FitzHugh-Nagumo Oscillators Research Articles

Pinning control of chimera states.

The position of the coherent and incoherent domain of a chimera state in a ring of nonlocally coupled oscillators is strongly influenced by the initial conditions, making nontrivial the problem of confining them in a specific region of the structure. In this paper we propose the use of spatial pinning to induce a chimera state where the nodes belonging to one domain, either the coherent or the incoherent, are fixed by the control action. We design two different techniques according to the dynamics to be forced in the region of pinned nodes, and validate them on FitzHugh-Nagumo and Kuramoto oscillators. Furthermore, we introduce a suitable strategy to deal with the effects of finite size in small structures.

Suppressing Activity of an Array of Coupled Fitzhugh-Nagumo Oscillators

An extremely simple method for stabilizing unstable steady states in an array of coupled neuronal FitzHugh– Nagumo type oscillators is described. A two-terminal electronic feedback controller has been developed. The feedback circuit, when coupled to an array of oscillators, damps the spiking neurons, thus does away with the effect of synchronization. Both, numerical simulations and hardware experiments with the electronic circuits have been performed. The results for an array of three mean-field coupled FitzHugh–Nagumo oscillators are presented.

Different synchronization characteristics of distinct types of traveling waves in a model of active medium with periodic boundary conditions

The model of a one-dimensional active medium, which cells are the FitzHugh–Nagumo oscillators, is studied for periodical boundary conditions. The medium possesses three different regimes in dependence on the parameter values. The regimes correspond to the self-sustained oscillations, excitable dynamics or bistability of the medium cells. Periodic boundary conditions provide the existence of traveling wave modes in all mentioned cases without any deterministic or stochastic excitation. The spatial waveforms and the character of oscillations in time can be similar in the different cases, but the properties of wave modes depend considerably on the medium regime. So, the dispersion characteristics and the synchronization phenomena are essentially different for bistable and excitable media on the one hand, and for the self-sustained oscillatory medium on the other hand. The local and distributed periodic influence on the medium are studied. The phenomenon of the traveling wave frequency locking is observed for all three regimes of the active medium. The comparison of synchronization effects in self-oscillatory, excitable and bistable regimes of the active medium is carried out.

Synchronization of Coupled Oscillator Dynamics

Abstract Synchronization is most significant phenomena to study the collective behaviour of coupled oscillators. Synchronization is said to occur if phase locking and consensus among corresponding states of coupled dynamical systems is achieved. In general, how to achieve exact conditions for synchronization is unclear. Therefore, it becomes essential to derive conditions on these coupled dynamical systems which lead to synchronization. Here, we aim to derive sufficient conditions for synchronization of selected benchmark oscillators which are linearly coupled. We have used Lyapunov approach to obtain sufficiency condition for synchronization of coupled oscillators. We study synchronization of Van der Pol oscillators and Fitzhugh Nagumo oscillators with all-to-all connectivity, for both uniformly and non-uniformly linearly coupled configurations. These results have been numerically simulated for both the above types of oscillators.

Two-terminal feedback circuit for suppressing synchrony of the FitzHugh–Nagumo oscillators

An extremely simple analog technique for desynchronization of neuronal FitzHugh-Nagumo- type oscillators is described. Two-terminal feedback circuit has been developed. The feedback circuit, when coupled to a network of oscillators, nullifies the volt- age at the coupling node and thus effectively decou- ples the individual oscillators. Both numerical sim- ulations and hardware experiments have been per- formed. The results for an array of three mean- field coupled FitzHugh-Nagumo-type oscillators are presented.

Asymmetric Interaction of a Pair FitzHugh–Nagumo Oscillators

A pair of diffusion connected FitzHugh–Nagumo oscillators with an asymmetric interaction are considered. The problem is investigated in the close to critical case, where the matrix of the linear part of the system has a pair of purely imaginary eigenvalues. The normal form is constructed and its coefficients are determined depending on the initial parameters. The source system may be in two different situations: stable singlefrequency oscillations with two different frequencies coexist or a single-frequency mode branches from the equilibrium. The obtained asymptotic results are supplemented by the numerical analysis.

Robustness of chimera states for coupled FitzHugh-Nagumo oscillators.

