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Systematic errors in connectivity inferred from activity in strongly recurrent networks


Understanding the mechanisms of neural computation and learning will require knowledge of the underlying circuitry. Because it is difficult to directly measure the wiring diagrams of neural circuits, there has long been an interest in estimating them algorithmically from multicell activity recordings. We show that even sophisticated methods, applied to unlimited data from every cell in the circuit, are biased toward inferring connections between unconnected but highly correlated neurons. This failure to ‘explain away’ connections occurs when there is a mismatch between the true network dynamics and the model used for inference, which is inevitable when modeling the real world. Thus, causal inference suffers when variables are highly correlated, and activity-based estimates of connectivity should be treated with special caution in strongly connected networks. Finally, performing inference on the activity of circuits pushed far out of equilibrium by a simple low-dimensional suppressive drive might ameliorate inference bias.

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Fig. 1: Structure and dynamics of the generative network.
Fig. 2: Quality of circuit inference using 108 spikes from a fully observed ring network, as a function of weight strength.
Fig. 3: Inference quality as a function of data volume.
Fig. 4: Results extend to different inference methods.
Fig. 5: Results extend to different networks.
Fig. 6: Circuit-to-activity map is inherently less invertible when correlations are strong.
Fig. 7: Inference bias due to unobserved neurons is exacerbated at strong weights.
Fig. 8: Sampling non-equilibrium data mitigates inference bias.

Data and code availability

The data and code that support the findings of this study are available from the corresponding author upon reasonable request.


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We thank R. Chaudhuri, B. Gerçek, I. Kanitscheider, T. Taillefumier, D. Schwab, M. Bethge and P. Dayan for helpful discussions. This project was funded in part by the ONR, the Howard Hughes Medical Institute through the Faculty Scholars Program and the Simons Collaboration on the Global Brain through the Simons Foundation.

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Correspondence to Ila R. Fiete.

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Extended data

Extended Data Fig. 1 Inference with matched models.

a, Using an Ising model for both generation and inference. Top: superposed inferred weights from each node to the rest (line marks zero). Bottom: pattern coherence, and inference error with different data volumes, against weight strength. b, Squared total, variance and bias errors against data volume at weak and strong weights. c, Using a generalized linear model with an exponential nonlinearity for both generation and inference (see Methods). Top: pattern coherence, and inference error with different data volumes, against weight strength. Bottom: Superposed inferred weights from each node to the rest.

Extended Data Fig. 2 Tuning firing thresholds of different generative networks to control the average network inter-spike-interval as recurrent weight strength is varied.

a, Firing thresholds to hold the average ISI of the ring network with local and non-local synapses, and the local ring network with rectifying LNP dynamics, at 16.0 ± 0.1 ms, and that of the random, fully-connected, symmetric network at 34.0 ± 0.1 ms. b, Firing rate coefficients α (see eq. (7)) to hold the average ISI of the ring network with GLM dynamics (logarithmic link function) at 16.0 ± 0.1 ms. c, Firing thresholds to hold the average ISI of the sparse, non-symmetric random network at 16.0 ± 0.1 ms.

Extended Data Fig. 3 Binning affects noise correlations and inference.

a, Noise correlations between neuron pairs (top: full matrix, bottom: superposed rows) for binned vs unbinned spikes from the ring network. The optimal bin-width groups causally related spikes together, and noise correlations at an intermediate r then reflect underlying weights. b, Inference error using inverse Ising with MPF on spike data binned at different widths. c, Fraction of spikes discarded when binarizing binned spike data for Inverse Ising inference.

Extended Data Fig. 4 Distribution of inference errors of individual weights in the ring circuit, at different recurrent weight strengths.

a, Histograms of the inference errors (relative to the length of the ground-truth weight vector). At weak weights, errors are random and normally distributed. As the weight increases, errors first shrink as noise weakens and SNR grows, then they become increasingly non-normal due to bias. b, Negentropy of the error distribution (see Methods) against weight strength.

