Recovering Sparse Neural Connectivity from Partial Measurements: A Covariance-Based Approach with Granger-Causality Refinement

Abstract

Inferring the connectivity of neural circuits from incomplete observations is a fundamental challenge in neuroscience. We present a covariance-based method for estimating the weight matrix of a recurrent neural network from sparse, partial measurements across multiple recording sessions. By accumulating pairwise covariance estimates across sessions where different subsets of neurons are observed, we reconstruct the full connectivity matrix without requiring simultaneous recording of all neurons. A Granger-causality refinement step enforces biological constraints via projected gradient descent. Through systematic experiments on synthetic networks modeling small brain circuits, we characterize a fundamental control-estimation tradeoff: stimulation aids identifiability but disrupts intrinsic dynamics, with the optimal level depending on measurement density. We discover that the ``incorrect'' linear approximation acts as implicit regularization -- outperforming the oracle estimator with known nonlinearity at all operating regimes -- and provide an exact characterization via the Stein--Price identity.

Figures (8)

• Figure 1: Problem setup: partial measurement across recording sessions.(Left) A network of $N=12$ neurons with directed weighted connectivity $W$. In Session 1, a subset of neurons (green circles) is measured while the rest (gray) are unobserved. Nodes with red borders are central pattern generator (CPG) drivers; triangle-shaped nodes are sensor neurons receiving extrinsic stimulation (orange arrows labeled "S"). (Right) Covariance accumulation across $K$ sessions: each session yields a partial covariance matrix $\hat{\Sigma}_k$ covering only co-observed neuron pairs. These are averaged element-wise to reconstruct the full accumulated covariance $\hat{\Sigma}$, from which $W$ is estimated via Eq. \ref{['eq:estimator']}.
• Figure 2: Method pipeline. Data from $K$ recording sessions yield partial covariance matrices, which are accumulated element-wise into a full covariance estimate $\hat{\Sigma}$. The connectivity matrix is estimated via least-squares inversion, then refined by projected gradient descent enforcing biological constraints (Algorithm \ref{['alg:pipeline']}).
• Figure 3: Scaling properties of the covariance estimator.(Left) Median recovery error (Frobenius distance / $N$) vs. recording duration $T$ for three network sizes ($N \in \{8, 12, 30\}$) with $66\%$ measurement. Shaded regions show 95% bootstrap confidence intervals over 17 random topologies (50 instances each). Error decreases with both $T$ and $N$, with the best result at $N=30$, $T=1000$: Granger-refined error of $0.053$. (Right) Error vs. network size $N$ at fixed recording durations $T \in \{100, 500, 1000\}$.
• Figure 4: Effect of Granger-causality refinement ($N=12$, $66\%$ measured, stim$=1.0$, 50 instances). (A--D) Weight matrix heatmaps from a representative topology: (A) true $W$, (B) raw covariance estimate $\hat{W}$, (C) Granger-refined $\hat{W}$, (D) absolute error. The covariance estimate captures the overall structure; Granger refinement sharpens it by enforcing sparsity and non-negativity. (E--G) Quantitative evaluation over 30 random topologies: (E) Frobenius error (estimate already has diagonal zeroed) reduced from $0.100$ to $0.094$ with Granger ($6\%$), (F) edge detection precision, (G) recall achieves perfect median $1.0$. Violin plots with individual topologies overlaid. (H) Ablation: each step adds knowledge, from chance ($0.54$) to Granger-refined ($0.09$), an $83\%$ total improvement.
• Figure 5: Stimulation-dynamics tradeoff reveals a critical interaction between stimulation and measurement density ($N=12$, $T=900$, 17 topologies $\times$ 50 instances). (Left) Schematic: extrinsic Gaussian stimulation of varying intensity applied to sensor nodes. (Right) Recovery error vs. stimulation gain $\sigma$ for three measurement fractions (33%, 66%, 100%). At low measurement, the effect is mild. At high measurement, zero stimulation fails (error $>4.0$) because CPG dynamics alone leave $\Sigma_{x,x}$ rank-deficient, while moderate stimulation ($\sigma \approx 0.5$) yields the best recovery ($\sim 0.03$). This confirms the control-estimation tradeoff: stimulation is necessary for identifiability but excessive noise degrades the covariance structure.