Rooted in dynamic systems theory, convergent cross mapping (CCM) has attracted increased attention recently due to its capability in detecting linear and nonlinear causal coupling in both random and deterministic settings. One limitation with CCM is that it uses both past and future values to predict the current value, which is inconsistent with the widely accepted definition of causality, where it is assumed that the future values of one process cannot influence the past of another. To overcome this obstacle, in our previous research, we introduced the concept of causalized convergent cross mapping (cCCM), where future values are no longer used to predict the current value. In this paper, we focus on the implementation of cCCM in causality analysis. More specifically, we demonstrate the effectiveness of cCCM in identifying both linear and nonlinear causal coupling in various settings through a large number of examples, including Gaussian random variables with additive noise, sinusoidal waveforms, autoregressive models, stochastic processes with a dominant spectral component embedded in noise, deterministic chaotic maps, and systems with memory, as well as experimental fMRI data. In particular, we analyze the impact of shadow manifold construction on the performance of cCCM and provide detailed guidelines on how to configure the key parameters of cCCM in different applications. Overall, our analysis indicates that cCCM is a promising and easy-to-implement tool for causality analysis in a wide spectrum of applications.

Background: Existing work suggested that AD pathology can affect the direction and intensity of information signaling in functional brain regions. The present study evaluates the time‐delayed effective connectivity of normal controls (NC) and patients with mild cognitive impairment (MCI) under motion‐detection tasks and explores identification of possible anomalies and deviated patterns in effective connectivity associated with AD pathology.Method: Our research focuses on task‐based EEG (64‐channel), where participants were asked to perform a motion direction discrimination task. The current dataset includes 56 consensus‐diagnosed, community‐dwelling African Americans with subjective cognitive complaints (ages 60‐90 years, 28 normal controls (NC) and 28 MCI patients) recruited through the Wayne State Institute of Gerontology and Michigan Alzheimer’s Disease Research Center.The successive motion‐task performance periods examined include: (I) stimulus onset to Go/No‐Go indication, (II) Go (or No‐Go) indication to motion‐stop, and (III) the Button‐Press period. For each period, we evaluated the time‐delayed effective connectivity across all the possible EEG region pairs using causalized convergent cross‐mapping (cCCM) of the current source density. For each region pair, the delay (within 50ms) that results in the largest cCCM value (which reflects the intensity of the information flow) is taken as the dominant delay. Within the NC/MCI group, the directed signaling path (with cCCM value > 0.7) that appears in more than 80% of the participants is defined as a common active path of the group.Result: Our analysis shows that: (i) Delayed effective connectivity is noticeably stronger than its zero‐delay counterpart, indicating the existence of non‐zero transmission delays in inter‐region information transfer; (ii) NC and MCI may share common active paths, but each group has their own, unique common paths; (iii) Both NC and MCI groups have fewer common active paths in the Go and Button‐Press periods than during the stimulus onset and No‐Go periods, which may imply that brain effective connectivity is more focused during task execution.Conclusion: Time‐delayed effective connectivity analysis may enable identification of possible signaling pathways in the brain and of dissimilarities in pathway usage between NC and MCI associated with AD pathology.Funding: NSF‐2032709/Li; NIH‐1R21AG046637‐01A1/Kavcic; NIH‐1R01AG054484‐01A1/Kavcic; NIH‐P30AG072931/Paulson; NIH‐P30AG024824/Yung.