Department of Mathematics, VU University Amsterdam, Amsterdam, The Netherlands.
PLoS One. 2020 Dec 11;15(12):e0242715. doi: 10.1371/journal.pone.0242715. eCollection 2020.
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.
物理系统的测量结果源自系统活动通过正向模型转换为传感器测量值。在许多情况下，正向模型的反演对传感器噪声或正向模型中的数值误差等扰动极其敏感。因此需要正则化，这会给系统活动的重建引入偏差。在空间扩展的复杂系统（如人脑）的相互作用重建中，这种情况尤其成问题。脑相互作用可以从非侵入性测量（如脑电图或脑磁图）中重建，其正向模型是线性和瞬时的，但具有较大的零空间和高条件数。这导致正向模型的不完全解混，从而产生虚假相互作用。这促使人们开发了仅对滞后，即延迟相互作用敏感的相互作用度量。这种措施的缺点是，它们只能检测到具有足够大滞后的相互作用，这会给重建的脑网络引入偏差。我们引入了三种用于空间扩展系统中线性相互作用的估计器，它们对所有滞后都具有均匀的敏感性。我们推导了估计器的一些基本性质和关系，并使用简单基准模型的数值模拟来评估它们的性能。
