Intrinsic noise and two-dimensional maps: Quasicycles, quasiperiodicity, and chaos

Parra‐Rojas, César; Challenger, Joseph D.; Fanelli, Duccio; McKane, Alan J.

doi:10.1103/physreve.90.032135

Cited by 9 publications

(17 citation statements)

References 37 publications

(103 reference statements)

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“…So, it must be clear that what is new here is not the stochastic oscillations (quasicycles) in biosystems, since there is a whole literature about that 28–34 . What is new here is the interaction of the SO with a critical point with an absorbing state.…”

Section: Discussionmentioning

confidence: 99%

“…For finite networks, finite size fluctuations (“demographic noise”) perturbs the almost unstable focus, fueling the SO. This kind of stochastic oscillation is known in the literature, sometimes called quasicycles 28–34 but, to produce them, one ordinarily needs to fine tune the system close to the bifurcation point. In contrast, for aSOC systems, there is a self-organization dynamics that tunes the system very close to the critical point 9,14,15,20,25,26 .…”

Section: Introductionmentioning

confidence: 99%

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Stochastic oscillations and dragon king avalanches in self-organized quasi-critical systems

Kinouchi

Brochini

Costa

et al. 2019

Sci Rep

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In the last decade, several models with network adaptive mechanisms (link deletion-creation, dynamic synapses, dynamic gains) have been proposed as examples of self-organized criticality (SOC) to explain neuronal avalanches. However, all these systems present stochastic oscillations hovering around the critical region that are incompatible with standard SOC. Here we make a linear stability analysis of the mean field fixed points of two self-organized quasi-critical systems: a fully connected network of discrete time stochastic spiking neurons with firing rate adaptation produced by dynamic neuronal gains and an excitable cellular automata with depressing synapses. We find that the fixed point corresponds to a stable focus that loses stability at criticality. We argue that when this focus is close to become indifferent, demographic noise can elicit stochastic oscillations that frequently fall into the absorbing state. This mechanism interrupts the oscillations, producing both power law avalanches and dragon king events, which appear as bands of synchronized firings in raster plots. Our approach differs from standard SOC models in that it predicts the coexistence of these different types of neuronal activity.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stochastic oscillations and dragon king avalanches in self-organized quasi-critical systems

Kinouchi

Brochini

Costa

et al. 2019

Sci Rep

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“…The method of random replacement employs the stochastic difference equation used previously [5][6][7], which is summarized in Eqs. (18) and (19).…”

Section: The Truncated and Censored Noise Distributionsmentioning

confidence: 99%

Suppressing escape events in maps of the unit interval with demographic noise

et al. 2016

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We explore the properties of discrete-time stochastic processes with a bounded state space, whose deterministic limit is given by a map of the unit interval. We find that, in the mesoscopic description of the system, the large jumps between successive iterates of the process can result in probability leaking out of the unit interval, despite the fact that the noise is multiplicative and vanishes at the boundaries. By including higher-order terms in the mesoscopic expansion, we are able to capture the non-Gaussian nature of the noise distribution near the boundaries, but this does not preclude the possibility of a trajectory leaving the interval. We propose a number of prescriptions for treating these escape events, and we compare the results with those obtained for the metastable behavior of the microscopic model, where escape events are not possible. We find that, rather than truncating the noise distribution, censoring this distribution to prevent escape events leads to results which are more consistent with the microscopic model. The addition of higher moments to the noise distribution does not increase the accuracy of the final results, and it can be replaced by the simpler Gaussian noise.

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“…Most of the existing methods to detect stochastic oscillations are based on the use of the power spectral density and the autocorrelation of the time series [ 30 , 31 ]. Power spectra are, for instance, used in [ 32 ] to explore the relationship between noisy cycles and quasi-cycles, and in [ 33 ] to assess oscillations of quasi-cycles in discrete-time models. On the other hand, the autocorrelation function, together with marginal distributions of population sizes, is used in [ 34 ] to distinguish between noisy cycles and quasi-cycles, and in [ 35 ] to quantify the effect of noise on a periodic signal.…”

Section: Introductionmentioning

confidence: 99%

A straightforward method to compute average stochastic oscillations from data samples

Júlvez

2015

BMC Bioinformatics

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BackgroundMany biological systems exhibit sustained stochastic oscillations in their steady state. Assessing these oscillations is usually a challenging task due to the potential variability of the amplitude and frequency of the oscillations over time. As a result of this variability, when several stochastic replications are averaged, the oscillations are flattened and can be overlooked. This can easily lead to the erroneous conclusion that the system reaches a constant steady state.ResultsThis paper proposes a straightforward method to detect and asses stochastic oscillations. The basis of the method is in the use of polar coordinates for systems with two species, and cylindrical coordinates for systems with more than two species. By slightly modifying these coordinate systems, it is possible to compute the total angular distance run by the system and the average Euclidean distance to a reference point. This allows us to compute confidence intervals, both for the average angular speed and for the distance to a reference point, from a set of replications.ConclusionsThe use of polar (or cylindrical) coordinates provides a new perspective of the system dynamics. The mean trajectory that can be obtained by averaging the usual cartesian coordinates of the samples informs about the trajectory of the center of mass of the replications. In contrast to such a mean cartesian trajectory, the mean polar trajectory can be used to compute the average circular motion of those replications, and therefore, can yield evidence about sustained steady state oscillations. Both, the coordinate transformation and the computation of confidence intervals, can be carried out efficiently. This results in an efficient method to evaluate stochastic oscillations.

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Intrinsic noise and two-dimensional maps: Quasicycles, quasiperiodicity, and chaos

Cited by 9 publications

References 37 publications

Stochastic oscillations and dragon king avalanches in self-organized quasi-critical systems

Stochastic oscillations and dragon king avalanches in self-organized quasi-critical systems

Suppressing escape events in maps of the unit interval with demographic noise

A straightforward method to compute average stochastic oscillations from data samples

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