FitzHugh-Nagumo Neural Model Research Articles

Novel Hopf Bifurcation Exploration and Control Strategies in the Fractional-Order FitzHugh–Nagumo Neural Model Incorporating Delay

In this article, we propose a new fractional-order delay-coupled FitzHugh–Nagumo neural model. Taking advantage of delay as a bifurcation parameter, we explore the stability and bifurcation of the formulated fractional-order delay-coupled FitzHugh–Nagumo neural model. A delay-independent stability and bifurcation conditions for the fractional-order delay-coupled FitzHugh–Nagumo neural model is acquired. By designing a proper PDp controller, we can efficaciously control the stability domain and the time of emergence of the bifurcation phenomenon of the considered fractional delay-coupled FitzHugh–Nagumo neural model. By exploiting a reasonable hybrid controller, we can successfully adjust the stability domain and the bifurcation onset time of the involved fractional delay-coupled FitzHugh–Nagumo neural model. This study shows that when the delay crosses a critical value, a Hopf bifurcation will arise. When we adjust the control parameter, we can find other critical values to enlarge or narrow the stability domain of the fractional-order delay-coupled FitzHugh–Nagumo neural model. In order to check the correctness of the acquired outcomes of this article, we present some simulation outcomes via Matlab 7.0 software. The obtained theoretical fruits in this article have momentous theoretical significance in running and constructing networks.

The study on dynamical behavior of FitzHugh–Nagumo neural model under the co-excitation of non-Gaussian and colored noise

The dynamical behavior of the one-dimensional FitzHugh–Nagumo (FN) neural model under the co-excitation of cross-correlation multiplicative non-Gaussian noise and additive colored noise are investigated in this paper. Firstly, the one-dimensional Langevin equation of the FN model can be obtained by using the adiabatic elimination method. Then, the approximate Fokker–Planck equation (AFPE) is derived with functional methods, and the expressions of steady-state probability density function (PDF) and the mean first-passage time (MFPT) are obtained. Finally, the effects of different parameters on PDF and MFPT are discussed as a reflection of the properties of the FN neural model. Moreover, under certain conditions, noise enhanced stability (NES) effects can be induced by noise.

How additive noise forms and shifts phantom attractors in slow–fast systems

We present a peculiar phenomenon of the formation and shift of a localized random distribution in slow–fast systems forced by additive white Gaussian noise. Location of such ‘phantom’ attractors in zones far from deterministic attractors is theoretically analysed by the method of freezing and averaging. This method is applied to the analysis of the Goldbeter enzymatic model and FitzHugh–Nagumo neural model, in which noise-induced formation and shifts of ‘phantom’ attractors are detected.

Sigmoidal synaptic learning produces mutual stabilization in chaotic FitzHugh–Nagumo model

This paper investigates the interaction between two coupled neurons at the terminal end of a long chain of neurons. Specifically, we examine a bidirectional, two-cell FitzHugh-Nagumo neural model capable of exhibiting chaotic dynamics. Analysis of this model shows how mutual stabilization of the chaotic dynamics can occur through sigmoidal synaptic learning. Initially, this paper begins with a bifurcation analysis of an adapted version of a previously studied FitzHugh-Nagumo model that indicates regions of periodic and chaotic behaviors. Through allowing the synaptic properties to change dynamically via neural learning, it is shown how the system can evolve from chaotic to stable periodic behavior. The driving factor between this transition is representative of a stimulus coming down a long neural pathway. The result that two chaotic neurons can mutually stabilize via a synaptic learning implies that this may be a mechanism whereby neurons can transition from a disordered, chaotic state to a stable, ordered periodic state that persists. This approach shows that even at the simplest level of two terminal neurons, chaotic behavior can become stable, sustained periodic behavior. This is achieved without the need for a large network of neurons.

Mode transition and energy dependence of FitzHugh-Nagumo neural model driven by high-low frequency electromagnetic radiation

The FitzHugh-Nagumo (FHN) neural model is widely used to study the dynamic characteristics of signal propagation, synchronization, stochastic resonance(SR), coherent resonance(CR), and bifurcation of neurons. Based on Helmholtz theorem and the FHN neuron model, the expression of Hamilton energy function of FHN neural model driven by high-low frequency(HLF) electromagnetic radiation is derived. The correctness and uniqueness of the analytical solution are verified by using the constraints, and the electrical activities and Hamilton energy function of neuron are discussed by numerical simulations. It is found that electrical activity mode of FHN neuron undergoes a succession transition of quiescent state, spiking state, bursting state, and mixed state by changing the parameters such as the intensity of the external forcing current, the amplitude and angular frequency of HLF signal. The electrical activities process of FHN neuron is accompanied by the storage and release of system energy, this result may provide an understanding of the coding and conversion of electrical activity from the perspective of the relevance and dependence of energy costs.

