Electrical neuroimaging based on biophysical constraints
Introduction
The noninvasive three-dimensional reconstruction of the generators of the brain's electromagnetic activity measured at the scalp has been termed brain electromagnetic tomography (BET). In similarity with anatomical (magnetic resonance imaging; MRI) and functional tomographies (positron emission tomography or PET, single photon emission computed tomography or SPECT, and functional MRI or fMRI), the construction of a brain electromagnetic tomography requires the solution of an inverse problem. However, this inverse problem lacks a unique solution. In spite of this serious difficulty, there is an active past and ongoing research on this field because of the extreme clinical and research importance of the problem.
A reliable electromagnetic three-dimensional tomography is, hitherto, the only possible approach for noninvasively studying a direct reflection of neuronal activity in human subjects with the temporal resolution required to trace the dynamic behavior of the human brain. In contrast to hemodynamic techniques, electrically reconstructed tomographic images are directly linked to neuronal processes. Because of their high temporal resolution, these images provide information about the short time lived neuronal networks subserving sensory and cognitive events.
Historically, the predominant approaches for solving the BET associated inverse problem operated on the assumption that only a discrete number of generators (usually dipoles) were active at a given time or over a period (e.g., Mosher et al., 1992, Murray et al., 2002, Scherg, 1990, Sekihara et al., 2002). However, since the functional activation images produced by these approaches cannot be considered tomographic reconstructions, they will not be considered in further detail here (though see Table 1).
A second family of distributed solutions to the BET uses the general theory developed for linear underdetermined inverse problems. Underdetermined means that the number of available measurements is smaller than the number of brain sites where the activity is sought after. This mathematical theory has been developed or extensively reviewed by Bertero et al. (1985), Groetch (1984), Parker (1994), and Tikhonov and Arsenin (1977) among others. Solutions to these problems are typically stated in terms of a so-called regularization operator (Tikhonov and Arsenin, 1977) fulfilling a double task: (1) picking one of the multiple possible solutions by introducing in the formulation of the problem some a priori information about the true solution and (2) providing stability to the solution, that is, small variations in the data should not lead to large variations in the source configuration.
So far the distributed solutions proposed to the BET inverse problem that opted for a regularization-based approach have largely relied on mathematically driven operators (see Grave de Peralta and Gonzalez, 1999 for several examples). While these operators could be reasonable for general academic problems, they lack a direct physiological or physical basis. This explains why the introduction of anatomically, physiologically, and functionally based a priori information is receiving increased attention Bablioni et al., 2003, Dale and Sereno, 1993, Fuchs et al., 1999, Hauk et al., 2002, Liu et al., 2002, Phillips et al., 2002.
This paper shows properties of neurophysiological generators that are specific to them and thus can be, but have not been, used as general constraints to the inverse problem. In particular, it is shown that neurophysiological currents are ohmic and can therefore be expressed as gradients of potential fields. This fact is used to reformulate the inverse problem in more restrictive terms, providing the basis for the noninvasive estimation of intracranial local field potentials (LFPs) from scalp recorded EEG data. The ohmic character of the currents is further used to pick a single solution to the inverse problem by imposing to the solution a spatial structure dictated by the physical laws that describe the propagation of electric potentials and fields in biophysical media. This paper gives a detailed derivation of these constraints together with a description of the steps required for their mathematical implementation. Aimed at a multidisciplinary readership, it combines the rigorous mathematical derivations with intuitive explanations about their physical or physiological meaning. The possibilities offered by this method to provide reasonable information about the spatio-temporal aspects of brain processing are illustrated in the analysis of ERPs recorded from a healthy subject performing a visuo-motor reaction time task for which fMRI results from the same experiment are available.
The paper is structured as follows. We first consider the principles leading from microscopic (neuronal level) to macroscopic measurements (LFPs) as well as the particular equation governing the electromagnetic fields and the quasi-stationary approximation. This section finishes with a mathematically oriented section devoted to the statement and solution of the inverse problem. The third section describes the constraints used for the source model (ELECTRA), as well as the method to obtain a unique solution (LAURA) based on constraints derived from the physiology and the physics of the problem. To this follows a fourth section describing the experimental design employed for both the fMRI and the ERP data experiment. The type of analysis performed for both kinds of data and the inverse solution results are presented in this section that also discusses the neurophysiological interpretation of the results in light of the available experimental evidence. A final general discussion focuses on providing the intuitive reasoning underlying the incorporation of biophysical constraints into the solution of the BET and its experimental support. The results obtained in the analysis of experimental data in this and previous papers that considered separately this source model or the regularization strategy are used to argue in favor of this type of solution. Future possible applications of this method are similarly introduced.
Section snippets
Microscopic and macroscopic fields
Brain function is investigated at two different scales: (1) A microscopic level encompassing the activity of a single or few neurons studied by single or multiunit recordings in animals and (2) A macroscopic level reflecting the activity of neuronal ensembles recorded by either intracranial LFPs in patients or animals or by scalp-recorded electric and magnetic fields.
At the origin of all these measurements are identical neural phenomena. During cell activation, large quantities of positive and
Biophysical Constraints to solve the inverse problem
The previous section discussed the general mathematical formalism for solving linear inverse problems. However, their solution can be drastically improved by considering the physical or technical details concerning the particular inverse problem we want to solve. Consequently, in this section we consider the inclusion of a priori information derived from the biophysical laws characterizing the generation and propagation of electromagnetic fields in volume conductor media. In particular, we will
Data analysis
The previous section describes the approach to obtain a unique solution to the bioelectric inverse problem that relies on physically derived a priori information about the generators and the fields they produce in biological media. As any solution to underdetermined inverse problem, this solution should provide accurate results if the actual generators fulfill the properties we are incorporating as a priori information and is likely to fail otherwise. In principle, the best manner to assess the
Inverse solution activation
Statistical analysis of the estimated LFP yields a clear temporal separation of three response periods (Fig. 1). A first period (approximately 55–90 ms poststimulus) showed focal activation at ipsilateral visual areas as well as bilateral superior parietal areas and precuneus (Fig. 1a). Significant activation was also observed at the right infero-temporal gyrus and right insula. During a second period lasting from 100 to 200 ms (Fig. 1b) we observed spreading of activity to contralateral visual
Discussion
In the preceding sections, we described properties of biophysical generators that can be used to single out a unique solution to the bioelectromagnetic inverse problem. In particular, we showed that existing experimental evidence supports that ohmic currents produce both scalp and intracranial LFPs. The ohmic character of the currents is here used to derive a formulation of the inverse problem (ELECTRA) that aims to noninvasively estimate intracranial LFPs from scalp-recorded data. This
Conclusions
In this paper we described properties of neural generators that can be used to obtain a unique solution to the bioelectromagnetic inverse problem. We showed that electric fields and potentials within the brain are predominantly due to ohmic currents. On this basis we reformulated the inverse problem in terms of a restricted source model that allows one to noninvasively estimate Local Field Potentials (LFPs) in depth from scalp data. Incorporating as a priori information the physical laws
Acknowledgements
Swiss National Foundation Grants 3152A0-100745 and the MHV Foundation Grant 3234-069264 supported this work. Part of this research was conducted under support of a IM2 white paper grant.
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