Relative Dynamic Causal Strength (rDCS)

• Relative Dynamic Causal Strength (rDCS) is an information-theoretic measure that quantifies causal influence in transient systems using Structural Causal Models.
• It compares the factual conditional distribution with a counterfactual obtained by cutting a causal edge and reinjecting independent reference marginals, thus being sensitive to deterministic source changes.
• Implemented in the TranCIT Python toolbox alongside TE and DCS, rDCS effectively detects event-driven causal effects in non-stationary neural data.

Relative Dynamic Causal Strength (rDCS) is an information-theoretic measure of causal influence for transient, non-stationary interactions, formulated in the language of Structural Causal Models (SCMs). In the transient neural-event setting, rDCS measures the average Kullback–Leibler divergence between the factual conditional distribution of an effect variable and a counterfactual conditional obtained by cutting the relevant causal edge and reinjecting an independent copy drawn from a reference marginal of the cause. In this form, it quantifies how much the event-related change in the cause and the anatomical connectivity jointly raise the information that the cause’s past has about the effect (Shao et al., 2022). The measure is also implemented in the Python package TranCIT, which includes Granger Causality, Transfer Entropy, Dynamic Causal Strength (DCS), and rDCS for detecting event-driven causal effects in transient neural data (Nouri et al., 30 Aug 2025).

1. Genealogy and motivating criteria

The conceptual basis of rDCS lies in a more general proposal for quantifying causal influence in directed acyclic graphs (DAGs). In that framework, one starts from a joint distribution with Markov factorization

$P(x_1,\ldots,x_n)=\prod_j P(x_j\mid pa_j),$

and seeks a nonnegative quantity for the strength of an edge or set of edges. The proposal is organized around five postulates: P0 (Causal Markov Condition), P1 (Mutual Information), P2 (Locality), P3 (Quantitative Markov), and P4 (Heredity). These postulates are used to argue that Average Causal Effect, mutual information, conditional mutual information, Transfer Entropy, Directed Information, and previously defined Information Flow measures fail on simple DAGs of $\leq 3$ nodes (Janzing et al., 2012).

In that same framework, causal strength is defined by “cutting” an edge and replacing the severed parent input by an independent draw from the parent’s marginal. If $S$ is a set of edges, the post-cut distribution is

$$P_S(x_1,\ldots,x_n)=\prod_{j=1}^n P_S(x_j\mid pa_j^{\bar S}),$$

with the altered local conditional obtained by averaging over the cut parents using their marginals, and the causal strength is

$C_S(P)=D_{\mathrm{KL}}(P\|P_S).$

For a two-node graph $X\to Y$, this reduces to $I(X;Y)$, and for a single edge one has the lower bound

$C_{X\to Y}\ge I(X;Y\mid PA_Y\setminus\{X\}).$

The time-series extension applies the same intervention logic to lagged arrows: one cuts the lag-$\tau$ edges $X_{t-\tau}\to Y_t$, feeds them with independent copies drawn from the marginal, and defines an rDCS quantity from the divergence between the original and cut distributions. Aggregation over $\leq 3$0 yields a profile of causal influence over time lags (Janzing et al., 2012).

2. SCM formulation for transient events

In the transient-event setting, rDCS is formulated for two simultaneously recorded neural signals, written as $\leq 3$1 for the putative cause and $\leq 3$2 for the putative effect, under a time-inhomogeneous VAR$\leq 3$3 SCM:

$\leq 3$4

where $\leq 3$5 and $\leq 3$6 are exogenous innovations. The associated SCM graph has arrows from each component of $\leq 3$7 and $\leq 3$8 into $\leq 3$9, and from $S$0 into $S$1 (Shao et al., 2022).

The defining intervention has two parts. First, the causal arrow $S$2 is removed and an independent copy of the reference past of $S$3 is reinjected. Second, the resulting counterfactual distribution of $S$4 is compared to the factual one. The post-intervention conditional is

$S$5

where $S$6 is an independent copy drawn from the marginal at a reference time $S$7. The rDCS itself is then

$S$8

Under the Gaussian VAR assumption, this simplifies to a closed-form expression (Shao et al., 2022).

