Lattice models of pulse-coupled oscillators
Albert Diaz-Guilera
One of the central points of our research has been the study of biological rhythms, together with other dynamical aspects related to the cooperative behavior of extended populations. Particularly, our analysis has been emphasized on the study of the mechanisms that are behind phenomena like synchronization of the activity of these populations as well as the formation of spatio-temporal structures. A characteristic example observed in nature is a population of fireflies.
To carry out these analysis we consider populations of oscillators, coupled through a nonlinear interaction, in low-dimensionality lattices. Two different types of model have to be distinguished:
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•Pulse-coupled oscillators: This type of systems is characteristic of excitable media. Neurons subjected to a constant stimulus or heart pacemaker cells are known examples of oscillators that interact through electrical pulses emitted when some kind of threshold is reached. We have carried out theoretical analysis as well as numerical simulations. Related to other lines of research (complex networks) we have also studied the effect of a random connectivity on the synchronization properties of the system. Mathematical description of Peskin model
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•Kuramoto model In this case one considers systems that interact in a continuous way. This is a good description when the evolution of the elements that build the system describe limit cycles of constant amplitude. We have analyzed the stationary properties of the model by using a dynamic formalism of Focker-Planck type. Mathematical description of Kuramoto model
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