Fig 1.
Causal modeling of neural time series.
Schematic depiction of causal modeling of the neural time series (left) by the Unrolled Causal Graph (middle), and then rolling back its edges (colored) to define the Rolled CFC-DPGM (right).
Fig 2.
Illustration of Steps 1–7 in the TPC Algorithm: Time Advance, Bootstrap, PC, Orient, Rolled CFC-DPGM, Robust edges and Pruning.
Fig 3.
Comparative study of CFC inference.
(a) CFC inference by GC, DPGM, and TPC, is compared on three examples of motifs and simulation paradigms; from left to right: Linear Gaussian, Non-linear Non-Gaussian, CTRNN. Table: 4-neurons motifs that define the Ground Truth CFC (row 1) are depicted along with inferred CFC over several simulation instances according to the three different methods (row 2–4). Each inferred CFC has an edge v → w that corresponds to an edge detected in any of the inference instances. The percentage (blue) next to each edge indicates the number of times the edge was detected out of all instances. (b) IFPR (green), TP rate (orange) and Combined Score (purple) of each method are shown for each motif.
Fig 4.
Comparative study over levels of noise and thresholding parameter.
Combined Score of the three methods of CFC inference—TPC (red), DPGM (blue), GC (gray), over varying noise levels in simulation η = 0.1, 0.5, 1.0, …, 3.5, for simulated motifs from Linear Gaussian, Non-linear Non-Gaussian and CTRNN paradigms (left to right), with thresholding parameter α = 0.01, 0.05, 0.1 (top to bottom).
Fig 5.
Interventional connectivity weights.
Inference of interventional connectivity weights by the TPC algorithm with max delay 1 msec for the example motifs from the three simulation paradigms: Linear Gaussian VAR, Non-linear Non-Gaussian VAR, CTRNN (left to right). Top row: Ground Truth CFC with excitatory (red) and inhibitory (green) connections; Bottom row: Estimated CFC labeled with edge weights (median [min,max] over all instances) and inferred nature whether excitatory (red) or inhibitory (green).
Table 1.
Comparison of CFC inference by GC, DPGM, PCMCI-GPDC and selVAR, and TPC on benchmarking datasets. For each dataset, each method’s Combined Score, True Positive Rate, and 1-False Positive Rate are reported (Higher value is better).
Fig 6.
Application to Neuropixels dataset.
Comparison and demonstration of the FC inferred for a benchmark of mice brain data from the Allen Institute’s Neuropixels dataset, by three methods for FC inference: Associative FC using Sparse Partial Correlation, and Causal FC using GC and TPC. The estimated FC is represented by its adjacency matrix with edge weights, which is symmetric for Associative FC and asymmetric for Causal FC. The mice were subject to different stimuli, among which we selected four stimuli categories with distinct characteristics: Natural Scenes, Static Gratings, Gabor Patches and Full-Field Flashes [43]. The neurons are clustered by the region of brain: Visual Cortex, Hippo-Campal Formation, and Thalamus, which are further divided into sub-regions. In the adjacency matrices, a non-zero entry in (i, j) represents the connection of neuron i → j.
Fig 7.
Graphical comparison of estimated CFC over stimuli.
This figure compares the distribution of graph measures of CFC obtained by TPC over different stimuli: natural scenes, static gratings, Gabor patches and flashes. The distribution for each graph measure and stimuli is shown by a boxplot.
Table 2.
Comparative summary of different approaches for causal modeling.
Fig 8.
Rolled CFC-DPGM (left) for neurons 1–4 with dynamics as in Example 5.1, and consequence of intervention on neurons labelled A and B by (i) Ablation of A and (ii) External modulation of B.