Pulse-coupled oscillators are popular and powerful models used widely to describe the dynamics of networks of various nature, including neuronal populations, cardiac tissues and many others [1–3]. The key features captured by these models are the periodical self-sustained activity of the individual units and the pulse-mediated interaction between them. Pulses here are brief signals which have a temporal duration much smaller than the oscillations period. In realistic systems, such pulses often have finite speed of propagation between the units, which leads to the emergence of coupling delays.
In the framework of pulse-coupled oscillators, the effect of a pulse depends on the dynamical state of the oscillator at the time of the pulse arrival. In the case of weak coupling the oscillators are usually described by their phases, and the influence of a pulse is determined with the help of the so-called phase resetting curves (PRCs, [4, 5]). A characteristic PRC of any oscillatory system corresponding to any type of stimulus can be computed numerically or measured experimentally [5–10]. Thus, pulse-coupled systems can be considered either as stand-alone models, or as approximations of more complex systems. Among the advantages of such models is their simplicity both for numerical implementation and theoretical analysis.
A number of important results have been obtained within the framework of pulse-coupled oscillators. The stability of synchronous [1, 2] and desynchronized [11] states have been studied. Emergence of coexistent stable clusters is reported in [12–15]. Splay states and their stability in globally pulse-coupled systems are reported in [16]. Rings of pulse-coupled oscillators are investigated in [17]. Recently, a novel dynamical phenomenon has been discovered in spiking oscillators subject to delayed pulse feedback [18, 19]. In such systems, under certain conditions on the oscillator’s PRC, regular spiking may destabilize and change to irregular, so-called “jittering” regimes with non-equal inter-spike intervals (ISIs). The number of different coexisting jittering regimes grows exponentially with the delay which makes the system highly multistable.
In the present paper, we make a first step to extend the concept of “jittering” to systems more complex than just a single oscillator. Namely, we discover and study jittering regimes in a system of two spiking oscillators with time-delay coupling.
Jittering regimes of a single spiking oscillator have been studied in detail in our previous papers [18, 19]. Here we reproduce the key results of those papers because the similar technique will be used in the present study. The basic model of a single spiking oscillator with delayed feedback is as follows:
The oscillator is described by its phase
Dynamics of the oscillators with pulse delayed coupling and derivation of the discrete maps. Dots depict the moments of spikes emission. (a) Single oscillator with delayed feedback and (b) its dynamics. Each spike produced arrives to the oscillator after delay time
Previously, the researchers concentrated on regular regimes of (1) when spikes are produced with constant inter-spike interval (ISI). However, recently we have shown that regular regimes may destabilize and give rise to irregular ones with distinct ISIs. These regimes were called “jittering”, and the corresponding scenario was called “multi-jitter bifurcation”. In the bifurcation point, numerous jittering regimes emerge at the same point which makes the system highly multistable. Namely, we have shown that the number of coexisting jittering regimes grows exponentially as the delay increases.
Analytical study of the multi-jitter bifurcation relies on the technique developed in [20] which allows to reduce (1) to a multidimensional discrete map. This map fully describes the dynamics of the oscillator and deals with the sequence of the ISIs denoted as
Here,
For regular spiking, when all the ISIs equal
The stability of the regular regime is determined by the roots of the characteristic equation
where
The multi-jitter bifurcation occurs at points with
As mentioned above, the criterion for the multi-jitter bifurcation is sufficient steepness of the PRC. Suppose there exists an interval of phases
Now let us consider a system of two pulse oscillators with mutual delayed coupling. For the sake of simplicity, let the system be symmetrical, so that the native frequencies, the coupling strength and the delays are the same. In this case the system of two oscillators is governed by the equations
Here,
The system of two delay-coupled oscillators have been intensively studied previously [2, 12, 16, 23]. Depending on the frequency mismatch and the coupling strength, the oscillators may either synchronize or demonstrate asynchronous behavior. If the oscillators are identical they always synchronize, but depending on the delay value two different synchronous regimes are possible: the in-phase and the antiphase synchronization. For the in-phase synchronization, the oscillators emit spikes simultaneously. In this case the stable synchronous manifold in the system phase space exists for which the states of the oscillators are equal. On this manifold, two oscillators behave as one oscillator with delayed feedback. This means that the system dynamics on the synchronous manifold is described by (1), and they should have the same properties. Thus, the in-phase jittering of two oscillators should be possible.
