Time-domain asymmetric dynamical connectivity - data structure storage and basic OLS algorithms

Describe the new feature or enhancement

This is related to a goal of mine to add generic “connectivity” function to MNE for the code-sprint.

Right now, there are a decent amount of “spectral connectivity” (possibly over time) metrics derived from raw EEG/MEG/iEEG data.There are alternative characterizations, such as dynamical system perspective, which models the data as:

x(t+1) = Ax(t)

where A is possibly an asymmetric matrix characterized by its eigenvalues. Note this is different then just looking at the “pearson correlation” (or covariances) over time, since correlation matrices are PSD and symmetric.

Since a linear dynamical system representation of the MEEG data is useful, it would be nice to have a “data structure” to store such a representation and also expose basic algorithms for estimating this model from the raw data. I’m opening this issue to have a discussion to see if this is feasible to add.

Describe your proposed implementation

One can leverage scikit-learn to add an option to say Raw, or Epochs.

One can add regularization as a result. The algorithm would essentially consist of a window_size and step_size, which takes a sliding window and estimates sequences of these A matrices.

To make this a “nice” and “useful” workflow, I would argue that we need a:

• general Connectivity container (e.g. generalize what Connectivity data structures mne already has)
• a simple API for computing the sequence of A matrices

Describe possible alternatives

In a related viewpoint, this is related to what is known as Dynamic Mode Decomposition (DMD), which is also useful, but the representation of the data is different, because there one is interested in not only the A matrix, but also the “DMD modes”. This is related to the linear Koopman operator in dynamical systems theory; DMD is a finite approximation of Koopman.

Example paper:

• https://arxiv.org/pdf/1409.5496.pdf

Additional comments

Example papers using A matrix:

• https://pubmed.ncbi.nlm.nih.gov/29060480/
• https://www.biorxiv.org/content/10.1101/862797v4.full

In addition https://arxiv.org/pdf/1802.08334.pdf talks about the convergence of such OLS estimates for A, giving some sort of theoretical guarantees on how “estimable” an A matrix is from MEEG data.

Comparisons to spectral connectivity

Spectral connectivity generally are defined as some correlation measure over frequency domain, and is generally symmetric. Here, I am proposing a time-domain and possibly ``asymmetric` “connectivity” derivative of the raw MEEG data.

Testing

Datasets can be tested against A matrices I’ve computed using numpy.

[agramfort]

I would start from https://github.com/mne-tools/mne-project-template

and start putting there the various connectivity measure we have in mne-python and elsewhere

after the first release of “mne-connectivity” we should start deprecating the mne.connectivity module.

my 2c

[adam2392]

Btw @alexrockhill spoke with @agramfort and he’s opposed to adding this type of functionality into MNE-python. He thinks that it might be useful for a sub-group of us to try to band together w/ scot, and ppl here https://github.com/mne-tools/mne-python/issues/9030#issuecomment-802110862 to basically implement a 3rd-party mne-connectivity stand-alone package that:

1. takes out the current mne.connectivity module
2. implements anything suggested here e.g. state-space / lds modeling

I’m pretty open to this idea if there’s enough ppl game. However, for my purposes I really only use lds modeling…