CITS: Nonparametric Statistical Causal Modeling for High-Resolution Neural Time Series

TL;DR

CITS (Causal Inference in Time Series), a nonparametric framework for inferring statistically causal structure from multivariate time series, is introduced and established as a theoretically grounded and empirically validated method for discovering interpretable statistically causal networks in neural time series.

Abstract

Identifying causal interactions in complex dynamical systems is a fundamental challenge across the computational sciences. Existing functional connectivity methods capture correlations but not causation. While addressing directionality, popular causal inference tools such as Granger causality and the Peter-Clark algorithm rely on restrictive assumptions that limit their applicability to high-resolution time-series data, such as the large-scale recordings now standard in neuroscience. Here, we introduce CITS (Causal Inference in Time Series), a nonparametric framework for inferring statistically causal structure from multivariate time series. CITS models dynamics using a structural causal model of arbitrary Markov order and statistical tests for lagged conditional independence. We prove consistency under mild assumptions and demonstrate superior accuracy over state-of-the-art baselines across simulated linear, nonlinear, and recurrent neural network benchmarks. Applying CITS to large-scale neuronal recordings from the mouse visual cortex, thalamus, and hippocampus, we uncover stimulus-specific causal pathways and inter-regional hierarchies that align with known anatomy while revealing new functional insights. We further highlight CITS ability in accurately identifying conditional dependencies within small inferred neuronal motifs. These results establish CITS as a theoretically grounded and empirically validated method for discovering interpretable statistically causal networks in neural time series. Beyond neuroscience, the framework is broadly applicable to causal discovery in complex temporal systems across domains.

Figures (6)

• Figure 1: Inference of the Unrolled Directed Acyclic Graph (DAG) for Neural Time Series and its Rolled Graph. (A) Example Markovian Structural Causal Model interactions with $\tau = 2$. (B) Formation of time-windowed samples for each $(v,t)$, where $0 \leq t \leq 2\tau$. (C) The invariant unrolled DAG extends from time $t-\tau$, with edges projecting into it originating from at most time $t-2\tau$, motivating conditional dependence tests within a $2\tau$ window.
• Figure 2: Comparative study of inferring the Rolled graph. Inference of Rolled graph for five simulation settings (left to right): Linear Gaussian Models 1 and 2, Non-linear Non Gaussian Models 1 and 2, Continuous Time Recurrent Neural Network (CTRNN). Row 1: The ground truth for each simulation paradigm is graphically represented. The performances of the five methods Granger Causality 1 (GC1), Granger Causality 2 (GC2), Peter-Clark (PC), Time-Aware PC (TPC) and CITS, are shown in terms of three metrics (right column): $1 -$ False Positive Rate (IFPR) (green), True Positive Rate (TPR) (orange) and Combined Score (CS) (purple). Row 2 shows the performance the three metrics for $alpha = 0.05$ and noise level $1.0$. The CS of the methods over varying noise levels in simulation $\eta = 0.1,0.5,1.0,\ldots,3.5$, with significance level $\alpha = 0.01, 0.05, 0.1$ are also demonstrated in rows 3-5 respectively.
• Figure 3: Comparison of Ground Truth and Estimated Causal Edge Weights Across Simulation Paradigms Top row: Ground truth edges for the simulation paradigms of Linear Gaussian 1, Linear Gaussian 2, Non-linear Non-Gaussian 1, Non-linear Non-Gaussian 2, and Continuous Time Recurrent Neural Network (CTRNN) (left to right). The ground truth edge weights are well-defined for linear paradigms. Bottom row: Estimated edge weights (median [min, max]).
• Figure 4: Comparison of Associative and Causal Functional Connectivity Methods on Neuropixels Mouse Brain Data. Four different methods for inferring functional connectivity (FC) were compared using benchmark mouse brain data from the Allen Institute's Neuropixels dataset. These methods include associative FC using Partial Correlation, and causal FC using Granger Causality 1 (GC1), Granger Causality 2 (GC2), Time-Aware PC (TPC), and CITS. FC is estimated and visualized as an adjacency matrix with edge weights, which is symmetric for associative methods and asymmetric for causal methods. In each matrix, a non-zero entry at position $(i, j)$ indicates a directed connection from neuron $i$ to neuron $j$.
• Figure 5: Examples of Neural Signal Dependencies within Inferred CFC Motifs. A. Common-Source neuronal motif identified by CITS. In the inferred CFC, for the motif of neuron 105 $\leftarrow$ 125 $\rightarrow$ 247, pairwise plots exhibit correlations between all pairs including spurious correlation between 105 and 247 (red) even though not directly connected in the motif (top row). As predicited, conditioned on the parent, 105 and 125's signals are uncorrelated. B. Multi-Parent neuronal motif identified by CITS. In a more complex motif of 106 $\leftarrow$ 105 $\leftrightarrow$ 118 $\rightarrow$ 109, pairwise plots exhibit correlations between all pairs (rows 1-2). As predicted, conditioned on both parents, neurons 106 and 109 become uncorrelated. This highlights CITS ability in correctly identifying these nodes as being not directly connected.

Theorems & Definitions (12)

• definition 1
• lemma 1
• theorem 1: CITS-Oracle Recovery Guarantee
• remark 1: Interpretation of Rolled Graph under Concurrent Effects
• theorem 2: CITS Consistency for Time-Unrolled and Rolled Graphs
• corollary 1
• theorem 2: CITS Consistency for Time-Unrolled and Rolled Graphs (restated)
• lemma 2
• proof
• corollary 2