Scalable Inference for Neuronal Connectivity from Calcium Imaging
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.
Summary:The authors consider the problem of neural connectivity inference from calcium imaging observations. Their model follows the approach of [7] (Mischenko et. al 2011), a network of spiking neurons coupled together through a matrix of weights that is to be inferred. The important difference is that the spiking neurons are modeled as leaky integrate-and-fire (LIF) neurons instead of generalized linear model. Similarly to [7], they propose an EM approach where at the E step the hidden variables (spikes, membrane potential, calcium concentration) are estimated and the M step is used to estimate the connectivity matrix W and other parameters. The main contribution of the paper is the efficient implementation of the E step using a generalized approximate message passing algorithm (GAMP), which scales linearly both with the number of timesteps and the number of neurons. They also present a sparse probit regression initialization approach for W, that arises naturally from the structure and dynamics of the LIF neurons. The algorithm is applied on synthetic data and the authors claim to achieve performance at least similar to that of previous methods (e.g. [7]) but with significantly less computation.Comments- Why are all the nonzero connectivity weights excitatory? The authors cite simplicity but it looks to me that allowing negative weights shouldn't change anything. Why isn't this the case?- The authors do not address the "common input" problem, according to which connectivity inference is limited from the fact that only subsets of the whole network can be observed at any given point in time. It seems to me that the methods proposed here could be connected with the approach presented in [24] (Keshri et al. 2013) to address this issue. It would be nice to see a discussion about this point, since a potential application of the GAMP algorithm to the case of unobserved units could bring even more substantial computational gains, and address the major limitation of [24].QualityThe method presented in the paper is technically sound and is based of the GAMP algorithm that has been very successful in the efficient statistical estimation of sparse/structured signals. ClarityThe paper is at times dense but mostly clearly written, with a lot of useful information presented in the appendix. There are several typos (see below).Originality & SignificanceThe approach of the authors is novel and very likely to be of use in future work since it establishes an efficient for approaching this problem. Since the dimensionality of the connectivity problem can be extremely high, computationally efficient methods will be of greater importance. Even though all the tools that are presented here have been presented before, the application is original since the authors consider a LIF based setup that allows them to apply the GAMP algorithm in a non-trivial way.Typos:line 21: "methods" possibly missing after "Bayesian inference"line 107: The second instance of \tilde{v}_i^k should be without the \tilde?line 107: The bias term b_{IF,i} should probably be moved outside the definition of q_i^k since q is subsequently expressed as q = Ws.line 109: eq (2) v_i^k < \mu --->  \tilde{v}_i^k < \muline 148: \tau_{CA} and \tau{y} in eq. (4) is the same noise variance?line 183: v^k_{T-1} should be changed to v^k_{N}line 312: add "to" before a global maximaline 404: smaller than the… ??