How to infer distributions in the brain from subsampled observations

Inferring the dynamics of a system from observations is a challenge, even if one can observe all system units or components. The same task becomes even more challenging if one can sample only a small fraction of the units at a time. As a prominent example, spiking activity in the brain can be accessed only for a very small fraction of all neurons in parallel. These limitations do not affect our ability to infer single neuron properties, but it influences our understanding of the global network dynamics or connectivity: Subsampling can hamper inferring whether a system shows scale-free topology or scale-free dynamics (criticality) [1, 2]. Criticality is a dynamical state that maximizes information processing capacity in models, and therefore is a favorable candidate state for brain function. Experimental approaches to test for criticality extract spatio-temporal clusters of spiking activity, called avalanches, and test whether they followed power laws. Avalanches can propagate over the entire system, thus observations are strongly affected by subsampling. We developed a formal ansatz to infer avalanche distributions in the full system from spatial subsampling using both analytical and numerical approaches. In the mathematical model subsampling from exponential distribution does not change the class of distribution, but only its parameters. In contrast, power law distributions, despite their alias “scale-free”, do not manifest as power laws under subsampling [2]. We study changes in distributions to derive “subsampling scaling” that allows to extrapolate the results from subsampling to a full system: P(s) = psubPsub(s/ psub) where P(s) is the original distribution, Psub is the one under subsampling, and psub = N M is the probability to sample a unit, N— number of sampled units, M—system size. In the model with critical avalanches, subsampling scaling collapses distributions for any N (Fig. 47B). However, for subcritical models, no distribution collapse is observed (Fig. 47D). Thus we demonstrate that subsampling scaling allows to distinguish critical from non-critical systems. With the help of this novel method we studied dissociated cortical cultures. For these we artificially subsampled recordings by considering only fraction of all 60 electrodes. We find that in the first days subsampling scaling does not collapse distributions well, whereas mature cultures (~from day 21) allow for a good collapse, indicating development toward criticality (Fig. 47C, E)