FitzHugh–Nagumo model. (Top) Snapshot of the spatial distribution of the fast variable in physical space, i.e., . The greyscale color key is shown in the bottom panel. (Bottom) Dynamics of state variables during one beat, i.e., and at the site indicated by the number 1 in the top panel.
Flower garden (original origin choice). The spiral wave tip trajectories in physical space for the FitzHugh–Nagumo model [Eq. (1)] as a function of parameters and . Phase was computed according to Eq. (2) and the instantaneous wave tip was identified using Eq. (3). The origin choice in Eq. (2) was .
Spatiotemporal dynamics of a stable spiral. (a) Greyscale image snapshot of fast variable depicts a spiral wave rotating counterclockwise. (b) Surface plot of snapshot of with -axis representing the value of . (c) Greyscale image of integral of over one period; the darker regions indicate lower integral values. (d) Surface plot of integral of over one period. (e) Contour plot of snapshot of zoomed in at the center of rotation, which is denoted by an asterisk. (f) Trajectories in state space for the four sites on contour (horizontal grey line) denoted by numbers 1–4. Since is the fast variable all trajectories are counterclockwise in state space. (g) Time series of the four sites labeled on (e). The thick vertical gray line indicates the time of snapshot in (e).
The spiral wave tip trajectories in the FitzHugh–Nagumo model [Eq. (1); and ] as a function of origin choice. (a) Phase was computed according to Eq. (2) and the instantaneous wave tip was identified using Eq. (3). The height of the plot, i.e., the vertical axis, represents the distance from defined as . (b) Spiral tip trajectory for . (c) Spiral tip trajectory for . (d) Schematic diagram illustrating choice of along the diagonal in state space.
Flower garden . The spiral wave tip trajectories in physical space for the FitzHugh–Nagumo model [Eq. (1)] as a function of parameters and . Phase was computed according to Eq. (2) and the instantaneous wave tip was identified using Eq. (3). The origin choice in Eq. (2) was in contrast to (0,0) in Fig. 2 for the same parameter values of and .
The dynamics of state variables ( and ) in state space for one site during spiral wave rotation as a function of and . The locations of (see Table I) are shown as asterisks. For clarity, the axis labels are only shown in the top left plot. The origin (0,0) axes are indicated by dashed lines.
Analytically derived spatiotemporal spiral wave tip trajectories for stable figure-of-eight reentry. Position of spiral wave tip for clockwise (counterclockwise) rotating spiral is shown in blue (red). (a) The tip location at each instant of time for each spiral wave identified as the center of rotation, i.e., site was (90,100) for counterclockwise and (110,100) for clockwise waves. (b) The tip locations identified using traditional methods which contain a rotational component, i.e., site was ( ) for counterclockwise and ( ) for clockwise waves. See text for discussion.
State space dynamics. [(a) and (b)] Example nullclines for two-state variable systems. The fast variable nullcline (black) corresponds to and the slow variable nullcline (gray) corresponds to . When the Laplacian term is zero (e.g., spatially homogenous state variable patterns) the fixed points of the system correspond to the intersection of the nullclines. For example, in excitable systems, there is a stable solution to the PDE [Eq. (4)] that represents all sites at a “quiescent” state that corresponds to a stable fixed point (depicted as closed circles). Depending on the shape of the slow variable nullcline, there may be additional fixed points. For example, (b) depicts three intersections of the nullclines including a marginally stable fixed point (half-filled circle) and an unstable fixed point (open circle). The dashed lines indicate the effect of adding a constant value to the fast variable equation which corresponds to the dynamics at . This addition causes a vertical shift in the fast variable nullcline and the result is depicted as a dashed line. This shift may act to slightly perturb the quiescent solution as in (a) or may act to fundamentally change the system fixed points as in (b), where the shift results in the elimination of two fixed points. (c) The “bad” choice of origin illustrates nonunique representation as indicated by the three intersection points of the outermost trajectory by the gray line originating at the origin.
The values of the solution to Eq. (5) at the center of rotation, i.e., for the FitzHugh–Nagumo equation [Eq. (1)], as a function of the two model parameters and .
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