Abstract
nferring 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 the 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. These avalanches can propagate over the entire system, thus
observations are strongly affected by subsampling. Therefore, we developed a formal
ansatz to infer avalanche distributions in the full system from subsampling using both
analytical approximation and numerical results.
In the mathematical model subsampling from exponential (or, more generally, negative
binomial distribution) does not change the class of distribution, but only its parameters.
In contrast, power law distributions, do not manifest as power laws under subsampling
[3]. We study changes in distributions to derive “subsampling scaling” that allows to
extrapolate the results from subsampling to a full system:
P
(
s
) =
p
sub
P
sub
(
s/p
sub
)
,
where
P
(
s
)
is an original distribution,
P
sub
– distribution in the subsampled system,
p
sub
probability to observe any particular event. In the model of critical avalanches
subsampling scaling collapses distributions for all number of sampled units (Figure 1. B).
However, for subcritical settings no distribution collapse is observed (Figure 1. D). With
the help of this novel discovery we studied dissociated cortical cultures. We artificially
subsampled recordings by considering only fraction of all electrodes. We observed that
in the first days subsampling scaling does not collapse distributions well, whereas mature
( from day 21) allow for a good collapse, indicating development toward criticality
(Figure 1. C, E) [4].
