Duality of edge time series

In a few recent papers (Zamani Esfahlani et al 2020 and Faskowitz et al 2020), we proposed what we called ``edge time series'' as a way of obtaining framewise contributions to the correlation-based measures of functional connectivity (FC). The approach is simple. Suppose $\mathbf{z}_i = [z_i(1),\ldots,z_i(T)]$ is a z-scored time series from brain region $i$. The correlation between regions $i$ and $j$ is given as:

$$ r_{ij} = \frac{1}{T - 1} \sum_t [z_i(t) \cdot z_j(t)] $$

The variable $r_{ij}$ is the standard bi-variate product-moment correlation and is often interpreted as a measure of functional connectivity. Our approach simply omits the summation, returning instead:

$$ r_{ij}(t) = z_i(t) \cdot z_j(t) $$

The value of $r_{ij}(t)$ is the instantaneous co-fluctuation between the activity of regions $i$ and $j$. It is positive when the two regions are deviating in the same direction with respect to their means. It is large when the deviations are large; it is close to zero if one or both regions are close to their mean. Importantly, it tells us when these co-fluctuations occur.

When we first proposed this measure, we only had these ideas in mind; instantaneous co-fluctuations were a convenient way of decomposing a correlation into its framewise contributions (so we could, in principle, filter our correlations by retaining only specific kinds of frames) and returning a measure of time-varying connectivity. What we overlooked was the fact that each edge time series, $\mathbf{r}_{ij}$, is defined mathematically the exact same way we define an interaction in elementary statistics.

We felt that this was interesting (it put our purely ad hoc measure of co-fluctuation time series on stronger statistical ground), but also opened up our eyes to new possibile ways of using these time series. One possibility, and the one that we focus on in the accompanying preprint, is to fully embrace edge time series as an interaction term in a multi-linear model that also contains terms for activations of regions $i$ and $j$, and to use the complete model to explain some time-behavior. That is, suppose $\mathbf{y}$ is some behavioral time series, we want to fit regression coefficients $\beta_i$, $\beta_j$, and $\beta_{ij}$ in the following equation:

$$ \mathbf{y} = \mathbf{z}_i \cdot \beta_i + \mathbf{z}_j \cdot \beta_j + \mathbf{z}_{ij} \cdot \beta_{ij} + \varepsilon $$

Importantly, if $\beta_{ij}$ was statistically significant, this would suggest that time-varying connectivity (as measured by co-fluctuation time series) contributes explanatory power not obviously captured by activity alone. This would represent a potentially important finding, as many studies have suggested that time-varying features of brain recordings (especially BOLD fMRI) reflect statistical artifacts, e.g. apparent time-varying connectivity might simply be sampling variability (for example, see Laumann et al 2017).

In our preprint, we use these models to explain time-varying behavior in zebrafish, worms, and humans and show that, in all cases, the interaction term -- time-varying connectivity -- exhibits non-trivial explanatory power.

This repository contains example code for fitting these types of models (really nothing more than multiple linear regression). It includes a data file, data.mat, that contains behavioral and parcel time series for one animal (zebrafish). The accompanying script, fit_models.m does just that -- it reads in the data and fits the above linear model for every pair of parcels.

The fit_models.m script is ``bare bones,'' in the sense that it provides a scalable and extendable skeleton for applying these types of models to other datasets.

If you use this code, please cite the preprint as:

Merritt, H., Mejia, A., & Betzel, R. (2024). The dual interpretation of edge time series: Time-varying connectivity versus statistical interaction. bioRxiv, 2024-08. link to preprint