Collective behavior in populations of coupled stochastic oscillators

Abstract

Study objectives: Theoretical studies of coupled oscillators have potential implications for such diverse fields as physics, chemistry, mathematics, and biology. One crucial question about the collective behavior of such populations is whether or not coupling causes the individual oscillators to eventually come into phase (ie, whether the population synchronizes). Previous studies have shown that populations of diffusively coupled identical oscillators tend to synchronize. Other studies have shown that populations of coupled oscillators with a range of natural frequencies may or may not synchronize, depending on the strength of coupling. Less is known about this question for populations of coupled stochastic ("noisy") oscillators. A stochastic oscillator is characterized by random variation in the period and amplitude of oscillation. Because all biochemical systems are to some extent noisy, the behavior of such systems is relevant to biology. We examine under what conditions populations of diffusively coupled stochastic oscillators tend to synchronize. Methods: This was a theoretical study using computer simulation. Using an algorithm from the theory of stochastic dynamical systems, the behavior of populations of coupled stochastic oscillators was simulated. A model chemical oscillator known as the "Brusselator" was chosen for simulation. Each oscillator was coupled equally to all the others by a process akin to chemical diffusion. Two main sets of simulations were conducted. In the first, the strength of the coupling parameter was gradually increased from zero through 9 successive values to a maximum. All other system parameters were held constant. In the second set of simulations, the number of oscillators was set at 20, 30, 40, 50, 60, 70, 80, 90, and 100. All other system parameters were held constant, including strength of coupling. In each simulation, the average behavior of the population was graphed against time. For populations with no synchrony, the average behavior is expected to fluctuate irregularly around a baseline with no clear oscillatory behavior. For synchronized populations, the average is expected to clearly oscillate around a baseline. Results: As the coupling parameter is increased, there is a clear trend from no synchrony in the population to a high degree of synchrony. For higher values of the coupling parameter, the degree of synchrony increases less. As the number of oscillators in the population is increased, there is a clear trend from less synchrony to more synchrony. The degree of change in synchrony is less than that observed as the coupling parameter is varied. Conclusion: Increased coupling and increased population size tend to increase synchrony in populations of coupled stochastic oscillators. For the conditions examined, the effect of coupling is stronger than that of population size. Coupled biochemical oscillators may regularize and synchronize their behavior by stronger coupling or increased population size.