  Fluorescent calcium imaging provides a potentially powerful tool for
inferring connectivity in neural circuits with up to thousands of neurons.
However, a key challenge in using calcium imaging for connectivity detection is
that current systems often have a temporal response and frame rate that can be
orders of magnitude slower than the underlying neural spiking process. Bayesian
inference methods based on expectation-maximization (EM) have been proposed to
overcome these limitations, but are often computationally demanding since the
E-step in the EM procedure typically involves state estimation for a
high-dimensional nonlinear dynamical system. In this work, we propose a
computationally fast method for the state estimation based on a hybrid of loopy
belief propagation and approximate message passing (AMP). The key insight is
that a neural system as viewed through calcium imaging can be factorized into
simple scalar dynamical systems for each neuron with linear interconnections
between the neurons. Using the structure, the updates in the proposed hybrid
AMP methodology can be computed by a set of one-dimensional state estimation
procedures and linear transforms with the connectivity matrix. This yields a
computationally scalable method for inferring connectivity of large neural
circuits. Simulations of the method on realistic neural networks demonstrate
good accuracy with computation times that are potentially significantly faster
than current approaches based on Markov Chain Monte Carlo methods.
