I am looking for a sparse MEG inverse solution that would be appropriate for input to Granger causality analysis. In particular, since Granger causality is usually implemented by linear means, would the output from a non-linear, sparse inverse solution such as TF-MxNE be appropriate here? I have not been able to determine this from Gramfort's 2013 NeuroImage paper or other sources (probably because of my own mathematical shortcomings). In particular, I cannot tell if the non-linearity in TF-MxNE is limited to the localizations (which would be acceptable) or if it applies to the corresponding timecourses as well (in which case I would expect it to disrupt linear Granger analysis).
I am using the non-parametric Granger causality methods of Dhamala, Rangarajan, and Ding (Physical Review Letters, NeuroImage, 2008, where Wilson's 1972 numerical spectral decomposition is used in place of MVAR estimation), and the ability of TF-MxNE to work with non-stationary data is very appealing.
I am looking for a sparse MEG inverse solution that would be appropriate
for input to Granger causality analysis. In particular, since Granger
causality is usually implemented by linear means, would the output from a
non-linear, sparse inverse solution such as TF-MxNE be appropriate here?
TF-MxNE achieves an adaptive non-stationary filtering of evoked data
built in the source localization algorithm.
Granger causality with AR models are not playing nice with filtered
data and work on single trial or raw data AFAIK.
It is therefore unclear what can happen if you apply GC after TF-MxNE.
I have not been able to determine this from Gramfort's 2013 NeuroImage paper
or other sources (probably because of my own mathematical shortcomings). In
particular, I cannot tell if the non-linearity in TF-MxNE is limited to the
localizations (which would be acceptable) or if it applies to the
corresponding timecourses as well (in which case I would expect it to
disrupt linear Granger analysis).
the non-linearity is also temporal as an entire time interval of data
is processed together.
I am using the non-parametric Granger causality methods of Dhamala,
Rangarajan, and Ding (Physical Review Letters, NeuroImage, 2008, where
Wilson's 1972 numerical spectral decomposition is used in place of MVAR
estimation), and the ability of TF-MxNE to work with non-stationary data is
very appealing.
I have tried to answer this question with a simulation, but am not completely sure I have done this right. I began with the "plot_simulate_evoked_data.py" example (http://martinos.org/mne/stable/auto_examples/plot_simulate_evoked_data.html#example-plot-simulate-evoked-data-py) and replaced the two wavelet-derived timecourses with a bivariate MVAR system which I generated using the "nitime" package. I commented out the IIR filtering but continued to use "generate_evoked" to create an evoked response which I then input to tf_mixed_norm (does this make sense? it feels like inputting something that is already in source space into the localizer algorithm). After ~250 iterations tf_mixed_norm reduces this system to two sources as expected, but both sources fall in the right hemisphere and an MVAR estimation of their timecourses no longer matches the system that I put it.
In general I would like to simulate a simple MVAR system and assign it to a number of locations in brain space. Since TF-MxNE expects a sensor space evoked input, I think I would need to project this onto the sensor array? I would then like to pass this through TF-MxNE and see if the structure and locations of the original MVAR system are preserved. Does this sound like an appropriate way to approach this problem or am I missing something important?
Thanks again,
Per Lysne
The University of New Mexico
PS: The non-parametric Granger causality method I mentioned below is implemented in Fieldtrip, although it may be an undocumented option.
I've been working through the mechanics of TF-MxNE this weekend, and I think I can answer two recent questions I have asked:
With regard to TF-MxNE and Granger Causality, it seems to me that this should work. The mathematics of the localizations themselves may be non-linear, but Eqns. 3 and 4 from Gramfort et al. (2013) appear to be completely linear. Therefore, a linear system of sources within the brain are linearly projected onto the sensors by the forward solution, and then Eqn. 3 (solved for X) brings an estimate of them back into brain space and the structure of the system should not be disturbed?
With regard to individual trial output, the estimate Z* from Eqn. 4 could be determined based on an average evoked response, but then applied to individual trials in sensor space to determine timecourses for these trials at the already established locations?
Does this make sense, or can you see a flaw with my reasoning?
Also, looking at Figure 6 panel d) in (2013), are these timecourses completely positive by coincidence, or is the magnitude taken? Around 50 and 150ms both approach zero and then abruptly turn upward as if the absolute value were being plotted?
Thanks again for your help,
-Per
PS: I agree, an MVAR system does assume stationary.
I've been working through the mechanics of TF-MxNE this weekend, and I think I can answer two recent questions I have asked:
With regard to TF-MxNE and Granger Causality, it seems to me that this should work. The mathematics of the localizations themselves may be non-linear, but Eqns. 3 and 4 from Gramfort et al. (2013) appear to be completely linear. Therefore, a linear system of sources within the brain are linearly projected onto the sensors by the forward solution, and then Eqn. 3 (solved for X) brings an estimate of them back into brain space and the structure of the system should not be disturbed?
the forward is always linear indeed. It's the inverse / source
estimation which is not.
With regard to individual trial output, the estimate Z* from Eqn. 4 could be determined based on an average evoked response, but then applied to individual trials in sensor space to determine timecourses for these trials at the already established locations?
what would make sense would be to use the spatiotemporal
atoms/coefficient estimated on the evoked and run a least square fit
for only these atoms on each trial.
It's however not implemented.
Does this make sense, or can you see a flaw with my reasoning?
Also, looking at Figure 6 panel d) in (2013), are these timecourses completely positive by coincidence, or is the magnitude taken?
Around 50 and 150ms both approach zero and then abruptly turn upward as if the absolute value were being plotted?
it's amplitude as these results were obtained with loose orientation.