Available via license: CC BY-NC-ND 4.0
Content may be subject to copyright.
Impact of noise on the instability of spiral waves in stochastic 2D mathematical
models of human atrial fibrillation
Euijun Song1,2
1Yonsei University College of Medicine, Seoul, Republic of Korea.
2Present: Independent Researcher, Gyeonggi, Republic of Korea.
Abstract. Sustained spiral waves, also known as rotors, are pivotal mechanisms in persistent
atrial fibrillation (AF). Stochasticity is inevitable in nonlinear biological systems such as the heart;
however, it is unclear how noise affects the instability of spiral waves in human AF. We present
a stochastic two-dimensional mathematical model of human AF and explore how Gaussian white
noise affects the instability of spiral waves. In homogeneous tissue models, Gaussian white noise
may lead to spiral-wave meandering and wavefront break-up. As the noise intensity increases, the
spatial dispersion of phase singularity (PS) points increases. This finding indicates the potential
AF-protective effects of cardiac system stochasticity by destabilizing the rotors. However, Gaussian
white noise is unlikely to affect the spiral-wave instability in the presence of localized scar or fibrosis
regions. The PS points are located at the boundary or inside the scar/fibrosis regions. Localized
scarring or fibrosis may play a pivotal role in stabilizing spiral waves regardless of the presence of
noise. This study suggests that fibrosis/scars are essential for determining the rotors in stochastic
mathematical models of AF, and further patient-derived modeling studies are required.
I. Introduction
Atrial fibrillation (AF) is the most common cardiac arrhythmia characterized by chaotic elec-
trical wave propagation, and is associated with mortality and morbidity [Iwasaki et al., 2011].
AF causes electrical and structural remodeling of the atrial tissues, evolving from paroxysmal
AF to persistent AF (PeAF). Although the mechanisms of PeAF are poorly understood, recent
studies suggest that PeAF is driven by sustained spiral waves (“rotors” or “reentrant drivers”)
localized within spatially compacted regions [Narayan et al., 2012]. The core of a spiral wave,
known as a spiral-wave tip, can be mathematically described as a phase singularity (PS) point
[Gray et al., 1998], which is the intersection of the depolarizing wavefront and the repolarizing
wave tail [Jalife, 2000]. Cardiac computational modeling approaches have been widely used to
study the complex spiral wave dynamics of human AF. In homogeneous atrial tissue models, the
electrical remodeling conditions of PeAF can sustain stable rotor meandering in spatially com-
pacted regions [Pandit et al., 2005]. In the presence of electrophysiological heterogeneities, rotors
are frequently found in fibrotic regions or at the boundaries between fibrotic and non-fibrotic tissues
[Deng et al., 2017,Roney et al., 2016,Zahid et al., 2016]. However, most computational models of
human AF numerically solve deterministic partial differential equations (PDEs), ignoring the sto-
chastic nature of complex biological systems.
E-mail address:drjunsong@gmail.com.
2020 Mathematics Subject Classification. 92C05, 92C30, 60H15.
Key words and phrases. Atrial fibrillation, Fibrosis, Spiral wave dynamics, Gaussian white noise, Stochastic partial
differential equation.
1
arXiv:2304.06439v1 [physics.bio-ph] 13 Apr 2023
2 E. SONG
Stochasticity is inevitable in complex biological systems such as gene regulatory networks, neu-
ronal networks, and cardiac systems, and plays an important role in the dynamic behavior of
nonlinear systems [Buric et al., 2007,Gammaitoni, 1995]. For example, noise-induced stochastic
resonance can be found in the FitzHugh–Nagumo model [Pikovsky and Kurths, 1997]. In two-
dimensional (2D) neuronal networks, certain thresholds of noise intensity can affect the formation
and instability of spiral waves [Ma et al., 2010,Yao et al., 2017]. However, it is unknown how noise
affects the spiral wave dynamics in human AF models. Does noise affect spiral-wave instability in
terms of the meandering of spiral-wave tips and wave break-up? Do PS points also localize near
fibrotic regions in stochastic AF models? We do not know the answers to these questions based on
experimental and clinical studies. Stochastic mathematical modeling of AF is essential for studying
the effect of noise on the spiral wave dynamics in human AF models.
