Computer Modelling of the Sinoatrial Node - Biopacemaking - page 122

I Ca,T , respectively), I f , I NaCa , and a sustained inward current ( I st ). Furthermore, the
acetylcholine-sensitive outward potassium current ( I K,ACh ) provides vagal control of
pacemaker rate. Each of these currents provides a potential target for pacemaker
regulation. Further insight into pacemaker mechanisms and control of pacemaker
activity may be obtained through the use of computer models of electrical activity of
SA nodal cells. The aim of the present review is to provide an overview of the
available single cardiac cell models, with particular emphasis on single SA nodal cell
models, and the tools that are available for non-experts in computer modelling to run
these models. This review is thus intentionally restricted to single cardiac cell models.
Propagation of pacemaker activity is addressed elsewhere in this issue in a review by
Joyner et al. ([40], Chapter 8).
2 Development of Cardiac Cell Models
Back in 1928, van der Pol and van der Mark [84] presented the first mathematical
description of the heartbeat, which is in terms of a relaxation oscillator. Their work
has given rise to a family of models of nerve and heart 'cells' (excitable elements) in
terms of their key properties, i.e. excitability, stimulus threshold, and refractoriness.
These models are relatively simple with a minimum number of equations and
variables [23, 83]. Because they are compact, these models have been widely used in
studies of the spread of excitation in tissue models consisting of large numbers of
interconnected 'cells' (e.g. [83]). A major drawback of this family of models is the
absence of explicit links between the electrical activity and the underlying
physiological processes like the openings and closures of specific ion channels.
Today's sophisticated cardiac cell models provide such links and are all built on the
framework defined by the seminal work of Hodgkin and Huxley [34], for which they
received the 'Nobel Prize in Physiology or Medicine' in 1963.
2.1 The Seminal Hodgkin-Huxley Model
Hodgkin and Huxley investigated the electrical activity of the squid giant axon, on
which they published a series of five (now classical) papers in 1952. In their
concluding paper, they summarized their experimental findings and presented “a
quantitative description of membrane current and its application to conduction and
excitation in nerve” [34]. This “quantitative description” included a mathematical
model derived from an electrical equivalent of the nerve cell membrane. They
identified sodium and potassium currents flowing across the giant axon membrane and
represented these in terms of the sum of conductive components, what we now
identify as ion channels, and membrane capacitance. Figure 1 shows the associated
electrical circuit diagram in its simplest form. The membrane current ( I m ) is a function
of the voltage across the cell membrane ( V m ), the equilibrium potential ( E m ) of the
ions carrying this membrane current, and the membrane conductance ( g m ). Because
no current is entering or leaving the electrical circuit, the sum of the capacitative
current and I m equals zero. As a consequence (Fig. 1), we have
C m × d V m /d t = - I m (1)