Liberation of a pinned spiral wave by a single stimulus in excitable media

The unpinning of a spiral wave from an anatomic obstacle by the application of a single stimulus near the core of the rotating wave was studied experimentally in a cell culture of cardiomyocyte monolayers as well as by computer simulations. It is shown that, with suitable positioning and timing, a single stimulus is sufficient for the successful unpinning of a pinned spiral wave. Successful unpinning is achieved when two conditions are fulfilled: (1) The stimulus is delivered in the vulnerable window of the rotating wave, and (2) the stimulus is delivered in a spatial zone in proximity to the obstacle, where the shape of the zone is defined by the phase of the anchored spiral wave. Two different scenarios for successful unpinning are discussed, which are distinguished by the distance to the stimuli applied to the obstacle.

Figure 1

Effect of the application of a single stimulus in a one-dimensional (1D) excitable system. (a) shows the intensity profile of a propagating wave, composed of three different states. The vulnerable window (VW), absolute refractoriness (AR), and resting state (RS) are indicated. (b) and (c) show the spatiotemporal profiles after the application of a stimulus in the RS and VW, respectively.

Figure 2

(Color online) Appearance of a pair of spiral tips caused by a single stimulus with delay, $T_s$, after the propagation of a planar wave. (a) shows three independent stimuli applied in the VW of a downward-propagating wave. The image shows a snapshot at $36\mathrm{ms}$ after the stimulus was applied. The trajectories of the spiral pairs, tracked for $300\mathrm{ms}$, are shown in (b). The applied stimuli correspond to $T_s$ equal to $93\mathrm{ms}$ (red, 1), $91\mathrm{ms}$ (blue, 2), and $87\mathrm{ms}$ (green, 3). Smaller stimulation delays led to smaller apex angles of wave tip trajectories, and larger $T_s$ led to larger apex angles. The apex angle $\phi$ is defined as the angle between the trajectories of the spiral pair, as sketched with black dashed lines.

Figure 3

Dependence of the apex angle of inhibition, $\alpha$, on the stimulus time $T_s$ and stimulus position $d_s$ as illustrated by filled circles (a) Left and right vertical dotted-striped lines define the transient of the AR and RS regions. The analytical dependence [corresponding to Eq. (11)] is given by the solid black line, whereas the variables corresponding to $\alpha$ smaller and larger than 180°, respectively, are shown in (b).

Figure 4

Dependence of the apex angle of the paired trajectory, $\phi$, and spiral-tip-displacement ${\Delta }_x$ after $300\mathrm{ms}$ with variation of the stimulus position in the VW. The angle $\phi$ is plotted with black squares corresponding to the left-hand axes. The spiral-tip-displacement is measured perpendicular to the conditioning wave after $300\mathrm{ms}$ of propagation, and is plotted with black filled circles highlighted by gray bars plotted from the origin of the applied stimulus. Left and right vertical dotted-striped lines define the transient of the AR and RS regions, which lead to inhibition of the stimulus and circular wave propagation, respectively. The striped arrows indicate the inflection point of displacement with respect to the conditioning wave.

Figure 5

Different scenarios after the application of a single stimulus near the obstacle pinned spiral wave. Field of view is $7.5\mathrm{cm}$. Pacing locations are given in spherical coordinates with respect to the obstacle center and the pinned spiral wave front, respectively, and are shown by white filled circles and emphasized with white arrows in frame A. The obstacle has a radius of $0.5\mathrm{cm}$. Frames are spaced every $30\mathrm{ms}$. Three scenarios are shown: (a) Unpinning via the induction of a pair of spirals due to pacing into the VW of the rotating spiral apart from the obstacle. Pacing location is at $(1.4\mathrm{cm}∕270°)$. (b) Unpinning via pacing at the border on the obstacle. Pacing location is at $(0.6\mathrm{cm}∕225°)$. (c) Temporary unpinning of the spiral wave via pacing at the border of the obstacle. Pacing location is at $(0.7\mathrm{cm}∕200°)$.

