Parallel transport of vectors and tensor densities along curves is defined using the covariant derivative. A geodesic is defined as such a curve, along which the tangent vector, when parallely transported, is collinear with the tangent vector defined at the endpoint. Affine parametrisation is introduced.

The curvature tensor is defined via the commutators of second covariant derivatives acting on tensor densities. It is shown that curvature is responsible for the path-dependence of parallel transport. Algebraic and differential identities obeyed by the curvature tensor are derived. The geodesic deviation is defined, and the equation governing it is derived.

The diffusion of ions in the extracellular space of the brain is normally assumed to have negligible effects on extracellular potentials. However, during periods of intense neural activity or in pathological conditions such as spreading depression, concentration gradients in brain tissue can become quite pronounced, and the effects of diffusion on electric potentials cannot be a priori neglected. We here present the theory for computing diffusion potentials, and we evaluate whether diffusion potentials can become “visible” within the frequency range considered in standard LFP recordings.

Maxwell’s equations in curved spacetime are presented, and Einstein’s equations with electromagnetic field included in the sources are derived. The attempt to unify electromagnetism with gravitation in the Kaluza–Klein theory is presented.

The derivation of the Einstein equations is presented following Einstein’s method. Hilbert’s derivation (from a variational principle) is also presented. The Newtonian limit of Einstein’s theory is discussed. A Bianchi type I solution of Einstein’s equations with a dust source is derived. A brief review of other theories of gravitation (Brans–Dicke, Bergmann–Wagoner, Einstein–Cartan and Rosen) is presented. The matching conditions for different metrics are derived. The weak-field approximation to general relativity is presented.

The Robertson–Walker metrics are presented as the simplest candidates for the models of our observed Universe. The Friedmann solutions of the Einstein equations (which follow when a R–W metric is taken as an ansatz), with and without the cosmological constant, are derived and discussed in detail. The Milne–McCrea Newtonian analogues of the Friedmann models are derived. Horizons in the R–W models are discussed following the classical Rindler paper. The conceptual basis of the inflationary models is critically reviewed.

This chapter contains detailed hints on how to solve the more difficult exercises and verify some other complicated calculations.

The local field potential (LFP) is the low-pass filtered extracellular potential recorded inside brain tissue. Unlike spikes, which reflect neuronal action potentials and thus the output of neurons, the LFP is believed to predominantly reflect the synaptic inputs to neurons. Here, we use computer simulations and approximate analytical formulas of LFPs from single neurons and populations of neurons to give a comprehensive overview of the various factors that can contribute to shaping the LFP and its frequency content. We consider the effects of neural morphology, intrinsic dendritic filtering, synaptic distributions, synchrony in synaptic inputs, the position of the recording electrode, and possible contributions from action potentials, calcium spikes, NMDA spikes, and active subthreshold dendritic ion channels.

The plane- and hyperbolically symmetric counterparts of the L–T models (i.e. the Ellis solutions), and generalisations of all three classes to charged dust source are derived and discussed. It is shown that the most natural interpretation of the plane-symmetric Ellis metric is an expanding or contracting family of 2-dimensional flat tori. The proof of the Ori theorem that for a spherically symmetric weakly charged dust ball shell crossings will block the bounce through the minimal radius is copied in detail. A subcase left out by Ori is discussed, but it will also lead to a shell crossing, only at the other side of the minimal radius. In this special case, a peculiar direction-dependent singularity is present: at the centre the matter density becomes negative for a short period before and after the bounce. The Datt–Ruban solution, its generalisation to charged dust source and the matching of both these solutions to, respectively, the Schwarzschild and Reissner–Nordstr\“{o}m solutions are presented and discussed. In the matched configuration the DR region stays inside the Schwarzschild or RN event horizon.

It is shown how the assumption of symmetry implies the Killing equations (more generally, invariance equations of arbitrary tensors are derived and discussed). It is also shown how to find the symmetry transformations of a manifold given the Killing vectors. The Lie derivative is introduced, and it is shown that the algebra of a symmetry group always has a finite dimension, not larger than n(n+1)/2, where nis the dimension of the manifold. Conformal symmetries are defined and it is shown that the algebra of the conformal symmetry group has dimension not larger than (n+1)(n+2)/2. The metric of a spherically symmetric 4-dimensional manifold is derived from the Killing equations, and its general properties are discussed. Explicit formulae for the conformal symmetries of a flat space of arbitrary dimension are given.

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.