Globally coupled stochastic two-state oscillators: fluctuations due to finite numbers

Symbolic dynamics of joint brain states during dyadic coordination

We propose a novel approach to investigate the brain mechanisms that support coordination of behavior between individuals. Brain states in single individuals defined by the patterns of functional connectivity between brain regions are used to create joint symbolic representations of brain states in two or more individuals to investigate symbolic dynamics that are related to interactive behaviors. We apply this approach to electroencephalographic data from pairs of subjects engaged in two different modes of finger-tapping coordination tasks (synchronization and syncopation) under different interaction conditions (uncoupled, leader-follower, and mutual) to explore the neural mechanisms of multi-person motor coordination. Our results reveal that dyads exhibit mostly the same joint symbols in different interaction conditions-the most important differences are reflected in the symbolic dynamics. Recurrence analysis shows that interaction influences the dwell time in specific joint symbols and the structure of joint symbol sequences (motif length). In synchronization, increasing feedback promotes stability with longer dwell times and motif length. In syncopation, leader-follower interactions enhance stability (increase dwell time and motif length), but mutual interaction dramatically reduces stability. Network analysis reveals distinct topological changes with task and feedback. In synchronization, stronger coupling stabilizes a few states, preserving a core-periphery structure of the joint brain states while in syncopation we observe a more distributed flow amongst a larger set of joint brain states. This study introduces symbolic representations of metastable joint brain states and associated analytic tools that have the potential to expand our understanding of brain dynamics in human interaction and coordination.

Self-organized spiral patterns at the edge of an order-disorder nonequilibrium phase transition

We present a spatially extended version of the Wood-Van den Broeck-Kawai-Lindenberg stochastic phase-coupled oscillator model. Our model is embedded in two-dimensional (2d) array with a range-dependent interaction. The Wood-Van den Broeck-Kawai-Lindenberg model is known to present a phase transition from a disordered state to a globally oscillatory phase in which the majority of the units are in the same discrete phase. Here we address a parameter combination in which such global oscillations are not present. We explore the role of the interaction range from a nearest neighbor coupling in which a disordered phase is observed and the global coupling in which the population concentrate in a single phase. We find that for intermediate interaction range the system presents spiral wave patterns that are strongly influenced by the initial conditions and can spontaneously emerge from the stochastic nature of the model. Our results present a spatial oscillatory pattern not observed previously in the Wood-Van den Broeck-Kawai-Lindenberg model and are corroborated by a spatially extended mean-field calculation.

Synchronization and fluctuations: Coupling a finite number of stochastic units

It is well established that ensembles of globally coupled stochastic oscillators may exhibit a nonequilibrium phase transition to synchronization in the thermodynamic limit (infinite number of elements). In fact, since the early work of Kuramoto, mean-field theory has been used to analyze this transition. In contrast, work that directly deals with finite arrays is relatively scarce in the context of synchronization. And yet it is worth noting that finite-number effects should be seriously taken into account since, in general, the limits N→∞ (where N is the number of units) and t→∞ (where t is time) do not commute. Mean-field theory implements the particular choice first N→∞ and then t→∞. Here we analyze an ensemble of three-state coupled stochastic units, which has been widely studied in the thermodynamic limit. We formally address the finite-N problem by deducing a Fokker-Planck equation that describes the system. We compute the steady-state solution of this Fokker-Planck equation (that is, finite N but t→∞). We use this steady state to analyze the synchronic properties of the system in the framework of the different order parameters that have been proposed in the literature to study nonequilibrium transitions.

Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic Markov processes

Single-cell experiments show that gene expression is stochastic and bursty, a feature that can emerge from slow switching between promoter states with different activities. In addition to slow chromatin and/or DNA looping dynamics, one source of long-lived promoter states is the slow binding and unbinding kinetics of transcription factors to promoters, i.e. the non-adiabatic binding regime. Here, we introduce a simple analytical framework, known as a piecewise deterministic Markov process (PDMP), that accurately describes the stochastic dynamics of gene expression in the non-adiabatic regime. We illustrate the utility of the PDMP on a non-trivial dynamical system by analysing the properties of a titration-based oscillator in the non-adiabatic limit. We first show how to transform the underlying chemical master equation into a PDMP where the slow transitions between promoter states are stochastic, but whose rates depend upon the faster deterministic dynamics of the transcription factors regulated by these promoters. We show that the PDMP accurately describes the observed periods of stochastic cycles in activator and repressor-based titration oscillators. We then generalize our PDMP analysis to more complicated versions of titration-based oscillators to explain how multiple binding sites lengthen the period and improve coherence. Last, we show how noise-induced oscillation previously observed in a titration-based oscillator arises from non-adiabatic and discrete binding events at the promoter site.

