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Conceived and designed the experiments: IY YO YJ HN KK. Performed the experiments: IY YO HN KK. Wrote the paper: IY HN KK.

The authors have declared that no competing interests exist.

Time delay is known to induce sustained oscillations in many biological systems such as electroencephalogram (EEG) activities and gene regulations. Furthermore, interactions among delay-induced oscillations can generate complex collective rhythms, which play important functional roles. However, due to their intrinsic infinite dimensionality, theoretical analysis of interacting delay-induced oscillations has been limited. Here, we show that the two primary methods for finite-dimensional limit cycles, namely, the center manifold reduction in the vicinity of the Hopf bifurcation and the phase reduction for weak interactions, can successfully be applied to interacting infinite-dimensional delay-induced oscillations. We systematically derive the complex Ginzburg-Landau equation and the phase equation without delay for general interaction networks. Based on the reduced low-dimensional equations, we demonstrate that diffusive (linearly attractive) coupling between a pair of delay-induced oscillations can exhibit nontrivial amplitude death and multimodal phase locking. Our analysis provides unique insights into experimentally observed EEG activities such as sudden transitions among different phase-locked states and occurrence of epileptic seizures.

In many natural systems, time delay can induce spontaneous breaking of continuous time translational symmetry and lead to self-sustained oscillations

In this study, we develop systematic reduction methods for coupled delay-induced oscillations. In analyzing dynamical models of biological systems, reduction methods are known to be considerably useful in facilitating mathematical treatments

In the present paper, we give a systematic derivation of the reduced dynamical equations for coupled delay-induced oscillations. In particular, we analyze a cortico-thalamic model

It has been argued that slow EEG rhythms are generated by mutual influence between the cortex and the thalamus with delays in transmission of electrical activities

Kim and Robinson's cortico-thalamic model of EEG oscillations was originally defined in spatially extended media

In the absence of mutual interaction, each local area can exhibit stable delay-induced limit-cycle oscillations, i.e., it behaves as a self-sustained oscillator. We focus on the case that such oscillators are symmetrically and diffusively coupled with each other through the

Though each individual oscillator has only a single variable,

The cortico-thalamic model Eq. (1) is described by coupled infinite-dimensional functional differential equations

Let us first consider a single oscillator and analyze the linear stability of its stationary state. We omit the oscillator index

Left panel shows stable and unstable regions of a cortico-thalamic model without coupling, Eq. (1). The solid bifurcation curve is obtained from Eq. (3). Points A, B, and C represent the parameter sets used for the analysis [ A:

Delay differential equations such as Eq. (1) should be considered as functional differential equations. The state of the system at time

Furthermore, if we focus on the situation that the natural frequencies are narrowly distributed around

The phase reduction method, which describes a limit-cycle oscillator using only its phase variable by eliminating the amplitude degrees of freedom, can be adopted even if the oscillator is far from the bifurcation point. The necessary condition is that the oscillator has a stable limit cycle and is only weakly perturbed by the coupling. Although this method is usually applied to a finite-dimensional limit cycle, we can extend it to a limit-cycle solution of delay differential equations by appropriately defining the phase.

We first define an asymptotic phase

For sufficiently weak perturbations, the PRC is proportional to the amplitude of the perturbation, so that the linear response coefficient, which we call

When we consider two-oscillator systems with symmetric coupling [

To confirm the validity of the reduced equations derived by the center manifold and the phase reduction methods, three comparisons were carried out around the point A in

First, the phase sensitivity function

(a) Phase sensitivity functions

Second, the phase coupling functions

Third, critical values of the coupling strength for phase synchronization were compared near the point A for values obtained by (i) direct numerical simulations of the original delay differential equations (1), (ii) numerical simulations of the coupled amplitude equations (4) obtained by the center manifold reduction, and (iii) analytical calculations from the asymmetric part of the phase coupling function. We prepared two oscillators exhibiting equal-amplitude limit cycles with a small frequency mismatch and introduced weak mutual coupling only through the

These results indicate that both the center manifold reduction and the phase reduction are appropriately accomplished near the point A in

We can also use the reduced phase equations to analyze collective behavior of a population of mutually interacting delay-induced oscillations. In the

