  Power laws arise in a variety of phenomena ranging from matter undergoing
phase transition to the distribution of word frequencies in the English
language. Usually, their presence is only apparent when data is abundant, and
accurately determining their exponents often requires even larger amounts of
data. As the scale of recordings in neuroscience becomes larger, an increasing
number of studies attempt to characterise potential power-law relationships in
neural data. In this paper, we aim to discuss the potential pitfalls that one
faces in such efforts and to promote a Bayesian interpolation framework for
this purpose. We apply this framework to synthetic data and to data from a
recent study of large-scale recordings in mouse primary visual cortex (V1),
where the exponent of a power-law scaling in the data played an important role:
its value was argued to determine whether the population's stimulus-response
relationship is smooth, and experimental data was provided to confirm that this
is indeed so. Our analysis shows that with such data types and sizes as we
consider here, the best-fit values found for the parameters of the power law
and the uncertainty for these estimates are heavily dependent on the noise
model assumed for the estimation, the range of the data chosen, and (with all
other things being equal) the particular recordings. It is thus challenging to
offer a reliable statement about the exponents of the power law. Our analysis,
however, shows that this does not affect the conclusions regarding the
smoothness of the population response to low-dimensional stimuli but casts
doubt on those to natural images. We discuss the implications of this result
for the neural code in the V1 and offer the approach discussed here as a
framework that future studies, perhaps exploring larger ranges of data, can
employ as their starting point to examine power-law scalings in neural data.
