FitzHugh-Nagumo Neurons Research Articles

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

Energy computation and multiplier-less implementation of the two-dimensional FitzHugh-Nagumo (FHN) neural circuit.

In this work, with the aim of reducing the cost of the implementation of the traditional 2D FHN neuron circuit, a pair of diodes connected in an anti-parallel direction is used to replace the usual cubic nonlinearity (implemented with two multipliers). Based on the stability of the model, the generation of self-excited firing patterns is justified. Making use of the famous Helmholtz theorem, a Hamilton function is provided for the estimation of the energy released during each electrical activity of the model. From the investigation of the 1D evolution of the maxima of the membrane potential of the model, it was recorded that the considered model is able to experience a period of doubling bifurcation followed by a crisis that enables the increasing of the volume of the attractor. This contribution ends with the realization of a neural circuit without analog multipliers for the validation of the obtained results.

Propagation of chaos in mean field networks of FitzHugh-Nagumo neurons

In this article, we are interested in the behavior of a fully connected network of $N$ neurons, where $N$ tends to infinity. We assume that the neurons follow the stochastic FitzHugh-Nagumo model, whose specificity is the non-linearity with a cubic term. We prove a result of uniform in time propagation of chaos of this model in a mean-field framework. We also exhibit explicit bounds. We use a coupling method initially suggested by A. Eberle (arXiv:1305.1233), and recently extended in (1805.11387), known as the reflection coupling. We simultaneously construct a solution of the $N$-particle system and $N$ independent copies of the non-linear McKean-Vlasov limit in such a way that, considering an appropriate semi-metric that takes into account the various possible behaviors of the processes, the two solutions tend to get closer together as $N$ increases, uniformly in time. The reflection coupling allows us to deal with the non-convexity of the underlying potential in the dynamics of the quantities defining our network, and show independence at the limit for the system in mean field interaction with sufficiently small Lipschitz continuous interactions.

Detecting the subthreshold signal in a neural network via statistical complexity measure

This paper proposes an information theory approach for detecting the subthreshold signal in a small-world network composed of Fitz Hugh-Nagumo (FHN) neurons. Statistical complexity measure (SCM) and normalized Shannon-entropy (NSE) have been defined based on the specific and nonconsecutive firing time intervals series, and employed to quantify the stochastic multiresonance (SMR) phenomena in this small-world neural network. The results show that there are several maxima of SCM and several minima of NSE at various optimal noise levels, which is regarded as the signature of the occurrence of SMR. This also implies that the subthreshold signal can be accurately detected across multiple levels of noise. More intriguingly, we have discovered that the degree of this neural network can induce the generation of multiple resonance-like behaviors. In addition, our findings demonstrate that the SCM outperforms the traditional signal-to-noise ratio (SNR) in accurately identifying firing dynamical regularity under certain noisy conditions. Furthermore, the SCM can capture subtle resonance-like behavior induced by the degree, which the SNR is unable to detect. Thus, the SCM could have potential power on detecting weak signals in neural networks.

Unstable wave patterns of information in neural network under light illumination and magnetic field

Functional neurons built from neural circuits are capable of perceiving and processing external signals such as light illumination and magnetic radiation, by converting the physical signals into modulated bioelectric signals called action potential with diverse forms and shapes. Through modulational instability (MI), modulated wave formation and pattern transition are studied in a chain memristive network of [Formula: see text] photosensitive neurons. Memristors and photocells are incorporated in a simple FitzHugh–Nagumo neuron to detect and process the external magnetic flux and light illumination. To determine regions of modulated wave formation, linear stability analysis is performed on a nonlinear envelope equation which resulted from the asymptotic expansion of the generic dynamical equations. The growth rate of MI is plotted and the distinct zones of stable/unstable MI are presented. We confirm the analytical result through numerical simulations whereby the initial plane wave solutions lead to the emergence of localized structures with traits of spiking, bursting and chaotic states. High-frequency photocurrent changes orderly localized patterns to chaotic-like patterns while high-frequency magnetic flux promotes pattern transition from bursting to 2-period spiking state and a 4-period spiking state. This could provide an adequate way to influence the behaviors of artificial neurons as well as potential mechanism of information coding in the nervous system.

Delay-induced self-oscillation excitation in the Fitzhugh–Nagumo model: Regular and chaotic dynamics

The stochastic FitzHugh–Nagumo model with time delayed-feedback is often studied in excitable regime to demonstrate the time-delayed control of coherence resonance. Here, we show that the impact of time-delayed feedback in the FitzHugh–Nagumo neuron is not limited by control of noise-induced oscillation regularity (coherence), but also results in excitation of the regular and chaotic self-oscillatory dynamics in the deterministic model. We demonstrate this numerically by means of simulations, linear stability analysis, the study of Lyapunov exponents and basins of attraction for both positive and negative delayed-feedback strengths. It has been established that one can implement a route to chaos in the explored model, where the intrinsic peculiarities of the Feigenbaum scenario are exhibited. For large time delay, we complement the study of temporal evolution by the interpretation of the dynamics as patterns in virtual space.

Temperature-induced logical resonance in the Hodgkin–Huxley neuron

Logical resonance has been demonstrated to be present in the FitzHugh–Nagumo (FHN) neuron, namely, the FHN neuron can operate as a reliable logic gate within an optimal parameter window. Here we attempt to extend the results to the more biologically realistic Hodgkin–Huxley (HH) model of neurons. In general, biological organisms have an optimal temperature at which the biological functions are most effective. In view of this, we examine if there is an optimal range of temperature where the HH neuron can work like a specific logic gate, and how temperature influences the logical resonance. Here we use the success probability P to measure the reliability of the specific logic gate. For AND logic gate, P increases with temperature T, reaches the maximum in an optimal window of T, and eventually decreases, which indicates the occurrence of the temperature-induced logical resonance phenomenon in the HH neuron. Moreover, single and double logical resonances can be induced by altering the frequency of the modulating periodic signal under the proper temperatures, suggesting the appearance of temperature-controlled transition of logical resonance. These results provide important clues for constructing neuron-based energy-efficient new-fashioned logical devices.

