Noise-induced macroscopic oscillations in a network of synaptically coupled quadratic integrate-and-fire neurons

Ratas, Irmantas; Pyragas, K.

doi:10.1103/physreve.100.052211

Cited by 43 publications

(32 citation statements)

References 46 publications

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“…Nonetheless, similarly to the case of linear variables, where for non-Gaussian cases one can benefit from corrections owned by a finite number of higher-order cumulants, closures with more than one circular cumulant are useful. We also show that in some physical systems the macroscopic variables driving collective dynamics (e.g., neuron firing rate [28,[31][32][33][34][35]) can depend on the Kuramoto-Daido order parameters in such a way that a careless representation of these variables in terms of circular cumulants can be always diverging. Further, we discuss how the issue of approximations with a finite number of circular cumulants should be handled in a regular way in the cases where the cumulants form a decaying geometric progression.…”

Section: Introductionmentioning

confidence: 89%

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Ott-Antonsen ansatz truncation of a circular cumulant series

Goldobin

Dolmatova

2019

Phys. Rev. Research

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The cumulant representation is common in classical statistical physics for variables on the real line and the issue of closures of cumulant expansions is well elaborated. The case of phase variables significantly differs from the case of linear ones; the relevant order parameters are the Kuramoto-Daido ones but not the conventional moments. One can formally introduce 'circular' cumulants for Kuramoto-Daido order parameters, similar to the conventional cumulants for moments. The circular cumulant expansions allow to advance beyond the Ott-Antonsen theory and consider populations of real oscillators. First, we show that truncation of circular cumulant expansions, except for the Ott-Antonsen case, is forbidden. Second, we compare this situation to the case of the Gaussian distribution of a linear variable, where the second cumulant is nonzero and all the higher cumulants are zero, and elucidate why keeping up to the second cumulant is admissible for a linear variable, but forbidden for circular cumulants. Third, we discuss the implication of this truncation issue to populations of quadratic integrate-and-fire neurons [E. Montbrió, D. Pazó, A. Roxin, Phys. Rev. X 5, 021028 (2015)], where within the framework of macroscopic description, the firing rate diverges for any finite truncation of the cumulant series, and discuss how one should handle these situations. Fourth, we consider the cumulant-based low-dimensional reductions for macroscopic population dynamics in the context of this truncation issue. These reductions are applicable, where the cumulant series exponentially decay with the cumulant order, i.e., they form a geometric progression hierarchy. Fifth, we demonstrate the formation of this hierarchy for generic distributions on the circle and experimental data for coupled biological and electrochemical oscillators. Our main conclusion for applications is that, if the first and second circular cumulants are nonzero, there must be infinitely many nonzero higher cumulants. However, these higher cumulants can be small, in which case one can construct approximate equations for the dynamics of a finite number of leading cumulants.

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Section: Introductionmentioning

confidence: 89%

“…In [20,21], corrections owned by the second cumulant allowed to achieve accurate results where the OA ansatz was significantly inaccurate. The cumulant approach can be also applicable for a theoretical analysis of the non-OA situations, e.g., in [24][25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Ott-Antonsen ansatz truncation of a circular cumulant series

Goldobin

Dolmatova

2019

Phys. Rev. Research

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“…Low dimensional rate models able to capture the synchronization dynamics of spiking networks have been recently introduced [48,49], but they are usually limited to homogenous populations. MF formulations for heterogeneous networks subject to extrinsic noise sources have been examined in the context of the circular cumulants expansion [40,[49][50][51]. However, as noticed in [51], this expansion has the drawback that any finite truncation leads to a divergence of the population firing rate.…”

Section: (C) and (D))mentioning

confidence: 99%

“…This approach is based on the expansion of the characteristic function in terms of pseudo-cumulants , thus avoiding the divergences related to the expansion in conventional moments or cumulants. The implementation and benefits of this formulation are demonstrated for heterogenous populations of QIF neurons in presence of extrinsic and endogenous noise sources, where the conditions for a LD of the membrane potentials [21] are violated as in [44, 45]. In particular, we will derive a hierarchy of low-dimensional MF models for noisy globally coupled populations and deterministic sparse random networks, with a particular emphasis on the emergence of fluctuation driven collective oscillations (COs).…”

Section: Introductionmentioning

confidence: 99%

A reduction methodology for fluctuation driven population dynamics

Goldobin

Volo

Torcini

2021

Preprint

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Lorentzian distributions have been largely employed in statistical mechanics to obtain exact results for heterogeneous systems. Analytic continuation of these results is impossible even for slightly deformed Lorentzian distributions, due to the divergence of all the moments (cumulants). We have solved this problem by introducing a pseudo-cumulants’ expansion. This allows us to develop a reduction methodology for heterogeneous spiking neural networks subject to extrinsinc and endogenous noise sources, thus generalizing the mean-field formulation introduced in [E. Montbrió et al., Phys. Rev. X 5, 021028 (2015)].

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“…One of the characteristic properties of these mean-field models is the ability to exactly describing the network dynamics but not approximately reduce the network. Thus, the mean-field model derived by these methods can obtain the exact dynamical mechanism of the network models [25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a Large-Scale Spiking Neural Network with Quadratic Integrate-and-Fire Neurons

2021

Neural Plasticity

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Since the high dimension and complexity of the large-scale spiking neural network, it is difficult to research the network dynamics. In recent decades, the mean-field approximation has been a useful method to reduce the dimension of the network. In this study, we construct a large-scale spiking neural network with quadratic integrate-and-fire neurons and reduce it to a mean-field model to research the network dynamics. We find that the activity of the mean-field model is consistent with the network activity. Based on this agreement, a two-parameter bifurcation analysis is performed on the mean-field model to understand the network dynamics. The bifurcation scenario indicates that the network model has the quiescence state, the steady state with a relatively high firing rate, and the synchronization state which correspond to the stable node, stable focus, and stable limit cycle of the system, respectively. There exist several stable limit cycles with different periods, so we can observe the synchronization states with different periods. Additionally, the model shows bistability in some regions of the bifurcation diagram which suggests that two different activities coexist in the network. The mechanisms that how these states switch are also indicated by the bifurcation curves.

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Noise-induced macroscopic oscillations in a network of synaptically coupled quadratic integrate-and-fire neurons

Cited by 43 publications

References 46 publications

Ott-Antonsen ansatz truncation of a circular cumulant series

Ott-Antonsen ansatz truncation of a circular cumulant series

A reduction methodology for fluctuation driven population dynamics

Dynamics of a Large-Scale Spiking Neural Network with Quadratic Integrate-and-Fire Neurons

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