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.

The membrane potential of a neuron varies widely across the spatial extent of a neuron. The membrane may have spatially distinct distributions of ion channels and synaptic inputs arrive at different dendritic locations and propagate to the cell body. The membrane potential varies along axons, as the action potential propagates. We therefore need neuron models that include spatial, as well as temporal, dimensions. The most common approach is compartmental modelling in which the spatial extent of a neuron is approximated by a series of small compartments, each assumed to be isopotential. In limited cases of simple neuron geometry, analytical solutions for the membrane potential at any point along a neuron can be obtained through the use of the cable theory. We describe both modelling approaches here. Two case studies demonstrate the power of compartmental modelling: (1) action potential propagation along axons; and (2) synaptic signal integration in pyramidal cell dendrites.

This chapter presents the first quantitative model of active membrane properties, the Hodgkin–Huxley model. This was used to calculate the form of action potentials in the squid giant axon. Our step-by-step account of the construction of the model shows how Hodgkin and Huxley used the voltage clamp method to produce the experimental data required to construct mathematical descriptions of how the sodium, potassium and leak currents depend on the membrane potential. Simulations of the model produce action potentials similar to experimentally recorded ones and account for the threshold and refractory effects observed experimentally. Whilst subsequent experiments have uncovered limitations in the Hodgkin–Huxley model descriptions of the currents carried by different ions, the Hodgkin–Huxley formalism is a useful and popular technique for modelling channel types.

There are many types of active ion channel beyond the squid giant axon sodium and potassium voltage-gated ion channels studied in , including channels gated by ligands such as calcium. This chapter presents methods for modelling the kinetics of any voltage-gated or ligand-gated ion channel. The formulation used by Hodgkin and Huxley of independent gating particles can be extended to describe many types of ion channel. This formulation is the foundation for thermodynamic models, which provide functional forms for the rate coefficients derived from basic physical principles. To improve on the fits to data offered by models with independent gating particles, the more flexible Markov models are introduced. When and how to interpret kinetic schemes probabilistically to model the stochastic behaviour of single ion channels will be considered. Experimental techniques for characterising channels are outlined, and an overview of the biophysics of channels relevant to modelling channels is given.

Richly illustrated in full colour and packed with examples from every major continent and wetland type, this third edition has been completely rewritten to provide undergraduates with a thoroughly accessible introduction to the basic principles. It divides the world’s wetlands into six principal types and presents six major causal environmental factors, arranged by importance and illustrated with clear examples, making it easy for instructors to plan tailored lectures and field trips and avoid overwhelming students with unnecessary detail. It retains its rigour for more advanced students, with sections on research methods and experiments, and over a thousand classic and contemporary references. Each chapter ends with questions that review the content covered and encourage further investigation. With expanded sections on topical issues such as sea level rise, eutrophication, facilitation and the latest approaches to restoration and conservation, the new edition of this prize-winning textbook is a vital resource for wetland ecology courses.

While the previous chapter covered probability on events, in this chapter we will switch to talking about random variables and their corresponding distributions. We will cover the most common discrete distributions, define the notion of a joint distribution, and finish with some practical examples of how to reason about the probability that one device will fail before another.