399 research outputs found
Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling
Small lattices of nearest neighbor coupled excitable FitzHugh-Nagumo
systems, with time-delayed coupling are studied, and compared with systems of
FitzHugh-Nagumo oscillators with the same delayed coupling. Bifurcations of
equilibria in N=2 case are studied analytically, and it is then numerically
confirmed that the same bifurcations are relevant for the dynamics in the case
. Bifurcations found include inverse and direct Hopf and fold limit cycle
bifurcations. Typical dynamics for different small time-lags and coupling
intensities could be excitable with a single globally stable equilibrium,
asymptotic oscillatory with symmetric limit cycle, bi-stable with stable
equilibrium and a symmetric limit cycle, and again coherent oscillatory but
non-symmetric and phase-shifted. For an intermediate range of time-lags inverse
sub-critical Hopf and fold limit cycle bifurcations lead to the phenomenon of
oscillator death. The phenomenon does not occur in the case of FitzHugh-Nagumo
oscillators with the same type of coupling.Comment: accepted by Phys.Rev.
Mean field approximation for noisy delay coupled excitable neurons
Mean field approximation of a large collection of FitzHugh-Nagumo excitable
neurons with noise and all-to-all coupling with explicit time-delays, modelled
by stochastic delay-differential equations is derived. The resulting
approximation contains only two deterministic delay-differential equations but
provides excellent predictions concerning the stability and bifurcations of the
averaged global variables of the exact large system.Comment: 15 pages, 3 figure
Mean field approximation of two coupled populations of excitable units
The analysis on stability and bifurcations in the macroscopic dynamics
exhibited by the system of two coupled large populations comprised of
stochastic excitable units each is performed by studying an approximate system,
obtained by replacing each population with the corresponding mean-field model.
In the exact system, one has the units within an ensemble communicating via the
time-delayed linear couplings, whereas the inter-ensemble terms involve the
nonlinear time-delayed interaction mediated by the appropriate global
variables. The aim is to demonstrate that the bifurcations affecting the
stability of the stationary state of the original system, governed by a set of
4N stochastic delay-differential equations for the microscopic dynamics, can
accurately be reproduced by a flow containing just four deterministic
delay-differential equations which describe the evolution of the mean-field
based variables. In particular, the considered issues include determining the
parameter domains where the stationary state is stable, the scenarios for the
onset and the time-delay induced suppression of the collective mode, as well as
the parameter domains admitting bistability between the equilibrium and the
oscillatory state. We show how analytically tractable bifurcations occurring in
the approximate model can be used to identify the characteristic mechanisms by
which the stationary state is destabilized under different system
configurations, like those with symmetrical or asymmetrical inter-population
couplings.Comment: 5 figure
Early warning signal for interior crises in excitable systems
The ability to reliably predict critical transitions in dynamical systems is
a long-standing goal of diverse scientific communities. Previous work focused
on early warning signals related to local bifurcations (critical slowing down)
and non-bifurcation type transitions. We extend this toolbox and report on a
characteristic scaling behavior (critical attractor growth) which is indicative
of an impending global bifurcation, an interior crisis in excitable systems. We
demonstrate our early warning signal in a conceptual climate model as well as
in a model of coupled neurons known to exhibit extreme events. We observed
critical attractor growth prior to interior crises of chaotic as well as
strange-nonchaotic attractors. These observations promise to extend the classes
of transitions that can be predicted via early warning signals.Comment: 6 pages, 4 figure
Regular and irregular patterns of self-localized excitation in arrays of coupled phase oscillators
We study a system of phase oscillators with nonlocal coupling in a ring that
supports self-organized patterns of coherence and incoherence, called chimera
states. Introducing a global feedback loop, connecting the phase lag to the
order parameter, we can observe chimera states also for systems with a small
number of oscillators. Numerical simulations show a huge variety of regular and
irregular patterns composed of localized phase slipping events of single
oscillators. Using methods of classical finite dimensional chaos and
bifurcation theory, we can identify the emergence of chaotic chimera states as
a result of transitions to chaos via period doubling cascades, torus breakup,
and intermittency. We can explain the observed phenomena by a mechanism of
self-modulated excitability in a discrete excitable medium.Comment: postprint, as accepted in Chaos, 10 pages, 7 figure
Cooperative behavior between oscillatory and excitable units: the peculiar role of positive coupling-frequency correlations
We study the collective dynamics of noise-driven excitable elements,
so-called active rotators. Crucially here, the natural frequencies and the
individual coupling strengths are drawn from some joint probability
distribution. Combining a mean-field treatment with a Gaussian approximation
allows us to find examples where the infinite-dimensional system is reduced to
a few ordinary differential equations. Our focus lies in the cooperative
behavior in a population consisting of two parts, where one is composed of
excitable elements, while the other one contains only self-oscillatory units.
Surprisingly, excitable behavior in the whole system sets in only if the
excitable elements have a smaller coupling strength than the self-oscillating
units. In this way positive local correlations between natural frequencies and
couplings shape the global behavior of mixed populations of excitable and
oscillatory elements.Comment: 10 pages, 6 figures, published in Eur. Phys. J.
Time-delayed feedback control of coherence resonance chimeras
This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 27, 114320 (2017) and may be found at https://doi.org/10.1063/1.5008385.Using the model of a FitzHugh-Nagumo system in the excitable regime, we investigate the influence of time-delayed feedback on noise-induced chimera states in a network with nonlocal coupling, i.e., coherence resonance chimeras. It is shown that time-delayed feedback allows for the control of the range of parameter values where these chimera states occur. Moreover, for the feedback delay close to the intrinsic period of the system, we find a novel regime which we call period-two coherence resonance chimera.
Coherence resonance chimeras in nonlocally coupled networks of excitable elements represent partial synchronization patterns composed of spatially separated domains of coherent and incoherent spiking behavior, which are induced by noise. These patterns are different from classical chimera states occurring in deterministic oscillatory systems and combine properties of the counter-intuitive phenomenon of coherence resonance, i.e., a constructive role of noise, and chimera states, i.e., the coexistence of spatially synchronized and desynchronized domains in a network of identical elements. Another distinctive feature of the particular type of chimera we study here is its alternating behavior, i.e., periodic switching of the location of coherent and incoherent domains. Applying time-delayed feedback, we demonstrate how to control coherence resonance chimeras by adjusting delay time and feedback strength. In particular, we show that feedback increases the parameter intervals of existence of chimera states and has a significant impact on their alternating dynamics leading to the appearance of novel patterns, which we call period-two coherence resonance chimera. Since the dynamics of every individual network element in our study is given by the FitzHugh-Nagumo (FHN) system, which is a paradigmatic model for neurons in the excitable regime, we expect wide-range applications of our results to neural networks.DFG, 163436311, SFB 910: Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und Anwendungskonzept
- …