Separating intrinsic interactions from extrinsic correlations in a network of sensory neurons

Ulisse Ferrari, Stéphane Deny, Matthew Chalk, Gašper Tkačik, Olivier Marre, and Thierry Mora

Phys. Rev. E 98, 042410 – Published 17 October 2018

Abstract

Correlations in sensory neural networks have both extrinsic and intrinsic origins. Extrinsic or stimulus correlations arise from shared inputs to the network and, thus, depend strongly on the stimulus ensemble. Intrinsic or noise correlations reflect biophysical mechanisms of interactions between neurons, which are expected to be robust to changes in the stimulus ensemble. Despite the importance of this distinction for understanding how sensory networks encode information collectively, no method exists to reliably separate intrinsic interactions from extrinsic correlations in neural activity data, limiting our ability to build predictive models of the network response. In this paper we introduce a general strategy to infer population models of interacting neurons that collectively encode stimulus information. The key to disentangling intrinsic from extrinsic correlations is to infer the couplings between neurons separately from the encoding model and to combine the two using corrections calculated in a mean-field approximation. We demonstrate the effectiveness of this approach in retinal recordings. The same coupling network is inferred from responses to radically different stimulus ensembles, showing that these couplings indeed reflect stimulus-independent interactions between neurons. The inferred model predicts accurately the collective response of retinal ganglion cell populations as a function of the stimulus.

Received 22 January 2018
Revised 24 May 2018

DOI:https://doi.org/10.1103/PhysRevE.98.042410

Physics Subject Headings (PhySH)

Biological neural networks Neural encoding Neuroscience, neural computation & artificial intelligence

Ising model

Interdisciplinary PhysicsStatistical Physics & ThermodynamicsPhysics of Living Systems

Authors & Affiliations

Ulisse Ferrari^1,*, Stéphane Deny², Matthew Chalk¹, Gašper Tkačik³, Olivier Marre¹, and Thierry Mora⁴

¹Sorbonne Université, INSERM, CNRS, Institut de la Vision, 17 rue Moreau, 75012 Paris, France
²Neural Dynamics and Computation Laboratory, Stanford University, Stanford, California 94305, USA
³Institute of Science and Technology Austria, Klosterneuburg, Austria
⁴Laboratoire de physique statistique, CNRS, Sorbonne Université, Université Paris-Diderot and École normale supérieure (PSL), 24 rue Lhomond, 75005 Paris, France

^*ulisse.ferrari@gmail.com

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Vol. 98, Iss. 4 — October 2018

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Images

Figure 1
Ex vivo recording of a retinal ganglion cell shows short-time-scale noise correlations. (a) Top: Activity of one ganglion cell over repeated presentations of the same stimulus. Bottom: firing rate of the cell across time. (b) Receptive field mosaic of the isolated OFF ganglion cells. (c) Empirical cross-covariance between two example cells (green trace) superimposed on the covariance between their average firing rate (black trace). The difference, which is large only at short time scales, is the noise correlation. (d) Zero-lag noise correlation as a function of the cell pair distance. The black point refers to the pair of neurons plotted in C. The dashed red line shows an exponential fit with a spatial scale of 0.11 mm.
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Figure 2
Time-dependent inference provides good estimates of the interaction network. (a) Comparison between the inferred interaction from the first (later used as training) and the second (later used as testing) halves of the repeated data set. The black line is the identity. (b) Comparison of the predicted noise correlations when the interaction matrix $J$ learned using the training set is applied to the testing set. The black line is the identity. (c) The behavior with distance of the inferred interactions scales similarly to that of noise correlations [see Fig. 1], although it goes to at shorter distances. The dashed red line is an exponential fit with spatial constant 0.08 mm.
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Figure 3
Predictions of the conditionally independent feed-forward model. (a) For an example cell, the model-predicted firing rate (red trace) is superimposed on the empirical one (green trace). By computing the correlation between the two traces ( $ρ = 0.87$ ) we can estimate the model performance. (b) Scatterplot of the model performance vs the cell reliability, computed as the correlation between two halves of the repeated data set. The black point refers to the cell in panel A. (c) Model-predicted stimulus covariances account for more than $84 %$ of the stimulus covariance (slope of a linear fit) but systematically underestimate their empirical value. The black line is the identity.
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Figure 4
Population model predicts empirical noise covariances and performs as well as the conditional-independent model on stimulus-dependent quantities. (a) The performance in estimating the cell firing rate [same as Fig. 3] of the population model is equivalent to that of the conditionally independent model. (b) Both models also have the same performance in predicting the spike count variance. (c) The population model predicts empirical noise covariances when applied to a testing set (blue points), while the conditionally independent model predicts zero-noise covariances (red). (d) The population model predicts the spiking activity of neurons in response to yet unseen stimuli and given the activity of other cells better than the conditionally independent model, as measured by the log-likelihood difference between the two models averaged over all cells and time bins (conditioned on the number of observed spikes in the bin). In particular, including network effects becomes more important as cells have a larger spiking activity.
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Figure 5
Inferred couplings generalize from one stimulus ensemble to another. (a) Scatterplot of the inferred couplings from the response to the two-bar stimulation ( $x$ axis) vs those from the response to the checkerboard. (b) Population model prediction of empirical noise covariances in response to the two bars when the couplings are inferred from repeated checkerboard stimulation.
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Figure 6
(a) Total response covariances predicted by the coupled GLM model ( $y$ axis) versus response covariances measured from the data ( $x$ axis). Each point corresponds to a single neuron pair. (b) Noise covariances predicted by the coupled GLM model ( $y$ axis) versus noise covariances measured from the data ( $x$ axis). (c) Firing rate of a single example cell [green; also plotted in Fig. 2], alongside the predictions of the coupled (“population”; blue) and uncoupled (“conditionally independent”; red) GLM models. (d) Correlation coefficient between the firing rate predicted by the coupled ( $y$ axis) and that by the uncoupled ( $x$ axis) GLM models and data, for each cell. The cell shown in (c) is indicated in black.
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