Figure 6

(Color online) Map of stimuli that affect a pinned spiral wave. Field of view is $7.5\mathrm{cm}$. The obstacle has a radius of $0.5\mathrm{cm}$. (a) Stimuli are applied around the obstacle at steps of 15° and $0.2\mathrm{cm}$ in spherical coordinates. Different colors and symbols denote the formation of different patterns. (b)–(f) examplify the formation of each pattern, marked in the respective colors shown in (a). The initial pinned wave and the stimulus applied at $t=0\mathrm{ms}$ are shown. In addition, the wave fronts of the propagating waves are shown with a step of $5\mathrm{ms}$ to illustrate the initial pattern formation up to $t=75\mathrm{ms}$. (b) Green (cross): Stimulus is applied in absolute refractoriness. No wave propagation is observed. (c) Light-blue (plus mark): Stimulus leads to circular wave propagation without affecting on the wave-tip trajectory of the pinned spiral wave. (d) Blue (asterisk): Creation of a free spiral, which does not lead to unpinning of the rotating wave. (e) Magenta (filled circle): Applied stimulus leads to spiral, pairs which in turn lead to unpinning of the spiral wave after the first rotation of their spiral tips. (f) Yellow (square): Applied stimulus leads to unpinning of the spiral wave immediately after stimulation.

Figure 7

(Color online) Close-up map of stimuli effects on a pinned spiral wave. The obstacle has a radius of $0.5\mathrm{cm}$. Stimuli, shown in radial coordinates with $r$ and $\varphi$, are applied in steps of 2.5° and $0.1\mathrm{cm}$ relative to the obstacle between 180° and 280°. The color coding corresponds to Fig. 6a. Yellow (square): Applied stimulus leads to unpinning of the spiral wave immediately after stimulation; magenta (filled circle), applied stimulus leads to spiral pairs which lead to unpinning of the spiral wave after the first rotation of their spiral tips; light blue (plus mark), stimulus leads to circular wave propagation without affecting the tip trajectory of the pinned spiral wave; green (cross), stimulus is applied in the region of absolute refractoriness (no wave propagation). Black (triangle), applied stimuli lead to temporary detachment of the spiral wave; blue (asterisk), lead to spiral pairs which lead not to unpinning. The figure also shows two voltage isolines $-84.9\mathrm{mV}$ and $-84.7\mathrm{mV}$ in red (right-hand side) and blue (left-hand side), respectively.

Figure 8

(Color online) Close-up representation of successful (a) and unsuccessful (b) scenarios on which spiral tip trajectories convert a pinned spiral wave to an unpinned spiral wave with the application of a single stimulus. Trajectories of the spiral tips are shown as a green dashed line and magenta dotted line, respectively. The trajectories are superimposed on the initial applied stimulus and pinned rotating wave. Pacing locations are given in spherical coordinates with respect to the obstacle center and the pinned spiral wave front, respectively. (a) shows successful unpinning. Pacing location is at $(1.4\mathrm{cm}∕270°)$. The magenta dotted spiral tip failed to propagate on the obstacle. (b) shows successful unpinning. Pacing location is at $(1.0\mathrm{cm}∕270°)$. Spiral tips drift away from the obstacle, and do not lead to unpinning of the spiral wave.

Figure 9

(Color online) Experiment on a cardiomyocyte monolayer. Snapshots of successful unpinning via low-frequency pacing near the obstacle in cardiomyocyte monolayers. The pacing frequency and field of view are $1.0\mathrm{Hz}$ and $5\mathrm{mm}$, respectively. The size of the obstacle is $\sim 0.45\mathrm{mm}$. Frames are spaced every $150\mathrm{ms}$ in (b) and (c). The unpinning scenario is divided into three parts: (a) Stable rotating spiral anchored at an obstacle with a period of rotation of $720\mathrm{ms}$. The white arrow indicates the direction of rotation of the spiral; (b) creation of a spiral pair by the application of the stimulus in the VW of the pinned spiral wave apart from the obstacle; (c) successful unpinning by propagation failure of one spiral arm on the obstacle, and annihilation of the other free spiral wave with the pinned spiral wave.

Figure 10

(Color online) Close up of propagating wave tips in cardiomyocyte monolayer leading to unpinning of a pinned spiral wave. White dashed arrows correspond to the direction of propagation of the free spiral waves. The upper and lower tips of each line define the end and the start of rotation of the free spiral wave, respectively. The apex angle between both arrows is approximately $60\text{degrees}$, and the length of straight propagation (white dashed lines) is approximately $1\mathrm{mm}$.