In this study, we present a stochastic 2D mathematical model of human AF by adding Gaussian
white noise to the conventional deterministic reaction-diffusion equation. We use the Courtemanche-
Ramirez-Nattel human atrial cell model [Courtemanche et al., 1998], which is widely utilized in 2D
and three-dimensional (3D) AF models, to study AF mechanisms and personalize ablation treatment
strategies [Azzolin et al., 2023,Boyle et al., 2019]. Using the stochastic mathematical model of AF,
we numerically simulate spiral waves on 2D isotropic, homogeneous atrial tissues by varying noise
intensity levels. To examine whether the PS points localize at fibrotic regions in the stochastic
AF model, we further explore the spiral wave dynamics in the presence of localized scar or fibrosis
regions (Figure 1). We show that Gaussian white noise can lead to spiral-wave meandering and
wavefront break-up in homogeneous atrial tissues, whereas localized scar or fibrotic regions can
stabilize spiral waves without generating wavefront break-up.
Figure 1. Stochastic two-dimensional computational modeling of human atrial fib-
rillation. Spiral waves were numerically simulated on atrial tissues with and without
localized scar and fibrosis regions (see Methods for details).
II. Methods
II.1. Stochastic 2D computational modeling of AF. We present a stochastic 2D mathe-
matical model of human AF using the Courtemanche-Ramirez-Nattel human atrial cell model
[Courtemanche et al., 1998]. The conventional deterministic 2D AF model can be described by
Impact of noise on the instability of spiral waves in stochastic 2D mathematical models of human atrial fibrillation 3
the following reaction-diffusion equation [Pandit et al., 2005,Xie et al., 2002]:
(1) ∂V
∂t =−Iion +Istim
Cm
+D∇2V
where V(t, x) (mV) is the transmembrane potential, Iion (pA) is the total ionic current, Istim (pA)
is the stimulus current, Cm= 100 pF is the membrane capacitance, and D(mm2/ms) is the diffusion
coefficient. The total ionic current is given by
Iion =INa +ICaL +IK r +IKs +Ito +IK1+IK ur +IN aK +IN CX +IpCa +IN a,b +IC a,b.
The biophysical details of each ionic current can be found in Courtemanche et al [Courtemanche et al., 1998].
Here, we add a Gaussian white noise term to obtain the following stochastic PDE [Jung and Mayer-Kress, 1995,
Shardlow, 2005]:
(2) ∂V
∂t =−Iion +Istim
Cm
+D∇2V+σξ (t, x)
where σ(mV) is the noise intensity and ξ(t, x) is the white noise satisfying
hξ(t, x)it= 0,ξ(t, x), ξ t0,xt=δt−t0
for each x∈R2. We can rewrite the above stochastic PDE (Eq. 2) in the differential form as
follows:
(3) dV =−Iion +Istim
Cm
+D∇2Vdt +σdW (t, x)
where W(t, x) is the Wiener process satisfying dW (t, x) = ξ(t, x)dt, which is the differential form
of the Brownian motion.
We numerically solve this stochastic PDE (Eq. 3) on a 2D isotropic, homogeneous domain of
area 75×75 mm2, consisting of a 2D lattice network of 300×300 atrial cells. We used the forward
Euler method with a fixed time step of ∆t= 0.02 ms and a space step of ∆x= ∆y= 0.25 mm, and
applied the Neumann (no-flux) boundary conditions. The Laplacian ∇2Vwas approximated using
the five-point stencil. The increment of the Wiener process was numerically implemented in the Itˆo
sense as
W(t+ ∆t, x)−W(t, x) = σ√∆t η
where η∼ N (0,1) is a Gaussian random number with a mean value of 0 and a standard deviation
of 1 [Guo et al., 2012,Yao et al., 2017].
