Abstract-A solution of the forward problem is an important component of any method for computing the spatio-temporal activity of the neural sources of magnetoencephalography (MEG) and electroencephalography (EEG) data. The forward problem involves computing the scalp potentials or external magnetic field at a finite set of sensor locations for a putative source configuration. We present a unified treatment of analytical and numerical solutions of the forward problem in a form suitable for use in inverse methods. This formulation is achieved through factorization of the lead field into the product of the moment of the elemental current dipole source with a "kernel matrix" that depends on the head geometry and source and sensor locations, and a "sensor matrix" that models sensor orientation and gradiometer effects in MEG and differential measurements in EEG. Using this formulation and a recently developed approximation formula for EEG, based on the "Berg parameters," we present novel reformulations of the basic EEG and MEG kernels that dispel the myth that EEG is inherently more complicated to calculate than MEG. We also present novel investigations of different boundary element methods (BEM's) and present evidence that improvements over currently published BEM methods can be realized using alternative error-weighting methods. Explicit expressions for the matrix kernels for MEG and EEG for spherical and realistic head geometries are included.Index Terms-Boundary element method (BEM), electroencephalogram (EEG), forward model, head modeling, realistic head model, spherical head model.
An array of biomagnetometers may be used to measure the spatio-temporal neuromagnetic field or magnetoencephalogram (MEG) produced by neural activity in the brain. A popular model for the neural activity produced in response to a given sensory stimulus is a set of current dipoles, where each dipole represents the primary current associated with the combined activation of a large number of neurons located in a small volume of the brain. An important problem in the interpretation of MEG data from evoked response experiments is the localization of these neural current dipoles. We present here a linear algebraic framework for three common spatio-temporal dipole models: i) unconstrained dipoles, ii) dipoles with a fixed location, and iii) dipoles with a fixed orientation and location. In all cases, we assume that the location, orientation, and magnitude of the dipoles are unknown. With a common model, we show how the parameter estimation problem may be decomposed into the estimation of the time invariant parameters using nonlinear least-squares minimization, followed by linear estimation of the associated time varying parameters. A subspace formulation is presented and used to derive a suboptimal least-squares subspace scanning method. The resulting algorithm is a special case of the well-known MUltiple SIgnal Classification (MUSIC) method, in which the solution (multiple dipole locations) is found by scanning potential locations using a simple one dipole model. Principal components analysis (PCA) dipole fitting has also been used to individually fit single dipoles in a multiple dipole problem. Analysis is presented here to show why PCA dipole fitting will fail in general, whereas the subspace method presented here will generally succeed. Numerically efficient means of calculating the cost functions are presented, and problems of model order selection and missing moments are discussed. Results from a simulation and a somatosensory experiment are presented.
SummaryGeneral formulas are presented for computing a lower bound on localization and moment error for electroencephalographic (EEG) or magnetoencephalographic (MEG) current source dipole models with arbitrary sensor array geometry. Specific EEG and MEG formulas are presented for multiple dipoles in a head model with 4 spherical shells. Localization error bounds are presented for both EEG and MEG for several different sensor configurations. Graphical error contours are presented for 127 sensors coveting the upper hemisphere, for both 37 sensors and 127 sensors covering a smaller region, and for the standard 10-20 EEG sensor arrangement. Both 1-and 2-dipole cases were examined for all possible dipole orientations and locations within a head quadrant. The results show a strong dependence on absolute dipole location and orientation. The results also show that fusion of the EEG and MEG measurements into a combined model reduces the lower bound. A Monte Carlo simulation was performed to check the tightness of the bounds for a selected case. The simple head model, the low power noise and the few strong dipoles were all selected in this study as optimistic conditions to establish possibly fundamental resolution limits for any localization effort. Results, under these favorable assumptions, show comparable resolutions between the EEG and the MEG models, but accuracy for a single dipole, in either case, appears limited to several millimeters for a single time slice. The lower bounds increase markedly with just 2 dipoles. Observations are given to support the need for full spatiotemporal modeling to improve these lower bounds. All of the simulation results presented can easily be scaled to other instances of noise power and dipole intensity.
Subjects avoided shock by pressing on one lever under an unsignalled condition, but by pressing a separate lever they changed the condition to signalled avoidance for 1-min periods. Signalled avoidance periods were identified by a correlated stimulus. All eight subjects responded to change the unsignalled schedule to a signalled one. Once contact with signalled avoidance was made, subjects continued responding to remain in that condition. Other tests showed that changeover responding was greater when the correlated stimulus was presented without the signal than when the signal was presented without the correlated stimulus. An analysis based upon shock and shock-free periods is presented.A considerable body of literature exists that describes the different behavioral characteristics generated by signalled and unsignalled Sidman avoidance. It has been found that subjects under discriminated avoidance consistently "wait" for the signal, (Hyman, 1969;Sidman, 1955;Ulrich, Holz, and Azrin, 1964) rather than avoid in its absence. It has also been shown that both response and shock rates are lower when signals are used. Based upon these findings it would be reasonable to assume that, given a choice, subjects would prefer signalled over unsignalled avoidance. The present research tested this assumption and focused on factors thought to affect relative aversiveness of signalled and unsignalled avoidance conditions. Specifically, this study attempted to determine whether a situation involved signalled free-operant avoidance was preferred to one involving unsignalled avoidance. A changeover procedure was introduced that allowed subjects to control the condition in effect. This procedure required that subjects avoid unsignalled shock by pressing on one lever (avoidance) but by pressing on a second lever (changeover) they could convert the schedule to signalled avoidance for 1-min periods. METHOD SubjectsEight experimentally naive female albino rats of the Sprague-Dawley strain (Holtzman) between 90 to 125 days old served. ApparatusAll subjects were tested in Foringer operant conditioning chambers modified so-that the grid bars were perpendicular to the levers. Two boxes, each with two levers, were enclosed in IAC acoustical chambers. The third was housed in a Foringer acoustical apparatus.
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