FitzHugh-Nagumo Neuron Model Research Articles

A biologically inspired approach to signal compression

Stimuli observed by our sensory systems are communicated to the brain by short duration electrical pulses known as action potentials, or spikes.1 Spike trains are generated by special sensory cells and then propagate to the brain along nerve fibers. Different neural systems use a number of different methods to encode information in spike trains. Mathematical and engineering approaches are increasingly assisting neuroscientists to understand and quantify these different neural coding mechanisms. Given that biology can solve many tasks far better than our current technology, new knowledge gained from this computational neuroscience approach should lead to novel bioinspired and biomimetic engineering. Despite many previous studies of information transmission in neural systems, one aspect, lossy compression, has been largely unexplored. Unlike lossless compression—e.g., zipping computer files—lossy compressionmeans deliberately discarding information in order to reduce storage or communication costs. Decompressing the result gives data that, although distorted, is close enough to the original to be useful in some way. Familiar examples include the JPEG and MP3 standards. The goal of an optimized data acquisition system is to provide only the minimum amount of information required to achieve some task. One suspects that biology has found many near-optimal solutions to this goal. The dual aims of my research are to use information-theoretic tools to model sensory neural systems as lossy compression schemes, and to design artificial compression methods inspired by this modeling. Any such method is likely to be useful in environments similar to those faced by biological systems, such as low-power distributed sensor networks making noisy measurements. The design of lossy compression methods involves two conflicting requirements: minimizing the distortion; and maximizing the amount of compression. My initial work has quantified the trade-off between distortion and compression in simple neural ‘rate-coding’ models.2 Although some neurons accurately encode stimulus features by the duration between spikes, this is impossible for timevarying analog stimuli, due to non-zero refractory times (minimum time between spikes) and random noise. Instead, the average spike rate encodes stimulus strength: a weak stimulus leads to only a few spikes, while a strong stimulus generates many. This means that some information is lost, since an average can be formed only from a finite number of spikes. Measuring the loss of information incurred in rate coding means defining a measure of distortion. This is one of the main unsolved questions in this sort of computational neuroscience research: What is an appropriate measure of distortion? In initial studies,2 I avoided this question simply by using a standard information-theoretic measure of distortion: mean square error (MSE). I have completed simulations in which I treat a single Fitzhugh-Nagumo neuron model as a scalar quantizer—a specific form of lossy compression, such as occurs in analog-to-digital converter (ADC) circuits—and measure the theoretically smallest MSE that can be achieved under various assumptions on timescales and noise. This approach uses results from a phenomenon called suprathreshold stochastic resonance3, 4 from the field of statistical physics. Three timescales determine MSE performance: the refractory time (minimum time between spikes); the duration of a counting window (biophysical memory); and the stimulus correlation time (how rapidly the stimulus varies). Together, these timescales define the achievable resolution of rate coding. If biologically relevant assumptions are made about these timescales, or they are known experimentally, quantitative comparisons of the compression performance of different sensory systems can be made. Part of the motivation for this work is to study the ability of sensory neurons to efficiently encode sensory stimuli even when signal-to-noise-ratios are very low.1 My lossy compression approach will be extended by studying more biologically realistic

Double Coherence Resonance in Neuron Models Driven by Discrete Correlated Noise

We study the influence of correlations among discrete stochastic excitatory or inhibitory inputs on the response of the FitzHugh-Nagumo neuron model. For any level of correlation, the emitted signal exhibits at some finite noise intensity a maximal degree of regularity, i.e., a coherence resonance. Furthermore, for either inhibitory or excitatory correlated stimuli, a double coherence resonance is observable. Double coherence resonance refers to a (absolute) maximum coherence in the output occurring for an optimal combination of noise variance and correlation. All of these effects can be explained by taking advantage of the discrete nature of the correlated inputs.

Stochastic resonance in the FitzHugh-Nagumo model from a dynamic mutual information point of view

Stochastic resonance(SR) in a FitzHugh-Nagumo neuron model is investigated based on a dynamic mutual information (DMI) between the input and the corresponding output signals. The DMI is expressed in terms of the (cross)power spectra of the input and output time series. Both stochastic-periodic and aperiodic SR are treated based on the DMI and our results are in good accord with the SR measured by the signal to noise ratio(SNR) for the case of the stochastic-periodic input and the power norm for the case of the aperiodic input.

High-frequency effects in the FitzHugh-Nagumo neuron model

The effect of a high-frequency signal on the FitzHugh-Nagumo excitable model is analyzed. We show that the firing rate is diminished as the ratio of the high-frequency amplitude to its frequency is increased. Moreover, it is demonstrated that the excitable character of the system, and consequently the firing activity, is suppressed for ratios above a given threshold value. In addition, we show that the vibrational resonance phenomenon turns up for sufficiently large noise strength values.

