^{1,a)}, John P. Wikswo

^{2}and Niels F. Otani

^{3}

### Abstract

Rotating spiral waves have been observed in numerous biological and physical systems. These spiral waves can be stationary, meander, or even degenerate into multiple unstable rotating waves. The spatiotemporal behavior of spiral waves has been extensively quantified by tracking spiral wave tip trajectories. However, the precise methodology of identifying the spiral wave tip and its influence on the specific patterns of behavior remains a largely unexplored topic of research. Here we use a two-state variable FitzHugh–Nagumo model to simulate stationary and meandering spiral waves and examine the spatiotemporal representation of the system’s state variables in both the real (i.e., physical) and state spaces. We show that mapping between these two spaces provides a method to demarcate the spiral wave tip as the center of rotation of the solution to the underlying nonlinear partial differential equations. This approach leads to the simplest tip trajectories by eliminating portions resulting from the rotational component of the spiral wave.

Spiral waves are the subject of intense investigation and occur in various nonlinear media.

^{1–7}The wave tip serves as an organizing center that often appears to meander in epicyclic patterns; each epicyclic “petal” typically represents one rotation of the meandering spiral. These patterns are usually represented in a “flower garden” arrangement

^{8–12}and the associated dynamics have important implications, e.g., dangerous cardiac arrhythmias are the result of the movement and stability of rapidly rotating spiral waves propagating in the heart.

^{13}The instantaneous wave-tip location and its trajectory are identified by methodologies whose theoretical basis and limitations have not been adequately addressed. Identifying the tip based on the spiral wave solution of the underlying equations eliminates one epicycle per spiral-wave rotation, i.e., all petals were “plucked” from each flower we “picked.” Just as Copernican astronomy eliminated the epicyclic descriptions of planetary orbits of the Ptolemaic system,

^{14}so our model shows that extensively studied epicycles of a meandering spiral-wave tip arise from inappropriate origin choice.

We would like to thank Michael Cross for valuable discussions and comments on the manuscript. This work was supported by the National Institutes of Health (Grant Nos. R01-HL63267 to R.A.G. and R01-HL58241 to J.P.W.) and the National Science Foundation (CAREER Award to R.A.G.).

I. INTRODUCTION

A. Identifying spiral wave tip trajectories

II. SIMULATION OF SPIRAL WAVES

III. THEORY

IV. NUMERICAL METHODS

V. RESULTS

VI. DISCUSSION

### Key Topics

- Partial differential equations
- 10.0
- Cardiac dynamics
- 6.0
- Trajectory models
- 6.0
- Heart
- 5.0
- Kinematics
- 4.0

## Figures

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.

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 .

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).

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.

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 .

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.

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.

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.

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.

## Tables

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 .

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 .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content