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

Abstract

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...

Self-Organized Criticality in the Brain

“…oscillations and scale-invariant avalanche statistics (e.g., [88]), while others have added an oscillating extinction rate to a continuous-time branching process using perturbative field theory [89]. Coherence-potential-like activity and its potential computational advantages have been explored by the Gong group [90][91][92].…”

Hopf Bifurcation in Mean Field Explains Critical Avalanches in Excitation-Inhibition Balanced Neuronal Networks: A Mechanism for Multiscale Variability

“…They focused on the analysis of population activity without considering the irregular spiking and E-I balanced input of individual units. Oscillation component can also be included in physical model of branching process (Pausch et al, 2020), while being away from the criticality in branching process means either explosion or being completely in salience, which is not biologically desirable. Thus, the biological reality of our model with synaptic filtering kinetics, E-I balance and irregular spiking time in individual neurons is well beyond these previous studies.…”

Noise can lead to exponential epidemic spreading despite $R_0$ below one

“…A branching process [1,2] is a stochastic process in which individuals randomly create copies of themselves or become extinct. This model has a wide range of applications and individuals might represent offspring [3][4][5], particles [6][7][8], active neurons [9][10][11][12][13], or infected individuals [9,[14][15][16] among others [17]. At the centre of many investigations of branching processes is the statistics of spells of activity, which are called avalanches [6][7][8][9][10][11][12][13] in most applications or outbreaks in the context of infectious diseases.…”

“…This model has a wide range of applications and individuals might represent offspring [3][4][5], particles [6][7][8], active neurons [9][10][11][12][13], or infected individuals [9,[14][15][16] among others [17]. At the centre of many investigations of branching processes is the statistics of spells of activity, which are called avalanches [6][7][8][9][10][11][12][13] in most applications or outbreaks in the context of infectious diseases. Avalanches, or outbreaks, are defined as the activity in the branching process which is initiated by a single individual and lasts until all subsequent individuals have become extinct.…”

“…More explicitly in a branching process, an existing individual waits until a branching event occurs. In many models, the waiting time between branching events is fixed [1,23,24], however in the present work, we consider only exponentially distributed waiting times [8,11,25]. At a branching event, an individual is replaced by its offspring, which is a random number K ∈ N 0 of individuals.…”