In this chapter we show that many optimization problems can be reduced to a network flow problem. This polynomially solvable problem is a powerful model, which has found a remarkable array of applications. Roughly stated, in a network flow problem, one is given a transportation network and is required to find the optimal way of sending some content through this network. The chapter covers basic primitives around a general flow formulation called the minimum-cost flow.

This chapter gives an introduction to complexity analysis, data representations, and reductions. In addition, the Knuth–Morris–Pratt algorithm is covered to give some taste of dynamic programming – a technique introduced in Chapters 4 and 6 and used extensively thereafter.

A pragmatic problem arising in the analysis of biological sequences is that collections of genomes, and especially collections of read sets consisting of material from many species, occupy too much space. This chapter explores techniques to efficiently compress such collections. Several algorithms related to Lempel–Ziv factorization are covered, as well as the prefix-free parsing technique to run-length encode the Burrows–Wheeler transform of a collection of genomes.

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.