^{1}, F. Javier de la Rubia

^{1}and Plamen Ch. Ivanov

^{2,a)}

### Abstract

There is evidence that spiral waves and their breakup underlie mechanisms related to a wide spectrum of phenomena ranging from spatially extended chemical reactions to fatal cardiac arrhythmias [A. T. Winfree, The Geometry of Biological Time (Springer-Verlag, New York, Year: 2001);J. Schutze, O. Steinbock, and S. C. Muller, Nature356, 45 (Year: 1992);S. Sawai, P. A. Thomason, and E. C. Cox, Nature433, 323 (Year: 2005);L. Glass and M. C. Mackey, From Clocks to Chaos: The Rhythms of Life (Princeton University Press, Princeton, Year: 1988);R. A. Gray et al. , Science270, 1222 (Year: 1995);F. X. Witkowski et al. , Nature392, 78 (Year: 1998)]. Once initiated, spiral waves cannot be suppressed by periodic planar fronts, since the domains of the spiral waves grow at the expense of the fronts [A. N. Zaikin and A. M. Zhabotinsky, Nature225, 535 (Year: 1970);A. T. Stamp, G. V. Osipov, and J. J. Collins, Chaos12, 931 (Year: 2002);I. Aranson, H. Levine, and L. Tsimring, Phys. Rev. Lett.76, 1170 (Year: 1996);K. J. Lee, Phys. Rev. Lett.79, 2907 (Year: 1997);F. Xie, Z. Qu, J. N. Weiss, and A. Garfinkel, Phys. Rev. E59, 2203 (Year: 1999)]. Here, we show that introducing periodic planar waves with long excitation duration and a period longer than the rotational period of the spiral can lead to spiral attenuation. The attenuation is not due to spiral drift and occurs periodically over cycles of several fronts, forming a variety of complex spatiotemporal patterns, which fall into two distinct general classes. Further, we find that these attenuation patterns only occur at specific phases of the descending fronts relative to the rotational phase of the spiral. We demonstrate these dynamics of phase-dependent spiral attenuation by performing numerical simulations of wave propagation in the excitable medium of myocardial cells. The effect of phase-dependent spiral attenuation we observe can lead to a general approach to spiral control in physical and biological systems with relevance for medical applications.

The dynamics of waves in excitable media

^{1–11}have been studied in physical, chemical, and biological systems under a variety of conditions including noise and inhomogeneities in the medium

^{12–14}and mechanical deformation.

^{15}Of particular interest is the problem of nonlinear wave interaction in the excitable medium of the heart muscle, as loss of wave stability and spiral wave breakup lead to spatiotemporal patterns associated with adverse cardiac events such as ventricular fibrillation and sudden cardiac death.

^{5,6,16,17}While different approaches to prevent spiral breakup have been proposed,

^{18–22}it is widely accepted that stable spiral waves cannot be suppressed by periodic planar wave fronts, since the frequency of the spiral is higher than the frequency of the fronts, and thus the domains of the spiral waves grow at the expense of the slower wave fronts.

^{7–11}Here, we focus on the attenuation of a single stable spiral wave. We show that it is possible to attenuate spiral waves by planar wave fronts with period longer than the rotational period of the spiral, and we address the problem of how to control spiral attenuation in excitable media. We find that when the fronts have long excitation duration, and are delivered at a specific phase relative to the rotational phase of the spiral, the spiral-front interaction is characterized by periodic patterns of spiral attenuation, which remain stable in time and over a broad range of physiologically meaningful parameter values. While spiral drift has been shown under similar conditions,

^{23}we do not aim to achieve spiral drift but to attenuate a stable spiral, i.e., to reduce the area covered by the spiral and the number of cells involved in the propagation of the spiral wave.

