Synchronization Rates in Classes of Relaxation Oscillators

Abstract

Abstract-Relaxation oscillators arise frequently in physics, electronics, mathematics, and biology. Their mathematical definitions possess a high degree of flexibility in the sense that through appropriate parameter choices relaxation oscillators can be made to exhibit qualitatively different kinds of oscillations. We study numerically four different classes of relaxation oscillators through their synchronization rates in one-dimensional chains with a Heaviside step function interaction and obtain the following results. Relaxation oscillators in the sinusoidal and relaxation regime both exhibit an average time to synchrony, , where is the chain length. Relaxation oscillators in the singular limit exhibit , where is a numerically obtained value less than 0.5. Relaxation oscillators in the singular limit with parameters modified so that they resemble spike oscillations exhibit log( ) in chains and log( ) in two-dimensional square networks of length . Finally, using a sigmoid interaction results in 2 , for relaxation oscillators in the sinusoidal and relaxation regimes, indicating that the form of the coupling is a controlling factor in the synchronization rate.

How to Achieve Fast Entrainment? The Timescale to Synchronization

“…Therefore, similar features observed in our results might be expected in other synchronization scenarios (see Supporting Information S1). The time to synchronization for a network of oscillators has been studied for several systems using analytical approaches [36]–[40] and numerical simulations [41]–[43]. Most synchronization studies focused on specific model oscillators at the network level and derived scaling laws associated with the number of oscillators.…”

“…Most synchronization studies focused on specific model oscillators at the network level and derived scaling laws associated with the number of oscillators. In [41] and [43], the synchronization rate of different conductance based models (Hodgkin-Huxley type models) was studied. Both studies showed that, a spike-like oscillator reached a synchronized state much more rapidly than a sinusoidal oscillator.…”

A NAD(P) reductase like protein is the salicylic acid receptor in the appendix of theSauromatum guttatuminflorescence

“…The fast transitions are suitable for triggering synchronization of cell energy metabolism and time keeping. Fast modulation of the threshold of relaxation oscillators can lead to synchronization 51 , 52 . The switch between the 2 phases of RL can act as a biological clock in which switching every 4–5 min can produce a pulse.…”

Wave Mediated Synchronization of Nonuniform Oscillatory Media

“…(1). Recent studies concerning synchronization rate as a function of system size in arrays of relaxation oscillators [24] and the treatment of an oscillator chain driven by a pacemaker [1], which has been further developed for certain classes of coupling [25], are also relevant to our study. The effects of natural frequency dispersion are likely to play an important role in population dynamics [8] and in neural systems [26].…”