Models of extracellular potentials are typically based on treating brain tissue as a continuous volume conductor. An important parameter, or sometimes variable, in volume-conductor theory is the conductivity. Here, we present both theoretical and experimental estimates of the conductivity of brain tissue. A common modeling approximation is to assume that the conductivity does not vary with position, is the same in all directions, and does not depend on the frequency of the electric signal. With references to both experimental and theoretical studies, we discuss whether these approximations are reasonable, and we introduce ways to relax these approximations in models.

The logical and observational problems of Newton’s theory of gravitation that led Einstein to think about general relativity are briefly presented. In particular, this was the anomalous orbital motion of Mercury and the failed attempts to explain it within Newton’s theory. The local equivalence of inertial and gravitational forces is demonstrated.

The need for differential geometry is explained by considering the construction of parallel straight lines running far from each other in Euclidean space. Generalisation of the notion of parallelism to curved surfaces is explained.

We here round off a book on biophysical foundations and computational modeling of electric and magnetic signals in the brain. We summarize some key insights from such modeling, and we clear up some common misconceptions about extracellular potentials. We address the main limitations with the standard modeling framework used to compute extracellular potentials, discussing the uncertainty in model parameters and its neglect of ephaptic interactions between active neurons. We identify what we believe are key areas of future applications and give an outlook for future modeling challenges.

The description of motion of a continuous medium in curved spacetime is introduced and related to the corresponding Newtonian description. Expansion, acceleration, shear and rotation of the medium are defined and interpreted. The Raychaudhuri equation and other evolution equations of hydrodynamical quantities are derived. A simple example of a singularity theorem is presented. Relativistic thermodynamics is introduced and it is shown that a thermodynamical scheme is guaranteed to exist only in such spacetimes that have an at least 2-dimensional symmetry group.

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 geometric optics approximation to Maxwell equations is derived. The redshift and the description of bundles of rays via expansion, shear and rotation are defined. Equations of propagation of these optical tensors are derived. The proofs of the Goldberg - Sachs theorem and of the reciprocity theorem are presented. The equations of the Fermi - Walker transport and of the position drift of light sources are derived.

The Lemaitre–Tolman class of cosmological models (spherically symmetric inhomogeneous metrics obeying the Einstein equations with a dust source) is derived and discussed in much detail, from the point of view of its geometry and its applications to cosmology. It is shown that these metrics can be used to describe the formation of cosmic voids and of galaxy clusters out of small perturbations of homogeneity at the time of emission of the cosmic microwave background radiation. Apparent horizons for central and noncentral observers, the formation of black holes, the existence and avoidance of shell crossings, the equations of redshift and the generation and meaning of blueshift are discussed. A simple example of a shell focussing singularity is derived. Among the cosmological applications are: solving the horizon problem without inflation, mimicking the accelerating expansion of the Universe by mass-density inhomogeneities in a decelerating model, drift of light rays, lagging cores of Big Bang, misleading conclusions drawn from observed mass distribution in redshift space.

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 covariant derivative is introduced via its postulated properties (the same as of ordinary derivative, plus the requirement that it produces tensor densities when acting on tensor densities). It is shown that tensor densities of arbitrary rank can be represented by sets of scalars – the projections on vector bases in the tangent space to the manifold. The coefficients of affine connection are defined using these bases, and the explicit formula for a covariant derivative of an arbitrary tensor density is derived.

The family of the Szekeres–Szafron, shell solutions of Einstein’s equations, is derived and discussed in detail. The discussion contains, among other things, the interpretation of the Szekeres coordinates, the invariant definitions of the whole family, the class II ($\beta,_z = 0$) family as a limit of the class I family, matching the Szekeres metric to the Schwarzschild metric (the class II S metric can be matched to Schwarzschild only inside the event horizon), conditions for absence of shell crossings, the description of the mass dipole, the apparent horizons, the Goode–Wainwright representation and a brief listing of recommended further reading on geometric and astrophysical properties of these solutions.