I. PHYSIOLOGICAL CONSIDERATIONS AND MODELING

II. RESULTS

III. SUMMARY

### Key Topics

- Wave attenuation
- 30.0
- Mechanical waves
- 8.0
- Excited states
- 4.0
- Lattice dynamics
- 4.0
- Cardiac dynamics
- 3.0

## Figures

Time evolution of the transmembrane potential of a ventricular myocite. After a superthreshold perturbation, the potential sharply increases from the resting state, of , to the excited state, with a plateau of positive potential of . The duration of the excitation ranges from to . The excited state is followed by a smooth decrease of the potential during the absolute refractory period, . The decrease in the transmembrane potential continues during the relative refractory period, , when a cell can be excited again but to a lower potential, and for shorter excitation duration compared to an excitation started during the resting state (dashed line).

Time evolution of the transmembrane potential of a ventricular myocite. After a superthreshold perturbation, the potential sharply increases from the resting state, of , to the excited state, with a plateau of positive potential of . The duration of the excitation ranges from to . The excited state is followed by a smooth decrease of the potential during the absolute refractory period, . The decrease in the transmembrane potential continues during the relative refractory period, , when a cell can be excited again but to a lower potential, and for shorter excitation duration compared to an excitation started during the resting state (dashed line).

Schematic presentation of the model. (a) Excitation threshold vs time past after the last excitation of a cell [also called diastolic interval (DI)]. For short DI, during the absolute refractory period, the cell cannot be excited and the excitation threshold is infinite. When the cell enters the relative refractory period, the excitation threshold is , and with increasing DI the threshold decreases linearly in agreement with experimental observations (Ref. ^{ 29 } ) until it reaches the value at the end of the relative refractory period. For long DI, during the resting state, the threshold remains constant and equal to (Ref. ^{ 27 } ) We choose and to maintain the movement of the spiral tip in our simulations within a small area in agreement with experimental observations (Ref. ^{ 29 } ). (b) Restitution curve—relation between the excitation duration [action potential duration (APD)] vs DI. There are no action potentials in the absolute refractory period. During the relative refractory period, the APD increases linearly with time, and in the resting state the APD is constant. We use the experimental restitution curve (denoted by ) and the conduction speed for guinea pig ventricular myocites (Ref. ^{ 29 } ) to calibrate the parameter values, so that the restitution curve in our simulations reproduces the experimental one.

Schematic presentation of the model. (a) Excitation threshold vs time past after the last excitation of a cell [also called diastolic interval (DI)]. For short DI, during the absolute refractory period, the cell cannot be excited and the excitation threshold is infinite. When the cell enters the relative refractory period, the excitation threshold is , and with increasing DI the threshold decreases linearly in agreement with experimental observations (Ref. ^{ 29 } ) until it reaches the value at the end of the relative refractory period. For long DI, during the resting state, the threshold remains constant and equal to (Ref. ^{ 27 } ) We choose and to maintain the movement of the spiral tip in our simulations within a small area in agreement with experimental observations (Ref. ^{ 29 } ). (b) Restitution curve—relation between the excitation duration [action potential duration (APD)] vs DI. There are no action potentials in the absolute refractory period. During the relative refractory period, the APD increases linearly with time, and in the resting state the APD is constant. We use the experimental restitution curve (denoted by ) and the conduction speed for guinea pig ventricular myocites (Ref. ^{ 29 } ) to calibrate the parameter values, so that the restitution curve in our simulations reproduces the experimental one.

Time evolution of the total number of excited cells from simulations on a square lattice of size . Time is presented in units of the simulation time step . Data show a variety of robust patterns of spiral attenuation that remain stable in time. Absent and reduced peaks correspond to attenuation of the spiral. We find that these patterns belong to two general classes. (i) Class I , where within a cycle of fronts we have consecutive spiral rotations followed by one spiral attenuation. Examples of Class I patterns are presented in (a) pattern 2:1—out of the collision of the spiral with two consecutive fronts there is first a spiral attenuation (denoted by B) followed by one surviving spiral (denoted by C); (b) pattern 3:2—for each cycle of three consecutive fronts there is first a spiral attenuation (B) followed by two surviving spirals (C and D); (c) pattern 4:3—for each cycle of four consecutive fronts there is a spiral attenuation (B) and three surviving spirals (C, D, and E). The Class I patterns in (a), (b), and (c) are obtained for the following parameter values: , , , , , respectively. (ii) Class II , where within a cycle of fronts there are spiral rotations and two spiral attenuations. Examples of Class II patterns are presented in (d) pattern 3:1—for each cycle of three fronts there are two spiral attenuations (B and D) and one surviving spiral (C); (e) pattern 5:3—for each cycle of five fronts there are two spiral attenuations (B and E) and three surviving spirals (C, D, and F); (f) pattern 7:5—for each cycle of seven fronts we have two attenuations (B and F) and five surviving spirals (C, D, E, G, and H). The Class II patterns in (d), (e), and (f) are obtained for the following parameter values: , , , , , respectively. In all panels, the instant in which a spiral attenuation is initiated is denoted by A, and the beginning of the next cycle is denoted by , repeating the spiral attenuation in . We find the same attenuation patterns independently of the size of the lattice and for a broad range of parameter values (Fig. 7 ).

