This technical note presents an inference framework for hierarchically conditioned (embedded) Hermitian Gaussian Graphical Models (GGM). Our methodology, although extendible to several GGMs applications, is mainly centered on the specific neuroscientific context encompassing the estimation of brain sources activityand connectivity from Magnetoencephalography or Electroencephalography (MEEG) signals. This will be conceived in a series of papers with two parts. Part I: Analysis of a single frequency component from a single subject. Part II: Analysis of the Cross-Spectra for multiple subjects in a population. The strategy is based on Maximum Posterior Analysis, via the Expectation Maximization (EM) algorithm, of the Hermitian GGMs Parameters and Hyperparameters. The Hyperparameters set, which comprises MEEG signal noise covariance and source precision (connectivity) matrices of the MEEG generative model, is informed with Priors of special interest in the field and whose implementation in GGM estimation still poses a technical challenge. The theory presented here encapsulates two essential contributions . First: The formulation of a target function for the connectivity estimator that resembles an effective GGM of source activity, which is merged to the maximization step of the iterative EM strategy. Second: A new procedure for the estimation of the source activity effective GGM. These theoretical contributions provide for an explicit and highly optimal connectivity estimation procedure .
Statistical analysis of functional images (signals) allows the assessment of neural connectivity patterns at mesoscopic scale. Scalp fields, recorded with Magneto-Electro Encephalography MEEG, strongly correlate to the average effect of synaptic activity, i.e. Primary Current Density (PCD) at the Gray Matter space. Nevertheless, what is observed at the sensor space with either technique is a superposition of projected fields, from the whole Gray-Matter PCD. That is the reason for a major pitfall of MEEG analysis methods, distorted reconstruction of source activity and its connectivity or "Leakage". There has been a quest for the modelling to counterattack Leakage of both activation and connectivity. It has been proven that current methods produce incorrect connectomes, also related to incorrect connectivity modelling: they disregard either System Theory and Bayesian Formalism. We introduce a system theoretic formalism of MEEG connectivity alongside its Bayesian inference: Hidden Hermitian Gaussian Graphical Model (H-HGGM). This is a source (state) GGM hidden by the MEEG observation equation, i.e. equivalent to a frequency domain Linear State Space Model (LSSM). It is unhidden due to the Type II Likelihood approximated representation: Expected Log-Likelihood (ELL). The essential contributions here are a theory for HGGM solvers, and its implementation for the subsequent Maximization step of the H-HGGM ELL. Its efficacy is demonstrated in high resolution EEG simulations and a Steady State Visual Evoked potential. Open source packages, to reproduce the results presented in this paper and to analyze external MEEG databases, are freely available online .
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This technical note presents an inference framework for hierarchically conditioned (embedded) Hermitian Gaussian Graphical Models (GGM). Our methodology, although extendible to several GGMs applications, is mainly centered on the specific neuroscientific context encompassing the estimation of brain sources activityand connectivity from Magnetoencephalography or Electroencephalography (MEEG) signals. This will be conceived in a series of papers with two parts. Part I: Analysis of a single frequency component from a single subject. Part II: Analysis of the Cross-Spectra for multiple subjects in a population. The strategy is based on Maximum Posterior Analysis, via the Expectation Maximization (EM) algorithm, of the Hermitian GGMs Parameters and Hyperparameters. The Hyperparameters set, which comprises MEEG signal noise covariance and source precision (connectivity) matrices of the MEEG generative model, is informed with Priors of special interest in the field and whose implementation in GGM estimation still poses a technical challenge. The theory presented here encapsulates two essential contributions . First: The formulation of a target function for the connectivity estimator that resembles an effective GGM of source activity, which is merged to the maximization
Statistical analysis of functional images (signals) allows the assessment of neural connectivity patterns at mesoscopic scale. Scalp fields, recorded with Magneto-Electro Encephalography MEEG, strongly correlate to the average effect of synaptic activity, i.e. Primary Current Density (PCD) at the Gray Matter space. Nevertheless, what is observed at the sensor space with either technique is a superposition of projected fields, from the whole Gray-Matter PCD. That is the reason for a major pitfall of MEEG analysis methods, distorted reconstruction of source activity and its connectivity or "Leakage". There has been a quest for the modelling to counterattack Leakage of both activation and connectivity. It has been proven that current methods produce incorrect connectomes, also related to incorrect connectivity modelling: they disregard either System Theory and Bayesian Formalism. We introduce a system theoretic formalism of MEEG connectivity alongside its Bayesian inference: Hidden Hermitian Gaussian Graphical Model (H-HGGM). This is a source (state) GGM hidden by the MEEG observation equation, i.e. equivalent to a frequency domain Linear State Space Model (LSSM). It is unhidden due to the Type II Likelihood approximated representation: Expected Log-Likelihood (ELL). The essential contributions here are a theory for HGGM solvers, and its implementation for the subsequent Maximization
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iterative EM strategy. Second: A new procedure for the estimation of the source activity effective GGM. These theoretical contributions provide for an explicit and highly optimal connectivity estimation procedure
H-HGGM ELL. Its efficacy is demonstrated in high resolution EEG simulations and a Steady State Visual Evoked potential. Open source packages, to reproduce the results presented in this paper and to analyze external MEEG databases, are freely available online