Stochastic resonance induced by novel random transitions of motion of FitzHugh–Nagumo neuron model

Stochastic resonance induced by novel random transitions of motion of FitzHugh–Nagumo neuron model

Emergence of chaotic attractor and anti-synchronization for two coupled monostable neurons

The dynamics of two coupled piece-wise linear one-dimensional monostable maps is investigated. The single map is associated with Poincare section of the FitzHugh-Nagumo neuron model. It is found that a diffusive coupling leads to the appearance of chaotic attractor. The attractor exists in an invariant region of phase space bounded by the manifolds of the saddle fixed point and the saddle periodic point. The oscillations from the chaotic attractor have a spike-burst shape with anti-phase synchronized spiking.

Stochastic resonance induced by novel random transitions of motion of FitzHugh–Nagumo neuron model

In contrast to the previous studies which have dealt with stochastic resonance induced by random transitions of system motion between two coexisting limit cycle attractors in the FitzHugh–Nagumo (FHN) neuron model after Hopf bifurcation and which have dealt with the phenomenon of stochastic resonance induced by external noise when the model with periodic input has only one attractor before Hopf bifurcation, in this paper we have focused our attention on stochastic resonance (SR) induced by a novel transition behavior, the transitions of motion of the model among one attractor on the left side of bifurcation point and two attractors on the right side of bifurcation point under the perturbation of noise. The results of research show: since one bifurcation of transition from one to two limit cycle attractors and the other bifurcation of transition from two to one limit cycle attractors occur in turn besides Hopf bifurcation, the novel transitions of motion of the model occur when bifurcation parameter is perturbed by weak internal noise; the bifurcation point of the model may stochastically slightly shift to the left or right when FHN neuron model is perturbed by external Gaussian distributed white noise, and then the novel transitions of system motion also occur under the perturbation of external noise; the novel transitions could induce SR alone, and when the novel transitions of motion of the model and the traditional transitions between two coexisting limit cycle attractors after bifurcation occur in the same process the SR also may occur with complicated behaviors types; the mechanism of SR induced by external noise when FHN neuron model with periodic input has only one attractor before Hopf bifurcation is related to this kind of novel transition mentioned above.

A CHARACTERIZATION OF SUPRATHRESHOLD STOCHASTIC RESONANCE IN AN ARRAY OF COMPARATORS BY CORRELATION COEFFICIENT

Suprathreshold Stochastic Resonance (SSR), as described recently by Stocks, is a new form of Stochastic Resonance (SR) which occurs in arrays of nonlinear elements subject to aperiodic input signals and noise. These array elements can be threshold devices or FitzHugh-Nagumo neuron models for example. The distinguishing feature of SSR is that the output measure of interest is not maximized simply for nonzero values of input noise, but is maximized for nonzero values of the input noise to signal intensity ratio, and the effect occurs for signals of arbitrary magnitude and not just subthreshold signals. The original papers described SSR in terms of information theory. Previous work on SR has used correlation based measures to quantify SR for aperiodic input signals. Here, we argue the validity of correlation based measures and derive exact expressions for the cross-correlation coefficient in the same system as the original work, and show that the SSR effect also occurs in this alternative measure. If the output signal is thought of as a digital estimate of the input signal, then the output noise can be considered simply as quantization noise. We therefore derive an expression for the output signal to quantization noise ratio, and show that SSR also occurs in this measure.

Robust stochastic resonance: signal detection and adaptation in impulsive noise.

Stochastic resonance (SR) occurs when noise improves a system performance measure such as a spectral signal-to-noise ratio or a cross-correlation measure. All SR studies have assumed that the forcing noise has finite variance. Most have further assumed that the noise is Gaussian. We show that SR still occurs for the more general case of impulsive or infinite-variance noise. The SR effect fades as the noise grows more impulsive. We study this fading effect on the family of symmetric alpha-stable bell curves that includes the Gaussian bell curve as a special case. These bell curves have thicker tails as the parameter alpha falls from 2 (the Gaussian case) to 1 (the Cauchy case) to even lower values. Thicker tails create more frequent and more violent noise impulses. The main feedback and feedforward models in the SR literature show this fading SR effect for periodic forcing signals when we plot either the signal-to-noise ratio or a signal correlation measure against the dispersion of the alpha-stable noise. Linear regression shows that an exponential law gamma(opt)(alpha)=cA(alpha) describes this relation between the impulsive index alpha and the SR-optimal noise dispersion gamma(opt). The results show that SR is robust against noise "outliers." So SR may be more widespread in nature than previously believed. Such robustness also favors the use of SR in engineering systems. We further show that an adaptive system can learn the optimal noise dispersion for two standard SR models (the quartic bistable model and the FitzHugh-Nagumo neuron model) for the signal-to-noise ratio performance measure. This also favors practical applications of SR and suggests that evolution may have tuned the noise-sensitive parameters of biological systems.