To reflect the electrical remodeling of PeAF, we reduced the L-type Ca2+ current (ICaL) by 70%,
the transient outward K+current (Ito) by 50%, and the ultrarapid delayed rectifier K+current
(IKur ) by 50%, and increased the inward rectifier K+current (IK1) by 100%, as described by Pan-
dit et al [Pandit et al., 2005]. Two diffusion coefficients were tested: D=0.1 and 0.05 mm2/ms,
which produce conduction velocities of 0.43 and 0.27 m/s, respectively. D=0.1 mm2/ms is a com-
monly used diffusion coefficient in 2D AF models that produces a physiological conduction velocity
[Pandit et al., 2005,Xie et al., 2002]. We also tested the noise intensity levels of σ=0, 1, 5, and
10 mV. Spiral waves were initiated by applying the standard S1-S2 cross-field protocol, and the
AF wave dynamics were studied for 5 s. The numerical simulation was performed using C++ code
with OpenMP parallelization. The atrial cell model is publicly available at the CellML Physiome
Project (https://models.physiomeproject.org).
II.2. Modeling scar and fibrosis regions. In addition to the 2D homogeneous model described
in the previous section, we simulated the AF wave dynamics on inhomogeneous models in the
presence of a localized scar or fibrotic region. The scar and fibrotic regions were applied to the
center of the 2D cardiac tissue with a radius of 10 mm (Figure 1). The scar was modeled as an
inexcitable and nonconductive region (D= 0). In the fibrotic regions, we reduced IK1by 50%,
ICaL by 50%, and the sodium current (INa ) by 40% to reflect the TGF-β1 fibrogenic signaling
4 E. SONG
effects; we also decreased the diffusion coefficient Dby 30% to represent gap junction remodeling
[Roney et al., 2016,Zahid et al., 2016].
II.3. Analysis of spiral waves. To analyze the spatiotemporal patterns of spiral waves, we gen-
erated 2D maps of the transmembrane potential with a sampling time step of 10 ms. PS points
were identified using the method proposed by Iyer and Gray [Iyer and Gray, 2001]. The phase
θ(t, x)∈[−π, π] at each node is calculated as,
θ(t, x) = arctan [V(t+τ, x)−Vmean (x), V (t, x)−Vmean (x)]
where τ= 30 ms is the time delay constant and Vmean is the mean of the action potential for
the whole fibrillation state. The PS points are identified if H∇θ·dr =±2π[Bray et al., 2001,
Iyer and Gray, 2001]. To evaluate the spatial distribution of the PS points, we computed the PS
spatial dispersion as the standard deviation of the PS points as follows:
PS spatial dispersion = sPikP Si−P S meank2
NP S
where P Siis the ith PS point, P Smean is the average location of the PS points, NP S is the number
of PS points, and k·k is the L2norm. The PS spatial dispersion vanishes if the spiral-wave tip is
consistent over time. The spiral wave rotation frequency is estimated as the maximum dominant
frequency of the action potential signals acquired at the node (50, 50). All signal analyses were
performed using MATLAB 2021b (MathWorks, Inc.).
III. Results
III.1. Noise-induced instability of spiral waves in homogeneous models. First, we numer-
ically simulated human AF on 2D homogeneous tissue models. Figure 2shows the transmembrane
potential maps and PS plots. With noise intensity levels of σ=0 and 1 mV, spiral waves were
localized near the center of the atrial tissue, indicating sustained stable rotor dynamics. The max-
imum distance between the PS points was <20 mm. When a noise intensity level of σ=5 mV was
applied, spiral waves continuously meandered (>30 mm) and wavefront break-up occurred. At a
noise intensity level of σ=10 mV, spiral waves were largely meandered (>60 mm), and wavefront
break-up also occurred. At a noise intensity level of σ=10 mV and a diffusion coefficient of D=0.1
mm2/ms, the spiral waves were spontaneously terminated at 4.5 s. All the other cases showed
sustained fibrillation states for >5 s. Sequences of the transmembrane potential maps are shown in
Supplementary Figure S1.
