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Register a new userSynchronization, Diversity, and Topology of Networks of Integrate and Fire Oscillators
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Albert Diaz-Guilera
Publication date
2000
fields
Physics
and research's language is
English
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We study synchronization dynamics of a population of pulse-coupled oscillators. In particular, we focus our attention in the interplay between networks topological disorder and its synchronization features. Firstly, we analyze synchronization time $T$ in random networks, and find a scaling law which relates $T$ to networks connectivity. Then, we carry on comparing synchronization time for several other topological configurations, characterized by a different degree of randomness. The analysis shows that regular lattices perform better than any other disordered network. The fact can be understood by considering the variability in the number of links between two adjacent neighbors. This phenomenon is equivalent to have a non-random topology with a distribution of interactions and it can be removed by an adequate local normalization of the couplings.
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We study the dynamics of networks with coupling delay, from which the connectivity changes over time. The synchronization properties are shown to depend on the interplay of three time scales: the internal time scale of the dynamics, the coupling delay along the network links and time scale at which the topology changes. Concentrating on a linearized model, we develop an analytical theory for the stability of a synchronized solution. In two limit cases the system can be reduced to an effective topology: In the fast switching approximation, when the network fluctuations are much faster than the internal time scale and the coupling delay, the effective network topology is the arithmetic mean over the different topologies. In the slow network limit, when the network fluctuation time scale is equal to the coupling delay, the effective adjacency matrix is the geometric mean over the adjacency matrices of the different topologies. In the intermediate regime the system shows a sensitive dependence on the ratio of time scales, and specific topologies, reproduced as well by numerical simulations. Our results are shown to describe the synchronization properties of fluctuating networks of delay-coupled chaotic maps.
The effects of nonlocal and reflecting connectivity are investigated in coupled Leaky Integrate-and-Fire (LIF) elements, which assimilate the exchange of electrical signals between neurons. Earlier investigations have demonstrated that non-local and hierarchical network connectivity often induces complex synchronization patterns and chimera states in systems of coupled oscillators. In the LIF system we show that if the elements are non-locally linked with positive diffusive coupling in a ring architecture the system splits into a number of alternating domains. Half of these domains contain elements, whose potential stays near the threshold, while they are interrupted by active domains, where the elements perform regular LIF oscillations. The active domains move around the ring with constant velocity, depending on the system parameters. The idea of introducing reflecting non-local coupling in LIF networks originates from signal exchange between neurons residing in the two hemispheres in the brain. We show evidence that this connectivity induces novel complex spatial and temporal structures: for relatively extensive ranges of parameter values the system splits in two coexisting domains, one domain where all elements stay near-threshold and one where incoherent states develop with multileveled mean phase velocity distribution.
The human cortex is never at rest but in a state of sparse and noisy neural activity that can be detected at broadly diverse resolution scales. It has been conjectured that such a state is best described as a critical dynamical process -- whose nature is still not fully understood -- where scale-free avalanches of activity emerge at the edge of a phase transition. In particular, some works suggest that this is most likely a synchronization transition, separating synchronous from asynchronous phases. Here, by investigating a simplified model of coupled excitable oscillators describing the cortex dynamics at a mesoscopic level, we investigate the possible nature of such a synchronization phase transition. Within our modeling approach, we conclude that -- in order to reproduce all key empirical observations, such as scale-free avalanches and bistability, on which fundamental functional advantages rely -- the transition to collective oscillatory behavior needs to be of an unconventional hybrid type, with mixed features of type-I and type-II excitability, opening the possibility for a particularly rich dynamical repertoire.
A lattice of three-state stochastic phase-coupled oscillators introduced by Wood it et al. exhibits a phase transition at a critical value of the coupling parameter $a$, leading to stable global oscillations (GO). On a complete graph, upon further increase in $a$, the model exhibits an infinite-period (IP) phase transition, at which collective oscillations cease and discrete rotational ($C_3$) symmetry is broken. In the case of large negative values of the coupling, Escaff et al. discovered the stability of travelling-wave states with no global synchronization but with local order. Here, we verify the IP phase in systems with long-range coupling but of lower connectivity than a complete graph and show that even for large positive coupling, the system sometimes fails to reach global order. The ensuing travelling-wave state appears to be a metastable configuration whose birth and decay (into the previously described phases) are associated with the initial conditions and fluctuations.
We investigate the stability of synchronized states in delay-coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics. Using a master stability approach, we find that the master stability function shows a discrete rotational symmetry depending on the number of groups. The coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum, which helps to simplify the evaluation of the master stability function. Our theory allows for the characterization of stability of different patterns of synchronized dynamics in networks with multiple delay times, multiple coupling functions, but also with multiple kinds of local dynamics in the networks nodes. We illustrate our results by calculating stability in the example of delay-coupled semiconductor lasers and in a model for neuronal spiking dynamics.
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