Oscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in recent years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. At the bifurcation point numerous regimes with non-equal interspike intervals emerge. We show 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 phasereduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit. PACS numbers: 87.19.ll, 05.45.Xt, 87.19.lr, 89.75 Interaction via pulse-like signals is important in neuron populations [1][2][3], biological [4,5], optical and optoelectronic systems [6]. Often, time delays are inevitable in such systems as a consequence of the finite speed of pulse propagation [7]. In this letter we demonstrate that the pulsatile and delayed nature of interactions may lead to novel and unusual phenomena in a large class of systems. In particular, we explore oscillatory systems with pulsatile delayed feedback which exhibit periodic regular spiking (RS). We show that this RS regime may destabilize via a scenario in which a variety of higher-periodical regimes with non-equal interspike intervals (ISIs) emerge simultaneously. The number of the emergent, so-called "jittering" regimes grows exponentially as the delay increases. Therefore we adopt the term "multi-jitter" bifurcation.Usually, the simultaneous emergence of many different regimes is a sign of degeneracy and it is expected to occur generically only when additional symmetries are present [2,8].However, for the class of systems treated here no such symmetry is apparent. Nevertheless, the phenomenon can be reliably observed when just a single parameter, for example the delay, is varied. This means that the observed bifurcation has codimension one [9]. In addition to the theoretical analysis of a simple paradigmatic model, we provide numerical evidence for the occurrence of the multi-jitter bifurcation in a realistic neuronal model, as well as an experimental confirmation in an electronic circuit.As a universal and simplest oscillatory spiking model in the absence of the feedback, we consider the phase oscillator dϕ/dt = ω, where ϕ ∈ R ( mod 1), and ω = 1 without loss of generality. When the oscillator reaches ϕ = 1 at some moment t, the phase is reset to zero and the oscillator produces a pulse signal. If this signal is sent into a delayed feedback loop [ Fig. 1(a)] the emitted pulses affect the oscillator after a delay τ at the time instant t * = t+τ . When the pulse is received, the phase of the oscillator undergoes an instantaneous shift by an amount ∆ϕ = Z(ϕ(t * − 0)), where Z(ϕ) is the phase resetting curve (PRC).Thus, the dynamics of the oscillator can be describ...
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 nonequal 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.
We carry out theoretical and experimental studies of cross-frequency synchronization of two pulse oscillators with time-delayed coupling. In the theoretical part of the paper we utilize the concept of phase resetting curves and analyze the system dynamics in the case of weak coupling. We construct a Poincaré map and obtain the synchronization zones in the parameter space for m:n synchronization. To challenge the theoretical results we designed an electronic circuit implementing the coupled oscillators and studied its dynamics experimentally. We show that the developed theory predicts dynamical properties of the realistic system, including location of the synchronization zones and bifurcations inside them.
We propose a model of an adaptive network of spiking neurons that gives rise to a hypernetwork of its dynamic states at the upper level of description. Left to itself, the network exhibits a sequence of transient clustering which relates to a traffic in the hypernetwork in the form of a random walk. Receiving inputs the system is able to generate reproducible sequences corresponding to stimulus-specific paths in the hypernetwork. We illustrate these basic notions by a simple network of discrete-time spiking neurons together with its FPGA realization and analyse their properties.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.
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