Time evolution of the total number of excited cells from simulations on a square lattice of size . Time is presented in units of the simulation time step . Data show a variety of robust patterns of spiral attenuation that remain stable in time. Absent and reduced peaks correspond to attenuation of the spiral. We find that these patterns belong to two general classes. (i) Class I , where within a cycle of fronts we have consecutive spiral rotations followed by one spiral attenuation. Examples of Class I patterns are presented in (a) pattern 2:1—out of the collision of the spiral with two consecutive fronts there is first a spiral attenuation (denoted by B) followed by one surviving spiral (denoted by C); (b) pattern 3:2—for each cycle of three consecutive fronts there is first a spiral attenuation (B) followed by two surviving spirals (C and D); (c) pattern 4:3—for each cycle of four consecutive fronts there is a spiral attenuation (B) and three surviving spirals (C, D, and E). The Class I patterns in (a), (b), and (c) are obtained for the following parameter values: , , , , , respectively. (ii) Class II , where within a cycle of fronts there are spiral rotations and two spiral attenuations. Examples of Class II patterns are presented in (d) pattern 3:1—for each cycle of three fronts there are two spiral attenuations (B and D) and one surviving spiral (C); (e) pattern 5:3—for each cycle of five fronts there are two spiral attenuations (B and E) and three surviving spirals (C, D, and F); (f) pattern 7:5—for each cycle of seven fronts we have two attenuations (B and F) and five surviving spirals (C, D, E, G, and H). The Class II patterns in (d), (e), and (f) are obtained for the following parameter values: , , , , , respectively. In all panels, the instant in which a spiral attenuation is initiated is denoted by A, and the beginning of the next cycle is denoted by , repeating the spiral attenuation in . We find the same attenuation patterns independently of the size of the lattice and for a broad range of parameter values (Fig. 7 ).

(Color) Color-coded representation of the spiral-front interaction corresponding to the Class I and Class II patterns shown in Fig. 3 . For increasing values of we have absolute refractory cells in red, relative refractory cells in orange and yellow, and excited cells in cyan, blue, and violet (highest values of ). Snapshots for each pattern represent the same stages of the dynamics in time, as indicated by the corresponding capital letters in the panels of Fig. 3 .

(Color) Color-coded representation of the spiral-front interaction corresponding to the Class I and Class II patterns shown in Fig. 3 . For increasing values of we have absolute refractory cells in red, relative refractory cells in orange and yellow, and excited cells in cyan, blue, and violet (highest values of ). Snapshots for each pattern represent the same stages of the dynamics in time, as indicated by the corresponding capital letters in the panels of Fig. 3 .

(Color) Color-coded representation of the time evolution for the Class I 4:3 pattern obtained for the same parameter values as in Fig. 3(c) . Snapshots represent the state of the lattice in intervals of five time steps . Snapshots 1, 7, and 12 correspond to D, E, and A in Fig. 3(c) . For increasing values of we have absolute refractory cells in red, relative refractory cells in orange and yellow, and excited cells in cyan, blue, and violet (highest values of ).

(Color) Color-coded representation of the time evolution for the Class I 4:3 pattern obtained for the same parameter values as in Fig. 3(c) . Snapshots represent the state of the lattice in intervals of five time steps . Snapshots 1, 7, and 12 correspond to D, E, and A in Fig. 3(c) . For increasing values of we have absolute refractory cells in red, relative refractory cells in orange and yellow, and excited cells in cyan, blue, and violet (highest values of ).