We quantitatively determined how noise changes the spiral wave rotation frequency and PS spatial
dispersion, as shown in Figure 3. Noise changed the spiral wave frequency by only approximately
<2.6% (Figure 3A). As the noise intensity level increased from 0 to 10 mV, the spiral wave rotation
frequencies were 8.0, 8.0, 7.8, and 7.8 Hz for the D=0.1 mm2/ms cases, and 7.8, 7.8, 7.8, and
7.6 Hz for the D=0.05 mm2/ms cases, respectively. However, noise dramatically increased the PS
spatial dispersion (Figure 3B). As the noise intensity level increased from 0 to 10 mV, the PS spatial
dispersions were 4.5, 5.0, 9.5, and 28.1 mm for the D=0.1 mm2/ms cases, and 3.1, 3.5, 14.1, and
23.1 mm for the D=0.05 mm2/ms cases, respectively. This result is consistent with the observations
of noise-induced spiral-wave meandering and wavefront breakup, as shown in Figure 2.
III.2. Effects of scar regions. Next, we examined the effect of localized scar regions on electrical
wave propagation in stochastic AF models. As shown in Figure 4, electrical waves were periodi-
cally propagated around the scar region with a radius of 10 mm. The PS points were identified
at the boundary of the scar region. There was no wavefront break-up, and the fibrillation states
Impact of noise on the instability of spiral waves in stochastic 2D mathematical models of human atrial fibrillation 5
Figure 2. Transmembrane potential maps and phase singularity (PS) plots for
stochastic 2D atrial fibrillation simulations on homogeneous tissues. The simulations
were performed for diffusion coefficients of D=0.1 and 0.05 mm2/ms, and noise
intensity levels of σ=0, 1, 5, and 10 mV. The PS points were computed during the
whole fibrillation state, and the action potential signals were acquired at the node
(50, 50).
Figure 3. Spiral wave rotation frequency (A) and phase singularity (PS) spatial
dispersion (B) values for stochastic 2D atrial fibrillation simulations on homogeneous
tissues, depending on the diffusion coefficients of D=0.1 and 0.05 mm2/ms and the
noise intensity levels of σ=0, 1, 5, and 10 mV.
were sustained for >5 s. This stable wave propagation pattern is known as an “anatomical reen-
try” rather than a “spiral wave,” which is usually defined in the absence of an anatomic obstacle
[Allessie et al., 1977]. Noise changed the spiral wave frequencies by only approximately <3.3% (Fig-
ure 5A). As the noise intensity level increased from 0 to 10 mV, the spiral wave rotation frequencies
were 6.0, 6.0, 6.0, and 6.2 Hz for the D=0.1 mm2/ms cases, and 4.0, 4.0, 4.0, and 4.0 Hz for the
D=0.05 mm2/ms cases, respectively. In all cases, the PS spatial dispersions were consistently 10.0
mm, which is almost exactly the radius of the scar region (Figure 5B).
6 E. SONG
Figure 4. Transmembrane potential maps and phase singularity (PS) plots for
stochastic 2D atrial fibrillation simulations in the presence of scar regions. The
simulations were performed for diffusion coefficients of D=0.1 and 0.05 mm2/ms,
and noise intensity levels of σ=0, 1, 5, and 10 mV. The PS points were computed
during the whole fibrillation state, and the action potential signals were acquired at
the node (50, 50).
Figure 5. Spiral wave rotation frequency (A) and phase singularity (PS) spatial
dispersion (B) values for stochastic 2D atrial fibrillation simulations in the presence
of scar regions, depending on the diffusion coefficients of D=0.1 and 0.05 mm2/ms
and the noise intensity levels of σ=0, 1, 5, and 10 mV.
III.3. Effects of fibrosis regions. Similarly, we examined how localized fibrosis regions affect
the spiral wave dynamics in stochastic AF models. As shown in Figure 6, when the diffusion
coefficient was D=0.1 mm2/ms, spiral waves meandered around the fibrosis region with a radius
of 10 mm, occasionally invading the fibrosis region when those cells were recovered from refractory
periods. When the diffusion coefficient was D=0.05 mm2/ms, spiral waves meandered inside the
fibrotic region. The PS points were identified at the boundary and inside the fibrotic region. All
cases showed sustained fibrillation states for >5 s, and there was no wavefront breakup. The noise
Impact of noise on the instability of spiral waves in stochastic 2D mathematical models of human atrial fibrillation 7
changed spiral wave frequencies by only approximately <3.2% (Figure 7A). As the noise intensity
level increased from 0 to 10 mV, the spiral wave rotation frequencies were 6.2, 6.2, 6.2, and 6.4
Hz for the D=0.1 mm2/ms cases, and 5.2, 5.2, 5.1, and 5.2 Hz for the D=0.05 mm2/ms cases,
respectively. In all cases, the PS spatial dispersions were consistently below 10.0 mm, implying the
spiral wave meandering inside the fibrotic region (Figure 7B).
