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

Quilee Simeon

q-bio.QMMar 19, 2026

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

Authors: Quilee Simeon

Organizations: Massachusetts Institute of Technology

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 & tables

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. ( 6 ).

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 1 ).

$N$	$T$	Chance	Estimate	Granger	Improvement
8	100	0.540	0.157	0.150	4%
8	1000	0.540	0.131	0.127	3%
12	100	0.542	0.234	0.154	34%
12	1000	0.542	0.112	0.095	15%
30	100	0.561	0.185	0.106	43%
30	1000	0.561	0.060	0.053	12%

Table 1: Median recovery error (Frobenius /

N

) across network sizes and durations. Bold indicates best method per row.

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.

Figure 6: Effect of measurement density on connectivity recovery for

N=12

T=900

. (Left) Schematic: three example measurement patterns at 33%, 66%, and 100% neuron observability (green = measured, gray = unobserved). (Right) Median recovery error vs. measurement fraction for the raw covariance estimator (blue) and Granger-refined estimator (green), with 95% bootstrap CI (shaded). Dashed line shows the chance baseline. Full observability (

100\%

) yields error

0.094

; at

33\%

the Granger-refined estimate (

0.307

) is

48\%

better than the raw estimate (

0.592

). 17 topologies

\times

50 instances per condition.

Figure 7: Robustness to nonlinearity mismatch for

N=12

T=900

66\%

measured. (Left) The four activation functions tested. (Right) Median recovery error (log scale) for the raw estimator (blue) and Granger-refined (green) with bootstrap error bars. Counterintuitively,

\tanh

yields the best recovery (

0.094

) because it bounds state values, improving

\kappa(\Sigma_{x,x})

. Unbounded nonlinearities (identity, ReLU) allow state drift that worsens conditioning. Dots show individual topologies (

n=17

Figure 8: Dynamics, CPG detection, and error analysis. (A–C) Time series: autonomous CPG-driven dynamics produce structured oscillations; stimulation adds noise on top. (D–E) PCA phase portraits: autonomous dynamics trace low-dimensional trajectories; stimulation fills out the state space (aiding identifiability). (F) Power spectra: CPG dynamics have concentrated spectral peaks; stimulation flattens the spectrum. (G) CPG detection via per-node variance: CPG nodes (red) have systematically higher variance than non-CPG nodes (blue). (H) Error decomposition: input correlation

E_{2}

(from CPG–state dependence) is

3.5\times

larger than model mismatch

E_{1}

. (I) True vs. estimated weights: Pearson

r=0.90

for the Granger-refined estimator.

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Abstract

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