It is known that relatively strong mutual coupling between limit cycles can induce amplitude death, a phenomenon which means disappearance of oscillations due to the stabilization of rest states

Let us consider a coupled pair of oscillators whose parameters are at the points A and B in

(a) Time series of delay-induced oscillations at the parameter sets A and B in

The condition for amplitude death can be derived from linear stability analysis of the fixed point of Eq. (4),

The center manifold reduction can also be used to analyze the amplitude death effect in a population of coupled delay-induced oscillations that are distributed around the point A in

The most outstanding property of delayed dynamical systems is that even a simple equation can exhibit complex behaviors due to the systems' infinite dimensionality

We consider a coupled pair of two identical oscillators at the point C in

(a) Limit cycle oscillation at the parameter set C in

The phase sensitivity function

In order to confirm the above prediction based on the estimated phase coupling function, we performed direct numerical simulations of the original model Eq. (1) with fixed coupling strength

In this study, we applied center manifold and phase reduction theories to a system of interacting delay-induced oscillations. We could successfully reduce the dynamics of the system to low-dimensional coupled ordinary differential equations without delay, namely, the coupled amplitude equations (5) and the coupled phase equations (6). Generally, analytical treatments of dynamical systems with delays had been restricted due to their infinite-dimensional nature. We demonstrated that the two principal reduction methods can provide analytical predictions for interacting delay-induced oscillations. We successfully derived the network version of the complex Ginzburg-Landau equation and the phase equation known in physics and applied mathematics and whose dynamics can be studied in detail. We then showed that the effect of external perturbations and couplings are well described by reduced equations that illustrate validity of the reduced equations. Furthermore, synchronization properties of identical and nonidentical oscillators were appropriately evaluated by the complex Ginzburg-Landau equation near the bifurcation point as well as by the phase equation for weak coupling regimes (

In the cortico-thalamic model that we have analyzed, the variable

Our results on the amplitude death phenomenon suggest that interactions between delay-induced oscillations with different frequencies may affect the oscillations' stabilities and modulate their amplitudes. In the case of EEG, oscillations with various frequencies ranging from a few Hz (

Regarding the phase reduction approach, the PRCs of delay differential equations have been measured in several studies, e.g., for models of circadian rhythms

The result of multimodal phase-locking implies that signals in different regions of the brain may easily change their phase relationships with others, even without transmission delays in the mutual coupling. For example, Roelfsema's experiment

It is widely known that real neuronal networks have delays in synaptic and axonal transmission, and mathematical models with delayed coupling have been extensively analyzed

For example, we may consider arrays of neuronal populations with local interactions between neighbors using the same reduction methods. The reduced equations would typically exhibit wave propagation phenomena, which may explain some features of the EEG dynamics observed in the brain. Moreover, taking into account detailed neuronal network structures as well as transmission delays within a population

There are many biological systems in which self-sustained oscillations arise from delayed feedback and mutually interact with other oscillations. Therefore, we believe that our analysis will lead to a deeper understanding of various biological oscillations with delay, such as EEG dynamics (e.g., epileptic seizure and information processing between cortical regions), blood pressure regulation, and somite segmentations in vertebrate embryos.

To carry out center manifold reduction of Eq. (1) in the vicinity of the Hopf bifurcation, we assume that all oscillators are close to the bifurcation point and the deviations of their bifurcation parameters from the critical value

We can further bring the above equation into the complex Ginzburg-Landau equation when the differences in

The phase sensitivity function

We prepare two oscillators with the same amplitude but whose frequencies are slightly different, and we then calculate the critical coupling strength for the phase synchronization. It is generally not easy to find appropriate values for parameters of the delay differential equation that satisfy such a condition, but in the present case, it is easily achieved using the results of the center manifold reduction ((Eqs. (3),( 5), and ( 16)). We choose the distances of the two oscillators from the respective bifurcation points as

The critical values of the coupling strength for the synchronization are obtained by three methods, (i) direct numerical simulations of the original delay differential equations (1), (ii) numerical simulations of the coupled amplitude equations (4) obtained by the center manifold reduction, and (iii) analytical calculations using the asymmetric part of the phase coupling function,

The condition for amplitude death can be derived from linear stability analysis of the fixed point of Eq. (4),

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