File:Des Fig01.png

Desynchronization is a process inverse to synchronization, where initially synchronized oscillating systems desynchronize as parameters change or they do so under the influence of an external force or feedback. Desynchronization is important, for example, in neuroscience and medicine, were pathologically strong synchronization of neurons may severely impair brain function as, e.g., in Parkinson's disease or epilepsy.

This article briefly overviews several methods for the control of (de)synchronization in oscillatory networks.

Ensemble of coupled oscillators

Figure 1: Synchronization in the Kuramoto model (<ref>eq_Kuramoto</ref>): order parameter \(R\) versus coupling strength (bottom plot) and the corresponding distributions (snapshots) of the phases in the plane \((\cos(\psi_j), \sin(\psi_{j}))\).

The widely accepted model capturing fundamental properties of the collective dynamics of interacting oscillators like oscillatory neurons is the Kuramoto model of globally coupled phase oscillators (Kuramoto 1984, Strogatz 2000, Acebron, et al. 2005)

<math eq_Kuramoto>

\dot{\psi}_j = \omega_j +\frac{C}{N} \sum_{k=1}^N \sin(\psi_k-\psi_j), \quad j = 1, 2, \dots, N. </math> The phases \(\psi_{j}\) characterize the oscillatory dynamics of the elements of the ensemble and increase by \(2\pi\) after each completed cycle. For oscillating neurons, for instance, the cycle can be defined as the time period between two successive spikes or bursts.

For the inhomogeneous ensemble (<ref>eq_Kuramoto</ref>), i.e., if the natural frequencies \(\omega_j\) are different, the phase oscillators remain desynchronized and oscillate with different frequencies if the coupling among the oscillators (parameter \(C\)) is sufficiently weak. The desynchronization-synchronization transition takes place in system (<ref>eq_Kuramoto</ref>) if the coupling among the oscillators increases. In the limit \(N \to \infty\) and if the natural frequencies \(\omega_j\) are randomly chosen from a symmetric probability density \(g(\omega)\), \(g(\Omega+\omega)=g(\Omega-\omega)\), where \(\Omega\) is the mean frequency, the critical coupling of spontaneous synchronization is given by (Kuramoto 1984, Strogatz 2000)

<math eq_Kur-Thresh>

C_{cr} = \frac{2}{\pi g(\Omega)}. </math> For \( C < C_{cr} \), the system relaxes to an incoherent state, where all oscillators are not synchronized, but for \(C > C_{cr}\), mutual synchronization occurs in a group of oscillators. This transition can be characterized by values of the {order parameter} \(R(t)\) calculated as

<math eq_Order_Par>

R(t)\exp[i\Psi(t)] = \displaystyle{\frac{1}{N}} \sum_{j = 1}^{N}\exp[i \psi_j(t)], </math> where \(\Psi(t)\) is the mean phase. The state of in-phase synchronization, were all phases are close to each other \(\psi_{j} \approx \psi_{k}\), is characterized by large values of \(R \approx 1\) (Fig. <ref>fig_Kuramoto</ref>). For a desynchronized state, where the phases are uniformly distributed on the circle \((0, 2\pi)\), the order parameter is small, \(R \approx 0\), and scales as \(R \sim 1/\sqrt{N}\) with the number of elements \(N\) -- so-called finite-size effect (Pikovsky, et al. 2001).

The problem of desynchronization consists in designing a method which can effectively desynchronize an ensemble of strongly synchronized oscillators by delivering a stimulation signal \(S(t)\) to the oscillators. The efficacy and the main properties of several desynchronization methods will be illustrated below on the exemplary phase ensemble (<ref>eq_Kuramoto</ref>).