Time evolution of the total number of excited cells in a square lattice of size for isolated fronts (without a spiral), isolated spiral (without fronts), linear superposition of fronts and spiral, and the Class II pattern 7:5, generated for the same parameter values as in Fig. 3(f) (arrows inclined to the right indicate one cycle of the 7:5 pattern). It is apparent that the 7:5 attenuation pattern cannot be a result of the linear superposition of periodic fronts and the spiral wave. This linear superposition is characterized by absence of attenuation, much higher average value of the number of excited cells, different profile of the periodic peaks, and shorter cycle (indicated by vertical arrows) compared to the 7:5 attenuation pattern, generated by the nonlinear interaction of the spiral wave and lower frequency fronts with maximum APD.

Time evolution of the total number of excited cells in a square lattice of size for isolated fronts (without a spiral), isolated spiral (without fronts), linear superposition of fronts and spiral, and the Class II pattern 7:5, generated for the same parameter values as in Fig. 3(f) (arrows inclined to the right indicate one cycle of the 7:5 pattern). It is apparent that the 7:5 attenuation pattern cannot be a result of the linear superposition of periodic fronts and the spiral wave. This linear superposition is characterized by absence of attenuation, much higher average value of the number of excited cells, different profile of the periodic peaks, and shorter cycle (indicated by vertical arrows) compared to the 7:5 attenuation pattern, generated by the nonlinear interaction of the spiral wave and lower frequency fronts with maximum APD.

Diagram of spiral attenuation patterns in parameter space vs , for a square lattice of and fixed parameter values and . We observe attenuation patterns for a broad range of parameter values where each pattern can be found along a single straight line, in accordance with Eqs. (2.3) and (2.4) . Patterns of Class I and Class II alternate in a series of parallel lines, where increases with increasing . To assess the intensity of the attenuation effect in different regions of the parameter diagram, we estimate for each cycle the ratio between the average number of excited cells when there is no spiral attenuation (large peaks in Fig. 3 ) and during spiral attenuation (reduced or absent peaks in Fig. 3 ). We find that this ratio is (i) characterized by a broad maximum in the central region of the parameter diagram and (ii) it exhibits a monotonic decrease in all directions of the parameter space for both classes of patterns, indicating a common behavior in the intensity of spiral attenuation.

Diagram of spiral attenuation patterns in parameter space vs , for a square lattice of and fixed parameter values and . We observe attenuation patterns for a broad range of parameter values where each pattern can be found along a single straight line, in accordance with Eqs. (2.3) and (2.4) . Patterns of Class I and Class II alternate in a series of parallel lines, where increases with increasing . To assess the intensity of the attenuation effect in different regions of the parameter diagram, we estimate for each cycle the ratio between the average number of excited cells when there is no spiral attenuation (large peaks in Fig. 3 ) and during spiral attenuation (reduced or absent peaks in Fig. 3 ). We find that this ratio is (i) characterized by a broad maximum in the central region of the parameter diagram and (ii) it exhibits a monotonic decrease in all directions of the parameter space for both classes of patterns, indicating a common behavior in the intensity of spiral attenuation.

Dependence of the attenuation patterns on the relative phase between the first released front and the spiral. Presented are only the patterns 2:1 (Class I) and 5:3 (Class II) for two sets of parameter values and , with the same symbols as in Fig. 7 . Our results show that, for each set of parameter values on the diagram in Fig. 7 , the patterns can appear only for specific values of the relative phase between the front and the spiral, indicating that the phase in which the front hits the spiral is crucial to achieve spiral attenuation.

Dependence of the attenuation patterns on the relative phase between the first released front and the spiral. Presented are only the patterns 2:1 (Class I) and 5:3 (Class II) for two sets of parameter values and , with the same symbols as in Fig. 7 . Our results show that, for each set of parameter values on the diagram in Fig. 7 , the patterns can appear only for specific values of the relative phase between the front and the spiral, indicating that the phase in which the front hits the spiral is crucial to achieve spiral attenuation.

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