The development of computational modelling in neuroscience

The field of computational modelling in neuroscience has been in existence for almost 100 years. During that time it has gone through several stages of development. From the 1930s, researchers in the field of mathematical biophysics conceived of mathematical and physical models of biophysical phenomena. In the more neural applications, there was work on how networks of nerve cells could store and retrieve information through the adaptive tuning of the thresholds of selected nerve cells (Shimbel, 1950), on linking psychophysical judgements with the underlying neural mechanisms (Landahl, 1939) and mathematical accounts of how nerve cells could act as logical elements and what the computational capabilities of a network of such elements were (McCulloch and Pitts, 1943). From the 1950s onwards there was interest in treating the brain as a uniform medium and calculating its modes of behaviour, such as the conditions under which it would support the propagation of waves of activity across it (Beurle, 1956). Much of this work laid the foundation for the field of artificial neural networks in the 1950s and 1960s.

Modelling the spatially distributed neuron

The basis of modelling the electrical properties of a neuron is the RC electrical circuit representation of passive membrane, consisting of a capacitor, leak resistor and a leak battery (Figure 2.14). Active membrane channels may be added by, for example, following the Hodgkin–Huxley approach. Further frameworks for modelling the myriad of ion channels found in neuronal membrane are covered in Chapter 5. If we are interested in voltage changes in more than just an isolated patch of membrane, we must consider how voltage spreads along the membrane. This can be modelled with multiple connected RC circuits. This approach is used widely and often referred to as multi-compartmental modelling or, more simply, compartmental modelling.

To understand the nervous system of even the simplest of animals requires an understanding of the nervous system at many different levels, over a wide range of both spatial and temporal scales. We need to know at least the properties of the nerve cell itself, of its specialist structures such as synapses, and how nerve cells become connected together and what the properties of networks of nerve cells are.

The complexity of nervous systems make it very difficult to theorise cogently about how such systems are put together and how they function. To aid our thought processes we can represent our theory as a computational model, in the form of a set of mathematical equations. The variables of the equations represent specific neurobiological quantities, such as the rate at which impulses are propagated along an axon or the frequency of opening of a specific type of ion channel. The equations themselves represent how these quantities interact according to the theory being expressed in the model. Solving these equations by analytical or simulation techniques enables us to show the behaviour of the model under the given circumstances and thus addresses the questions that the theory was designed to answer. Models of this type can be used as explanatory or predictive tools.

This field of research is known by a number of largely synonymous names, principally computational neuroscience, theoretical neuroscience or computational neurobiology.

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.

Introduction

Software development and maintenance under EMBOSS is made easy. EMBOSS has powerful inbuilt functionality which any native application can make use of with little or no additional coding, saving you a great deal of effort. It includes extensive C programming libraries for extending the core functionality and developing new applications. Well-defined processes are in place for key aspects such as quality assurance testing, installation, maintenance and support. General aspects are handled by the EMBOSS developers, leaving you to support the parts specific to your own software.

Inbuilt functionality

Major inbuilt areas of functionality include:

• Support for biological datatypes

• Support for common file formats

• Simple database configuration

• Command line handling

• Command line qualifiers

• Sequence and sequence feature specification.