Lag-invariant detection of interactions in spatially-extended systems using linear inverse modeling

Measurements on physical systems result from the systems' activity being converted into sensor measurements by a forward model. In a number of cases, inversion of the forward model is extremely sensitive to perturbations such as sensor noise or numerical errors in the forward model. Regularization is then required, which introduces bias in the reconstruction of the systems' activity. One domain in which this is particularly problematic is the reconstruction of interactions in spatially-extended complex systems such as the human brain. Brain interactions can be reconstructed from non-invasive measurements such as electroencephalography (EEG) or magnetoencephalography (MEG), whose forward models are linear and instantaneous, but have large null-spaces and high condition numbers. This leads to incomplete unmixing of the forward models and hence to spurious interactions. This motivated the development of interaction measures that are exclusively sensitive to lagged, i.e. delayed interactions. The drawback of such measures is that they only detect interactions that have sufficiently large lags and this introduces bias in reconstructed brain networks. We introduce three estimators for linear interactions in spatially-extended systems that are uniformly sensitive to all lags. We derive some basic properties of and relationships between the estimators and evaluate their performance using numerical simulations from a simple benchmark model.