Dynamics of spatiotemporal line defects and chaos control in complex excitable systems

Abstract

Spatiotemporal pattern formation governs dynamics and functions in various biological systems. In the heart, excitable waves can form complex oscillatory and chaotic patterns even at an abnormally higher frequency than normal heart beats, which increase the risk of fatal heart conditions by inhibiting normal blood circulation. Previous studies suggested that line defects (nodal lines) play a critical role in stabilizing those undesirable patterns. However, it remains unknown if the line defects are static or dynamically changing structures in heart tissue. Through in vitro experiments of heart tissue observation, we reveal the spatiotemporal dynamics of line defects in rotating spiral waves. We combined a novel signaling over-sampling technique with a multi-dimensional Fourier analysis, showing that line defects can translate, merge, collapse and form stable singularities with even and odd parity while maintaining a stable oscillation of the spiral wave in the tissue. These findings provide insights into a broad class of complex periodic systems, with particular impact to the control and understanding of heart diseases.

Figure 1

Schematic illustration of spatially discordant alternans in high-frequency oscillatory cardiac tissue. The three time series illustrate the temporal intracellular calcium release at three spatially different positions. Positions x₁ and x₃ are spatially separated by the nodal line (x₂). The black arrows illustrate the direction of wave propagation originating from the electrode. The three frequency spectra on the right side show the respective amplitudes |F _x,y(f )|. In case of period-1 oscillations (nodal lines, x₂) no frequency- peak at f _1/2 is detected and f is the rotational frequency of the spiral wave. Thus, a spatial amplitude map at f _1/2 will reveal the nodal lines as low amplitude (low energy) bands in the tissue. (For a more detailed example that compares normal wave conduction, spatially discordant and spatially concordant alternans, see the supplemental material, Fig. S1).

Figure 2

Analysis of spatially concordant alternans. (A to C) show three typical calcium signaling examples that are obtained using the signal oversampling technique (black lines). The average is shown as a red line that is used to quantify CA-ALTM and ΔCT (see Eqs (1) and (2)). A shows an example observed at a nodal line. (B and C) show examples of low (CA-ALTM ~ 100%) and strong (CA-ALTM ~ 50%) alternans, respectively. (D) shows an illustration of the cytosolic transient amplitude (CTA) and cytosolic transient duration/interval (CTD/CTI) in a typical single pixel recorded cytosolic calcium signal of high-frequency paced waves. CTD and CTI are determined by considering the intensity threshold at 50%. (E and F) show a free and an obstacle-anchored discordant alternating wave obtained by multidimensional Fourier analysis, respectively. The nodal lines are analyzed in line segments perpendicular to the nodal line orientation along a line segment of approximately 5 mm (see arrows in E) and (F). (G and H) show the CA-ALTM and ΔCT calculated along the line segment by Eq. (1) and by Eq. (2), respectively. The black circles and white squares correspond to the examples shown in E and F, respectively.

Figure 3

Spatiotemporal cobweb dynamics in concordant alternating spiral waves. (A and B) show the double period (f _1/2) of the cytosolic calcium signaling along the line segments shown in Fig. 2A. The corresponding spatial cobweb dynamics are shown below in (C and D), respectively. The black circles indicate the restitution curve of the respective wave dynamics. The white circles highlight the slope of the intrinsic frequency f at which the spiral wave rotates around the wave singularity, and the obstacle, respectively, where f can be obtained as the interpolation to f ⁻¹ = CTI(CTD = 0) or f ⁻¹ = CTD(CTI = 0).

Figure 4

Experimentally observed discordantly alternating spiral waves in confluent cardiomyocyte tissue. The upper and lower images show the phase and amplitude of the spiral waves at the rotating frequency f (left) and the half-frequency f _1/2 (right) in each panel, respectively. (A) shows a non-alternating (period-1) obstacle-anchored spiral wave, (B) shows an obstacle-anchored spiral wave, and (C) shows a free discordantly alternating (period-2) spiral wave. The amplitude is normalized for each spiral wave independently. The discordant alternating spirals show an increase in energy (amplitude) at f _1/2 compared to the rotating frequency f of the spiral. Nodal lines (black regions) are detected at f _1/2 as low-amplitude stripes that separate the high amplitude patches showing the spatially alternating wave dynamics.

Figure 5

Spatiotemporal nodal line dynamics in a cardiomyocyte tissue. (A to D) show the amplitude (f _1/2) and phase space (f ) of the spatiotemporal evolving nodal line dynamics of the free spiral wave (see Fig. 4C). A and B show the merging of a nodal line (white arrow). (C) shows the anisotropic merging after a nodal line break, which temporally induces a locally confined spiral wave (white curved arrow). (D) shows wave-curvature dominant dynamics induced by the linear decay of the spiral frequency f. The temporal change of f is shown in (E) within a time window of 30 minutes. (F) shows the estimated restitution curve of the cytosolic calcium dynamics extracted from A (CTD < 0.2 sec) and spontaneous wave activity (CTD > 0.2 sec) before the creation of the concordant alternans spiral wave. The latter is shown as an intensity time series in the inlet.