Complex Oscillations in the Delayed FitzHugh–Nagumo Equation

Krupa, Maciej; Touboul, Jonathan

doi:10.1007/s00332-015-9268-3

Cited by 29 publications

(11 citation statements)

References 51 publications

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“…The FHN neuron model (for biophysical details, see [55]) has been extensively used to investigate many complex dynamical behaviors occurring in neuroscience [56][57][58][59][60]. For our investigation of SISR, we consider the following version of the FHN neuron model written on the slow timescale τ as…”

Section: Model Equationmentioning

confidence: 99%

Coherent neural oscillations induced by weak synaptic noise

Yamakou

Jost

2018

Nonlinear Dyn

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We analyze the effect of synaptic noise on the dynamics of the FitzHugh-Nagumo (FHN) neuron model. In our deterministic parameter regime, a limit cycle solution cannot emerge through a singular Hopf bifurcation, but such a limit cycle can nevertheless arise as a stochastic effect, as a consequence of weak synaptic noise in a regime of strong timescale separation (ε → 0) between the slow and fast variables of the model. We investigate the mechanism behind this phenomenon, known as self-induced stochastic resonance (SISR) (Muratov et al. in Physica D 210:227-240, 2005), by using multiple-time perturbation techniques and by analyzing the escape mechanism of the random trajectories from the stable manifolds of the model equation. Even though SISR occurs only in the limit as the singular parameter ε → 0, decreasing ε does not increase the coherence of the oscillations in the FHN model, but rather increases the interval of the noise amplitude σ for which coherence occurs. This is in contrast to the dynamical system studied in Mura- 2005). Moreover, the phenomenon is robust under parameter tuning. Numerical simulations exhibit the results predicted by the theoretical analysis. In fact, our analysis together with that in Yamakou and Jost (Weak noise-induced transitions with inhibition and modulation of neural oscillations, 2017) reveals that the FHN model can support different stochastic resonance phenomena. While it had been known (Pikovsky and Kurths in Phys Rev Lett 78:775-778, 1997) that coherence resonance can occur when the slow variable is subjected to noise, we show that, when noise is added to the fast variable, two other types, inverse stochastic resonance and SISR, may emerge in the same weak noise limit and that the transition between these phenomena can be induced by varying a simple parameter.

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Section: Model Equationmentioning

confidence: 99%

Coherent neural oscillations induced by weak synaptic noise

Yamakou

Jost

2018

Nonlinear Dyn

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“…In Ref. [31], the authors investigated Eqs. (1) with c < 0 and determined the possible routes to chaos [32].…”

Section: Introductionmentioning

confidence: 99%

Short-time-delay limit of the self-coupled FitzHugh-Nagumo system

et al. 2016

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We analyze the FitzHugh-Nagumo equations subject to time-delayed self-feedback in the activator variable. Parameters are chosen such that the steady state is stable independent of the feedback gain and delay τ . We demonstrate that stable large-amplitude τ -periodic oscillations can, however, coexist with a stable steady state even for small delays, which is mathematically counterintuitive. In order to explore how these solutions appear in the bifurcation diagram, we propose three different strategies. We first analyze the emergence of periodic solutions from Hopf bifurcation points for τ small and show that a subcritical Hopf bifurcation allows the coexistence of stable τ -periodic and stable steady-state solutions. Second, we construct a τ -periodic solution by using singular perturbation techniques appropriate for slow-fast systems. The theory assumes τ = O(1) and its validity as τ → 0 is investigated numerically by integrating the original equations. Third, we develop an asymptotic theory where the delay is scaled with respect to the fast timescale of the activator variable. The theory is applied to the FitzHugh-Nagumo equations with threshold nonlinearity, and we show that the branch of τ -periodic solutions emerges from a limit point of limit cycles.

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“…We also show the existence of one, two, or three different modes of spiking as a result of pitchfork cycle and torus (Neimark-Sacker) bifurcations. We want to emphasize that, by BendixsonDulac theorem, it can be seen that the single neuron of the FHN model does not admit periodic solutions for the ranges chosen in our paper, unlike the system in [28] whose dynamics, in the absence of delays, is characterized by a canard explosion and relaxation oscillations. Thus the periodic solutions are due to coupling of the neurons, or time delay.…”

Section: Introductionmentioning

confidence: 94%

“…There are many technological and biological systems which exhibit such behavior. Some examples of delayed systems are coupled lasers [26], high-speed milling, population dynamics, epidemiology [27], and gene expression, Krupa and Touboul [28] investigated a self-coupled delayed FHN system to understand the effect of introducing delays in a system whose dynamics, in the absence of delays, is characterized by a canard explosion and relaxation oscillations. They have shown that delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting, canard and chaos.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation structure of two coupled FHN neurons with delay

Tehrani

Razvan

2015

Mathematical Biosciences

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This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Homoclinic, fold, and pitchfork bifurcations of limit cycles are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf bifurcation points can be obtained from the intersection points of two branches of Hopf bifurcation. The dynamical behavior of the system may exhibit one, two, or three different periodic solutions due to pitchfork cycle and torus bifurcations (Neimark-Sacker bifurcation in the Poincare map of a limit cycle), of which detection was impossible without exact and systematic dynamical study. In addition, Hopf, double-Hopf, and torus bifurcations of the non trivial rest points are found. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behaviors are clarified.

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Complex Oscillations in the Delayed FitzHugh–Nagumo Equation

Cited by 29 publications

References 51 publications

Coherent neural oscillations induced by weak synaptic noise

Coherent neural oscillations induced by weak synaptic noise

Short-time-delay limit of the self-coupled FitzHugh-Nagumo system

Bifurcation structure of two coupled FHN neurons with delay

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