Undersampled Critical Branching Processes on Small-World and Random Networks Fail to Reproduce the Statistics of Spike Avalanches

The power-law size distributions obtained experimentally for neuronal avalanches are an important evidence of criticality in the brain. This evidence is supported by the fact that a critical branching process exhibits the same exponent t~3=2. Models at criticality have been employed to mimic avalanche propagation and explain the statistics observed experimentally. However, a crucial aspect of neuronal recordings has been almost completely neglected in the models: undersampling. While in a typical multielectrode array hundreds of neurons are recorded, in the same area of neuronal tissue tens of thousands of neurons can be found. Here we investigate the consequences of undersampling in models with three different topologies (two-dimensional, small-world and random network) and three different dynamical regimes (subcritical, critical and supercritical). We found that undersampling modifies avalanche size distributions, extinguishing the power laws observed in critical systems. Distributions from subcritical systems are also modified, but the shape of the undersampled distributions is more similar to that of a fully sampled system. Undersampled supercritical systems can recover the general characteristics of the fully sampled version, provided that enough neurons are measured. Undersampling in two-dimensional and small-world networks leads to similar effects, while the random network is insensitive to sampling density due to the lack of a well-defined neighborhood. We conjecture that neuronal avalanches recorded from local field potentials avoid undersampling effects due to the nature of this signal, but the same does not hold for spike avalanches. We conclude that undersampled branching-process-like models in these topologies fail to reproduce the statistics of spike avalanches.

Work supported by Coordenac¸a˜o de Aperfeic¸oamento de Pessoal de Nı´vel Superior (CAPES), Financiadora de Estudos e Projetos (FINEP) grant 01.06.1092.00, Pro´-Reitoria de Po´s-Graduac¸a˜o da Universidade Federal do Rio Grande do Norte (UFRN), Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq)/Ministe´rio da Cieˆncia, Tecnologia e Inovac¸a˜o (MCTI), CNPq Universal Grants 481351/2011-6, 473554/2011-9 and 480053/2013-8, Programa de Apoio a Nu´cleos Emergentes PRONEM 003/2011 FAPERN/CNPq and PRONEM 12/2010 FACEPE/CNPq, Pew Latin American Fellows Program in the Biomedical Sciences, and Centro de Pesquisa, Inovac¸a˜o e Difusa˜o (CEPID-Neuromat). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.