Phase-locking patterns in a resonate and fire neural model with periodic drive

In this paper we studied a resonate and fire relaxation oscillator subject to time dependent modulation to investigate phase-locking phenomena occurring in neurophysiological systems. The neural model (denoted LFHN) was obtained by linearization of the FitzHugh-Nagumo neural model near an hyperbolic fixed point and then by introducing an integrate-and-fire mechanism for spike generation. By employing specific tools to study circle maps, we showed that this system exhibits several phase-locking patterns in the presence of periodic perturbations. Moreover, both the amplitude and frequency of the modulation strongly impact its phase-locking properties. In addition, general conditions for the generation of firing activity were also obtained. In addition, it was shown that for moderate noise levels the phase-locking patterns of the LFHN persist. Moreover, in the presence of noise, the rotation number changes smoothly as the stimulation current increases. Then, the statistical properties of the firing map were investigated too. Lastly, the results obtained with the forced LFHN suggest that such neural model could be used to fit specific experimental data on the firing times of neurons.

Ergodicity and spike rate for stochastic FitzHugh–Nagumo neural model with periodic forcing

Abstract We discuss ergodicity on a Poincare section and estimate the average spike rate for a time periodically forced stochastic FitzHugh–Nagumo model with degenerate noise. Stochastic FitzHugh–Nagumo (SFHN) model is a prototype stochastic neural oscillator, describing the generation and propagation of action potentials or spikes in an excitable neuron at the intracellular level. Neuronal spikes play significant role in neural information coding of various nervous systems, they are described in terms of an infinitesimal probability that spikes occur, known as spike rate. Estimation of this spike rate is a subtle task for time continuous stochastic processes such as solutions of SFHN model and, in particular, time periodically forced SFHN model. Using the regularity of the ergodic periodic measure, we estimate the average spike rate in terms of the probability density of two-point motions of the membrane potential via Rice’s formula.

The mean first-passage time in simplified FitzHugh–Nagumo neural model driven by correlated non-Gaussian noise and Gaussian noise

In this paper, the mean first-passage time (MFPT) in simplified FitzHugh–Nagumo (FHN) neural model driven by correlated multiplicative non-Gaussian noise and additive Gaussian white noise is studied. Firstly, using the path integral approach and the unified colored-noise approximation (UCNA), the analytical expression of the stationary probability distribution (SPD) is derived, and the validity of the approximation method employed in the derivation is checked by performing numerical simulation. Secondly, the expression of the MFPT of the system is obtained by applying the definition and the steepest-descent method. Finally, the effects of the multiplicative noise intensity D, the additive noise intensity Q, the noise correlation time [Formula: see text], the cross-correlation strength [Formula: see text] and the non-Gaussian noise deviation parameter q on the MFPT are discussed.

Vibrational and Stochastic Resonance in Time-Delayed Fitzhugh-Nagumo Neural Model Subject to Colored Multiplicative Noise

Vibrational and Stochastic Resonance in Time-Delayed Fitzhugh-Nagumo Neural Model Subject to Colored Multiplicative Noise

Effect of correlation in FitzHugh–Nagumo model with non-Gaussian noise and multiplicative signal

The effect of correlation in FitzHugh–Nagumo neural model induced by non-Gaussian noise and multiplicative signal is studied. Based on the corresponding Fokker–Planck equation, the explicit expressions of the stationary probability distribution function, the mean first passage time and signal-to-noise ratio are obtained, respectively. By analyzing the influence of different parameters, we observe that the system undergoes a succession of phase transition-like phenomena as correlation strength λ, correlation time τ and multiplicative noise intensity D are increased. And the mean first passage time exhibits a maximum, which identifies the noise-enhanced stability effect when correlation strength λ < 0. Furthermore, inhibition phenomenon and double stochastic resonance occur in FitzHugh–Nagumo neural model under different values of system parameters.

Stochastic Resonance in FitzHugh-Nagumo Neural Model

Stochastic resonance is a non-linear phenomenon where the output response of the dynamic system reaches the maximum value under the joint action of a certain intensity of noises and external incentives. In this paper, the phenomenon of stochastic resonance in a FitzHugh-Nagumo neural (FHN) model is studied. For the case that the frequency of the HF signal is much higher than that of the LF signal, under the adiabatic approximation condition, the expression of the signal-to-noise ratio (SNR) with respect to the LF signal is obtained. It is shown that, the SNR is a non-monotonous function of the amplitude and frequency of the HF signal. In addition, the SNR varies non-monotonically with increasing the intensities of the multiplicative and additive noise, with increasing the delayed-time as well as increasing the system parameters of the FHN model. The influence of the correlation time of the colored multiplicative noise and the influence of the coupling strength between the multiplicative and additive noise on the SNR is discussed.

