Modeling spatiotemporal covariance for magnetoencephalography or electroencephalography source analysis Sergey M. Plis, * J. S. George, S. C. Jun, J. Paré-Blagoev, D. M. Ranken, C. C. Wood, and D. M. Schmidt Applied Modern Physics Group, Los Alamos National Laboratory, MS-D454, Los Alamos, New Mexico 87545, USA Received 6 July 2006; revised manuscript received 1 November 2006; published 30 January 2007 We propose a new model to approximate spatiotemporal noise covariance for use in neural electromagnetic source analysis, which better captures temporal variability in background activity. As with other existing formalisms, our model employs a Kronecker product of matrices representing temporal and spatial covariance. In our model, spatial components are allowed to have differing temporal covariances. Variability is represented as a series of Kronecker products of spatial component covariances and corresponding temporal covariances. Unlike previous attempts to model covariance through a sum of Kronecker products, our model is designed to have a computationally manageable inverse. Despite increased descriptive power, inversion of the model is fast, making it useful in source analysis. We have explored two versions of the model. One is estimated based on the assumption that spatial components of background noise have uncorrelated time courses. Another version, which gives closer approximation, is based on the assumption that time courses are statistically independent. The accuracy of the structural approximation is compared to an existing model, based on a single Kronecker product, using both Frobenius norm of the difference between spatiotemporal sample covariance and a model, and scatter plots. Performance of ours and previous models is compared in source analysis of a large number of single dipole problems with simulated time courses and with background from authentic magnetoencephalography data. DOI: 10.1103/PhysRevE.75.011928 PACS numbers: 87.57.Ra, 87.80.Tq, 02.30.Zz I. INTRODUCTION The physical and physiological consequences of the cor- related activity of substantial populations of neurons can be detected with noninvasive measurement techniques, includ- ing electroencephalography EEG and magnetoencephalog- raphy MEG. These macroscopic electrophysiological tech- niques can resolve the time course of neural population activation with millisecond temporal resolution. Neural elec- tromagnetic NEM responses are governed by the same physical processes that give rise to electric and magnetic fields in other systems: vector currents established by poten- tial differences along cellular structures give rise to an elec- tric field aligned with the current and an orthogonal magnetic field that encircles the current element. Because many differ- ent sensors typically detect signal contributions from a given source, data sets often contain identifiable patterns of spatial covariance associated with sources of interest as well as background processes. Because neural activation typically proceeds with a characteristic time course, spatial covariance components often exhibit structured temporal covariance and correlation. The unique strengths of neural electromagnetic methods stem from their capacity to define the dynamics of neural population activity. Even a single electrode pasted to the scalp may disclose a complex temporal wave-form consist- ing of a series of peaks and valleys. The first 50 years of work with EEG involved little quantitative effort to localize the sources of observed topographies in the surface potential data. Inspection or simple quantification of temporal wave- form features served as the basis of diagnostic procedures in clinical neurology as well as experimental studies of cogni- tive processing. The development of MEG and the recogni- tion that many observed field distributions could be ex- plained by a simple forward model lead to advances in procedures that have subsequently been applied to EEG data. Basic and clinical neuroscience are very motivated to iden- tify the anatomical sources of observed functional activity, as evident in the explosion of interest in functional magnetic resonance imaging fMRI. Suitable geometric models of neural sources, coupled with physical “forward models” de- scribing the relationships between sources, detectors, and the tissue medium, and adequate optimization strategies, enable useful localization of neural electromagnetic sources, in spite of the ill-posed, ambiguous nature of the inverse problem. Even if the objective of analysis is to describe the dynamics of neural activation, this is most effective in the context of an adequate model of the underlying neural sources. Ongoing spontaneous activity recorded at the surface of the human head using MEG or EEG, typically is character- ized by regions of relatively large amplitude oscillatory pat- terns that vary as a function of position on the head and state of the subject. The signals associated with responses to indi- vidual stimuli or other punctuate cognitive or control pro- cesses are typically much smaller and require specialized ex- perimental paradigms and signal processing techniques to pull the signals out of the noise. In order to enhance the consistent aspects of the neural response while suppressing the contribution of other physiological processes or environ- mental noise, most investigators employ sensory-evoked re- sponse or event-related paradigms, averaging temporal se- quences time locked to the stimulus or a behavioral response. The central limit theorem lends support to the common assumption that the averaged background data are Gaussian distributed, even though the distribution of a single trial *E-mail address: pliz@lanl.gov PHYSICAL REVIEW E 75, 011928 2007 1539-3755/2007/751/01192813 ©2007 The American Physical Society 011928-1