The standard two-step scheme for modeling extracellular signals is to first compute the neural membrane currents using multicompartment neuron models (step 1) and next use volume-conductor theory to compute the extracellular potential resulting from these membrane currents (step 2). Here, we present the volume-conductor theory used in step 2. The neural output from step 1 can be represented in terms of (i) a set of point sources, (ii) a set of line sources, (iii) a current-source density, or (iv) one or several current dipoles. We derive equations for the extracellular potential under the approximations (i–iv), discuss the validity and applicability of the different approximations, and explain how they are related. We also discuss how to model the effects that the electrode itself can have on the measured extracellular potential.

The electrocorticographic (ECoG) signal is the electric potential recorded above the cortical surface and reflects the combined activity of large populations of neurons. As ECoG recordings are closer to the neuronal sources than the EEG recordings and further away than LFP recordings, approximations used when modeling LFPs and EEG signals can not a priori be used to model ECoG signals. Here, we give a brief overview of the challenges involved when modeling the ECoG signal and give an overview of previous modeling studies.

The electroencephalographic (EEG) signal is the electric potential recorded on the scalp, and it is believed to originate from the combined activity of large populations of neurons. In forward models of EEG signals, one typically (i) represents neuronal sources in terms of effective current dipoles, (ii) defines a head model, which is a specification of the conductivity profile for the medium between the sources and the recording position (brain tissue, cerebrospinal fluid, skull, scalp), and (iii) uses volume-conductor theory to compute the resulting electric potential at the scalp. In this chapter, we introduce the key theory and computational frameworks for modeling EEG signals. We illustrate how biophysically detailed models of neurons can be reduced to approximate equivalent dipoles, and we discuss further ways to simplify neural simulations in order to reduce the computational cost. Using a combination of computational modeling and analytical approximations, we analyze how various factors are involved in shaping the EEG signal.

The magnetoencephalographic signal is the magnetic field recorded outside the scalp of the head. It is believed to originate from the combined activity of large populations of neurons. In forward models of MEG signals, the neural output is typically represented in terms of equivalent current dipoles. We here go through the mathematical equations for computing MEG signals from current dipoles, present computer simulations of MEG signals using simplified models of the head geometry, and show how the predicted MEG signals depend on the chosen head model and orientations of the dipoles. We also present a formalism for modeling the magnetic field inside brain tissue.

Basic concepts of electromagnetic theory; Coulomb gauge; intensity of electromagnetic field. Electrons in an electromagnetic field: from the Lagrangian to the Hamiltonian; canonical momentum. Interaction Hamiltonian. Semiclassical approximation; weak-field limit. Electric dipole approximation. Calculation of the optical susceptibility by using the density matrix approach. From optical susceptibility to absorption coefficient. Momentum of an electron in a periodic crystal.