GEC directionality interpretation

[davidaquilue]

Hello,

I have been working on determining how GEC's asymmetry can be leveraged in the context of Alzheimer's Disease. I have been testing out neuronumba's implementation and the pipeline is quite useful.

Just wanted to mention that the way the code is set up in .../src/neuronumba/observables/lagged_cov.py, with

@staticmethod
def _calc_COV_emp(tss: np.ndarray, timelag: int = 1):
     """
     Parameters
     ----------
     tss : non-perturbed timeseries, in format (n_roi, n_timesteps)
     timelag : the number of timesteps of your timelag, default = 1

     Returns
     -------
     time-lagged cov matrix in format(n_roi, n_roi)
     """
     n_roi = tss.shape[0]
     EC = np.zeros((n_roi, n_roi))
     for i in range(n_roi):
         for j in range(n_roi):
             correlation = signal.correlate(tss[i, :] - tss[i, :].mean(), tss[j, :] - tss[j, :].mean(), mode='full')
             lags = signal.correlation_lags(tss[i, :].shape[0], tss[j, :].shape[0], mode='full')
             EC[i, j] = correlation[lags == timelag] / tss.shape[1]
     return EC

as a consequence of how signal.correlate works, returns a shifted covariance matrix whose {i,j}-th element answers the following question: How does signal j predict i.

I am unsure whether this is the desired output. I would assume that the expected nature of GEC_{i,j} is how dynamic's in node i affect node j, which would make use of the transposed output from the _calc_COV_emp() funciton.

Would love to hear your input and whether this has already been discussed previously.

Salut!

David

[ajunca]

Hi David,

It has not been discussed. I've been checking if everything looks consistent, and I think it looks, but you are right, now it is saying j influence i. And probably it would make sense to invert it so i influence j. Have you notice any issue apart of this directionality?

The problem is that if we decide that should be inverted I fear it would be not so easy as just transpose the _calc_COV_emp() result. We have the empirical cov but also I think the Jacobian should be changed.

We can go for transposing the GEC end result, eg: return save_SC.T, and it will work, but then we should also transpose the "starting" SC for the GEC (in some edge case the user is using non symmetric starting SC?).

So, If I'm not missing anything important, I see two main directions:

• We do a proper documentation on the code saying that the direction is from j to i.
• We transpose the GEC results and the input (and maybe I'm missing something else?). Something like:

 def fitGEC(self, timeseries, FC_emp, starting_SC):                                                                                                                                                             
      # external API uses i→j 
      # Internal math uses j→i                                                                                                                                                      
      starting_SC = starting_SC.T                           

      # ... all internal fitting unchanged (j→i throughout) ...

      return save_SC.T  # back to i→j for the caller

[davidaquilue]

Hello Albert,

Thanks for your answer, actually, the Issue was motivated from me believing that the xcov implementation in Matlab computed the tranpose of what the scipy.correlate function computes, but after deeper evaluation, both of them are indeed consistent.

After commenting it with some colleagues, it seems that indeed, the GEC matrix obtained from Matlab implementations (as well as with neuronumba) should be interpreted as j influencing i. So, I would go ahead with your first main direction to keep it as it is but document its behavior.

In line with the documentation topic, after reading around the repo, I believe it would significantly help new users to generally include the documentation of the repo, I will open another Issue to discuss it.