3. Relation to Transfer Entropy and Dynamic Causal Strength

The main comparisons in the transient-event literature are with Transfer Entropy (TE) and Dynamic Causal Strength (DCS). TE is defined as

$S$9

and is equivalent to the KL divergence between the full conditional and a reduced density obtained by marginalizing over $$P_S(x_1,\ldots,x_n)=\prod_{j=1}^n P_S(x_j\mid pa_j^{\bar S}),$$0. DCS is

$$P_S(x_1,\ldots,x_n)=\prod_{j=1}^n P_S(x_j\mid pa_j^{\bar S}),$$1

where the post-intervention term is obtained by cutting $$P_S(x_1,\ldots,x_n)=\prod_{j=1}^n P_S(x_j\mid pa_j^{\bar S}),$$2 and reinjecting an independent copy of $$P_S(x_1,\ldots,x_n)=\prod_{j=1}^n P_S(x_j\mid pa_j^{\bar S}),$$3 (Shao et al., 2022).

rDCS differs from DCS by also “knocking out” the event-related shift in the cause marginal and using the reference cause marginals $$P_S(x_1,\ldots,x_n)=\prod_{j=1}^n P_S(x_j\mid pa_j^{\bar S}),$$4. In the transient-event setting, the reported limitations are specific. TE’s reduced model is not an SCM intervention; TE is sensitive to strong synchronization and can vanish if $$P_S(x_1,\ldots,x_n)=\prod_{j=1}^n P_S(x_j\mid pa_j^{\bar S}),$$5 is almost deterministically tied to $$P_S(x_1,\ldots,x_n)=\prod_{j=1}^n P_S(x_j\mid pa_j^{\bar S}),$$6; and TE is insensitive to deterministic exogenous inputs. DCS, by contrast, is insensitive to time-varying or driving shifts in the cause: it is invariant under additive shifts in $$P_S(x_1,\ldots,x_n)=\prod_{j=1}^n P_S(x_j\mid pa_j^{\bar S}),$$7 and therefore fails to pick up purely deterministic perturbations. rDCS is introduced precisely to restore sensitivity to deterministic changes at the source (Shao et al., 2022).

Measure	Defining comparison	Reported transient-event behavior
TE	Full conditional vs marginalized reduced density	Not an SCM intervention; can vanish under strong synchronization; insensitive to deterministic exogenous inputs
DCS	Factual conditional vs cut model with independent copy of $$P_S(x_1,\ldots,x_n)=\prod_{j=1}^n P_S(x_j\mid pa_j^{\bar S}),$$8	Invariant under additive shifts in $$P_S(x_1,\ldots,x_n)=\prod_{j=1}^n P_S(x_j\mid pa_j^{\bar S}),$$9; fails on purely deterministic perturbations
rDCS	Factual conditional vs cut model using reference cause marginals	Sensitive to deterministic changes at the source

The earlier causal-strength analysis supplies additional counterexamples that motivate this distinction. In an XOR gate with confounder, $C_S(P)=D_{\mathrm{KL}}(P\|P_S).$0 although cutting $C_S(P)=D_{\mathrm{KL}}(P\|P_S).$1 changes the joint. In a chain $C_S(P)=D_{\mathrm{KL}}(P\|P_S).$2 with deterministic copy $C_S(P)=D_{\mathrm{KL}}(P\|P_S).$3, $C_S(P)=D_{\mathrm{KL}}(P\|P_S).$4 but the edge strength remains nonzero. In perfect-copy bidirectional chains, Transfer Entropy and Directed Information can fail, whereas the intervention-based relative-entropy construction registers the influence of the relevant arrows (Janzing et al., 2012).

4. Estimation from peri-event data

For peri-event multi-trial data, the computational pipeline begins with event detection in the putative cause channel $C_S(P)=D_{\mathrm{KL}}(P\|P_S).$5, extraction of peri-event trials of length $C_S(P)=D_{\mathrm{KL}}(P\|P_S).$6 aligned so that the detected event time maps to $C_S(P)=D_{\mathrm{KL}}(P\|P_S).$7, and selection of a reference window before the event to estimate stationary marginals of $C_S(P)=D_{\mathrm{KL}}(P\|P_S).$8. For each peri-event time point, one constructs a dataset of $C_S(P)=D_{\mathrm{KL}}(P\|P_S).$9 across trials, selects the VAR order $X\to Y$0 via BIC over the pooled multi-trial dataset (Boroumand–BIC), and fits a time-varying VAR$X\to Y$1 at each $X\to Y$2:

$X\to Y$3

Using the reference window, one then estimates the joint marginal $X\to Y$4, approximates the counterfactual density by integrating the conditional Gaussian over that marginal, and computes the empirical expectation of $X\to Y$5. Significance can optionally be tested by permutation, by shuffling $X\to Y$6 across trials (Shao et al., 2022).

