Here is a list of resources related to computational neuroscience modelling. Most of these are resources that, at the time of writing, are available as open source software, but we cannot say for how long they will continue to be available in this way. Please refer to our web site, compneuroprinciples.org, for more up-to-date information.

Simulators

If the solution of a computational model is the evolution of a quantity, such as membrane potential or ionic concentration, over time and space, it constitutes a simulation of the system under study. Often simulated quantities change continuously and deterministically, but sometimes quantities can move between discrete values stochastically to represent, for example, the release of a synaptic vesicle or the opening and closing of an ion channel.

The process of describing and implementing the simulations of complex biophysical processes efficiently is an art in itself. Fortunately, for many of the models described in this book, in particular, models of the electrical and chemical activity of neurons, and to an extent the models of networks, this problem has been solved. An abundance of mature computer simulation packages exists, and the problem is in choosing a package and learning to use it.

Synaptic input

So far we have considered neuronal inputs in the form of electrical stimulation via an electrode, as in an electrophysiological experiment. Many neuronal modelling endeavours start by trying to reproduce the electrical activity seen in particular experiments. However, once a model is established on the basis of such experimental data, it is often desired to explore the model in settings that are not reproducible in an experiment. For example, how does the complex model neuron respond to patterns of synaptic input? How does a model network of neurons function? What sort of activity patterns can a network produce? These questions, and many others besides, require us to be able to model synaptic input. We discuss chemical synapses in most detail as they are the principal mediators of targeted neuronal communication. Electrical synapses are discussed in Section 7.7.

Numerical integration methods

Most of the mathematical models presented in this book involve differential equations describing the evolution in time and space of quantities such as membrane potential or calcium concentration. The differential equations are usually too complex to allow an analytical solution that would enable the explicit calculation of a value of, say, voltage at any particular time point or spatial position. The alternative is to derive algebraic expressions that approximate the differential equations and allow the calculation of quantities at specific, predefined points in time and space. This is known as numerical integration. Methods for defining temporal and spatial grid points and formulating algebraic expressions involving these grid points from the continuous (in time and space) differential equations are known as finite difference and finite element methods.

It is not our intention here to provide full details of these numerical integration methods. Instead, we will outline some of the simplest methods to illustrate how they work. This includes the Crank–Nicholson method (Crank and Nicholson, 1947), which is widely used as a basis for solving the cable equation. Further details on these methods as applied to neural models can be found in Carnevale and Hines (2006) and Mascagni and Sherman (1998).

Up until this point, the focus has been on adding details to our neuron models. In this chapter we take the apparently paradoxical step of throwing away a lot of what is known about neurons. Given all the painstaking work that goes into building detailed models of neurons, why do this? There are at least two reasons:

1. (1) We wish to explain how a complicated neural model works by stripping it down to its bare essentials. This gives an explanatory model in which the core mechanisms have been exposed and so are easier to understand.

2. […]

What is this book about?

This book is about how to construct and use computational models of specific parts of the nervous system, such as a neuron, a part of a neuron or a network of neurons. It is designed to be read by people from a wide range of backgrounds from the biological, physical and computational sciences. The word ‘model’ can mean different things in different disciplines, and even researchers in the same field may disagree on the nuances of its meaning. For example, to biologists, the term ‘model’ can mean ‘animal model’; to physicists, the standard model is a step towards a complete theory of fundamental particles and interactions. We therefore start this chapter by attempting to clarify what we mean by computational models and modelling in the context of neuroscience. Before giving a brief chapter-by-chapter overview of the book, we also discuss what might be called the philosophy of modelling: general issues in computational modelling that recur throughout the book.

Theories and mathematical models

In our attempts to understand the natural world, we all come up with theories. Theories are possible explanations for how the phenomena under investigation arise, and from theories we can derive predictions about the results of new experiments.

Over 100 types of ion channel are known. Each type of channel has a distinct response to the membrane potential, intracellular ligands, such as calcium, and extracellular ligands, such as neurotransmitters. The membrane of a single neuron may contain a dozen or more different types, with the density of each type depending on its location in the membrane.

Ionic concentrations and electrical response

Most work in computational neuroscience involves the construction and application of computational models for the electrical response of neurons in experimental and behavioural conditions. So far, we have presented the fundamental components and techniques of such models. Already we have seen that differences in particular ionic concentrations between the inside and outside of a cell are the basis of the electrical response. For many purposes our electrical models do not require knowledge of precise ionic concentrations. They appear only implicitly in the equilibrium potentials of ionic species such as sodium and potassium, calculated from the Nernst equation assuming fixed intra-and extracellular concentrations (Chapter 2). The relative ionic concentrations, and hence the equilibrium potentials, are assumed to remain constant during the course of the electrical activity our models seek to reproduce.