Figure 1:
(a) Isopotential map and tip trajectory of spiral wave solution of
equations (2) and (3), in a 20 
20 mm medium with 
and 
, with the 10 voltage isolines spanning -89.4 to 32.4 mV in
equal steps of 13.5 mV. The isolines represent the V(t) waveform
2.67 s after its initiation from a broken wavefront, and the solid
line ending in 
is the trajectory of the tip during the
preceding 190 ms.
(b, c) The tip trajectory for spiral wave solutions of
(2, 3) produced by a cut plane wave
(b) and a twin stimulus protocol (c).
Figure 1(a) illustrates a spiral wave solution of the model, as the
spatial distribution of isolines of membrane potential V, at an
instant 2.67 s after the spiral wave was initiated by cutting a broken
plane wave. The spiral rotated with an initial period of approximately
170 ms, and over the first 1 s the period decreased to
100-110 ms. If the medium were large enough, the distance
between successive wavefronts far from the tip, i.e. the wavelength of
the spiral, would be about 4 cm. We define the tip of the spiral
by the intersection of the 
and the f=0.5 isolines,
where f is the 
inactivation gating variable. The
qualitative behaviour of the trajectory of the tip does not depend on
the precise choice of these values. The trajectory of the tip of the
spiral is not stationary, but meanders, and its motion is nonuniform,
moving by a jump-like alternation between fast and very slow phases,
with about 5 jumps per full rotation. This motion resembles an
irregular, nearly biperiodic process, with the ratio of the two periods
close to 1:5.
The multi-lobed pattern of the trajectory of the tip takes time to
develop, and itself develops with time. Figure 1(b,c) follows the tip
trajectory for spirals initiated from a broken wavefront (b), by a twin
pulse protocol (c). In both cases the tip trajectory follows a
transient over 3-4 rotations, in which there is an extended ``core'',
with a length of about a centimeter, that evolves into an irregular,
biperiodic motion around a core contained within 4 mm square. Both
the extended transient and the biperiodic motions are composed of
almost linear segments (when the velocity of the tip movement is fast,
about 0.15 m/s) broken by sharp turns of up to 170 
(when
the tip trajectory is almost stationary).
Figure 2: :
(a) Isochronal map of a spiral wave solution of equations
(2) and (3), initiated by the twin pulse
protocol in a 30 30 mm medium with 
ms
and 
mm. Thin lines are the excitation front
position, drawn at every 10 ms during one rotation 1.2 to 1.29
s. The front lines are defined as loci of 
and
f;SPMgt;0.5, where f is the inactivation gating variable. The
thick line is the trajectory of the spiral tip during this rotation.
A-D and O mark the position of recording sites.
(b) V(t) at sites A-D during the first 12 rotations of the
spiral after its initiation. Voltage scale is from -100 mV to
100 mV with marks in 50 mV, marks on time axis are in 100 ms.
(c,d) Membrane potential V(t), upper graphs, and principal
currents 
(solid), 
(long dashed), 
(medium
dashed), 
(dotted), lower graphs, at points A (c) and
O (d) of Figure 2(a), 1.5 to 1.8 s after the
initiation of spiral wave from broken wavefront. Voltage is in
mV, current in nA, marks on time axis through 100 ms.
The rotation of the spiral wave can be monitored by following an isoline on the wavefront, and the trajectory of the tip of the spiral, as illustrated in figure 2(a). The area enclosed by the tip trajectory is analogous to the core of a rigidly rotating spiral, and is not invaded by the action potential. Characteristics of the V(t) observed at different sites in the medium during the evolution of a rotating spiral wave are (figure 2(b))
),
are reduced by more than an order of magnitude, and V(t) appears more as
slow oscillations than action potentials.
The membrane potential and principal currents ( , ,
, ) far from, and within the core of the spiral, sites
``A'' and ``O'' of figure 2(a), are illustrated in
figure 2(c,d), respectively. Within the core (d) the membrane
potential remains between -45 and -5 mV; this persistent
depolarization inactivates the inward current (on the graph,
it is indistinguishable from zero), thus blocking propagation into the
core. The trajectory of the tip maps out an area of the conduction
block. Note that the change of action potential waveform in
different sites is a general feature of spiral waves in excitable
media; see e.g. (Plesser et al. 1990) for analogous waveform changes in
the Belousov-Zhabotinsky reaction.
Figure 3:
(a-f) Successive 150 ms pieces of the tip trajectory,
starting 1.5 s after spiral initiation from a broken plane wave, with
, , in a medium 
mm. (g-i) Isochronal map during of biperiodic tip meander,
2.480-2.520 s since spiral was initiated from a broken plane wave,
(g) in isotropic medium, (h) in anisotropic medium with
longitudinal axis of fibres in the vertical direction, and (i) in
anisotopic medium, with longitudinal axis of fibres in the horizontal
direction. The isochrons are labeled in ms relative to 2480 ms.
, , medium 20 20 mm.
Figure 3(a-f) shows that as the asymmetric, multilobed trajectory continues to evolve, the area it encloses continues to decrease, and the pattern changes from having two or three sharp turns, to having five. The trajectory during any rotation differs from the trajectory during the preceding rotations. The spiral wave solutions have been followed for up to 3 s, and so, once established, the slowly evolving, almost biperiodic meander pattern shown in figures 1-3 appears to be stable.
Anisotropy in conduction velocity will distort the spiral wave and tip trajectory; figure 3(g-i) shows voltage isochrons in isotropic and anisotropic media: in the isotropic medium the isochrons for the small, multilobed tip trajectory of figure 3(a-f) appear to emerge from a compact core, while in the anisotropic medium the isochrons appear to move around a linear core.
Figure 4:
Tip trajectories under feedback controlled, resonant driving. When the
wavefront of the spiral wave (depolarization through -10 mV) reached a
recording site in the bottom left hand corner, a 2 ms, 4 V/s
depolarizing perturbation was added after a fixed delay. Each
trajectory is for a different delay, from 0 to 100 ms, and corresponds
to applying the perturbation at a different phase of the spiral. All
trajectories start in the same place in the center, move toward
boundaries and annihilate. The dots mark point on trajectories
corresponding to the moments of stimulation. ,
, medium 20 20 mm.
The tip of the spiral wave solutions of figures 1-3 moves irregularly in a complicated trajectory, but does not move out of the medium: if the medium is large enough to contain the early transient motion around an almost linear core then the spiral wave remains in the medium. Small amplitude, spatially uniform repetitive stimulation under feedback control can be used to produce directed movement of a rigidly rotating spiral wave and so to push the spiral out the medium (Biktashev & Holden 1994). Figure 4 shows five tip trajectories produced by repetitive stimulation applied at five different fixed delays after the wavefront reached the bottom left hand corner of the medium. The delay determines the initial direction of drift. The curved drift trajectories all reach the medium boundaries. A repetitive perturbation of 15% the amplitude of the single shock defibrillation threshold produces a directed motion with a velocity of about 0.4 cm/s. Here we mean by defibrillation threshold the minimal amplitude of a brief pulse F(t) in (2), which is sufficient to eliminate the re-entrant activity, about 12.5 V/s.