Chimera states are complex spatio-temporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics. This counterintuitive phenomenon was first observed in systems of identical oscillators with symmetric coupling topology. Can one overcome these limitations? To address this question, we discuss the robustness of chimera states in networks of FitzHugh-Nagumo oscillators. Considering networks of inhomogeneous elements with regular coupling topology, and networks of identical elements with irregular coupling topologies, we demonstrate that chimera states are robust with respect to these perturbations and analyze their properties as the inhomogeneities increase. We find that modifications of coupling topologies cause qualitative changes of chimera states: additional random links induce a shift of the stability regions in the system parameter plane, gaps in the connectivity matrix result in a change of the multiplicity of incoherent regions of the chimera state, and hierarchical geometry in the connectivity matrix induces nested coherent and incoherent regions.

Stochastic switching in delay-coupled oscillators.

A delay is known to induce multistability in periodic systems. Under influence of noise, coupled oscillators can switch between coexistent orbits with different frequencies and different oscillation patterns. For coupled phase oscillators we reduce the delay system to a nondelayed Langevin equation, which allows us to analytically compute the distribution of frequencies and their corresponding residence times. The number of stable periodic orbits scales with the roundtrip delay time and coupling strength, but the noisy system visits only a fraction of the orbits, which scales with the square root of the delay time and is independent of the coupling strength. In contrast, the residence time in the different orbits is mainly determined by the coupling strength and the number of oscillators, and only weakly dependent on the coupling delay. Finally we investigate the effect of a detuning between the oscillators. We demonstrate the generality of our results with delay-coupled FitzHugh-Nagumo oscillators.

Stability analysis and rich oscillation patterns in discrete-time FitzHugh–Nagumo excitable systems with delayed coupling

In this paper, we give a detailed study of the stable region in discrete-time FitzHugh–Nagumo delayed excitable Systems, which can be divided into two parts: one is independent of delay and the other is dependent on delay. Two different new states are to be observed, which are new steady states (equilibria-the excitable FitzHugh–Nagumo) or limit cycles/higher periodic orbits (the FitzHugh–Nagumo oscillators) as the origin loses its stability, and usually, one is synchronized and the other asynchronized. We also find out that there exist critical curves through which there occur fold bifurcations, flip bifurcations, Neimark–Sacker bifurcations and even higher-codimensional bifurcations etc. It is also shown that delay can play an important role in rich dynamics, such as the occurrence of chaos or not, by means of Lyapunov exponents, Lyapunov dimensions, and the sensitivity to the initial conditions. Multistability phenomena are also discussed including the coexistence of synchronized and asynchronized oscillators, or synchronized/asynchronized oscillators and multiple stable nontrivial equilibria etc.

Numerically induced bursting in a set of coupled neuronal oscillators

We present our numerical observations on dynamics in the system of two linearly coupled FitzHugh–Nagumo oscillators close to the destabilization of the state of rest. Under the considered parameter values the system, if integrated sufficiently accurately, converges to small-scale periodic oscillations. However, minor numerical inaccuracies, which occur already at the default precision of the standard Runge–Kutta solver, lead to a breakup of periodicity and an onset of large-scale aperiodic bursting.

Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

The behavior of neurons can be modeled by the FitzHugh-Nagumo oscillator model, consisting of two nonlinear differential equations, which simulates the behavior of nerve impulse conduction through the neuronal membrane. In this work, we numerically study the dynamical behavior of two coupled FitzHugh-Nagumo oscillators. We consider unidirectional and bidirectional couplings, for which Lyapunov and isoperiodic diagrams were constructed calculating the Lyapunov exponents and the number of the local maxima of a variable in one period interval of the time-series, respectively. By numerical continuation method the bifurcation curves are also obtained for both couplings. The dynamics of the networks here investigated are presented in terms of the variation between the coupling strength of the oscillators and other parameters of the system. For the network of two oscillators unidirectionally coupled, the results show the existence of Arnold tongues, self-organized sequentially in a branch of a Stern-Brocot tree and by the bifurcation curves it became evident the connection between these Arnold tongues with other periodic structures in Lyapunov diagrams. That system also present multistability shown in the planes of the basin of attractions.

Chimera States in Networks of Nonlocally Coupled Hindmarsh–Rose Neuron Models

We have identified the occurrence of chimera states for various coupling schemes in networks of two-dimensional and three-dimensional Hindmarsh–Rose oscillators, which represent realistic models of neuronal ensembles. This result, together with recent studies on multiple chimera states in nonlocally coupled FitzHugh–Nagumo oscillators, provide strong evidence that the phenomenon of chimeras may indeed be relevant in neuroscience applications. Moreover, our work verifies the existence of chimera states in coupled bistable elements, whereas to date chimeras were known to arise in models possessing a single stable limit cycle. Finally, we have identified an interesting class of mixed oscillatory states, in which desynchronized neurons are uniformly interspersed among the remaining ones that are either stationary or oscillate in synchronized motion.

