Numerical Methods for Neuronal Modeling 14.1 Introduction

In this chapter we will discuss some practical and technical aspects of numerical methods that can be used to solve the equations that neuronal modelers frequently encounter. We will consider numerical methods for ordinary diierential equations (ODEs) and for partial diierential equations (PDEs) through examples. A typical case where ODEs arise in neuronal modeling is when one uses a single lumped-soma compartmental model to describe a neuron. Arguably the most famous PDE system in neuronal modeling is the phenomenological model of the squid giant axon due to Hodgkin and Huxley. The diierence between ODEs and PDEs is that ODEs are equations in which the rate of change of an unknown function of a single variable is prescribed, usually the derivative with respect to time. In contrast, PDEs involve the rates of change of the solution with respect to two or more independent variables, such as time and space. The numerical methods we will discuss for both ODEs and PDEs involve replacing the derivatives in the diierential equations with nite diierence approximations to these derivatives. This reduces the diierential equations to algebraic equations. The two major classes of nite diierence methods we will discuss are characterized by whether the resulting algebraic equations explicitly or implicitly deene the solution at the new time value. We will see that the method of solution for explicit and implicit methods will vary considerably, as will the properties of the solutions of the resulting nite