FitzHugh-Nagumo Neuron Model Research Articles

Mathematical and Experimental Model of Neuronal Oscillator Based on Memristor-Based Nonlinearity

This article presents a mathematical and experimental model of a neuronal oscillator with memristor-based nonlinearity. The mathematical model describes the dynamics of an electronic circuit implementing the FitzHugh–Nagumo neuron model. A nonlinear component of this circuit is the Au/Zr/ZrO2(Y)/TiN/Ti memristive device. This device is fabricated on the oxidized silicon substrate using magnetron sputtering. The circuit with such nonlinearity is described by a three-dimensional ordinary differential equation system. The effect of the appearance of spontaneous self-oscillations is investigated. A bifurcation scenario based on supercritical Andronov–Hopf bifurcation is found. The dependence of the critical point on the system parameters, particularly on the size of the electrode area, is analyzed. The self-oscillating and excitable modes are experimentally demonstrated.

Dynamical analysis of an improved FitzHugh-Nagumo neuron model with multiplier-free implementation

Dynamical analysis of an improved FitzHugh-Nagumo neuron model with multiplier-free implementation

Intermittent chaotic spiking in the van der Pol–FitzHugh–Nagumo system with inertia

The van der Pol–FitzHugh–Nagumo neuron model with inertia was shown to exhibit a chaotic mixed-mode dynamics composed of large-amplitude spikes separated by an irregular number of small-amplitude chaotic oscillations. In contrast to the standard 2D van der Pol–FitzHugh–Nagumo model driven by noise, the inter-spike intervals distribution displays a complex arrangement of sharp peaks related to the unstable periodic orbits of the chaotic attractor. For many ranges of parameters controlling the excitability of the system, we observe that chaotic mixed-mode states consist of lapses of nearly regular spiking interleaved by others of highly irregular one. We explore here the emergence of these structures and show their correspondence to the intermittent transitions to chaos. In fact, the average residence times in the nearly-periodic firing state, obey the same scaling law – as a function of the control parameter – than the one at the onset of type I intermittency for dynamical systems in the vicinity of a saddle node bifurcation. We hypothesize that this scenario is also present in a variety of slow-fast neuron models characterized by the coexistence of a two-dimensional fast manifold and a one-dimensional slow one.

Neuromorphic Oscillators and Electronic Implementation of One Novel MFNN Model

Exploring the equivalent mathematical and physical representation of biological neurons is one of the most essential tasks for building the complex neural systems and networks. In this paper, we focus on constructing a novel memristive FitzHugh-Nagumo Neuron (MFNN) model to simulate the nonlinear oscillation behaviors and neuromorphic dynamics of some biological neurons. Then, the relationship between ranges of circuit parameter values and pinched hysteresis loop are presented. Moreover, its behaviors and neuromorphic nonlinear dynamics are analyzed and several essential curves are depicted, such as phase portraits, time-domain waveforms, Lyapunov exponent spectrums, Poincare maps, the bifurcation diagrams, and so on. Furthermore, the most important characteristics are the generation of memristive oscillators and bursting firing phenomenon. There is no doubt that the good agreement among theoretical analysis, simulation and experimental results verifies the practicability and flexibility of the configured MFNN model.

An enhanced FitzHugh–Nagumo neuron circuit, microcontroller-based hardware implementation: Light illumination and magnetic field effects on information patterns

This contribution introduced and investigated an improved photosensitive memristive FitzHugh–Nagumo (FHN) neural circuit. The mathematical equations of the model have been derived from Kirchhoff’s electrical circuit law. Based on Helmholtz’s theorem, the Hamilton statistical function that enables us to obtain the physical energy of the neuron necessary to sustain electrical activity has been established. Therefore, the energy dependency on the light intensity and the magnetic field has been provided. Using two-parameter diagrams, bifurcation diagrams, Lyapunov exponents, and time series, the window of parameters in which regular and irregular patterns of the neuron occur have been characterized. Besides, the proposed FHN neuron model is implemented using a simple microcontroller platform and various firing patterns are acquired experimentally to validate the numerical results. Furthermore, light and field-induced pattern formation in a chain regular lattice network made of 100 photosensitive memristive neurons is being studied. Numerical experiments are carried out to identify the effects of photocurrent and magnetic flux parameters on information coding patterns translated by the membrane potential from voltage output. The spatiotemporal patterns revealed that the network presents localized wave patterns with some features of synchronization, sensitive to parameter change. This confirms that information coding and other collective behaviors of the network could be efficiently controlled by taming the intensity of light illumination and magnetic field exposure.