Pattern competition as a driver of diversity-induced resonance

Diversity-induced resonance, the emergence of coherent spatiotemporal patterns at intermediate parameter disorder, is a well-known phenomenon in lattices of excitable elements. Here we study the pattern events behind diversity-induced resonance in a lattice of coupled FitzHugh-Nagumo oscillators. Starting out with the observation that maximal spiral wave counts occur at intermediate values of parameter diversity, we analyze the competition between spiral and target wave patterns in the asymptotic collective state. We devise stylized numerical “in silico” competition experiments of (individual) patterns to understand the regulating parameters of the competing pattern events occurring stochastically in the full (“in vivo”) numerical simulation. Our analysis shows that pattern competition is a principal driving mechanism behind this form of diversity-induced resonance and that different types of competition take place: some follow the frequency composition of target and spiral waves, others are dictated by the statistics of parameter distributions. In particular, the increase and decrease of spiral wave counts with increasing diversity are statistically regulated by the number of oscillatory elements in the lattice, rather than by the frequency differences between target and spiral waves.

Noise-induced effects in an active medium with periodic boundary conditions

A model of an active medium with periodic boundary conditions in which the elementary cell is represented by a FitzHugh-Nagumo oscillator has been studied. Depending on the system parameters, the elementary cell can occur in either auto-oscillatory or excitable state. In both cases, an autonomous medium in the absence of noise performs sustained oscillations and exhibits the phenomenon of multistability. A method for diagnostics of the character of medium with the aid of external noise is proposed, specific features in behavior of the system near the point of transition from the excitable to auto-oscillatory state are considered, and the phenomena of coherent resonance and noise-induced switching are described.

Explosive synchronization transitions in complex neural networks

It has been recently reported that explosive synchronization transitions can take place in networks of phase oscillators [Gómez-Gardeñes et al. Phys. Rev. Lett. 106, 128701 (2011)] and chaotic oscillators [Leyva et al. Phys. Rev. Lett. 108, 168702 (2012)]. Here, we investigate the effect of a microscopic correlation between the dynamics and the interacting topology of coupled FitzHugh-Nagumo oscillators on phase synchronization transition in Barabási-Albert (BA) scale-free networks and Erdös-Rényi (ER) random networks. We show that, if natural frequencies of the oscillations are positively correlated with node degrees and the width of the frequency distribution is larger than a threshold value, a strong hysteresis loop arises in the synchronization diagram of BA networks, indicating the evidence of an explosive transition towards synchronization of relaxation oscillators system. In contrast to the results in BA networks, in more homogeneous ER networks, the synchronization transition is always of continuous type regardless of the width of the frequency distribution. Moreover, we consider the effect of degree-mixing patterns on the nature of the synchronization transition, and find that the degree assortativity is unfavorable for the occurrence of such an explosive transition.

Multi-chimera states in FitzHugh-Nagumo oscillators

Chimera states are spatio-temporal patterns of synchrony and disorder observed in homogeneous oscillatory media with nonlocal coupling. They are characterized by coexistence of distinct spatial regions with regular synchronized and irregular incoherent motion. Initially discovered for phase oscillators, chimera states have been also found in systems of nonlocally coupled discrete maps [1], time-continuous chaotic systems [2] and have been recently realized in experiments [3]. We demonstrate the existence of chimera states for neural oscillators [4]. In detail, we investigate the cooperative dynamics of nonlocally coupled neural populations modeled by FitzHugh-Nagumo systems, where each individual system displays oscillatory local dynamics described by a fast activator and slow inhibitor variable. Applying a phase-reduction technique we show that off-diagonal nonlocal coupling is a crucial factor for the appearance of chimera states. In the considered configuration, this off-diagonal coupling is realized by a rotational coupling matrix that mixes input of the activator and inhibitor and that is conveniently described by the coupling phase as a single control parameter. We analyze the stability of chimera states in the parameter space of the system and discuss mechanisms of transitions between different chimera types. Surprisingly, we find that for increasing coupling strength classical chimera states undergo transitions from one to multiple domains of incoherence. Then patches of synchronized dynamics appear within the incoherent domain giving rise to a multi-chimera state. We find that, depending on the coupling strength and range, different multi-chimeras arise in a transition from classical chimera states. The additional spatial modulation is due to strong coupling interaction and thus cannot be observed in simple phase-oscillator models.