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.

Research on constrained localization of ultrasound geometric distribution based on FHN neurons.

The autopilot positioning process is mainly affected by three aspects: the first is the spatial geometric distribution of positioning sensors; the second is the screening of spurious observations; and the third is the equivalent ranging error. A constrained positioning method based on the geometric distribution of FitzHugh-Nagumo (FHN) neurons is proposed. To reduce the geometric accuracy factor, a Horizontal Dilution Of Precision value algorithm with a weight factor was proposed by considering the spatial geometric distribution of base stations and the geometric relationship of anchor points. This paper proposes a geometric constraint data processing method for the error of the pseudo-observation value. Finally, considering the significant weak signal perception ability of the biological nervous system, and the stochastic resonance phenomenon caused by noise can enhance the ability of the neuronal system to detect weak signals, an ultrasonic receiving method based on the stochastic resonance characteristics of FHN neuronal system is proposed, to enhance the signal and reduce noise. The results show that under the optimized base station layout and data geometric constraint processing, the ultrasonic wave based on FHN neuron improves the accuracy of spurious observations, reduces the calculation amount of geometric constraint processing, and reduces the positioning error by 66.67%, which provides a new direction for improving the positioning accuracy.

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.

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.

Complex Dynamics of Coupled Neurons Through a Memristive Synapse: Extreme Multistability and Its Control With Selection of the Desired State

In this contribution, a new configuration involving the coupling of a 2D Fitzhugh-Nagumo (FN) neuron with a 3D Hindmarsh-Rose (HR) neuron via a memristive synapse is investigated. The self-excited dynamics of the coupled neurons is revealed after the analysis of the equilibria of the model. During the numerical investigation of the model, resting activity, periodic spikes, periodic and chaotic bursts are found. More interestingly, the coupled neurons display the striking and rare phenomenon of homogeneous extreme multistability. It corresponds to the coexistence of an infinite number of firing activities of the same nature but located at different levels in the phase space. Furthermore, the selection of the desired electrical activity dynamics is also underlined through the noninvasive control scheme. Finally, an electronic circuit of the coupled neuron is designed and investigated in the Pspice environment to further support the obtained results.

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.

Enhancement of time-frequency energy for the classification of motor imagery electroencephalogram based on an improved FitzHugh-Nagumo neuron system.

Brain-computer interface (BCI) based on motor imagery (MI) electroencephalogram (EEG) has become an essential way for rehabilitation, because of the activation and interaction of motor neurons between the brain and rehabilitation devices in recent years. However, due to the discrepancies between individuals, the frequency ranges can be different even for the same rhythm component of EEG recordings, which brings difficulties to the extraction of features for MI classification. Typical algorithms for MI classification such as common spatial patterns (CSP) require multi-channel analysis and lack frequency information. With the development of BCI, the single-channel BCI system has become indispensable for simplicity of use. However, the currently available single-channel detection methods have low classification accuracy. To address this issue, two novel frameworks based on an improved two-dimensional nonlinear FitzHugh-Nagumo (FHN) neuron system are proposed to extract features of the single-channel MI. To evaluate the effectiveness of the proposed methods, this research utilized an open-access database (BCI competition IV dataset 2a), an offline database, and a 10-fold cross-validation procedure. Experimental results showed that the improved nonlinear FHN system can transfer the energy of noise into MI, thereby effectively enhancing the time-frequency energy. Compared with the traditional methods, the proposed methods can achieve higher classification accuracy and robustness.

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.

Revelation and experimental verification of quasi-periodic bursting, periodic bursting, periodic oscillation in third-order non-autonomous memristive FitzHugh-Nagumo neuron circuit

It has been declared that constructing physical hardware circuits with the reproduction of abundant electrical activities of neurons is significant in neuron-based engineering applications. To this end, a novel third-order non-autonomous memristive FitzHugh-Nagumo (FHN) neuron circuit is designed by employing a first-order generalized memristive-diode-bridge (MDB) emulator and an AC voltage source. The memristive FHN neuron circuit can generate abundant electrical activities since the involvement of the MDB emulator. In theoretical analysis and numerical simulation, the corresponding circuit state equations and normalized system are formulated to analyze the characteristics of the equilibrium point and investigate the dynamical behaviors related to the internal resistor of the MDB emulator. Then phase portrait, time-domain waveform, bifurcation diagram, and Lyapunov exponent spectra are numerically simulated, from which abundant non-chaotic firing activities of quasi-periodic bursting, periodic bursting, and periodic oscillation are revealed. Additionally, the 0–1 test is utilized to further distinguish the above three non-chaotic behaviors. In hardware experiment, a breadboard hardware circuit is constructed and experimental measurements are executed. It is demonstrated that the experimental results well support the numerical simulations. Studying these non-chaotic behaviors is very important to understand the intrinsic nature of neuronal firing activities. Most notably, the numerical revelation and experimental verification of quasi-periodic bursting behavior have been rarely reported in the previously published literature.

Dynamical analysis of a multiple time delays FitzHugh–Nagumo neuron system with chemical and electrical coupling

Dynamical analysis of a multiple time delays FitzHugh–Nagumo neuron system with chemical and electrical coupling