Maximum likelihood estimators for truncated and censored power-law distributions show how neuronal avalanches may be misevaluated

Abstract

The coordination of activity amongst populations of neurons in the brain is critical to cognition and behavior. One form of coordinated activity that has been widely studied in recent years is the so-called neuronal avalanche, whereby ongoing bursts of activity follow a power-law distribution. Avalanches that follow a power law are not unique to neuroscience, but arise in a broad range of natural systems, including earthquakes, magnetic fields, biological extinctions, fluid dynamics, and superconductors. Here, we show that common techniques that estimate this distribution fail to take into account important characteristics of the data and may lead to a sizable misestimation of the slope of power laws. We develop an alternative series of maximum likelihood estimators for discrete, continuous, bounded, and censored data. Using numerical simulations, we show that these estimators lead to accurate evaluations of power-law distributions, improving on common approaches. Next, we apply these estimators to recordings of in vitro rat neocortical activity. We show that different estimators lead to marked discrepancies in the evaluation of power-law distributions. These results call into question a broad range of findings that may misestimate the slope of power laws by failing to take into account key aspects of the observed data.

The avalanche-like behaviour of large-scale haemodynamic activity from wakefulness to deep sleep

“…slow waves of activity present in reduced preparations [24], spacing between units in electrode arrays [67] and finite size and/or volume conduction effects in EEG data [34]. Besides finite size effects [68], we can expect departures from theoretical predictions since in fMRI data the characteristic times of signal transmission are related to the unusual hydrodynamics of the vascular system, and not directly related to the sequences of neural spikes. We must note, however, that scaling parameters α ≈ 2 were reported in previous work addressing the statistics of observables related to sleep physiology (e.g.…”

Irregular spiking of pyramidal neurons organizes as scale-invariant neuronal avalanches in the awake state

“…Similarly, the upper cut-off in avalanche size distributions has sometimes been included into statistical fits. However, this upper cut-off arises from finite-size effects and needs to be disregarded for fitting ( Yu et al, 2014 ), otherwise, statistical tests ( Langlois et al, 2014 ) can be misdirected to fit the cut-off only ( Clauset et al, 2009 ; Dehghani et al, 2012 ).…”

Criticality in the brain: A synthesis of neurobiology, models and cognition

“…This is a flawed assumption for most experimental data, which inevitably derive from a finite number of sensors, and may bias model selection. Recent work has revisited this assumption, developing methods that test the likelihood of a power law with a simple cut-off (Langlois et al, 2014;Shew et al, 2015). Second, as mentioned above, the range of many data tested for power law scaling often span less than two orders of magnitude, yielding data that is particularly sparse in the right-hand tail (precisely where power law scaling is most clearly expressed).…”