Modeling brain functional connectivity at rest

Well organized spatio-temporal low-frequency fluctuations (< 0.1 Hz), observed in blood-oxygen-level-dependent (BOLD) signal during rest, have been used to map several consistent resting state networks (RSNs) in the brain [1-3]. It has been hypothesized that these correlated fluctuations reflect synchronized variations in neural activity of particular brain areas, which are dynamically coupled to one another forming functional connections within networks of brain. Furthermore, it has been suggested that resting state functional connectivity (FC) is strongly shaped by underlying anatomical connectivity (AC). However, although RSNs reflect anatomical connections between brain areas comprising the networks in focus, FC cannot be understood in those terms alone [4]. Here, we combine experimental and modeling approach to investigate dynamics underlying correlated behavior of distant cortical regions and formation of the so called functional networks. We aim to address complicated interplay between network structure, dynamics of its components and emerging global behavior, as key ingredients of the networks complexity [5]. We study how functional connectivity arise from anatomical connections and compare obtained data with the networks simulated on the empirically derived FC networks from resting state fMRI data. We compare two distinctive networks: one with 90 brain regions defined using the Automated Anatomical Labeling (AAL) template [6], and another with 100 regions organized into seven distinctive resting state functional networks [7]. We choose to model local network dynamics by excitable FitzHugh-Nagumo oscillators subject to uncorrelated white Gaussian noise and time-delayed interactions to account for the finite speed of the signal propagation along the axons. We discuss FC between brain regions without apparent anatomical connections, exploring dynamics that underlie these correlations.

Realistic model of compact VLSI FitzHugh–Nagumo oscillators

In this article, we present a compact analogue VLSI implementation of the FitzHugh–Nagumo neuron model, intended to model large-scale, biologically plausible, oscillator networks. As the model requires a series resistor and a parallel capacitor with the inductor, which is the most complex part of the design, it is possible to greatly simplify the active inductor implementation compared to other implementations of this device as typically found in filters by allowing appreciable, but well modelled, nonidealities. We model and obtain the parameters of the inductor nonideal model as an inductance in series with a parasitic resistor and a second order low-pass filter with a large cut-off frequency. Post-layout simulations for a CMOS 0.35 μm double-poly technology using the MOSFET Spice BSIM3v3 model confirm the proper behaviour of the design.

Predictability of spatio-temporal patterns in a lattice of coupled FitzHugh–Nagumo oscillators

In many biological systems, variability of the components can be expected to outrank statistical fluctuations in the shaping of self-organized patterns. In pioneering work in the late 1990s, it was hypothesized that a drift of cellular parameters (along a 'developmental path'), together with differences in cell properties ('desynchronization' of cells on the developmental path) can establish self-organized spatio-temporal patterns (in their example, spiral waves of cAMP in a colony of Dictyostelium discoideum cells) starting from a homogeneous state. Here, we embed a generic model of an excitable medium, a lattice of diffusively coupled FitzHugh-Nagumo oscillators, into a developmental-path framework. In this minimal model of spiral wave generation, we can now study the predictability of spatio-temporal patterns from cell properties as a function of desynchronization (or 'spread') of cells along the developmental path and the drift speed of cell properties on the path. As a function of drift speed and desynchronization, we observe systematically different routes towards fully established patterns, as well as strikingly different correlations between cell properties and pattern features. We show that the predictability of spatio-temporal patterns from cell properties contains important information on the pattern formation process as well as on the underlying dynamical system.

NETWORK SYNCHRONIZATION BY DYNAMIC DIFFUSIVE COUPLING

The problem of controlled network synchronization for a class of nonlinear observable systems interconnected via dynamic diffusive coupling is studied. We construct a dynamic diffusive coupling combining a nonlinear observer and an output feedback controller. Sufficient conditions on the systems to be interconnected, on the network topology, and on the coupling strength that guarantee (global) synchronization are derived. Moreover, using the notion of semipassivity, we prove that under some mild assumptions, the solutions of interconnected semipassive systems are ultimately bounded. The results are illustrated by computer simulations of coupled FitzHugh–Nagumo oscillators.