Renormalized time scale for anticipating and lagging synchronization

Abstract

We consider fluctuations of the dissipated energy in nonlinear driven diffusive systems subject to bulk dissipation and boundary driving. With this aim, we extend the recently-introduced macroscopic fluctuation theory to nonlinear driven dissipative media, starting from the fluctuating hydrodynamic equations describing the system mesoscopic evolution. Interestingly, the action associated to a path in mesoscopic phase-space, from which large-deviation functions for macroscopic observables can be derived, has the same simple form as in non-dissipative systems. This is a consequence of the quasi-elasticity of microscopic dynamics, required in order to have a nontrivial competition between diffusion and dissipation at the mesoscale. Euler-Lagrange equations for the optimal density and current fields that sustain an arbitrary dissipation fluctuation are also derived. A perturbative solution thereof shows that the probability distribution of small fluctuations is always gaussian, as expected from the central limit theorem. On the other hand, strong separation from the gaussian behavior is observed for large fluctuations, with a distribution which shows no negative branch, thus violating the Gallavotti-Cohen fluctuation theorem as expected from the irreversibility of the dynamics. The dissipation large-deviation function exhibits simple and general scaling forms for weakly and strongly dissipative systems, with large fluctuations favored in the former case but heavily supressed in the latter. We apply our results to a general class of diffusive lattice models for which dissipation, nonlinear diffusion and driving are the key ingredients. The theoretical predictions are compared to extensive numerical simulations of the microscopic models, and excellent agreement is found. Interestingly, the large-deviation function is in some cases non-convex beyond some dissipation. These results show that a suitable generalization of macroscopic fluctuation theory is capable of describing in detail the fluctuating behavior of nonlinear driven dissipative media.

Anticipated and zero-lag synchronization in motifs of delay-coupled systems

“…In the presence of transmission delays, anticipated synchronization [9][10][11][12][13][14] is another remarkable collective behaviour that is observed in monkeys performing visual discrimination tasks. 6,7 Multi-electrode recordings showed a dominant directional influence between areas of sensorimotor cortex (as assessed by Granger causality), accompanied by either a negative or a positive phase lag between them.…”

“…Indeed, an inspection shows that the spectrum of A þ B has only negative real part as long as 2K À K 0 > 0: (19) Hence, Eqs. (14) and (19) provide the conditions for anticipated synchronization in the small-d limit.…”

Inhibitory autapse mediates anticipated synchronization between coupled neurons

“…f (S) is a vector function that describes the autonomous dynamical system, K is the coupling matrix and the delayed term R(t−t d ) is the self-feedback [13]. The solution R(t) = S(t + t d ), characterizes the anticipated synchronization and has been verified in a variety of theoretical [13][14][15][16][17][18][19] and experimental [20][21][22] studies. The striking aspect of this solution is that in a time t the Receiver R predicts the state of the Sender S in a future time…”

Exploring the Phase-Locking Mechanisms Yielding Delayed and Anticipated Synchronization in Neuronal Circuits

“…Based on previous work on anticipated synchronization in the framework of Equation (1) (Pyragienè and Pyragas, 2013; Hayashi et al, 2016; Dima et al, 2018), we studied the effect that an inhibitory connection and the external current plays in determining the frequency of the receiver system when the sender and the receiver are uncoupled (the receiver free-running frequency ω R ). In fact, we find that both the inhibitory conductance and the external stimuli modify the receiver internal dynamics.…”

“…AS has also been reported in a chain consisting of a sender and two receivers with switching parameters (Pyragienè and Pyragas, 2015), between two Hodgkin-Huxley neurons with different depolarization parameters (Simonov et al, 2014) and in the presence of an inhibitory loop mediated by an interneuron with a free-running frequency greater than the others (Matias et al, 2017). It has also been shown that AS may appear between two neuron models directly coupled provided that the mean frequency of the free receiver is greater than the mean frequency of the sender with (Hayashi et al, 2016) and without the explicit time-delay (Pyragienè and Pyragas, 2013; Dima et al, 2018). AS has been verified in a system in which the delayed feedback has been replaced by a simple, low-order all-pass filter (Pyragiene and Pyragas, 2017).…”