Exploring wave propagation dynamics in heterogeneous excitable media: effects of intermittent cellular inactivation

Figures

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  Figure 1. Uninterrupted planar wave propagation through a region containing a low density (<4%) of intermittently inactive cells. Initial wavefront at t = 0. Minor local distortions are observed within the heterogeneous zone (dashed rectangle: -7<X<0 and |Y|<200), yet the wave rapidly restores planar coherence. See Supplementary Movie at https://physics.bgu.ac.il/~yaacov/File/Scenario1.mp4 for the complete spatiotemporal evolution.

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  Figure 2. Wavefront fragmentation induced by a high density (>18%) of intermittently inactive cells. The advancing front is severed upon entering the heterogeneous region (dashed rectangle: -7<X<0 and |Y|<200), generating two free ends that curl and migrate. Partial reconnection occurs in some instances, but persistent distortion remains. See Supplementary Movie at https://physics.bgu.ac.il/~yaacov/File/Scenario2.mp4 for the complete spatiotemporal evolution.

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  Figure 3. Emergence and persistence of spiral waves in the intermediate regime (4- 18% intermittently inactive cells). Multiple random wave-breaks within the heterogeneous zone (dashed rectangle) generate fragments of sufficient curvature to initiate ectopic self-sustaining spiral waves visible after ~720 time units. See Supplementary Movie at https://physics.bgu.ac.il/~yaacov/File/Scenario3.mp4 for the complete spatiotemporal evolution.

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  Figure 4. Temporal evolution of the Normalized Wavefront Length across the three dynamic propagation regimes. The index is normalized against the constant length of the undisturbed planar wave prior to its interaction with the heterogeneous region (t< 600 TU, baseline = 1.0). Top (Scenario 1): In the low-density regime (<4% intermittently inactive cells), the wavefront experiences a minor transient truncation before rapidly healing and recovering full spatial coherence. Middle (Scenario 2): At high densities (>18%), the wavefront is severely blocked and severed, causing an abrupt initial drop. Subsequent curling of the free ends temporarily increases the total active length before collision and partial reattachment lead to a gradual decay. Bottom (Scenario 3): In the intermediate arrhythmogenic regime (4%-18%), initial localized wave breaks evolve into persistent spiral waves. The sustained surge of the normalized wavefront length far beyond the baseline, accompanied by pronounced aperiodic oscillations, explicitly reflects the continuous generation, rotation, and chaotic meandering of self-sustaining ectopic wavefronts.

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  Figure 5. Parametric bifurcation map illustrating the probability of occurrence of the propagation regimes as a function of the density of intermittently inactive cells. The primary y-axis specifically plots the probability of emergence for Scenario 3 (persistent spiral waves). For each density value, 25 independent realizations were performed with different random spatial configurations of the intermittently inactive region. Each realization was classified into one of three regimes: uninterrupted planar propagation (Scenario 1), wavefront fragmentation (Scenario 2), or persistent spiral wave activity (Scenario 3) based on visual inspection alongside quantitative measures of normalized wavefront length (see Section 3.1). Data points represent the mean probability across the 25 realizations, and error bars indicate the standard error of the mean (SEM). Shaded regions denote the specific density domains where Scenarios 1 and 2 are dominant (occurring with 100% probability).

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  Figure 6. Autonomous spiral-wave activity under repeated planar-wave stimulation in Scenario 3 conditions. A train of incoming planar waves, applied at an inter-pulse interval (IPI) of 300 time units, interacts with the heterogeneous region (dashed rectangle: -7< X< 0 and |Y|< 200). Under this relatively high-frequency stimulation, spiral activity is repeatedly generated near the boundary of the intermittent region but remains largely confined to its vicinity, reflecting the influence of partial refractoriness in the surrounding tissue. See Supplementary Movie at https://physics.bgu.ac.il/~yaacov/File/serieswave.mp4 for the complete spatiotemporal evolution.

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  Figure 7. Autonomous spiral-wave activity under repeated planar-wave stimulation in Scenario 3 conditions. A train of incoming planar waves, applied at an inter-pulse interval (IPI) of 1000 time units, interacts with the heterogeneous region (dashed rectangle: -7< X< 0 and |Y|< 200). Under this lower-frequency stimulation, the surrounding tissue has sufficient time to recover excitability, allowing spiral waves to drift and progressively invade larger regions of the excitable medium. See Supplementary Movie at https://physics.bgu.ac.il/~yaacov/File/serieswave2.mp4 for the complete spatiotemporal evolution.

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  Figure 8. Temporal evolution and phase-space trajectories of the variables V (black) and W (red) at two representative spatial locations. (A, B) At a point distant from the heterogeneous region (X = -300, Y = 0), the system exhibits regular periodic oscillations and a well-defined limit cycle. (C, D) - At a point near the boundary of the intermittent region (X = 10, Y = 0), the dynamics display pronounced stochastic oscillations and a dispersed, chaotic phase space trajectory, indicative of a loss of dynamic stability and increased arrhythmogenic potential. The boundary of the intermittent region is defined by: -7<X<0 and |Y|<200.