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要旨:
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).