A continuous-time persistent random walk model for flocking

A classical random walker is characterized by a random position and velocity. This sort of random walk was originally proposed by Einstein to model Brownian motion and to demonstrate the existence of atoms and molecules. Such a walker represents an inanimate particle driven by environmental fluctuations. On the other hand, there are many examples of so-called "persistent random walkers," including self-propelled particles that are able to move with almost constant speed while randomly changing their direction of motion. Examples include living entities (ranging from flagellated unicellular organisms to complex animals such as birds and fish), as well as synthetic materials. Here we discuss such persistent non-interacting random walkers as a model for active particles. We also present a model that includes interactions among particles, leading to a transition to flocking, that is, to a net flux where the majority of the particles move in the same direction. Moreover, the model exhibits secondary transitions that lead to clustering and more complex spatially structured states of flocking. We analyze all these transitions in terms of bifurcations using a number of mean field strategies (all to all interaction and advection-reaction equations for the spatially structured states), and compare these results with direct numerical simulations of ensembles of these interacting active particles.

Stochastic Kuramoto oscillators with discrete phase states

We present a generalization of the Kuramoto phase oscillator model in which phases advance in discrete phase increments through Poisson processes, rendering both intrinsic oscillations and coupling inherently stochastic. We study the effects of phase discretization on the synchronization and precision properties of the coupled system both analytically and numerically. Remarkably, many key observables such as the steady-state synchrony and the quality of oscillations show distinct extrema while converging to the classical Kuramoto model in the limit of a continuous phase. The phase-discretized model provides a general framework for coupled oscillations in a Markov chain setting.

Arrays of two-state stochastic oscillators: Roles of tail and range of interactions

We study the role of the tail and the range of interaction in a spatially structured population of two-state on-off units governed by Markovian transition rates. The coupling among the oscillators is evidenced by the dependence of the transition rates of each unit on the states of the units to which it is coupled. Tuning the tail or range of the interactions, we observe a transition from an ordered global state (long-range interactions) to a disordered one (short-range interactions). Depending on the interaction kernel, the transition may be smooth (second order) or abrupt (first order). We analyze the transient, which may present different routes to the steady state with vastly different time scales.

Synchronization of coupled noisy oscillators: Coarse graining from continuous to discrete phases

The theoretical description of synchronization phenomena often relies on coupled units of continuous time noisy Markov chains with a small number of states in each unit. It is frequently assumed, either explicitly or implicitly, that coupled discrete-state noisy Markov units can be used to model mathematically more complex coupled noisy continuous phase oscillators. In this work we explore conditions that justify this assumption by coarse graining continuous phase units. In particular, we determine the minimum number of states necessary to justify this correspondence for Kuramoto-like oscillators.

Phase transitions and entropies for synchronizing oscillators

We study a generic model of coupled oscillators. In the model there is competition between phase synchronization and diffusive effects. For a model with a finite number of states we derive how a phase transition occurs when the coupling parameter is varied. The phase transition is characterized by a symmetry breaking and a discontinuity in the first derivative of the order parameter. We quantitatively account for how the synchronized pulse is a low-entropy structure that facilitates the production of more entropy by the system as a whole. For a model with many states we apply a continuum approximation and derive a potential Burgers' equation for a propagating pulse. No phase transition occurs in that case. However, positive entropy production by diffusive effects still exceeds negative entropy production by the shock formation.

Reaching Consensus by Allowing Moments of Indecision

Group decision-making processes often turn into a drawn out and costly battle between two opposing subgroups. Using analytical arguments based on a master equation description of the opinion dynamics occurring in a three-state model of cooperatively interacting units, we show how the capability of a social group to reach consensus can be enhanced when there is an intermediate state for indecisive individuals to pass through. The time spent in the intermediate state must be relatively short compared to that of the two polar states in order to create the beneficial effect. Furthermore, the cooperation between individuals must not be too low, as the benefit to consensus is possible only when the cooperation level exceeds a specific threshold. We also discuss how zealots, agents that remain in one state forever, can affect the consensus among the rest of the population by counteracting the benefit of the intermediate state or making it virtually impossible for an opposition to form.

Synchronization and phase redistribution in self-replicating populations of coupled oscillators and excitable elements

We study the dynamics of phase synchronization in growing populations of discrete phase oscillatory systems when the division process is coupled to the distribution of oscillator phases. Using mean-field theory, linear-stability analysis, and numerical simulations, we demonstrate that coupling between population growth and synchrony can lead to a wide range of dynamical behavior, including extinction of synchronized oscillations, the emergence of asynchronous states with unequal state (phase) distributions, bistability between oscillatory and asynchronous states or between two asynchronous states, a switch between continuous (supercritical) and discontinuous (subcritical) transitions, and modulation of the frequency of bulk oscillations.

Arrays of stochastic oscillators: Nonlocal coupling, clustering, and wave formation

We consider an array of units each of which can be in one of three states. Unidirectional transitions between these states are governed by Markovian rate processes. The interactions between units occur through a dependence of the transition rates of a unit on the states of the units with which it interacts. This coupling is nonlocal, that is, it is neither an all-to-all interaction (referred to as global coupling), nor is it a nearest neighbor interaction (referred to as local coupling). The coupling is chosen so as to disfavor the crowding of interacting units in the same state. As a result, there is no global synchronization. Instead, the resultant spatiotemporal configuration is one of clusters that move at a constant speed and that can be interpreted as traveling waves. We develop a mean field theory to describe the cluster formation and analyze this model analytically. The predictions of the model are compared favorably with the results obtained by direct numerical simulations.