Figure 6. Transmembrane potential maps and phase singularity (PS) plots for
stochastic 2D atrial fibrillation simulations in the presence of fibrosis regions. The
simulations were performed for diffusion coefficients of D=0.1 and 0.05 mm2/ms,
and noise intensity levels of σ=0, 1, 5, and 10 mV. The PS points were computed
during the whole fibrillation state, and the action potential signals were acquired at
the node (50, 50).
Figure 7. Spiral wave rotation frequency (A) and phase singularity (PS) spatial
dispersion (B) values for stochastic 2D atrial fibrillation simulations in the presence
of fibrosis regions, depending on the diffusion coefficients of D=0.1 and 0.05 mm2/ms
and the noise intensity levels of σ=0, 1, 5, and 10 mV.
8 E. SONG
IV. Discussion
In this study, we numerically simulated stochastic 2D models of human AF and explored the
effects of Gaussian white noise on the instability of spiral waves. In homogeneous atrial tissue
models, the electrical remodeling condition of the PeAF can generate stable rotor dynamics in the
absence of noise. However, Gaussian white noise can lead to spiral-wave meandering and wavefront
breakup without significantly altering the spiral wave frequencies (Figures 2and 3). This finding
indicates the potential AF-protective effect of the stochasticity of cardiac systems by destabilizing
rotors. In contrast, Gaussian white noise is unlikely to affect spiral-wave instability in the presence
of localized scar and fibrosis regions, and the PS points are located at the scar or fibrosis areas
(Figures 4–7). Thus, scarring or fibrosis may play a pivotal role in stabilizing spiral waves regardless
of the Gaussian white noise. The overall results suggest that tissue heterogeneities such as scars
and fibrosis are essential for determining the rotors in stochastic 2D mathematical AF models, and
further patient-derived stochastic 3D modeling studies are needed.
The pathophysiological importance of fibrosis in AF has been extensively studied. In patient-
derived 3D computational models, the PS points are associated with fibrotic regions [Roney et al., 2016,
Zahid et al., 2016]; this spatial relationship between fibrosis and the rotor is robust to the model
parameter variability [Deng et al., 2017]. In addition, fibroblast–myocyte coupling can affect the
spiral wave dynamics and extracellular electrograms [Ashihara et al., 2012,Zlochiver et al., 2008];
however, this coupling effect was not incorporated in this study. The DECAAF clinical study
also demonstrated that the degree of atrial tissue fibrosis is associated with the catheter abla-
tion outcomes in AF [Marrouche et al., 2014]. In contrast, a recent non-invasive electrophysiology
mapping system found that rotors are not directly associated with fibrosis in patients with AF
[Sohns et al., 2017]. This discrepancy between computational and clinical studies may be attrib-
uted to the model parameter uncertainty and the absence of stochasticity. Our stochastic AF
modeling approach must be further tested to examine the noise-induced instability of spiral waves
in patient-derived 3D AF models that reflect patient-specific anatomy and electrophysiology.
Although D=0.1 mm2/ms is a widely used diffusion coefficient value in 2D AF models [Pandit et al., 2005,
Xie et al., 2002], we also tested D=0.05 mm2/ms to examine the spiral wave dynamics in a severely
remodeled condition in PeAF. The results from the two diffusion coefficients were similar, except
for the spiral wave frequencies. In homogeneous tissues, the spiral wave frequency is known to
be primarily dependent on the inverse of the action potential duration and not on the conduction
velocity if the curvature effects are negligible [Qu et al., 2000,Zimik et al., 2020]. In the present
study (Figure 3), the spiral wave frequencies in the homogeneous models were 7.6–8.0 Hz, not being
significantly affected by the diffusion coefficient and noise intensity level. However, in the presence
of scar or fibrosis, the spiral wave frequencies in the D=0.05 mm2/ms cases were consistently lower
than those in the D=0.1 mm2/ms cases (Figures 5and 7). As the PS points were identified at the
boundary and inside the scar/fibrosis region, the spiral wave frequency may be mainly dependent
on the conduction velocity. Although the spiral wave frequency was not the primary focus of the
present study, further studies are needed to systemically examine whether noise alters the spiral
wave frequency under various electrophysiological conditions [Mulimani et al., 2022].