Extended Data Fig. 5 Power-law decay of variance error of inferring the ring circuit, with increasing data volume.

a, The fitted exponents α of \({\Delta }_{v}^{2} \sim {D}^{\alpha }\) when using a generalized linear model for inference. Error-bands are 95% confidence intervals using 20 data points. The theoretical exponent is -1. b, The exponent α of the decay of total inference error Δ2 ~ Dα when using the Ising model for both data generation and inference. Error-bands are 95% confidence intervals using 20 data points. Here inference error is almost entirely due to variance, thus decays as the power law.

Extended Data Fig. 6 Circuit inference using logistic regression is not improved by l1 regularization.

a, Example ring network weight profiles inferred using logistic regression with zero, optimal and excessive regularization penalties. When weights are weak, regularization reduces some noise and marginally improves inference. At high weights, regularization suppresses both the spurious off-diagonal stripes and the true coupling shape, so is not helpful. b, Inference error vs weight strength using logistic regression with and without l1 regularization. c, Optimal l1 penalties (that produce the lowest inference errors) at each weight. Regularization improves inference in the strong and weak weight regimes, but barely.

Extended Data Fig. 7 Circuit inference using neural CCGs.

(a-d) Inferring the strong weight (r = 0.025) ring circuit using short-lag peaks in neural CCGs. a, Pearson’s cross-correlation of a connected and an unconnected neuron pair. b, Left: matrix of absolute lags of the CCG peaks. This partly reveals the circuit: directly connected neurons exhibit short lags. Right: binary matrix connecting neuron pairs with short lags. c, Top: CCG-based weight matrix: weight is set to the peak cross-correlation if the neurons are connected (lag < τ), zero otherwise. Bottom: avg. weight profile. GLM fares better. d, Combining CCG and GLM. Top: matrix of GLM-inferred weights if neuron pairs have short CCG lag, zero otherwise. Bottom: avg. weight profile. Relative to pure GLM (Δ = 0.23), this method (Δ = 0.32) removes some biases but introduces others. (e-j) Spike CCGs from the sparse, non-symmetric, strong weight (rRSA = 0.1) random network. e, Unconnected pair. CCG has no sharp features around 0, indicating no direct connection. f, Unconnected pair. Broad symmetric peak, indicating multiple indirect influences through mutual connections, is discounted. g, Connected pair. Sharp asymmetric dip at 0 reveals direct (inhibitory) connection. h, Unconnected pair. CCG is indistinguishable from the previous, and passes the criterion. i, Connected pair, but no CCG features. j, Connected pair, but broad symmetric peak is discounted.

Extended Data Fig. 8 Results of inference using a GLM on data generated by a linear-nonlinear-Poisson model with a rectifying linear response (see Methods).

a, Pattern coherence against weight strength for the generative LNP network. b, Inferred weight matrices (top) and superposition of rows (bottom, line marks zero), at several weight strengths. c, Inference error and bias fraction against weight strength. Optimal inference is at the point of pattern onset.

Extended Data Fig. 9 Activity and inference in the random balanced networks that are fully or sparsely connected.

a, Top: waterfall plots of neural fields of the full network at weak weights (when activity decays) and strong weights (when activity is chaotic), in response to a brief uniform feed-forward pulse. Bottom: corresponding inferences. (b-d) Inference on the sparse network. b, Left: true and inferred weights (using logistic regression) for the network with no noisy drive, at rRSB = 0.6. Some zero weights are inferred to be non-zero, while some nonzero weights are underestimated. Right: true and inferred weights (using only logistic regression, and augmented with CCG information) for the same condition, but when the network is noise-driven. c, Inference error (using the two methods) vs recurrent weight strength, and data volume, on the noise-driven network. d, Example CCG (top) and its time-derivative (bottom) of a connected neuron pair in the noise-driven network.

Extended Data Fig. 10 Entropy of spiking activity states of neural circuits.

a, Entropy of the distributions of 22-neuron spike sub-states from the true local ring circuit W and the non-local circuit \({\bf{W}}^{\prime}\). b, Entropies of the spike sub-states of the two circuits computed with different data fractions across weight strengths. At all weights, the computed entropies converge as the data approaches the total volume. c: Example slice of plot b at the weakest weights, where entropy convergence takes the longest.

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Das, A., Fiete, I.R. Systematic errors in connectivity inferred from activity in strongly recurrent networks. Nat Neurosci 23, 1286–1296 (2020).

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