Combined effects of correlated bounded noises and weak periodic signal input in the modified FitzHugh–Nagumo neural model

We study the dynamics of neurons via a bistable modified stochastic FitzHugh–Nagumo model having two stable fixed points separated by one unstable fixed point. Due to the ability of a neuron to detect and enhance weak information transmission, we show numerically that starting from the resting potential, we get firing activities (spiking) when operating slightly beyond the supercritical Hopf bifurcation. For real biological systems which are sometimes embedded in the complex environment, we observe that a gradual increase or decrease noise intensities did not result in a gradual change of the membrane potential distribution thanks to noise induced transition phenomena. We shown analytically that for zero correlation between two sine Wiener noises, additive noise has no effect on the transition between monostable and bistable phase on the neural model. We adapted a general expression of the signal-to-noise ratio for a general two-state theory extended in the asymmetric case and non-Gaussian noises in our model to study the influence of noise strength in stochastic resonance. Our investigation revealed that in the evolution of excitable system, neurons may use noises to their advantage by enhancing their sensitivity near a preferred phase to detect external stimuli or affect the efficiency and rate of information processing.

Stochastic resonance in FHN neural system driven by non-Gaussian noise

Stochastic resonance (SR) is studied in the FitzHugh-Nagumo (FHN) neural system subject to multiplicative non-Gaussian noise, additive Gaussian white noise and a periodic signal. Using the path integral approach and the two-state theory, the expression of the signal-to-noise ratio (SNR) is derived. The simulation results show that conventional SR and double SR occur in the FHN neural model under different values of system parameters. The effects of the additive and multiplicative noise intensities on SNR are different. Moreover, the addition of non-Gaussian noise is conductive to the enhancement of the response to the output signal of the FHN neural system.

Vibrational and Stochastic Resonance in the FitzHugh—Nagumo Neural Model with Multiplicative and Additive Noise

The vibrational resonance and stochastic resonance phenomena in the FitzHugh—Nagumo (FHN) neural model, driven by a high-frequency (HF) signal and a low-frequency (LF) signal and by coupled multiplicative and additive noises, is investigated. For the case that the frequency of the HF signal is much higher than that of the LF signal, under the adiabatic approximation condition, the expression of the signal-to-noise ratio (SNR) with respect to the LF signal is obtained. It is shown that the SNR is a non-monotonous function of the amplitude and frequency of the HF signal. In addition, the SNR varies non-monotonically with the increasing intensities of the multiplicative and additive noise as well as with the increasing system parameters of the FHN model. The influence of the coupling strength between the multiplicative and additive noises on the SNR is discussed.

Vibrational and Stochastic Resonance in FitzHugh-Nagumo Neural Model with Time-Delay and Colored Additive Noise

The vibrational resonance (VR) and stochastic resonance (SR) phenomena in time-delayed FitzHugh-Nagumo (FHN) neural model, driven by one high-frequency (HF) signal and one low-frequency (LF) signal, with coupled multiplicative and colored additive noise, is investigated. For the case that the frequency of the HF signal is much higher than that of the LF signal, under the adiabatic approximation condition, the expression of the signal-to-noise ratio (SNR) with respect to the LF signal is obtained. It is shown that, the SNR is a non-monotonous function of the amplitude and frequency of the HF signal. In addition, the SNR varies non-monotonically with increasing the intensities of the multiplicative and additive noise, with increasing the delayed-time as well as increasing the system parameters of the FHN model. The influence of the correlation time of the colored additive noise and the coupling strength between the multiplicative and additive noise on the SNR is discussed.

Effect of time delay in FitzHugh–Nagumo neural model with correlations between multiplicative and additive noises

We study the effect of time delay in the FitzHugh–Nagumo neural model with correlations between multiplicative and additive noise terms. Based on the corresponding Fokker-Planck equation, the explicit expressions of the stationary probability distribution function (SPDF), the mean first passage time (MFPT) and the signal-to-noise ratio (SNR) are obtained, respectively. Research results show that: (i) the system undergoes a succession of two phase transitions (i.e., the reentrance phenomenon) as the noise correlation parameter is increased and a (single) phase transition as the time delay is increased. (ii) The MFPT as a function of the multiplicative noise intensity exhibits a maximum. This maximum for MFPT identifies the noise enhanced stability (NES) effect, the noise correlation parameter intensifies the NES effect while the time delay, and the additive noise intensity weakens it. (iii) The existence of a maximum in the SNR as a function of the multiplicative noise intensity is the identifying characteristic of the stochastic resonance (SR) phenomenon, the noise correlation parameter enhances the SR while the time delay, and the additive noise intensity weaken it.