  One of the primary goals of systems neuroscience is to relate the structure
of neural circuits to their function, yet patterns of connectivity are
difficult to establish when recording from large populations in behaving
organisms. Many previous approaches have attempted to estimate functional
connectivity between neurons using statistical modeling of observational data,
but these approaches rely heavily on parametric assumptions and are purely
correlational. Recently, however, holographic photostimulation techniques have
made it possible to precisely target selected ensembles of neurons, offering
the possibility of establishing direct causal links. Here, we propose a method
based on noisy group testing that drastically increases the efficiency of this
process in sparse networks. By stimulating small ensembles of neurons, we show
that it is possible to recover binarized network connectivity with a number of
tests that grows only logarithmically with population size under minimal
statistical assumptions. Moreover, we prove that our approach, which reduces to
an efficiently solvable convex optimization problem, can be related to
Variational Bayesian inference on the binary connection weights, and we derive
rigorous bounds on the posterior marginals. This allows us to extend our method
to the streaming setting, where continuously updated posteriors allow for
optional stopping, and we demonstrate the feasibility of inferring connectivity
for networks of up to tens of thousands of neurons online. Finally, we show how
our work can be theoretically linked to compressed sensing approaches, and
compare results for connectivity inference in different settings.
