Mathematical models of the electrophysiology of cardiac cells have advanced considerably since the first model described by four differential equations published in the 1960s (Noble, 1962). In that model, much less was known even about basic cellular physiology: calcium currents had not yet been discovered, and the action potential in the model was maintained strictly as a balance between inward sodium and outward potassium currents. Nevertheless, the description of Purkinje cell transmembrane currents heralded the beginning of a new era. Since then, a large number of models (for a review, see Fenton & Cherry, 2008) with many more differential equations (up to nearly 100) have been developed to describe in detail many more currents and ion concentrations, to represent different types of cells, and to incorporate increasingly complex intracellular calcium dynamics. Ironically, although Purkinje cells were the subject of the first myocyte model and also of the first ‘second-generation’ myocyte model in 1985 that included varying potassium and sodium concentrations as well as pump and exchanger currents (DiFrancesco & Noble, 1985), Purkinje modelling has received little attention since that time until recently.
In a recent article in The Journal of Physiology Sampson et al. (2010) show they are one of several groups seeking to bring Purkinje modelling up to date with modelling efforts for other types of cardiac cells. Not only does their model include Markov formalism to describe several important ion channels, but it also incorporates channels known to be important in Purkinje cells but not necessarily in ventricular cells, such as both L- and T-type calcium currents and multiple isoforms of the sodium channel. Where possible, the current formulations are derived from human ion channel expression systems, a process that makes representing the biophysical mechanisms underlying channel function more straightforward. Their work facilitates the development of new hypotheses that can be tested experimentally. The authors’ inclusion of a working code as Supplemental data may hasten adoption of the model and is a good practice many modellers now adopt.
Models like that of Sampson et al. provide a useful update in terms of improving the fidelity and reliability of Purkinje electrophysiology modelling. Nevertheless, many steps remain to be taken. At the subcellular level, intracellular calcium processes are known to be different from those in ventricular cells because of the absence of or significant changes in T tubules in Purkinje cells (Boyden et al. 1989). To date, we know of no model that takes into account the effects of this structural difference as well as the different mechanical properties of Purkinje cells, which, unlike working ventricular myocytes, need not contract strongly.
More work will also be needed to identify and to address additional limitations of this and other recent Purkinje models. For example, many models are tested in a relatively narrow range of pacing rates. However, it is likely that model behaviour at both fast and slow rates will be important for Purkinje fibres: at rapid rates they can exhibit alternans that may be proarrhythmic, and slower rate dynamics may be important for understanding the development of triggered activity like early afterdepolarizations as well as the backup ventricular pacemaking function of Purkinje cells. So far the only Purkinje model known to exhibit alternans during rapid pacing is the original Noble model, which ironically also includes the least amount of detail. In addition, the model of Sampson et al., like many other models, is tested only for isolated cells, but when multiple coupled cells are simulated, it is possible for the action potential amplitude to decrease enough to affect the timing and dynamics of ion currents, which potentially can alter action potential shape, duration (Cherry & Fenton, 2007; Decker et al. 2009) and dynamics dramatically (Bueno-Orovio et al. 2008).
Some additional modelling issues that have yet to be addressed are specific to Purkinje fibres. While much work has been done to characterize the heart's anatomical structure, to date less emphasis has been placed on creating realistic Purkinje network structures. The fibres form a complex network with branches that subdivide and cross, producing a complex structure that has not yet been fully described. The depth to which they penetrate is not fully understood, although it is known to vary among species. The specific properties of the fibres’ insertion into the ventricular myocardium may be complex and have yet to be modelled in detail.
Although it is still necessary to characterize Purkinje cell behaviour under a broader range of physiological conditions, to understand how Purkinje fibres are arranged within the ventricles, and to analyse how they interact with ventricular myocardium, the study of Sampson et al. provides much needed improvements and updates to Purkinje cell modelling, and their model is likely to lead to novel hypotheses regarding Purkinje cell behaviour. After 48 years of mathematical modelling of cardiac electrophysiology, both anatomical and physiological models have become much richer and more detailed, but much work remains to be done. Perhaps the most important tasks at hand are to ensure that models are robust and reliable and to make effective use of the ever-increasing amount of modelling details to improve our understanding of the behaviour of the whole heart.