Autapse-induced logical resonance in the FitzHugh–Nagumo neuron

It was demonstrated that the chaos-driven FitzHugh–Nagumo (FHN) neuron can be considered as a logic system to implement the reliable logical operations through the mechanism of logical resonance. Autapse (meaning the self-synapse) widely exists in various kinds of neurons, and it significantly affects the neuronal dynamics and functionalities. However, the effects of autapse on logical resonance have not been reported yet. Here, we explore the effects of autapse on the reliability of AND & NAND logical operations based on the autaptic FHN neuron model with time-varying coupling intensity. The numerical results demonstrate that there are the optimal ranges of parameters (including autaptic time delay, amplitude, frequency and phase fluctuation of autaptic coupling intensity) at which the reliability of logical operations can be maximized. Namely, autapse-induced logical resonance can be realized in the autaptic FHN neuron model. More interestingly, multiple logical resonances can be obtained by regulating autaptic time delay, phase fluctuation of autaptic coupling intensity, as well as frequency ratio between and autaptic coupling intensity and external periodic driving force. Finally, an intuitive interpretation for autapse-induced logical resonance is given based on the motion of the particle in the potential landscape.

A New Embodied Motor-Neuron Architecture

This brief introduces a novel bioinspired motor neuron dynamical unit where the neural dynamics, responsible for the generation of the action potentials, is embedded into the actuator dynamics, which here plays the role of, and substitutes, the recovery variable into the classical neuron equations. A recently introduced nullcline-based control strategy, over servomotors embedded into piecewise linearly approximated FitzHugh–Nagumo neuron models, is here applied to the synchronization of two embodied motor-neuron units in the form of either a continuously active proportional gain or an event-driven strategy. In view of the application to such problems as the generation of adaptive control laws for distributed oscillatory networks, at the basis of bioinspired walking machines, the advantages in terms of the reduced-order dynamical equations and ease of synchronization are presented both through simulations and with experiments devoted to control networked Dynamixel MX-28AT servomotors via an microcontroller unit (MCU) board.

Strange nonchaotic dynamics in a discrete FitzHugh-Nagumo neuron model with sigmoidal recovery variable.

We report the appearance of strange nonchaotic attractors in a discrete FitzHugh-Nagumo neuron model with discontinuous resetting. The well-known strange nonchaotic attractors appear in quasiperiodically forced continuous-time dynamical systems as well as in a discrete map with a small intensity of noise. Interestingly, we show that a discrete FitzHugh-Nagumo neuron model with a sigmoidal recovery variable and discontinuous resetting generates strange nonchaotic attractors without external force. These strange nonchaotic attractors occur as intermittency behavior (locally unstable behavior in laminar flow) in the periodic dynamics. We use various characterization techniques to validate the existence of strange nonchaotic attractors in the considered system.

Boosting learning ability of overdamped bistable stochastic resonance system based physical reservoir computing model by time-delayed feedback

Physical reservoir computing (RC), which can be implemented by various physical systems, is a low-cost neuromorphic framework with a fast learning capability. In the previous studies, an overdamped bistable system-based RC (OBRC) inspired by the FitzHugh-Nagumo neuron model has been proposed to construct an outstanding physical RC system. Benefitting from the stochastic resonance effect, the OBRC requires less power and has stronger noise robustness than many conventional physical RC systems. However, compared with conventional physical RC systems, its learning ability is not superior. To boost the performance of the OBRC, we propose an OBRC with time-delayed feedback (TOBRC). In this work, the TOBRC is implemented in a physical setting with time-multiplexing nodes design and simulated on a conventional computer. Moreover, we adopt a powerful optimization algorithm to automatically determine the optimal hyperparameters for both the OBRC and TOBRC; thus, a more precise quantitative discussion on the upper limit of the system can be made. To compare the TOBRC and OBRC, we conducted short-term memory and parity check tasks to assess the short-term memory ability and nonlinearity, which are the two core abilities of physical RC for learning. The results prove that the short-term memory ability and nonlinearity of the proposed TOBRC are 6.46 and 2.15 times higher than those of the OBRC, respectively. Moreover, the TOBRC outperforms the OBRC under different noise conditions. On the MNIST handwritten digit recognition benchmark, the TOBRC exhibited a lower error rate than the OBRC; it was comparable with that of advanced physical RC systems. Our study confirms that the TOBRC can exhibit excellent learning ability in practical problems.