Let us study the stability of the in-phase synchronous regime. For this sake, consider small perturbation of this regime so that the spikes are emitted by the oscillators not simultaneously, but with small time lags
For the in-phase synchronization,
Here,
Destabilization of the in-phase synchronous regime corresponds to |
which, for
appear simultaneously. These points correspond to the emergence of in-phase jittering regimes of two oscillators.
The other basic synchronous regime of two identical oscillators is antiphase synchronization when the oscillators emit spikes one after another with the same inter-spike intervals. To study the stability of the anti-phase synchronization let us first derive a discrete map governing the system dynamics in the neighborhood of this regime. For this sake we first introduce convenient notations. Denote
Similarly with the case of one oscillator, this phase can be found as
Here,
which calculates the next CSI based on
The anti-phase synchronization corresponds to the case when all the CSIs equal
Linearization of (12) leads to the characteristic equation
Here,
The set Λ
For
For
Thus, the characteristic equation (14) can have critical roots only for
Our analytical study suggests that both in-phase and antiphase synchronous regimes of two oscillators may destabilize and give rise to jittering regimes. Now we seek to confirm our theoretical predictions by numerical simulations. For this sake we consider the PRC in the form
Here,
The PRC for
First, let us study the synchronous solutions of (5) with PRC (16). For the in-phase synchronous regimes, the period is given by (3), which can be rewritten in parametric form as
Varying
The bifurcation diagram of two oscillators with mutual delayed coupling. The observed ISIs of the established dynamical regimes are plotted versus the time delay. Blue lines stand for near-in-phase, red for near-antiphase regimes. Theoretically predicted regular regimes are plotted by thin lines, solid for stable and dashed for unstable regimes. Diamonds depict saddle-node or period-doubling bifurcations, while stars depict multi-jitter bifurcations. Thick dots depict the numerical results. Note that almost everywhere the stable branches predicted by the theory are covered by numerically observed dots.
For anti-phase synchronous regimes , the period is given by (13), which also can be rewritten in parametric form as
All the branches of the antiphase synchronous regimes can be obtained by varying
To compare the theoretical predictions with the numerical results we plot the latter in the same figure. To obtain these results we numerically integrated system (5) and plotted all the ISIs observed after the transient. For each value of
Let us deeper study the emergent jittering regimes. The examples of them are depicted in Fig. 4. Note that each observed regime contains just two distinct ISIs, the short and the long one. Such regimes were previously named “bipartite” and were shown to play crucial role in the development of the multi-jitter instability for one oscillator with feedback [18, 19]. The two ISIs may form various sequences giving rise to multiple jittering regimes.
Examples of jittering regimes of two oscillators: (a) antiphase jittering at
Figure 4(a) shows the in-phase jittering regime observed at
We have shown that jittering regimes can emerge in a system of two pulse oscillators with mutual delayed coupling. In the case of no frequency detuning, such the system has two basic regular regimes: in-phase and antiphase synchronous spiking. Multiple branches of these two regimes exist in various intervals of the delays. If the PRC of the oscillators has slope = -1 at certain phases, regular regimes may destabilize and give rise to jittering regimes with distinct ISIs. The bifurcation points correspond to the situations when the input pulses hit the oscillators at these critical phase.
As one may see, there is a lot in common between the scenarios of development of the multi-jitter instability in a pair of coupled oscillators and in one oscillator with feedback. First of all, the condition for the bifurcation is the same: the slope of the PRC at the phase at which the oscillators receive pulses must be equal -1. Secondly, the emergent jittering regimes are bipartite, and the corresponding bifurcation diagram has a typical form with multiple loops (Fig. 3). This similarity is not surprising in the case of in-phase jittering, when the both oscillators behave as one and their dynamics is described by map (2), the same as in the case of one oscillator with feedback. However, the behavior of the oscillators in the antiphase regime is quite different and described by map (12). Curiously enough this map also demonstrates multi-jitter bifurcation.
The similarity of the properties of different systems allows to suppose that multi-jitter bifurcation may be an universal scenario of destabilization of regular regimes of networks with delayed pulse coupling. Examination of this hypothesis and definition of the conditions for the multi-jitter bifurcation in networks with various configurations should be a direction of the further study.