Intracellular molecular signalling plays a crucial role in modulating ion channel dynamics, synaptic plasticity and, ultimately, the behaviour of the whole cell. In this chapter, we investigate ways of modelling intracellular signalling systems. We focus on calcium, as it plays an extensive role in many cell functions. Included are models of intracellular buffering systems, ionic pumps and calcium-dependent processes. This leads us to outline other intracellular signalling pathways involving more complex enzymatic reactions and cascades. We introduce the well-mixed approach to modelling these pathways and explore its limitations. Rule-based modelling can be used when full specification of a signalling network is infeasible. When small numbers of molecules are involved, stochastic approaches are necessary and we consider both population-based and particle-based methods for stochastic modelling. Movement of molecules through diffusion must be considered in spatially inhomogeneous systems.

This chapter introduces the physical principles underlying the models of electrical activity of neurons. Starting with the neuronal cell membrane, we explore how its permeability to different ions and the maintenance by ionic pumps of concentration gradients across the membrane underpin the resting membrane potential. We show how these properties can be represented by an equivalent electrical circuit, which allows us to compute the response of the membrane potential over time to input current. We conclude by describing the integrate-and-fire neuron model, which is based on the equivalent electrical circuit.

So far, we have been discussing how to model accurately the electrical and chemical properties of neurons and how these cells interact within the networks of cells forming the nervous system. The existence of a correct structure is essential for proper functioning of the nervous system, and we now discuss modelling of the development of the nervous system. Most existing models of developmental processes are not as widely accepted as, for example, the Hodgkin–Huxley model of nerve impulse propagation. They are designed on the basis of usually unverified assumptions to test a particular theory for neural development. Our aim is to cast light on the different types of issues that arise when constructing a model of development through discussing several case examples of models applied to particular neural developmental phenomena. We look at models constructed at the levels of individual neurons and of ensembles of nerve cells.

When modelling networks of neurons, generally it is not possible to represent each neuron of the real system in the model. It is therefore essential to carry out appropriate simplifications for which many design questions have to be asked. These concern how each neuron should be modelled, the number of neurons in the model network and how the neurons should interact. To illustrate how these questions are addressed, networks using various types of model neuron are described. In some cases, the properties of each model neuron are represented directly in the model, and in others the averaged properties of a population of neurons. We then look at several large-scale models intended to model specific brain areas. In some of these models, the neurons are based on the neurons reconstructed from extensive anatomical and physiological measurements. The advantages and disadvantages of these different types of models are discussed.

This chapter covers a spectrum of models for both chemical and electrical synapses. Different levels of detail are delineated in terms of model complexity and suitability for different situations. These range from empirical models of voltage waveforms to more detailed kinetic schemes, and to complex stochastic models, including vesicle recycling and release. Simple static models that produce the same postsynaptic response for every presynaptic action potential are compared with more realistic models incorporating short-term dynamics that produce facilitation and depression of the postsynaptic response. Different postsynaptic receptor-mediated excitatory and inhibitory chemical synapses are described. Electrical connections formed by gap junctions are considered.

Modelling a neural system involves the selection of the mathematical form of the model’s components, such as neurons, synapses and ion channels, plus assigning values to the model’s parameters. This may involve matching to the known biology, fitting a suitable function to data or computational simplicity. Only a few parameter values may be available through existing experimental measurements or computational models. It will then be necessary to estimate parameters from experimental data or through optimisation of model output. Here we outline the many mathematical techniques available. We discuss how to specify suitable criteria against which a model can be optimised. For many models, ranges of parameter values may provide equally good outcomes against performance criteria. Exploring the parameter space can lead to valuable insights into how particular model components contribute to particular patterns of neuronal activity. It is important to establish the sensitivity of the model to particular parameter values.

The nervous system consists of not only neurons, but also of other cell types such as glial cells. They can be modelled using the same principles as for neurons. The extracellular space (ECS) contains ions and molecules that affect the activity of both neurons and glial cells, as does the transport of signalling molecules, oxygen and cell nutrients in the irregular ECS landscape. This chapter shows how to model such diffusive influences involving both diffusion and electrical drift. This formalism also explains the formation of dense nanometre-thick ion layers around membranes (Debye layers). When ion transport in the ECS stems from electrical drift only, this formalism reduces to the volume conductor theory, which is commonly used to model electrical potentials around cells in the ECS. Finally, the chapter outlines how to model ionic and molecular dynamics not only in the ECS, but also in the entire brain tissue comprising neurons, glial cells and blood vessels.

In this book, we have aimed to explain the principles of computational neuroscience by showing how the underlying mechanisms are being modelled, together with presenting critical accounts of examples of their use. In some chapters, we have placed the modelling work described in its historical context where we felt this would be interesting and useful. We now make some brief comments about where the field of computational neuroscience came from and where it might be going.

Candidate models for how neurons or networks operate must be validated against experimental data. For this, it is necessary to have a good model for the measurement itself. For example, to compare model predictions from cortical networks with electrical signals recorded by electrodes placed on the cortical surface or the head scalp, the so-called volume conductor theory is required to make a proper quantitative link between the network activity and the measured signals. Here we describe the physics and modelling of electric, magnetic and other measurements of brain activity. The physical principles behind electric and magnetic stimulation of brain tissue are the same as those covering electric and magnetic measurements, and are also outlined.

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, as well as their measurable signals. It is designed to be read by people from a wide range of backgrounds from the neurobiological, 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, this term can mean ‘animal model’. In particle physics, the ‘standard model’ is a step towards a complete theory of fundamental particles and interactions. We therefore attempt to clarify what we mean by modelling and computational models in the context of neuroscience. We discuss what might be called the philosophy of modelling: general issues in computational modelling that recur throughout the book.

Plasticity in the nervous system describes its ability to adapt to change, in response to exposure to new information, fluctuations in the internal environment or external injury. In each case, computational models at different levels of detail are required. Given that memory traces are stored in modifiable synapses, to model the storage and retrieval of information requires models of the modifiable synapse and of a network of neurons. We discuss the processing ability of the network as a whole, given a particular mechanism for synaptic modification, modelled in less detail. Neurons also exhibit homeostatic plasticity, the ability to maintain their firing activity in response to a fluctuating environment. This can involve modulation of intrinsic membrane currents, as well as synaptic plasticity. It must work in concert with synaptic plasticity for learning and memory to enable neural networks to retain and recall stored information whilst still being responsive to new information.

In this chapter, a range of models with fewer details than those in previous chapters is considered. These simplified neuron models are particularly useful for incorporating into networks, as they are computationally more efficient and sometimes they can be analysed mathematically. Reduced compartmental models can be derived from large compartmental models by lumping together compartments. Additionally, the number of gating variables can be reduced whilst retaining much of the dynamical flavour of a model. These approaches make it easier to analyse the function of the model using the mathematics of dynamical systems. In the yet simpler integrate-and-fire model, first introduced inand elaborated on in this chapter, there are no gating variables, with action potentials being produced when the membrane potential crosses a threshold. At the simplest end of the spectrum, rate-based models communicate via firing rates rather than via individual spikes.