Complex regimes in electronic neuron-like oscillators with sigmoid coupling

The new version of electronic implementation of FitzHugh–Nagumo neuron model was proposed together with a new circuit for synapse (sigmoid activation function). The proposed neuron and synapse models provide better representation of activation function in biological neurons including possibility to model excitatory and inhibitory connections. Various regimes in two FitzHugh–Nagumo neurons were studied numerically and in hardware experiment. Different scenarios of oscillation emergence were investigated, including saddle-node cycle bifurcation leading to appearance of highly nonlinear limit cycles of large amplitude. Long living transients near these bifurcations were found those are of particular interest for modeling some metastable phenomena in living systems like sleep and epilepsy.

Stability Analysis for a Fractional-Order Coupled FitzHugh–Nagumo-Type Neuronal Model

The aim of this work is to describe the dynamics of a fractional-order coupled FitzHugh–Nagumo neuronal model. The equilibrium states are analyzed in terms of their stability properties, both dependently and independently of the fractional orders of the Caputo derivatives, based on recently established theoretical results. Numerical simulations are shown to clarify and exemplify the theoretical results.

Resilience in Multiplex Networks by Addition of Cross-Repulsive Links

A multiplex network of identical dynamical units becomes resilient against parameter perturbation by adding selective linear diffusive cross-coupling links. A parameter drift at any instant in one or multiple network nodes can destroy synchrony, causing failure and even collapse in the network performance. We introduced [PRE 95, 062204(2017)] a recovery strategy by selective addition of cross-coupling links to save synchrony in the network from the edge of failure due to parameter mismatch (small or large) in any nodes. This concept is extended to 2-layered multiplex networks when the emergent synchrony becomes resilient against a small or large parameter drifting. In addition, the stability of the synchronous state is enhanced from local stability to global stability of synchrony. By the addition of cross-coupling, the network revives complete synchrony in all the nodes except the perturbed nodes, which emerges into a type of generalized synchrony with all the unperturbed nodes. The generalized synchrony is manifested simply by a linear amplitude response in the state variable(s) of the perturbed node(s) by a scaling factor proportional to the mismatch. A set of systematic rules has been derived from the linear flow matrix of the dynamical system representing the nodes dynamics that helps find the connectivity matrix of the cross-coupling links. Lyapunov function stability condition is used to determine the cross-coupling link strength that, in turn, establishes global stability of synchrony of the multiplex network. We verify the efficacy of our proposed coupling scheme with analytical results and numerical simulations of two examples of multiplex networks. In the first example, we use non-local connectivity in each layer, with nodal dynamics of the FitzHugh-Nagumo neuron model.

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.

An alternative approach for the circuit synthesis of the fractional-order FitzHugh-Nagumo neuron model

An alternative approach for the circuit synthesis of the fractional-order FitzHugh-Nagumo neuron model

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.

Multi-scale noise transfer and feature frequency detection in SSVEP based on FitzHugh–Nagumo neuron system

Objective. The steady-state visual evoked potential (SSVEP) is one of the most commonly used control signals for brain–computer interfaces (BCIs) due to its excellent interactive potential, such as high tolerance to noises and robust performance across users. In addition, it has a stable cycle, obvious characteristics and minimal training requirements. However, the SSVEP is extremely weak and companied with strong and multi-scale noise, resulting in a poor signal-to-noise ratio in practice. Common algorithms for classification are based on the principle of template matching and spatial filtering, which cannot obtain satisfied performance of SSVEP detection under the multi-scale noise. Therefore, using linear methods to extract SSVEP with obvious nonlinear and non-stationary characteristics, the useful signal will be attenuated or lost. Approach. To address this issue, two novel frameworks based on a two-dimensional nonlinear FitzHugh–Nagumo (FHN) neuron system are proposed to extract feature frequency of SSVEP. Results. In order to evaluate the effectiveness of the proposed methods, this research recruit 22 subjects to participate the experiment. Experimental results show that nonlinear FHN neuron model can force the energy of noise to be transferred into SSVEP and hence amplifying the amplitude of the target frequency. Compared with the traditional methods, the FHN and FHNCCA methods can achieve higher classification accuracy and faster processing speed, which effectively improves the information transmission rate of SSVEP-based BCI.

