Detecting the temporal structure of intermittent phase locking

Abstract

This study explores a method to characterize temporal structure of intermittent phase locking in oscillatory systems. When an oscillatory system is in a weakly synchronized regime away from a synchronization threshold, it spends most of the time in parts of its phase space away from synchronization state. Therefore characteristics of dynamics near this state (such as its stability properties/Lyapunov exponents or distributions of the durations of synchronized episodes) do not describe system’s dynamics for most of the time. We consider an approach to characterize the system dynamics in this case, by exploring the relationship between the phases on each cycle of oscillations. If some overall level of phase locking is present, one can quantify when and for how long phase locking is lost, and how the system returns back to the phase-locked state. We consider several examples to illustrate this approach: coupled skewed tent maps, which stability can be evaluated analytically, coupled Rössler and Lorenz oscillators, undergoing through different intermittencies on the way to phase synchronization, and a more complex example of coupled neurons. We show that the obtained measures can describe the differences in the dynamics and temporal structure of synchronization/desynchronization events for the systems with similar overall level of phase locking and similar stability of synchronized state.

Synchronized Beta-Band Oscillations in a Model of the Globus Pallidus-Subthalamic Nucleus Network under External Input

“…Synchronous dynamics at rest is very intermittent in both basal ganglia (Park et al, 2010; Ratnadurai-Giridharan et al, 2016) and cortex (Ahn and Rubchinsky, 2013). Thus, matching synchrony patterns in the model and experiment is an appropriate comparison tool, as was discussed in earlier studies (Ahn et al, 2011; Park et al, 2011; Rubchinsky et al, 2014). It ensures some similarity between large areas of the phase space of the model and real systems.…”

“…This analysis includes not only analysis of the average strength of the beta-band synchrony (presumably associated with the severity of hypokinetic symptoms), but also analysis of temporal patterns of synchrony. From a dynamical systems' perspective, matching synchrony patterns in the model and real data helps to match the phase spaces of the model and real system (see Park et al, 2011; Dovzhenok et al, 2013 for a discussion of this issue and Ahn et al, 2011; Rubchinsky et al, 2014 for a more theoretical perspective).…”

“…These phases were used to study temporal patterns of synchronization via the first-return maps. The details of the first-return map construction are available in Park et al (2010), Ahn et al (2011), and Rubchinsky et al (2014) and we describe the key steps below.…”

“…We define the transition rates for transitions between the aforementioned four regions of the map as the number of data points leaving a region, divided by the total number of data points in that region (see Park et al, 2010; Ahn et al, 2011). The rates of transitions between these regions were used to characterize the dynamics of the phase return map in earlier experimental (Park et al, 2010) and modeling studies (Park et al, 2011).…”

Analytical expression for zero Lyapunov exponent of chaotic noised oscillators

“…in physical or physiological systems. Indeed, many physiological and physical systems are known to demonstrate the phase synchronization regimes [57][58][59][60][61][62][63][64] or the pre-transitional behavior [32,35,[65][66][67][68]. The found value of zero Lyapunov exponent may be used in these tasks, e.g., as the quantity characterizing the degree of the synchronization [69] to reveal the particularities of the system dynamics depending on the control parameter values.…”