This chapter discusses record keeping, like maintaining a lab notebook. Historically, lab notebooks were analog, pen-and-paper affairs. With so much work being performed on the computer and with most scientific instruments creating digital data directly, most record-keeping efforts are digital. Therefore, we focus on strategies for establishing and maintaining records of computer-based work. Keeping good records of your work is essential. These records inform your future thoughts as you reflect on the work you have already done, acting as reminders and inspiration. They also provide important details for collaborators, and scientists working in large groups often have predefined standards for group members to use when keeping lab notebooks and the like. Computational work differs from traditional bench science, and this chapter describes practices for good record-keeping habits in the more slippery world of computer work.

The Bianchi classification of 3-dimensional Lie algebras is introduced by the Schucking method: mapping the structure constants of the algebras into the set of 3×3 matrices, and then considering all the inequivalent combinations of eigenvalues and eigenvectors. A general 4-dimensional metric with a symmetry algebra of Bianchi type is derived. The general metric of a spatially homogeneous and isotropic (= Robertson–Walker, R–W) spacetime is derived. The possible Bianchi types of R–W spacetimes are demonstrated.

It is shown that the Riemann tensor can be calculated in a simpler way when the metric is represented by a basis of differential forms. The formulae for the basis components of the Christoffel symbols (called Ricci rotation coefficients) and of the Riemann tensor are derived. A still-easier way to calculate the Riemann tensor, by using algebraic computer programs, is briefly advertised.

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 the volume-conductor theory to compute the extracellular potential resulting from these membrane currents (step 2). We here give a brief introduction to the multicompartment modeling of neurons in step 1. The formalism presented, which has become the gold standard within the field, combines a Hodgkin-Huxley-type description of membrane mechanisms with the cable theory description of the membrane potential in dendrites and axons.

Readers who do not have strong schooling in physics can consult this book chapter for an introduction to key concepts such as ion fluxes, electric fields, electric potentials, and electric currents as well as for definitions of the ohmic, electrodiffusive, and capacitive currents that govern the electrodynamics of brain tissue. Building on the biophysical principles and approximations introduced here, we explain how the electric potential surrounding neurons can be computed based on the principles of current conservation and electroneutrality, and wegive a brief overview of modeling schemes designed to perform such computations on computers. Towards the end of the chapter, we show how the standard theory for computing extracellular potentials relates to Maxwell’s equations and list the approximations that we typically make when we apply these equations in a complex medium like brain tissue.

This is a very brief listing of topics that belong to the general area of relativity but were not included in this book.

Solutions of the Einstein and Einstein–Maxwell equations for spherically symmetric metrics (those of Schwarzschild and Reissner–Nordstr\“{o}m) are derived and discussed in detail. The equations of orbits of planets and of bending of light rays in a weak field are derived and discussed. Two methods to measure the bending of rays are presented. Properties of gravitational lenses are described. The proof (by Kruskal) that the singularity of the Schwarzschild metric at r = 2m is spurious is given. The relation of the r = 2m surface to black holes is discussed. Embedding of the Schwarzschild spacetime in a 6-dimensional flat Riemann space is presented. The maximal extension of the Reissner–Nordstr\“{o}m metric (by the method of Brill, Graves and Carter) is derived. Motion of charged and uncharged particles in the Reissner–Nordstr\“{o}m spacetime is described.

The metric tensor and the (pseudo-)Riemannian manifolds are defined. The results of the earlier chapters are specialised to this case, in particular the affine connection coefficients are shown to reduce to the Christoffel symbols. The signature of a metric, the timelike, null and spacelike vectors are defined and the notion of a light cone is introduced. It is shown that in two dimensions the notion of curvature agrees with intuition. It is also shown that geodesic lines extremise the interval (i.e. the ‘distance’). Mappings between Riemann spaces are discussed. Conformal curvature (= the Weyl tensor) is defined and it is shown that zero conformal curvature on a manifold of dimension >=4 implies that the metric is proportional to the flat one. Conformal flatness in three dimensions and the Cotton–York tensor are discussed. Embeddings of Riemannian manifolds in Riemannian manifolds of higher dimension are discussed and the Gauss–Codazzi equations derived. The Petrov classification of conformal curvature tensors in four dimensions with signature (+ - - -) is introduced at an elementary level.

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). In this chapter, we introduce ways to implement this scheme in computer simulations based on designated software such as LFPy, the NEURON simulator, or the Arbor simulator. We also introduce various methods for reducing the computational cost of simulating the extracellular potentials of large networks of neurons as well as introduce heuristic approximate signal prediction methods.

Spinors are defined, their basic properties and relation to tensors are derived. The spinor image of the Weyl tensor is derived and it is shown that it is symmetric in all four of its spinor indices. From this, the classification of Weyl tensors equivalent to Petrov’s (by the Penrose method) is derived. The equivalence of these two approaches is proved. The third (Debever’s) method of classification of Weyl tensors is derived, and its equivalence to those of Petrov and Penrose is demonstrated. Extended hints for verifying the calculations (moved to the exercises section) are provided.

When a neuron fires an action potential, it causes a rapid fluctuation in the extracellular potential. This fluctuation is referred to as a spike and is normally “visible” only close to the neuron it originates from. Spikes are typically studied experimentally by high-pass filtering the extracellular potential. Here, we use computer simulations and approximate analytical formulas of spikes to explore how the amplitude and shape of spikes depend on various factors such as (i) the morphology of the neuron, (ii) the presence of active ion channels in the neuron’s dendrites, (iii) the part of the neuron (soma vs. dendrite) where the spike is recorded, (iv) the distance from the neuron the spike is recorded, and (v) the location in the neuron that the action potential is initiated. We also briefly discuss how the presence of the electrode can affect spike recordings as well as how to analyze data containing overlapping spikes from several neurons simultaneously.