A shotgun sampling solution for the common input problem in neural connectivity inference

Inferring connectivity in neuronal networks remains a key challenge in statistical neuroscience. The “common input” problem presents the major roadblock: it is difficult to reliably distinguish causal connections between pairs of observed neurons from correlations induced by common input from unobserved neurons. Since available recording techniques allow us to sample from only a small fraction of large networks simultaneously with sufficient temporal resolution, naive connectivity estimators that neglect these common input effects are highly biased. This work proposes a “shotgun” experimental design, in which we observe multiple subnetworks briefly, in a serial manner. We develop Bayesian methods based on generalized linear models (GLM) and “spike-and-slab” sparse priors to perform network inference given this type of data, and demonstrate in simulation that this shotgun experimental design can in fact eliminate the biases induced by common input effects. We find that observing as many different pairs of neurons as possible over the course of the serial experiment leads to the best inference results. We further develop methods for handling data with lower temporal resolution (as would be available via e.g. calcium fluorescence imaging). Finally, we develop approximate methods for performing inference in very large networks. Advances in large-scale multi-neuronal recordings have made it possible to study the simultaneous activity of large ensembles of neurons. Experimentalists now routinely record from hundreds, or even thousands, of neurons simultaneously in a wide range of preparations. These new multineuronal recording techniques in principle provide the opportunity to discern the functional architecture of neuronal networks. As has been discussed at length elsewhere, the ability to accurately estimate large, possibly time-varying neural connectivity diagrams would open up an exciting new range of fundamental research questions in systems and computational neuroscience. Therefore, connectivity estimation can be considered one of the central problems in statistical neuroscience. Perhaps the biggest challenge for inferring neural connectivity from functional data — and indeed in network analysis more generally — is the presence of hidden nodes which are not observed directly [1, 2, 3]. Despite swift progress in simultaneously recording activity in massive populations of neurons, it is still beyond the reach of current technology to monitor a complete network of spiking neurons at high temporal resolution (though see e.g. [4] for some impressive recent progress). Since estimation of functional connectivity relies on the analysis of the inputs to target neurons in relation to their observed spiking activity, the inability to monitor all inputs can result in persistent errors in the connectivity estimation due to model misspecification. More specifically, “common input” errors — in which correlations due to shared inputs from unobserved neurons are mistaken for direct, causal connections — plague most naive approaches to connectivity estimation. Developing ∗This work was supported by an NSF CAREER grant, a McKnight Scholar award, and by the U.S. Army Research Laboratory and the U. S. Army Research Office under contract number W911NF-12-1-0594. We thank D. Soudry for helpful conversations. 1 ar X iv :1 30 9. 37 24 v1 [ qbi o. N C ] 1 5 Se p 20 13 a robust approach for incorporating the latent effects of such unobserved neurons remains an area of active research in connectivity analysis [1, 2, 3]. In this paper we propose an experimental design which greatly ameliorates these common-input problems. The idea is simple: if we cannot observe all neurons in a network simultaneously, maybe we can instead observe many overlapping subnetworks in a serial manner over the course of a long experiment, and then use statistical techniques to patch the full estimated network back together1. (The analogy we have in mind is to “shotgun” genetic sequencing [6].) While it would be challenging to purposefully sample from many distinct but overlapping subnetworks using multi-electrode recording arrays, imaging methods (such as calcium fluorescence imaging [7, 8]) make this approach highly experimentally feasible. The remainder of this paper focuses on the development of statistical methods for inferring connectivity given this type of “shotgun” data. A Bayesian approach is natural, due to the massive degree of missingness in the data (we assume that only a small fraction of the network is observed at any single time) and since a great deal of prior information about neuronal connectivity can be exploited here. We begin by describing the basic statistical model and how to sample from the associated posterior distributions. We then provide simulated results demonstrating that the shotgun experimental design can largely eliminate the biases induced by common input effects, as desired. We close by discussing approximate inference methods that scale to the case of very large networks, where standard Bayesian inference methods become infeasible. Generalized linear model We use a rather standard discrete-time generalized linear network model (GLM) [9, 10, 11]: p(ηi,t = 1) = f θT i Xt + N ∑ j=1 Wi,jηj,t−1 + bi  (1) where ηi,t is a binary variable indicating whether the i-th neuron spiked at time t, f is a logistic function, Xt is a stimulus given to all the neurons at time t, θi is the spatio-temporal filter of neuron i, bi is a scalar offset, andW is the connectivity matrix we wish to infer. The diagonal elementsWi,i of the connectivity matrix are typically negative, corresponding to the post-spike filter accounting for the cell’s own (refractory) post-spike effects, while the off-diagonal terms Wi,j represent the connection weights from neuron j to neuron i. To simplify notation we have only included history effects with a single time lag; generalizing to multi-lag history effects is straightforward.