Spiking behavior in a noise-driven system combining oscillatory and excitatory properties.

We show that firing activity (spiking) can be regularized by noise in a FitzHugh-Nagumo (FHN) neuron model when operating slightly beyond the supercritical Hopf bifurcation (in the "canard" region). We also provide the conditions for imperfect phase locking between interspike intervals and low amplitude quasiharmonic oscillations. For the imperfect phase locking no need exists of an external signal as it follows from the FHN intrinsic dynamics.

Simulation of circuits demonstrating stochastic resonance

In certain dynamical systems, the addition of noise can assist the detection of a signal and not degrade it as normally expected. This is possible via a phenomenon termed stochastic resonance (SR), where the response of a nonlinear system to a subthreshold periodic input signal is optimal for some non-zero value of noise intensity. We investigate the SR phenomenon in several circuits and systems. Although SR occurs in many disciplines, the sinusoidal signal by itself is not information bearing. To greatly enhance the practicality of SR, an (aperiodic) broadband signal is preferable. Hence, we employ aperiodic stochastic resonance (ASR) where noise can enhance the response of a nonlinear system to a weak aperiodic signal. We can characterize ASR by the use of cross-correlation-based measures. Using this measure, the ASR in a simple threshold system and in a FitzHugh–Nagumo neuronal model are compared using numerical simulations. Using both weak periodic and aperiodic signals, we show that the response of a nonlinear system is enhanced, regardless of the signal.

ASYMPTOTIC SYNCHRONIZATION IN NETWORKS OF LOCALLY CONNECTED OSCILLATORS

In this paper, we describe the asymptotic behavior of a network of locally connected oscillators. The main result concerns asymptotic synchronization. The presented study is stated in the framework of neuronal modeling of visual object segmentation using oscillatory correlation. The practical motivations of the synchronization analysis are based on neurophysiological experiments which led to the assumptions that existence of temporal coding schemes in the brain by which neurons, with oscillatory dynamics, coding for the same coherent object synchronize their activities, while neurons coding for different objects oscillate with nonzero phase lags. The oscillator model considered is the FitzHugh–Nagumo neuron model. We restrict our study to the mathematical analysis of a network of such neurons. We firstly show the motivations and suitability of choosing FitzHugh–Nagumo oscillator, mainly for stimulus coding purposes, and then we give sufficient conditions on the coupling parameters which guarantee asymptotic synchronization of oscillators receiving the same external stimulation (input). We have used networks of such oscillators to design a layered architecture for object segmentation in gray-level images. Due to space limitations, description of this architecture and simulation results are briefly referred to by the end of the paper.

Mechanism of stochastic resonance enhancement in neuronal models driven by 1/f noise.

Noise can assist neurons in the detection of weak signals via a mechanism known as stochastic resonance (SR). In a previous study [Phys. Lett. A 243, 281 (1998)], we showed that when colored noise with 1/f(beta) spectrum is added to the FitzHugh-Nagumo (FHN) neuronal model, the optimal noise variance for SR could be minimized with beta approximately 1. In this study, we investigate analytically how the noise color (beta) affects the SR profile in a linearized version of the FHN model. We demonstrate that the aforementioned effect of 1/f noise is related to the dynamical characteristics of the model neuron, i.e., the refractory period, the low-pass filtering effect of the membrane capacitance, and the high-pass filtering effect of the recovery variable.

Synthetic analysis of periodically stimulated excitable and oscillatory membrane models

Many excitable neuronal membranes become oscillatory when stimulated by large enough dc currents. In this paper we investigate how the transition from excitable to oscillatory regimes affects the response of the membrane to periodic pulse trains. To this end, we examine how the dynamics of periodically stimulated FitzHugh-Nagumo neuron model changes as the system switches from excitability to oscillation. We show that, despite the important change in the asymptotic dynamics of the unperturbed model, $p:q$ phase-locking (i.e., the model membrane discharges q times in p interstimulus intervals and q input-output intervals repeat periodically) regions in the stimulus period-stimulus amplitude parameter plane (Arnold tongues) change continuously when the model changes from excitable to oscillatory. We provide further evidence for the continuous change of the Arnold tongues by using an analytically tractable one-dimensional map that approximates the Poincar\'e map of the forced system. We argue that the smooth change in the Arnold tongues results from the fact that, despite the qualitative difference between the asymptotic dynamics of unforced excitable and oscillatory regimes, other aspects of the dynamics such as the wave form of individual action potentials, are similar in the two regimes.

