74 research outputs found
Phase response function for oscillators with strong forcing or coupling
Phase response curve (PRC) is an extremely useful tool for studying the
response of oscillatory systems, e.g. neurons, to sparse or weak stimulation.
Here we develop a framework for studying the response to a series of pulses
which are frequent or/and strong so that the standard PRC fails. We show that
in this case, the phase shift caused by each pulse depends on the history of
several previous pulses. We call the corresponding function which measures this
shift the phase response function (PRF). As a result of the introduction of the
PRF, a variety of oscillatory systems with pulse interaction, such as neural
systems, can be reduced to phase systems. The main assumption of the classical
PRC model, i.e. that the effect of the stimulus vanishes before the next one
arrives, is no longer a restriction in our approach. However, as a result of
the phase reduction, the system acquires memory, which is not just a technical
nuisance but an intrinsic property relevant to strong stimulation. We
illustrate the PRF approach by its application to various systems, such as
Morris-Lecar, Hodgkin-Huxley neuron models, and others. We show that the PRF
allows predicting the dynamics of forced and coupled oscillators even when the
PRC fails
Stability threshold approach for complex dynamical systems
Acknowledgments This paper was developed within the scope of the IRTG 1740/TRP 2011/50151-0, funded by the DFG/FAPESP, and supported by the Government of the Russian Federation (Agreement No. 14.Z50.31.0033 with the Institute of Applied Physics RAS). The first author thanks Dr Roman Ovsyannikov for valuable discussions regarding estimation of the mistake probability.Peer reviewedPreprintPublisher PD
Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback
Interaction via pulses is common in many natural systems, especially
neuronal. In this article we study one of the simplest possible systems with
pulse interaction: a phase oscillator with delayed pulsatile feedback. When the
oscillator reaches a specific state, it emits a pulse, which returns after
propagating through a delay line. The impact of an incoming pulse is described
by the oscillator's phase reset curve (PRC). In such a system we discover an
unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic
regular spiking solution bifurcates with several multipliers crossing the unit
circle at the same parameter value. The number of such critical multipliers
increases linearly with the delay and thus may be arbitrary large. This
bifurcation is accompanied by the emergence of numerous "jittering" regimes
with non-equal interspike intervals (ISIs). Each of these regimes corresponds
to a periodic solution of the system with a period roughly proportional to the
delay. The number of different "jittering" solutions emerging at the
bifurcation point increases exponentially with the delay. We describe the
combinatorial mechanism that underlies the emergence of such a variety of
solutions. In particular, we show how a periodic solution exhibiting several
distinct ISIs can imply the existence of multiple other solutions obtained by
rearranging of these ISIs. We show that the theoretical results for phase
oscillators accurately predict the behavior of an experimentally implemented
electronic oscillator with pulsatile feedback
Multistable jittering in oscillators with pulsatile delayed feedback
Oscillatory systems with time-delayed pulsatile feedback appear in various
applied and theoretical research areas, and received a growing interest in the
last years. For such systems, we report a remarkable scenario of
destabilization of a periodic regular spiking regime. In the bifurcation point
numerous regimes with non-equal interspike intervals emerge simultaneously. We
show that this bifurcation is triggered by the steepness of the oscillator's
phase resetting curve and that the number of the emerging, so-called
"jittering" regimes grows exponentially with the delay value. Although this
appears as highly degenerate from a dynamical systems viewpoint, the
"multi-jitter" bifurcation occurs robustly in a large class of systems. We
observe it not only in a paradigmatic phase-reduced model, but also in a
simulated Hodgkin-Huxley neuron model and in an experiment with an electronic
circuit
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Interval stability for complex systems
Stability of dynamical systems against strong perturbations is an important problem of nonlinear dynamics relevant to many applications in various areas. Here, we develop a novel concept of interval stability, referring to the behavior of the perturbed system during a finite time interval. Based on this concept, we suggest new measures of stability, namely interval basin stability (IBS) and interval stability threshold (IST). IBS characterizes the likelihood that the perturbed system returns to the stable regime (attractor) in a given time. IST provides the minimal magnitude of the perturbation capable to disrupt the stable regime for a given interval of time. The suggested measures provide important information about the system susceptibility to external perturbations which may be useful for practical applications. Moreover, from a theoretical viewpoint the interval stability measures are shown to bridge the gap between linear and asymptotic stability. We also suggest numerical algorithms for quantification of the interval stability characteristics and demonstrate their potential for several dynamical systems of various nature, such as power grids and neural networks
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