Field-theoretic approach to the universality of branching processes

Garcia-Millan, Rosalba; Pausch, Johannes; Walter, Benjamin; Pruessner, Gunnar

doi:10.1103/physreve.98.062107

Cited by 19 publications

(74 citation statements)

References 34 publications

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“…Only to zeroth order in A, at A = 0 , do the factorial moments become time-homogeneous and reduce to those calculated in Ref. 16 , g…”

Section: Second Momentsupporting

confidence: 49%

“…g n (t 0 , t) = g (0) n (t 0 , t) + Ag (1) www.nature.com/scientificreports/ The n-th factorial moment at A = 0 , given by was calculated in closed form from the diagrams as a matter of combinatorics 16 . The function g…”

Section: Second Momentmentioning

confidence: 99%

“…3a,b. For A = 0 , the maximum of Cov(N(t 1 ), N(t 2 )) occurs at t 1 = t 2 = t and equals (2q 2 /r + 1)(exp −rt − exp −2rt) 16 . Furthermore, its maximum rises with increasing t, provided that rt < log(2) .…”

Section: Covariance the Autocorrelation Functionmentioning

confidence: 99%

“…The first term, which is independent of A, is the probability of survival of the critical branching process with constant extinction rate 16 (Fig. 4).…”

Section: Survival Probabilitymentioning

confidence: 99%

“…(19), it can be expected that the avalanche duration distribution P T is affected by the oscillations. Considering that the avalanche duration is equal to the time of death T of an avalanche that started at t = 0 , it can be derived from the survival probability P T (t) = − dP s (t) dt 16 :…”

Section: Survival Probabilitymentioning

confidence: 99%

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Time-dependent branching processes: a model of oscillating neuronal avalanches

Pausch

Garcia-Millan

Pruessner

2020

Sci Rep

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Recently, neuronal avalanches have been observed to display oscillations, a phenomenon regarded as the coexistence of a scale-free behaviour (the avalanches close to criticality) and scale-dependent dynamics (the oscillations). ordinary continuous-time branching processes with constant extinction and branching rates are commonly used as models of neuronal activity, yet they lack any such timedependence. In the present work, we extend a basic branching process by allowing the extinction rate to oscillate in time as a new model to describe cortical dynamics. By means of a perturbative field theory, we derive relevant observables in closed form. We support our findings by quantitative comparison to numerics and qualitative comparison to available experimental results. In the brain, electrical signals propagate between neurons of the cortical network through action potentials, which are spikes of polarisation in the membrane of the neuron's axon. These spikes have an amplitude of about 100 mV, typically last about 1 ms 1 and can be recorded using micro-electrodes 2,3. In order to study the signaling in larger regions of neurons, multielectrode arrays, comprising about 60 electrodes spread across ≈ 4 mm 2 , are used to capture the collective occurrence of spikes as local field potentials (LFPs). In this setting, the electrodes are extracellular and each is sensitive to electrical signals from several surrounding neurons 4-8. The data of the LFP recordings are processed by putting them into time bins of a few milliseconds duration and by introducing a voltage threshold. In addition, a refractory period is imposed to avoid counting large voltage excursions more than once. The details of processing can differ slightly between experiments 5-9. However, after processing, the data is a time series of two values for each electrode: on (detected signal above threshold) and off (no detected signal or signal below threshold). A neuronal avalanche is then defined as a set of uninterrupted signals detected across the micro-electrode array. Each avalanche is both preceded and succeeded by at least one time bin where none of the electrodes detected a signal, defining the avalanche duration as the number of time bins where the avalanche unfolds. Which and how many electrodes detect a signal varies during the avalanche 5,6. The duration of avalanches typically ranges between a few milliseconds and 30 ms 10. A prominent observable is the avalanche size, which is the total number of recorded signals during the avalanche. If there is only one electrode detecting a signal in each time bin of an avalanche, its size and duration are equal. However, the size of an avalanche is usually larger than its duration due to the simultaneous detection of signals by different electrodes. Histograms of the avalanche size show an apparent power-law distribution of sizes, with common small avalanches and rare large avalanches 5,6 , the fingerprint of scale-free phenomena. The exponent of the power law was observed to be −3/2 in Ref. 5. However, its exac...

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“…Only to zeroth order in A, at A = 0 , do the factorial moments become time-homogeneous and reduce to those calculated in Ref. 16 , g…”

Section: Second Momentsupporting

confidence: 49%

Section: Second Momentmentioning

confidence: 99%

Section: Covariance the Autocorrelation Functionmentioning

confidence: 99%

“…The first term, which is independent of A, is the probability of survival of the critical branching process with constant extinction rate 16 (Fig. 4).…”

Section: Survival Probabilitymentioning

confidence: 99%

Section: Survival Probabilitymentioning

confidence: 99%

See 3 more Smart Citations

Time-dependent branching processes: a model of oscillating neuronal avalanches

Pausch

Garcia-Millan

Pruessner

2020

Sci Rep

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Some harmonic functions for killed Markov branching processes with immigration and culling

Vidmar

2021

Stochastics

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For a continuous-time Bienaymé-Galton-Watson process, X, with immigration and culling, 0 as an absorbing state, call X q the process that results from killing X at rate q ∈ (0, ∞), followed by stopping it on extinction or explosion. Then an explicit identification of the relevant harmonic functions of X q allows to determine the Laplace transforms (at argument q) of the first passage times downwards and of the explosion time for X. Strictly speaking, this is accomplished only when the killing rate q is sufficiently large (but always when the branching mechanism is not supercritical or if there is no culling). In particular, taking the limit q ↓ 0 (whenever possible) yields the passage downwards and explosion probabilities for X. A number of other consequences of these results are presented.

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The concealed voter model is in the voter model universality class

Garcia-Millan¹

2020

J. Stat. Mech.

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The Concealed Voter Model (CVM) is an extension of the original Voter Model, where two different opinions compete until consensus is reached. In the CVM, agents express an opinion not only publicly, they also hold a private opinion, which they may disclose or change. In this paper I derive the critical exponents of both models via renormalised field theory and show that both belong to the compact directed percolation universality class.

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Field-theoretic approach to the universality of branching processes

Cited by 19 publications

References 34 publications

Time-dependent branching processes: a model of oscillating neuronal avalanches

Time-dependent branching processes: a model of oscillating neuronal avalanches

Some harmonic functions for killed Markov branching processes with immigration and culling

The concealed voter model is in the voter model universality class

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