Bursting behaviour of the FitzHugh-Nagumo neuron model subject to quasi-monochromatic noise

Noise-induced firing activity in the form of bursts of spikes is usually described by means of excitable systems that combine deterministic subthreshold oscillations with noise. In this paper, spontaneous bursting activity is obtained by submitting the well-known FitzHugh-Nagumo (FHN) model to quasi-monochromatic noise (QMN). In the regime of interest, the noise variable performs fast random oscillations with a fluctuating amplitude that varies in a slower time scale. Interspike interval histograms (ISIHs) for the voltage variable of the FHN model are obtained which, in some cases, show an imperfect phase locking between interspike intervals and both the period of the fundamental oscillatory component and the time scale of the random modulation of the noise. QMN seems to be a good candidate to describe the intrinsic noisy oscillations that lead to bursting in real neurons.

Aperiodic stochastic resonance with correlated noise

We examine the influence of the noise correlation on aperiodic stochastic resonance. To this end, we compute the discharge rate of a FitzHugh-Nagumo neuron model in response to a slow sub-threshold aperiodic input in the presence of additive correlated noise. Aperiodic stochastic resonance is observed for each level of noise correlation; i.e., there exists a noise amplitude that maximizes the covariance between the input signal and the model's discharge rate. Both the maximal covariance and the optimal noise level depend on the noise correlation. The former remains almost constant for low noise correlations, and steadily decreases at larger correlations. The latter displays a U-shaped curve when plotted against the noise correlation, indicating that aperiodic stochastic resonance occurs at lower noise amplitudes for an appropriate range of noise correlation. We show that the results are consistent with the evaluation of the discharge rate of the model through a quasistatic assumption. Our results suggest that the interplay of the noise correlation with the time scales of the neuron model can lead to an improvement of the aperiodic stochastic resonance effect, in that for an appropriate level of correlation maximal covariance can be attained with lower noise amplitude.

Enhancement of stochastic resonance in a FitzHugh-Nagumo neuronal model driven by colored noise

Abstract We investigate the stochastic resonance in a FitzHugh-Nagumo neuronal model driven by colored noise with a 1 f β spectrum (0 ≤ β ≤ 2). A numerical simulation shows that the noise intensity needed to maximize the coherence between input and output signals is the smallest when β ≈ 1. We also demonstrate analytically that this phenomenon is never seen in a nondynamical threshold system.

Aperiodic stochastic resonance.

Stochastic resonance (SR) is a phenomenon wherein the response of a nonlinear system to a weak periodic input signal is optimized by the presence of a particular level of noise. Recently, we presented a method and theory for characterizing SR-type behavior in excitable systems with aperiodic (i.e., broadband) input signals [Phys. Rev. E 52, R3321(1995)]. We coined the term aperiodic stochastic resonance (ASR) to describe this general type of behavior. In that earlier study, we demonstrated ASR in the FitzHugh-Nagumo neuronal model. Here we demonstrate ASR in three additional systems: a bistable-well system, an integrate-and-fire neuronal model, and the Hodgkin-Huxley (HH) neuronal model. We present computational and theoretical results for each system. In the context of the HH model, we develop a general theory for ASR in excitable membranes. This work clearly shows that SR-type behavior is not limited to systems with periodic inputs. Thus, in general, noise can serve to enhance the response of a nonlinear system to a weak input signal, regardless of whether the signal is periodic or aperiodic. \textcopyright{} 1996 The American Physical Society.

The detection threshold, noise and stochastic resonance in the Fitzhugh-Nagumo neuron model

In a class of experiments in sensory biology sub- or slightly supra-threshold stimuli are applied at the periphery while making action potential recordings at a higher level in the nervous system. In many cases the records are largely noise but with a small coherence with the stimulus. The detection threshold for increasing stimulus intensity is frequently measured, and this threshold can often be controlled by manipulating some additional biological parameter, for example, the temperature or inherent noise intensity. How does this parameter change the detection threshold? In this note, we test three popular paradigms on an electronic Fitzhugh-Nagumo neuron model and compare the results with a simple threshold theory.

The global bifurcation structure of the BVP neuronal model driven by periodic pulse trains

The response characteristics of the BVP (Bonhoeffer-van der Pol or FitzHugh-Nagumo) neuronal model to periodic pulse trains were investigated. The global bifurcation structure of model relative to stimulus intensity and period were analyzed using a one-dimensional mapping called the phase transition curve (PTC) extended by Maginu. The PTC clarified how periodic and chaotic responses bifurcate and revealed in particular several examples of chaotic responses bifurcating through period-doubling bifurcations, as well as the coexistences at the same parameter values, of two different periodic orbits, or of a chaotic and a periodic responses.