3 - Expressing conceptual models in mathematical terms

Summary

But this critical act is not always unattended with the saddest and most fatal casualties.

Computational neuronal modeling usually focuses on voltage and current in excitable cells, but it is often necessary to represent other processes such as chemical reactions, diffusion, and the behavior of electronic instrumentation. These phenomena seem quite different from each other, and each has evolved its own distinct “notational shorthand.” As these specialized notations have particular advantages for addressing domain-specific problems, NEURON has provisions that allow users to employ each of them as appropriate (see Chapter 9). Apparent differences notwithstanding, there are fundamental parallels among these notations that can be exploited at the computational level: all are equivalent to sets of algebraic and differential equations. In this chapter, we will explore these parallels by examining the mathematical representations of chemical reactions, electrical circuits, and cables.

Chemical reactions

A natural first step in thinking about voltage-dependent or ligand-gated channel models or elaborate cartoons of dynamic processes is to express them with chemical reaction notation (i.e. kinetic schemes) (Fig. 3.1). Kinetic schemes focus attention on conservation of material (in a closed set of reactions, material is neither created or destroyed) and flow of material from one state to another.