Spiral wave drift under optical feedback in cardiac tissue

Spiral waves occur in various types of excitable media and their dynamics determine the spatial excitation patterns. An important type of spiral wave dynamics is drift, as it can control the position of a spiral wave or eliminate a spiral wave by forcing it to the boundary. In theoretical and experimental studies of the Belousov-Zhabotinsky reaction, it was shown that the most direct way to induce the controlled drift of spiral waves is by application of an external electric field. Mathematically such drift occurs due to the onset of additional gradient terms in the Laplacian operator describing excitable media. However, this approach does not work for cardiac excitable tissue, where an external electric field does not result in gradient terms. In this paper, we propose a method of how to induce a directed linear drift of spiral waves in cardiac tissue, which can be realized as an optical feedback control in tissue where photosensitive ion channels are expressed. We illustrate our method by using the FitzHugh-Nagumo model for cardiac tissue and the generic model of photosensitive ion channels. We show that our method works for continuous and discrete light sources and can effectively move spiral waves in cardiac tissue, or eliminate them by collisions with the boundary or with another spiral wave. We finally implement our method by using a biophysically motivated photosensitive ion channel model included to the Luo-Rudy model for cardiac cells and show that the proposed feedback control also induces directed linear drift of spiral waves in a wide range of light intensities.

Figure 1

Spiral waves dynamics under optical feedback. The shapes of spiral waves in highly (a) and weakly (d) excitable media without application of light. The wave in (a) or (d) rotates clockwise and the path of the spiral tip (yellow) is a circle, which is the core of the spiral wave. The color in (a), (d) shows the spatial distribution of the variable $u$ with the red color corresponding to $u_{\mathrm{excited}}$ and the blue color corresponding to $u_{\mathrm{rest}}$. Tip trajectories under the optical feedback given by Eq. (7) in highly (b) and weakly (e) excitable media. (c), (f) The same for the optical feedback given by Eq. (8). $g_o=3.90625\times {10}^{-3}$, continuous light sources.

Figure 2

The drift of spiral waves when the optical feedback uses only depolarizing or hyperpolarizing current. (a), (c) Application of the depolarizing current given by Eq. (9). (b), (d) Application of the hyperpolarizing current given by Eq. (10). $g_o=3.90625\times {10}^{-3}$, continuous light sources. (a), (b) Highly excitable media. (c), (d) Weakly excitable media.

Figure 3

Tip trajectories of spiral waves under discrete spot lights (LEDs). (a)–(c) Highly excitable media. (d)–(f) Weakly excitable media. (a), (d) $\mathrm{LEDs}=32\times 32, g_o=0.04$. (b), (e) $\mathrm{LEDs}=16\times 16, g_o=0.16$. (c), (f) $\mathrm{LEDs}=8\times 8, g_o=0.64$. The feedback is given by Eq. (7). Time delay $\tau =0.01T$.

Figure 4

Tip trajectories of spiral waves with longer time delay than that in Figs. 3 and 3: $\mathrm{LEDs}=16\times 16, g_o=0.16$. (a), (b) Highly excitable media. (c), (d) Weakly excitable media. (a), (c) Time delay $\tau =0.02T$. (b), (d) Time delay $\tau =0.05T$. The feedback is given by Eq. (7).

Figure 5

The effect of the total light intensity on the drift speeds and directions. The light intensity was changed by increasing the number of LEDs. The feedback is given by Eq. (7). (a), (c) Relationships between the drift speeds and the LED numbers. (b), (d) Relationships between the drift angles and the LED numbers. (a), (b) Highly excitable media. (c), (d) Weakly excitable media. $g_o=0.04, \mathrm{LEDs}:16\times 16,18\times 18,...,30\times 30,32\times 32$.

Figure 6

The effect of the total light intensity on the drift speeds and directions in $16\times 16$ LED arrays and the feedback is given by Eq. (7). The light intensity was changed by increasing $g_o$. (a), (c) Relationships between the drift speeds and $g_o$. (b), (d) Relationships between the drift angles and $g_o$. (a), (b) Highly excitable media. (c), (d) Weakly excitable media.

Figure 7

Spiral waves drift and annihilate at the boundary under the optical feedback control Eq. (7) with $g_o=0.12$, continuous light sources. (a), (b) Highly excitable media. (c), (d) Weakly excitable media. (a), (c) The initial patterns of the spirals are the same as in 1 and 1.

Figure 8

Dynamics of two counter-rotating spiral waves under the optical feedback Eq. (7) in highly excitable media. (a)–(c) Two counter-rotating spiral waves move apart. $g_o=2, \mathrm{LEDs}=32\times 32$. (a) $t=0$. (b) Tip trajectories. (c) $t=400$. (d)–(f) Annihilation of spiral waves. $g_o=18, \mathrm{LEDs}=32\times 32$. (d) $t=0$. (e) Tip trajectories. (f) Annihilation at $t=57$.

Figure 9

Dynamics of two counter-rotating spiral waves under the optical feedback Eq. (7) in weakly excitable media. (a)–(c) Two counter-rotating spiral waves move apart. $g_o=1, \mathrm{LEDs}=32\times 32$. (a) $t=0$. (b) Tip trajectories. (c) $t=150$. (d)–(f) Annihilation of spiral waves. $g_o=5, \mathrm{LEDs}=32\times 32$. (d) $t=0$. (e) Tip trajectories. (f) Annihilation at $t=52$.

Figure 10

Spiral wave dynamics in the Luo-Rudy model with continuous light sources. Color coding shows the potential $V$. (a) The shapes of rigidly rotating spiral wave and its core without application of light. (b)–(f) End time patterns and tip trajectories of spiral with different light intensity. (b) The spiral drift with long transient time when the light is small. $I_o=3\times {10}^{-5}, {\left(I_{\mathrm{light}}\right)}_{max}=0.048\mathrm{mW}/{\mathrm{mm}}^2, t=2500\mathrm{ms}$. (c) The spiral drift with $I_o=5\times {10}^{-5}, {\left(I_{\mathrm{light}}\right)}_{max}=0.08\mathrm{mW}/{\mathrm{mm}}^2, t=1500\mathrm{ms}$. (d) The spiral annihilates at the boundary when the drift becomes stronger. $I_o=10\times {10}^{-5}, {\left(I_{\mathrm{light}}\right)}_{max}=0.16\mathrm{mW}/{\mathrm{mm}}^2, t=1242\mathrm{ms}$. (e) The spiral drift faster with $I_o=20\times {10}^{-5}, {\left(I_{\mathrm{light}}\right)}_{max}=0.32\mathrm{mW}/{\mathrm{mm}}^2, t=675\mathrm{ms}$. (f) The drift with large light intensity. $I_o=80\times {10}^{-5}, {\left(I_{\mathrm{light}}\right)}_{max}=1.28\mathrm{mW}/{\mathrm{mm}}^2, t=152\mathrm{ms}$.