The relevance of these computations to propagation during re-entrant arrhythmias in ventricular tissue can be assessed by quantitative features of the re-entry -- the period and waveform of the action potential, the size of the medium within which re-entry can be initiated, the rapidly decaying transient and then slow ageing of the re-entrant wave (i.e. change of period and core shape due to slow processes), the irregular, jump-like motion of the re-entrant wave, and the velocity at which it can be moved by resonant perturbation. The relevance can be questioned, on the grounds that macro- and micro-anatomical detail is ignored, as are the effects of heterogeneity. We would argue that the reasonable correspondence between the computations and observations of re-entrant propagation means that cardiac tissue, in spite of its cellularity, three-dimensional anatomy and heterogeneity, behaves as a reaction diffusion system, and so methods for controlling spiral waves in such systems may be applicable to controlling propagation during re-entrant ventricular arrhythmias.
A key feature of re-entrant waves is their spatial scale - their
wavelength, and the mininum size of the medium within which they can be
initiated, or persist. The spatial scaling of reaction-diffusion models
is determined by the diffusion coefficent D, which was selected to give
a conduction velocity of 
. Canine ventricular
conduction velocities range from 0.14-0.25 m/s (transverse) to
0.5-0.8 m/s (longitudinal) - see Pressler et al. (1995). The spatial
scaling of the re-entrant waves is simply proportional to the
conduction velocity, and so will be smaller in media with reduced
conduction velocity, say in simulations of ischaemic tissue.
Although 
of a solitary action potential
of the single cell model (580 V/s) compares well with the values
of 
V/s obtained by Taniguchi et al. (1994) for guinea-pig
isolated ventricular myocytes, the computed value of for the propagating wavefronts are too high.
Taniguchi et al. obtained for
propagating action potentials as 
V/s (longitudinal
propagation) and 
V/s (transverse propagation). of repetitive actity are less, and are reduced
as the rate increases. However, the computations with the Oxsoft model
consistently give higher for
propagating action potentials than seen in ventricular tissue with the
same conduction velocity. This due to a mismatch between the behaviour
of the fast inward current system of the model and in ventricular
cells.
The period of the spiral wave of about 100-110 ms in our model corresponds to 120-240 ms in the human, 110-165 ms in the rabbit, 120-130 ms in the pig in situ, 120-250 ms in the sheep and 100-200 ms in the dog for epicardial tissue slices (Pertsov et al. 1993). Computations by Courtemanche & Winfree (1991) with the modified Beeler-Reuter model give a longer (190-300 ms) period.
The persistence of the spiral wave solution (i.e. their stability and lack of breakdown) is consistent with the experimental observations of Davidenko et al. (1993). The earlier computations of Panfilov & Holden (1990) for a Purkinje-fibre kinetics and Courtemanch & Winfree (1991) for Beeler-Reuter kinetics showed spontaneous spiral breakdown.
The transient, extended tip trajectories seen on initiating a spiral wave in figure 1(b,c) means that although a large medium is required to initiate a re-entrant wave, once established, it can survive in a smaller medium. A transient change in medium properties (say, a decrease in action potential duration produced by a transient ischaemic episode) could allow the creation of re-entry in the transient abnormal tissue, and this could persist even when the tissue properties returned to normal, even though the size of the tissue was smaller than the critical mass (Zipes et al. 1975).
The slow ageing, i.e. change of the re-entry features from one period to another, is analogous to that seen in atrial tissue models (Holden & Zhang 1995), in that it is due to the slow dynamics of some of the current, pump and concentration components. However, the key characteristic of meander in this ventricular model is the alternation between fast, almost linear, motions and slow, sharp turns. This gives a core with a large perimeter but small area, and results from the dynamic interplay between the large current generating the fast rate of the action potential depolarization exciting the medium close to the tip of the spiral, forcing a high curvature, while the long duration of the action potential prevents previously active medium from being re-excited and so implies a spatially extended core with a small curvatures. This provides a general mechanism for `linear' conduction blocks in homogeneous excitable media with ventricular-like action potentials. This correlates with `linear', `Z-shaped' and `hypocycloidal' conduction blocks observed in cellular automata (Fast et al. 1990), modified FitzHugh-Nagumo (Krinsky et al. 1992) and Beeler-Reuter (Efimov et al. 1995) media. For instance, a `Z-shaped' core is similar to trajectory observed in first couple of rotations in our simulations. These extended arcs of conduction block would be enhanced by anisotropy in conduction velocity, as seen in figure 3(h,i) as `linear conduction block'.
A re-entrant wave in such an anisotropic circuit can have an excitable gap (in which recovery of excitability has occured) before the tissue is re-excited by the re-entrant wave. During the slow, sharp turn, the gap ahead is large; during the faster, linear motion the gap is smaller. However, in any case the excitable gap anywhere far from the core is virtually absent (see figure 2(b,c)).
We have proposed resonant drift under feedback control as a means of defibrillating cardiac (Biktashev & Holden 1994) and atrial (Biktashev & Holden 1995) tissues. This method exploits the stability and symmetry of a rigidly rotating spiral (i.e. with a circular core): the key idea is that stable spiral waves can be displaced by spatially uniform perturbations, and so appropriately timed perturbations could be used to drive a spiral wave out of a medium. The nonuniform tip velocity, and the spiky pattern of meander, both complicate the response of the ventricular model spiral wave to perturbation. The trajectories obtained by feedback controlled, resonant driving of the position of the spiral wave solution shown in figure 4 show that resonant drift can be used to control the position of reentrant sources in this model, and so feedback controlled resonant drift of re-entrant activity of a free (unpinned) spiral or scroll wave may be produced in ventricular tissue. However, even in two-dimensional tissue slices, the effects of local heterogeneities, that pin or bind re-entrant sources, may be dominant (Vinson et al. 1994).
In summary, a two-dimensional partial differential equation model of ventricular tissue, with a detailed description of membrane excitability, generates persistent re-entrant waves with an appropriate waveform and period, and with extended cores that would be apparent in mapping studies as lines or arcs of unidirectional conduction block. These re-entrant waves can be moved by resonant drift , with a velocity that would force a re-entrant wave to a medium boundary in an epicardial tissue slice preparation within 10 s, if the re-entry was not trapped by localized heterogeneities. Resonant drift under feedback control can be used to move re-entrant sources even when their pattern of rotation is not a rigid rotation around a circular core, but is around a line or arc of unidirectional conduction block.