Quantifying The Complex Spatiotemporal Chaos of Cardiac Fibrillation in Ionic Models Across Parameter Regimes

Abstract

Quantifying the complexity of cardiac systems is fundamental to understanding the onset of rhythm disorders, from mild arrhythmias to life-threatening fibrillation. In this work, we investigate how chaos shows up and evolves in simplified cardiac models by calculating the Lyapunov exponent (LE) across different parameter sets. We show that both temporal and spatial LE estimators can be effectively applied to action potential duration (APD) data, even without full access to state variables. Specifically, the spatial-temporal algorithm and Wolf's algorithm are used in quantifying the complexity, with experiments demonstrating their distinct behaviors on various single-spiral patterns. We also identify the minimum data length and sampling density necessary to achieve robust and accurate estimation. Overall, our results suggest that these APD-based methods can be applied not only to simulation data but also to future clinical or experimental data, particularly when observations are limited, such as when only APD data are available for analysis.

Figures (15)

• Figure 1: Schematic overview of the methodology for estimating LEs in cardiac systems, exemplified using the FK model with the multiple spiral wave pattern. Panels (a)–(c) (highlighted in red) illustrate the primary workflow, while (d)–(h) provide detailed descriptions of the underlying computational components and algorithmic implementations. (a) Simulation: The FK model is applied to simulate cardiac electrical activity using WebGL acceleration. A specific measuring point i is selected for later demonstration. (b) Data$_i$: Time series are recorded for the three state variables and the calculated APD. APD is calculated for later steps. An initial transient period of 520 s is skipped, and the remaining measurement time is divided into 5 zones ($\Gamma_1(\bm{k})$ to $\Gamma_5(\bm{k})$). (c) LE Calculation: Data is used in two separate ways: as a full set of state variables and as a partial observation where only APD is known. (d) Phase Space Reconstructor (PSR): With proper embedded dimension and lagging, APD could reconstruct the original phase space by lag-embedding, shown as $\text{APD}_{m,\tau}$. The APD sequence is chromatically encoded as a schematic abstraction of its temporal trajectory. (e) Temporal Calculator (TC): For each region, Temporal LE (TLE) is calculated at different points and then averaged to obtain the final $\lambda^T$. (f) Spatial Calculator (SC): For each region, all measuring points are used inside the algorithm to compute one final spatial LE (SLE). (g) TLE: Based on Wolf's algorithm, TLE is calculated by observing the divergence of similar pairs within the time series (h) SLE: Using the spatial-temporal algorithm, all measuring points in the tissue are utilized to calculate the SLE by observing divergence of similar pairs in space.
• Figure 2: Spiral wave initiation and spatial sampling points for LE analysis. (A) Propagating wave originated by a stimulus (S1) applied along the left edge of the tissue. (B) Spiral wave initiated from a second stimulus (S2) applied behind the first wave in the lower half of the tissue. (C) Example of spatial sampling points used for LE calculation, indicated by the red dots ($8 \times 8$ grid).
• Figure 3: Determining embedding dimension using the percentage of false nearest neighbors method with input data $X \equiv \text{APD}$ and lag $\tau = 1$. Each curve corresponds to a different value of $\tau_d$ in the FK model, from 0.406 (light blue) to 0.411 (dark blue) ms with a spacing of 0.005 ms (see Table \ref{['Table FK para']}). The percentage drops to near zero at $m = 4$, identifying it as the minimum sufficient embedding dimension. Varying other parameters achieved similar results.
• Figure 4: LEs ($\lambda_{\text{single spiral}}$ as shown in Eq. \ref{['Eq TLE_single_spiral']} and Eq. \ref{['Eq SLE_single_spiral']}) for different single spiral wave dynamics of the FK model. Top: LEs for six different spiral wave dynamics obtained from the state variables ($uvw$, see Eq. \ref{['FK Voltages']}) with Wolf’s algorithm ($\lambda^T(uvw)$, blue), from APD signals with Wolf’s algorithm ($\lambda^T(\text{APD})$, orange), and from APD signals with the Spatio-temporal algorithm ($\lambda^S(\text{APD})$, green). Bottom: Simulation snapshots with white curves indicating the spiral core trajectories: (A) circular, (B) epicycloidal, (C) cycloidal, (D) hypocycloidal, (E) hypermeandering, and (F) linear. Parameters values are listed in Table \ref{['table: single_spiral_para']}.
• Figure 5: Heatmaps of LEs calculated from APD signals using Wolf’s algorithm ($\lambda^T (\bm{\Gamma}(\bm{k}))$ at measuring points in Fig. \ref{['fig:initial condition']}C, without averaging in Eq. \ref{['Eq TLE_single_spiral']}). Panels A–F correspond to spiral wave dynamics shown in Fig. \ref{['fig: single_spiral']}. Colormap represents local Tempoarl LE value ranging from near zero (blue) to strongly positive (red), highlighting spatial heterogeneity in the degree of chaoticity across the tissue.