This study has several limitations. We adopted Gaussian white noise, which is neither structurally
correlated nor bounded. Because Gaussian noise is inappropriate for many real complex biological
systems, the impact of non-Gaussian noise must be investigated [Bobryk and Chrzeszczyk, 2005,
Yao et al., 2017]. The action potential and spiral wave dynamics are also sensitive to the model
parameter uncertainty/variability [Mirams et al., 2016,Qu et al., 2000]. The effects of various AF
remodeling conditions, tissue anisotropy, and electrophysiological heterogeneity should be systemi-
cally investigated further. We only tested the noise intensity levels of σ=0, 1, 5, and 10 mV because
of the large computational time. It is worthwhile to determining whether there is a critical σvalue
where transitioning of the instability of spiral waves occurs. In addition, various sizes of atrial tissue
Impact of noise on the instability of spiral waves in stochastic 2D mathematical models of human atrial fibrillation 9
and scar/fibrosis regions should be tested because wavelength and tissue size affect the spontaneous
termination of cardiac fibrillation [Qu, 2006].
Acknowledgments
This research received no external funding. This study did not produce new animal/clinical data.
The author would like to thank the anonymous reviewers for their valuable comments and Editage
for English language editing.
Conflict of interest
The author has no conflicts of interest to declare.
Authorship contribution
Euijun Song: Conceptualization, Methodology, Formal analysis, Software, Investigation, Visu-
alization, Writing – original draft.
References
[Allessie et al., 1977] Allessie, M. A., Bonke, F. I., and Schopman, F. J. (1977). Circus movement in rabbit atrial
muscle as a mechanism of tachycardia. iii. the ”leading circle” concept: a new model of circus movement in cardiac
tissue without the involvement of an anatomical obstacle. Circ Res, 41(1):9–18.
[Ashihara et al., 2012] Ashihara, T., Haraguchi, R., Nakazawa, K., Namba, T., Ikeda, T., Nakazawa, Y., Ozawa, T.,
Ito, M., Horie, M., and Trayanova, N. A. (2012). The role of fibroblasts in complex fractionated electrograms during
persistent/permanent atrial fibrillation: implications for electrogram-based catheter ablation. Circ Res, 110(2):275–
84.
[Azzolin et al., 2023] Azzolin, L., Eichenlaub, M., Nagel, C., Nairn, D., Sanchez, J., Unger, L., Dossel, O., Jadidi, A.,
and Loewe, A. (2023). Personalized ablation vs. conventional ablation strategies to terminate atrial fibrillation and
prevent recurrence. Europace, 25(1):211–222.
[Bobryk and Chrzeszczyk, 2005] Bobryk, R. V. and Chrzeszczyk, A. (2005). Transitions induced by bounded noise.
Physica A: Statistical Mechanics and its Applications, 358(2-4):263–272.
[Boyle et al., 2019] Boyle, P. M., Zghaib, T., Zahid, S., Ali, R. L., Deng, D., Franceschi, W. H., Hakim, J. B., Murphy,
M. J., Prakosa, A., Zimmerman, S. L., Ashikaga, H., Marine, J. E., Kolandaivelu, A., Nazarian, S., Spragg, D. D.,
Calkins, H., and Trayanova, N. A. (2019). Computationally guided personalized targeted ablation of persistent atrial
fibrillation. Nat Biomed Eng, 3(11):870–879.
[Bray et al., 2001] Bray, M. A., Lin, S. F., Aliev, R. R., Roth, B. J., and Wikswo, J. P., J. (2001). Experimental and
theoretical analysis of phase singularity dynamics in cardiac tissue. J Cardiovasc Electrophysiol, 12(6):716–22.
