Forward and inverse modeling
From a methodological standpoint, MEG/EEG source modeling is referred to as an ‘inverse problem’, an ubiquitous concept, well-known to physicists and mathematicians in a wide variety of scientific fields: from medical imaging to geophysics and particle physics (Tarantola, 2004). The inverse problem framework helps conceptualize and formalize the fact that, in experimental sciences, models are confronted to observations to draw specific scientific conclusions and/or estimate some parameters that were originally unknown. Parameters are quantities that might be changed without fundamentally violating and thereby invalidating the theoretical model. Predicting observations from a model with a given set of parameters is called solving the forward modeling problem. The reciprocal situation where observations are used to estimate the values of some model parameters is the inverse modeling problem.
In the context of brain functional imaging in general, and MEG/EEG in particular, we are essentially interested in identifying the neural sources of external signals observed outside the head (non invasively). These sources are defined by their locations in the brain and their amplitude variations in time. These are the essential unknown parameters that MEG/EEG source estimation will reveal, which is a typical incarnation of an inverse modeling problem.
Forward modeling in the context of MEG/EEG consists in predicting the electromagnetic fields and potentials generated by any arbitrary source model, that is, for any location, orientation and amplitude parameter values of the neural currents. In general, MEG/EEG forward modeling considers that some parameters are known and fixed: the geometry of the head, conductivity of tissues, sensor locations, etc. This will be discussed in the next section.
As an illustration, take a single current dipole as a model for the global activity of the brain at a specific latency of an MEG averaged evoked response. We might choose to let the dipole location, orientation and amplitude as the set of free parameters to be inferred from the sensor observations. We need to specify some parameters to solve the forward modeling problem consisting in predicting how a single current dipole generates magnetic fields on the sensor array in question. We might therefore choose to specify that the head geometry will be approximated as a single sphere, with its center at some given coordinates.
Modeling illustrated: (a) Some unknown brain activity generates variations of magnetic fields and electric potentials at the surface of the scalp. This is illustrated by time series representing measurements at each sensor lead. (b) Modeling of the sources and of the physics of MEG and EEG. As naively represented here, forward modeling consists of a simplification of the complex geometry and electromagnetic properties of head tissues. Source models are presented with colored arrow heads. Their free parameters – e.g., location, orientation and amplitude – are adjusted during the inverse modeling procedure to optimize some quantitative index. This is illustrated here in (c), where the residuals – i.e., the absolute difference between the original data and the measures predicted by a source model – are minimized.