Artificial spiking neuron based on a single-photon avalanche diode and a microcavity laser

We present an experimental realization and characterization of artificial spiking neuron based on an optoelectronic pair “microcavity laser-single photon avalanche diode” operating in few photon regime. We show that basic properties of biological neurons, such as an existence of the threshold and the refractory period, the insensitivity to the effect of the stimuli strength above the threshold, and the dependence of the neuron fire rate of the stimuli strength, can be realized with such a type of artificial neuron. To compare, we present corresponding results of the numerical simulation in the framework of the FitzHugh–Nagumo neuron model.

Noise induced suppression of spiral waves in a hybrid FitzHugh-Nagumo neuron with discontinuous resetting.

A modified FitzHugh-Nagumo neuron model with sigmoid function-based recovery variable is considered with electromagnetic flux coupling. The dynamical properties of the proposed neuron model are investigated, and as the excitation current becomes larger, the number of fixed points decreases to one. The bifurcation plots are investigated to show the chaotic and periodic regimes for various values of excitation current and parameters. A N×N network of the neuron model is constructed to study the wave propagation and wave re-entry phenomena. Investigations are conducted to show that for larger flux coupling values, the spiral waves are suppressed, but for such values of the flux coupling, the individual nodes are driven into periodic regimes. By introducing Gaussian noise as an additional current term, we showed that when noise is introduced for the entire simulation time, the dynamics of the nodes are largely altered while the noise exposure for 200-time units will not alter the dynamics of the nodes completely.

Chimera states in a thermosensitive FitzHugh-Nagumo neuronal network

Temperature can affect the action potential of a neuron such that the firing pattern is changed under its variation. To incorporate the effects of the temperature, recently, a thermosensitive neuron was proposed in which a thermistor was added to a simple FitzHugh-Nagumo neuron model. Here, thermosensitive neurons are studied to realize the consequence of the thermistor on the neurons’ collective behavior. For specific coupling strengths, the chimera state is developed. The parameter region of the occurrence of the chimera state is obtained by computing the strength of incoherence and shows that the occurrence of the chimera is highly dependent on the coefficient of temperature change in the thermistor. Since, in reality, the neurons are not identical and also the thermistors are not precisely the same, the network is further considered with non-identical neurons. For this purpose, the coefficient of temperature changes in thermistors is not fixed and selected randomly from a normal distribution. The results show that for distributions with a small standard deviation, the behavior of the network changes slightly. While with increasing the standard deviation, the synchronization is achieved for higher coupling strengths, and the region of chimera is extended. Furthermore, although in the asynchronous network the neurons have different attractors, by the appearance of coherent groups and creation of the chimera, the attractors become identical.

Positive role of fractional Gaussian noise in FitzHugh–Nagumo neuron model

Abstract Determining the complex mechanisms of process information in the neural activity is found to be especially challenging. The noise with a 1/f power spectrum has been observed in nervous system, but its functional significance in the neuron activity remains unclear. Persistent effort has been made to determine the factors for efficient processing of information and for promoting the mutual information between stimulus and spike train output. Establishing the Fitzhugh–Nagumo (FHN) model coupling fractional Gaussian noise (fGn) as a special form of stochastic differential equation, our study is to certify the stochastic resonance (SR) effect in the process of neuron activity. We proved that the nonmonotonic SR effect about FHN neuron model driven by fGn occurred under the sufficient conditions based on the principle of forbidden interval. The simulated results show that appropriate intensity of fGn can enhance the increase of the mutual information. Compared with the Hurst parameters of fGn, the increase is more dependent on the noise intensity of fGn.