[Buric et al., 2007] Buric, N., Todorovic, K., and Vasovic, N. (2007). Influence of noise on dynamics of coupled
bursters. Phys Rev E Stat Nonlin Soft Matter Phys, 75(6 Pt 2):067204.
[Courtemanche et al., 1998] Courtemanche, M., Ramirez, R. J., and Nattel, S. (1998). Ionic mechanisms underlying
human atrial action potential properties: insights from a mathematical model. Am J Physiol, 275(1):H301–21.
[Deng et al., 2017] Deng, D., Murphy, M. J., Hakim, J. B., Franceschi, W. H., Zahid, S., Pashakhanloo, F., Trayanova,
N. A., and Boyle, P. M. (2017). Sensitivity of reentrant driver localization to electrophysiological parameter vari-
ability in image-based computational models of persistent atrial fibrillation sustained by a fibrotic substrate. Chaos,
27(9):093932.
[Gammaitoni, 1995] Gammaitoni, L. (1995). Stochastic resonance and the dithering effect in threshold physical sys-
tems. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics, 52(5):4691–4698.
[Gray et al., 1998] Gray, R. A., Pertsov, A. M., and Jalife, J. (1998). Spatial and temporal organization during cardiac
fibrillation. Nature, 392(6671):75–8.
[Guo et al., 2012] Guo, W., Du, L.-C., and Mei, D.-C. (2012). Transitions induced by time delays and cross-correlated
sine-wiener noises in a tumor–immune system interplay. Physica A: Statistical Mechanics and its Applications,
391(4):1270–1280.
10 E. SONG
[Iwasaki et al., 2011] Iwasaki, Y. K., Nishida, K., Kato, T., and Nattel, S. (2011). Atrial fibrillation pathophysiology:
implications for management. Circulation, 124(20):2264–74.
[Iyer and Gray, 2001] Iyer, A. N. and Gray, R. A. (2001). An experimentalist’s approach to accurate localization of
phase singularities during reentry. Ann Biomed Eng, 29(1):47–59.
[Jalife, 2000] Jalife, J. (2000). Ventricular fibrillation: mechanisms of initiation and maintenance. Annu Rev Physiol,
62(1):25–50.
[Jung and Mayer-Kress, 1995] Jung, P. and Mayer-Kress, G. (1995). Noise controlled spiral growth in excitable media.
Chaos, 5(2):458–462.
[Ma et al., 2010] Ma, J., Tang, J., Zhang, A., and Jia, Y. (2010). Robustness and breakup of the spiral wave in a
two-dimensional lattice network of neurons. Science China Physics, Mechanics and Astronomy, 53:672–679.
[Marrouche et al., 2014] Marrouche, N. F., Wilber, D., Hindricks, G., Jais, P., Akoum, N., Marchlinski, F., Khol-
movski, E., Burgon, N., Hu, N., Mont, L., Deneke, T., Duytschaever, M., Neumann, T., Mansour, M., Mahnkopf,
C., Herweg, B., Daoud, E., Wissner, E., Bansmann, P., and Brachmann, J. (2014). Association of atrial tissue
fibrosis identified by delayed enhancement mri and atrial fibrillation catheter ablation: the decaaf study. JAMA,
311(5):498–506.
[Mirams et al., 2016] Mirams, G. R., Pathmanathan, P., Gray, R. A., Challenor, P., and Clayton, R. H. (2016). Uncer-
tainty and variability in computational and mathematical models of cardiac physiology. J Physiol, 594(23):6833–6847.
[Mulimani et al., 2022] Mulimani, M. K., Zimik, S., and Pandit, R. (2022). An in silico study of electrophysiological
parameters that affect the spiral-wave frequency in mathematical models for cardiac tissue. Frontiers in Physics, 9.
[Narayan et al., 2012] Narayan, S. M., Krummen, D. E., Shivkumar, K., Clopton, P., Rappel, W. J., and Miller, J. M.
(2012). Treatment of atrial fibrillation by the ablation of localized sources: Confirm (conventional ablation for atrial
fibrillation with or without focal impulse and rotor modulation) trial. J Am Coll Cardiol, 60(7):628–36.