A more general implementation strategy, described for causal strength estimation from data, fits a structural-equation model, recovers both the conditional maps and the implied noise samples, splits the data into two halves, generates virtual samples under the cut by independently shuffling the parent variables in the cut set, and recomputes the affected child via the learned structural equation. The divergence $X\to Y$7 can then be estimated nonparametrically using the k-nearest-neighbor KL estimator:

$X\to Y$8

where $X\to Y$9 is the distance from the $I(X;Y)$0-th sample point to its $I(X;Y)$1-th nearest neighbor in the original sample and $I(X;Y)$2 is the corresponding distance in the cut sample. Repeating the random permutation and averaging is proposed as a variance-reduction step (Janzing et al., 2012).

5. Assumptions, guarantees, and interpretive issues

The transient-event formulation is built on explicit modeling assumptions. The VAR$I(X;Y)$3 form is assumed to be correctly specified locally around events. The innovations $I(X;Y)$4 are independent, zero-mean Gaussians except that $I(X;Y)$5 may have a time-varying mean around events. The reference time $I(X;Y)$6 is chosen well before event onset, when $I(X;Y)$7 is stationary. Identifiability requires faithfulness and no hidden confounders between $I(X;Y)$8 and $I(X;Y)$9 (Shao et al., 2022).

The reported guarantees are correspondingly specific. rDCS is invariant under linear transformations of $C_{X\to Y}\ge I(X;Y\mid PA_Y\setminus\{X\}).$0, in the sense that any nonsingular linear re-parametrization of $C_{X\to Y}\ge I(X;Y\mid PA_Y\setminus\{X\}).$1 leaves the value unchanged. It is zero under disconnectedness: if the causal arrow $C_{X\to Y}\ge I(X;Y\mid PA_Y\setminus\{X\}).$2 is removed, rDCS vanishes identically. It satisfies stationarity consistency: if $C_{X\to Y}\ge I(X;Y\mid PA_Y\setminus\{X\}).$3 is stationary and the reference time has identical marginals, then $C_{X\to Y}\ge I(X;Y\mid PA_Y\setminus\{X\}).$4 and the value is constant over time. It is also sensitive to deterministic inputs, increasing with the magnitude of a deterministic mean shift in the cause innovation $C_{X\to Y}\ge I(X;Y\mid PA_Y\setminus\{X\}).$5 (Shao et al., 2022).

Several interpretive issues follow directly from these assumptions. Event alignment matters: aligning trials on the cause’s detected event onset is recommended to avoid selection bias. Reference-window choice also matters because the counterfactual explicitly uses the reference marginal of the cause. A plausible implication is that rDCS is best understood not as a generic conditional dependence statistic, but as an intervention-defined divergence whose value depends on the chosen SCM and the factual-versus-reference comparison encoded in that model.

6. Empirical behavior and software ecosystem

The transient-event study reports both simulation and neurophysiological results. In synchronous oscillators, TE dips during stronger synchrony, DCS is stable but small, and rDCS increases during a transient noise reduction that increases synchrony. In deterministic perturbation events modeled as VAR(4) with mean-shifted innovation on $C_{X\to Y}\ge I(X;Y\mid PA_Y\setminus\{X\}).$6, TE and DCS remain flat across the event, whereas rDCS sharply rises and falls, matching the known event waveform. In alignment tests, rDCS recovers ground-truth coupling best when aligned by cause, in line with the selection-bias discussion. In rodent sharp-wave ripple data, rDCS$C_{X\to Y}\ge I(X;Y\mid PA_Y\setminus\{X\}).$7 exhibits a pronounced peak just before ripple peak ($C_{X\to Y}\ge I(X;Y\mid PA_Y\setminus\{X\}).$8 to $C_{X\to Y}\ge I(X;Y\mid PA_Y\setminus\{X\}).$9), reflecting the determinant role of CA3 sharp waves in driving CA1 ripples; TE and DCS also detect direction but show weaker temporal modulation and miss the rapid event-related increase. The quantitative summary given is a signal-to-noise of the rDCS peak $\tau$0 above baseline versus $\tau$1 for TE and DCS, and a directionality index $\tau$2 that is significantly positive ($\tau$3, bootstrap) only for rDCS when aligned by CA3 (Shao et al., 2022).

The main accessible implementation reported in the provided material is TranCIT, the Transient Causal Interaction Toolbox. TranCIT is described as an open-source Python package for quantifying transient causal interactions from non-stationary neural signals. Its analysis pipeline includes Granger Causality, Transfer Entropy, and the SCM-based DCS and rDCS. The package is reported to capture causality in high-synchrony regimes where traditional methods fail and to identify the known transient information flow from hippocampal CA3 to CA1 during sharp-wave ripple events in real-world data. A plausible implication is that TranCIT situates rDCS within a comparative workflow in which intervention-based and non-intervention-based measures can be evaluated side by side on the same transient datasets (Nouri et al., 30 Aug 2025).