical sense. With the help of Gauss decoupling approximation, two substitute systems of FHN systems’ response to synaptic noise are established by ignoring the cumulants higher than the second order, and the feasibilities of using the substitute systems for response analysis are demonstrated through error analysis. Then, the deterministic analyses of FHN systems with synaptic noise are carried out by means of the two substitute systems. Numerical results show that whether it is the neuronal system or the network system, the synaptic noise can effectively regulate its dynamics, and induce the mode transitions of discharge activity. Based on the class II excitability of FHN neurons, the system activity has a nonlinear dependence on the noise parameter and the other variables of interest. Particularly, the synaptic noise not only makes the neuronal system transition from low-level to high-level narrow-amplitude oscillation by competing with the input signal, but also contributes to the response or detection of this system to weak input signals. This study reveals the deterministic characteristics of FHN systems with synaptic noise, which can provide a reference for the large-scale analysis.

An approximate solution for stochastic Fitzhugh–Nagumo partial differential equations arising in neurobiology models

In this paper, approximate solutions for stochastic Fitzhugh–Nagumo partial differential equations are obtained using two‐dimensional shifted Legendre polynomial (2DSLP) approximation. The problem's suitability and solvability are confirmed. The convergence analysis for the proposed methodology and the error analysis in the norm are carried out. Using Maple software, an algorithm is created and implemented to arrive at the numerical solution. The solution thus obtained is compared with the exact solution and the solution obtained using the explicit order RK1.5 method.

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.

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.

Synchronization evaluation of memristive photosensitive neurons in multi-neuronal systems

At the forefront of brain-computer interface (BCI) technology exploration, the synchronization phenomenon in neural networks demonstrates significant potential for application. This paper focuses on the memory characteristics and nonlinear dynamic behavior of memristors, an emerging electronic component. By innovatively integrating memristors, phototubes, and FitzHugh-Nagumo (FHN) circuits, we construct a memristive photosensitive neuron (MPN) model, aiming to explore its unique role in neural network synchronization. Utilizing the memristors' inherent memory capabilities and nonlinear properties, the MPN model mimics the dynamic behavior of biological neurons. Experiments show that synchronization is highly sensitive to the memristor's initial values, revealing intricate synchronization regulation mechanisms. Furthermore, we find that chaotic electrical currents, as environmental disturbances, have dual effects—either promoting or inhibiting MPN network synchronization—depending on specific parametric conditions. To simulate a realistic biological neural network and achieve efficient coupling among multiple MPN units, this paper adopts a Hopfield network structure. The results indicate that this structure significantly enhances the synchronization stability of the system, reduces sensitivity to initial conditions, and mitigates the adverse effects of chaotic currents. This research not only enhances our understanding of neural network synchronization but also provides novel theoretical support and technical pathways for the development of BCI technology, indicating broad application prospects in neuroscience, rehabilitative medicine, and other fields.

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.

Fractional-order heterogeneous neuron network based on coupled locally-active memristors and its application in image encryption and hiding

Synaptic crosstalk significantly influences neural firing in the brain. Locally-active memristors can effectively emulate neural network synapses and have a significant importance in neural network research. This paper designs a tristable locally-active memristive model and presents a novel fractional-order (FO) heterogeneous neuron network. This neural network consists of Hindmarsh-Rose (HR) neuron and FitzHugh-Nagumo (FHN) neuron, which are connected by coupling FO locally-active memristors. The research found that changes in the order of different dimensions have a significant effect on the neural network firing through the three-parameter bifurcation diagram. Moreover, it is found that the locally-active memristor as a synapse can affect the coexistence firing behavior of the network. The complex dynamics have been studied numerically by using phase diagrams, Lyapunov exponent spectrum, bifurcation diagram and extreme multistability can be found. In particular, the system can generate a complex bursting behavior in the presence of an external current. In order to verify the accuracy of the simulation, the phase diagram of FO heterogeneous neuron network is implemented by STM32 microcontroller, and results of the experiments are in great agreement with results of the numerical simulations. Finally, an image encryption and hiding method based on FO heterogeneous neuron network and discrete wavelet transform (DWT) is proposed. The experimental results demonstrate that the encryption and hiding scheme has excellent security and strong robustness.

On discrete FitzHugh–Nagumo reaction–diffusion model: Stability and simulations

This research paper focuses on the analysis of a discrete FitzHugh–Nagumo reaction–diffusion system. We begin by discretizing the FitzHugh–Nagumo reaction–diffusion model using the second-order and L1-difference approximations. Our study examines the local stability of the equilibrium points of the system. To identify conditions that ensure the global asymptotic stability of the steady-state solution, we employ various techniques, with a primary focus on the direct Lyapunov method. Theoretical results are supported by numerical simulations that demonstrate the practical validity of the asymptotic stability conclusions. Our findings provide new insights into the stability characteristics of discrete FitzHugh–Nagumo reaction–diffusion systems and contribute to the broader understanding of such systems in mathematical biology.

Firing patterns and fast–slow dynamics in an N-type LAM-based FitzHugh–Nagumo circuit

The diversity of firing patterns and their bifurcation mechanisms of a neuron circuit are crucial for exploiting bionic applications. This paper constructs an N-type locally active memristor (LAM)-based FitzHugh–Nagumo (FHN) circuit by replacing the tunnel-diode in the original FHN circuit. Numerical simulations and hardware measurements reveal that the N-type LAM-based FHN circuit can reproduce rich neuromorphic firing patterns of quasi-periodic/periodic bursting behaviors and chaotic/periodic spiking behaviors. The bursting and spiking behaviors are triggered by a low-frequency stimulus and a high-frequency one, respectively. Besides, the fold and Hopf bifurcation sets are depicted. Then, the bifurcation mechanisms for the quasi-periodic and periodic bursting behaviors via Hopf/Hopf and Hopf/fold bifurcations are theoretically deduced by time-domain waveform and equilibrium trajectory. The numerical results and hardware experiments exhibit the effectiveness of the proposed memristive FHN circuit in reproducing abundant firing patterns of bursting and spiking behaviors.

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.

Dynamics of delayed and diffusive FitzHugh–Nagumo network

Dynamics of delayed and diffusive FitzHugh–Nagumo network

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.

Weak synaptic connections may facilitate spiral wave formation under source-sink interactions

This study explores the interaction between two distinct sites, termed the source and the sink, to analyze the possibility of spiral wave formation. To this aim, a grid of memristive FitzHugh–Nagumo elements is designed to simulate biological excitable media, such as the myocardium. The source, characterized by high excitation levels with a gradual increase in the recovery variable, is primed to generate an excitatory wavefront. In contrast, the sink remains quiescent in excitation yet elevated in the recovery state, thus tending to absorb excitation as it is temporarily unexcitable but soon recoverable. Existing literature on spiral wave formation primarily focuses on rotor formation under source–sink adjacency. This work, on the other hand, examines the potential for re-entrant behavior under spatial separation between the source and the sink. Within the context of eight-nearest neighbor coupling, the results indicate that the time delay for the excitation wave to reach the refractory state of the sink, due to the increased distance, may still facilitate re-entry if the intensity of connections is sufficiently weak. However, beyond a certain threshold of the source–sink distance under weak connections, wave breakage may occur without resulting in re-entry. For the birth of a spiral wave rotor, the tip of the excitatory wavefront must converge with its refractory back, specifically at the outermost contour in the final stage of refractoriness, referred to as the wavetail. The formation of a spiral rotor cannot arise from a wavefront encountering any site earlier in the stage of refractoriness.