[Pandit et al., 2005] Pandit, S. V., Berenfeld, O., Anumonwo, J. M., Zaritski, R. M., Kneller, J., Nattel, S., and Jalife,
J. (2005). Ionic determinants of functional reentry in a 2-d model of human atrial cells during simulated chronic
atrial fibrillation. Biophys J, 88(6):3806–21.
[Pikovsky and Kurths, 1997] Pikovsky, A. S. and Kurths, J. (1997). Coherence resonance in a noise-driven excitable
system. Physical Review Letters, 78(5):775–778.
[Qu, 2006] Qu, Z. (2006). Critical mass hypothesis revisited: role of dynamical wave stability in spontaneous termi-
nation of cardiac fibrillation. Am J Physiol Heart Circ Physiol, 290(1):H255–63.
[Qu et al., 2000] Qu, Z., Xie, F., Garfinkel, A., and Weiss, J. N. (2000). Origins of spiral wave meander and breakup
in a two-dimensional cardiac tissue model. Ann Biomed Eng, 28(7):755–71.
[Roney et al., 2016] Roney, C. H., Bayer, J. D., Zahid, S., Meo, M., Boyle, P. M., Trayanova, N. A., Haissaguerre, M.,
Dubois, R., Cochet, H., and Vigmond, E. J. (2016). Modelling methodology of atrial fibrosis affects rotor dynamics
and electrograms. Europace, 18(suppl 4):iv146–iv155.
[Shardlow, 2005] Shardlow, T. (2005). Numerical simulation of stochastic pdes for excitable media. Journal of Com-
putational and Applied Mathematics, 175(2):429–446.
[Sohns et al., 2017] Sohns, C., Lemes, C., Metzner, A., Fink, T., Chmelevsky, M., Maurer, T., Budanova, M., Solntsev,
V., Schulze, W. H. W., Staab, W., Mathew, S., Heeger, C., Reissmann, B., Kholmovski, E., Kivelitz, D., Ouyang,
F., and Kuck, K. H. (2017). First-in-man analysis of the relationship between electrical rotors from noninvasive
panoramic mapping and atrial fibrosis from magnetic resonance imaging in patients with persistent atrial fibrillation.
Circ Arrhythm Electrophysiol, 10(8):e004419.
[Xie et al., 2002] Xie, F., Qu, Z., Garfinkel, A., and Weiss, J. N. (2002). Electrical refractory period restitution and
spiral wave reentry in simulated cardiac tissue. Am J Physiol Heart Circ Physiol, 283(1):H448–60.
[Yao et al., 2017] Yao, Y., Deng, H., Yi, M., and Ma, J. (2017). Impact of bounded noise on the formation and
instability of spiral wave in a 2d lattice of neurons. Sci Rep, 7(1):43151.
[Zahid et al., 2016] Zahid, S., Cochet, H., Boyle, P. M., Schwarz, E. L., Whyte, K. N., Vigmond, E. J., Dubois, R.,
Hocini, M., Haissaguerre, M., Jais, P., and Trayanova, N. A. (2016). Patient-derived models link re-entrant driver
localization in atrial fibrillation to fibrosis spatial pattern. Cardiovasc Res, 110(3):443–54.
[Zimik et al., 2020] Zimik, S., Pandit, R., and Majumder, R. (2020). Anisotropic shortening in the wavelength of elec-
trical waves promotes onset of electrical turbulence in cardiac tissue: An in silico study. PLoS One, 15(3):e0230214.
[Zlochiver et al., 2008] Zlochiver, S., Munoz, V., Vikstrom, K. L., Taffet, S. M., Berenfeld, O., and Jalife, J. (2008).
Electrotonic myofibroblast-to-myocyte coupling increases propensity to reentrant arrhythmias in two-dimensional
cardiac monolayers. Biophys J, 95(9):4469–80.
Impact of noise on the instability of spiral waves in stochastic 2D mathematical models of human atrial fibrillation 11
Supplementary materials
Figure S1. Sequences of transmembrane potential maps for stochastic 2D atrial
fibrillation simulations on homogeneous tissues.
