Introduction

In magnetoencephalography (MEG) and electroencephalography (EEG), scalp potentials and extracranial magnetic fields generated by electrical activity in the brain are detected non-invasively (Berger, 1929; Cohen, 1972) (for an overview of the methodology see, e.g., Hamalainen et al., 1993; Niedermeyer and Lopes da Silva, 1999; Michel et al., 2009; Hansen et al., 2010). MEG and EEG signals are superpositions of contributions from sources at different locations in the brain. Source estimation (also known as inverse modeling) refers to the problem of determining the spatiotemporal patterns of neural activity on the basis of the recorded signals (Figure 7.1). The specific goal in source estimation can be stated in two closely related ways: (a) to identify the locations of the sources of the measured signals as a function of time, or (b) to disentangle the contributions from different brain regions in the measured time-varying signals. The often used term source localization refers to the former, whereas spatiotemporal imaging emphasizes the latter, reflecting the use of MEG and EEG source estimation in the analysis of the dynamical activity in networks of brain areas.

A given source in the brain generates a characteristic spatial pattern of signals in arrays of MEG and EEG sensors. These patterns can be calculated by using a forward model (see Chapter 6). In source estimation, the measured spatial patterns of signals are analyzed in order to make inferences about the distribution of the sources in the brain.

Most of what we know about the biology of the brain has been obtained using a large variety of measurement techniques, from the intracellular electrode recordings used by Hodgkin and Huxley to understand the initiation of action potentials in squid axons to functional magnetic resonance imaging (fMRI), used to explore higher cognitive functions. To extract meaningful information from these measurements, one needs to relate them to neural activity, but this relationship is usually not trivial. For example, electroencephalograms (EEG) measure the summed electrical activity of many neurons, and relating the electrical signals of the electrodes to neural activity in specific brain areas requires a deep understanding of how these signals are formed. Therefore, the interpretation of measurements relies not only on an understanding of the physical measurement devices (what physical quantity is measured), but also on our current understanding of the brain (the relationship between the measured quantity and neural activity).

The biophysics of neurons is explained in great detail in a number of books. This book deals with the biophysical and mathematical principles of neural activity measurement, and provides models of experimental measures. We believe this should be useful for at least three broad categories of scientists: (1) neuroscientists who use these techniques in their own experimental protocols and need to interpret the results precisely, (2) computational neuroscientists who use the experimental results for their models, (3) scientists who want to develop new techniques or enhance existing techniques.

Introduction

The electroencephalogram (EEG) represents potential differences recorded from the scalp as function of time (Niedermayer and Lopes da Silva, 1987). The generators of the EEG consist of time-varying ionic currents generated in the brain by biochemical sources. These current sources also generate a small but measurable magnetic induction field, which can be recorded with magnetoencephalographic (MEG) equipment (Hämäläinen et al., 1993). When EEG and MEG are studied in the time or frequency domain, several rhythms can be discriminated that contain valuable information about the collective behavior of the living human brain as a neural network. In this chapter EEG and MEG are discussed in the spatial domain. We consider that these signals are recorded from multiple sensors with known positions and study the spatial distribution of EEG and MEG (in the sequel abbreviated as MEEG) in relation to the spatial distribution of the underlying sources.

More precisely, we consider the mathematical problem of predicting the spatial distribution of MEEG, from several physiological assumptions on the current sources. This problem is commonly named the “forward problem.” Solutions of the forward problem that are fast, accurate and practical are indispensable ingredients for the solution of the “inverse problem” or “backward problem,” which is the problem of extracting as much information as possible about the cerebral current sources, on the basis of MEEG data. Both the forward and the inverse problems are formulated within the framework of a certain mathematical model, wherein the underlying physiological assumptions are precisely formulated.

Introduction

Extracellular electric potentials, such as local field potentials (LFPs) or the electroencephalogram (EEG), are routinely measured in electrophysiological experiments. LFPs are recorded using micrometer-size electrodes, and sample relatively localized populations of neurons, as these signals can be very different for electrodes separated by 1 mm (Destexhe et al., 1999a) or by a few hundred micrometers (Katzner et al., 2009). In contrast, the EEG is recorded from the surface of the scalp using millimeter-scale electrodes and samples much larger populations of neurons (Niedermeyer and Lopes da Silva, 1998). LFPs are subject to much less filtering compared to EEG, because EEG signals must propagate through various media, such as cerebrospinal fluid, dura mater, cranium, muscle and skin. LFP signals are also filtered, because the recording electrode is separated from the neuronal sources by portions of cortical tissue. Besides these differences, EEG and LFP signals display the same characteristics during wake and sleep states (Steriade, 2003).

The observation that action potentials have a limited participation in the genesis of the EEG or LFPs dates from early studies. Bremer (1938, 1949) was the first to propose that the EEG is not generated by action potentials, based on the mismatch of the time course of EEG waves with action potentials. Eccles (1951) proposed that LFP and EEG activities are generated by summated postsynaptic potentials arising from the synchronized excitation of cortical neurons. Intracellular recordings from cortical neurons later demonstrated a close correspondence between EEG/LFP activity and synaptic potentials (Klee et al., 1965; Creutzfeldt et al., 1966a, 1966b).

Introduction

The most popular technique for investigating the functional organization and plasticity of the cortex involves the use of a single microelectrode. It offers the advantage of recording action potentials and subthreshold activity directly from cortical neurons with high spatial (point) and temporal (millisecond) resolution sufficient to follow real-time changes in neuronal activity at any location along a volume of cortex, with the disadvantage that recordings are invasive to the cortex. In order to assess the functional representation of a sensory organ (e.g. a finger, a whisker), neurons are recorded from different cortical locations and the functional representation of the organ is then defined as the cortical region containing neurons responsive to stimulation of that organ (i.e. neurons that have receptive fields localized at the sensory organ). A change in the spatial distribution of neurons responsive to a given sensory organ and/or in their amplitude of response is typically taken as evidence for plasticity in the functional representation of that sensory organ (Merzenich et al., 1984). As a cortical functional representation could comprise thousands to millions of neurons distributed over a volume of cortex, the use of a single microelectrode to map a functional representation and its plasticity requires many recordings across a large cortical region, recordings that can only be obtained in a serial fashion and require many hours to complete, thus the animal is typically anesthetized.