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Comparison of Complexity and Predictability of a Cellular Automaton Model in Excitable Media Cardiac Wave Propagation Compared with a FitzHugh-Nagumo Model

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Background: Excitable media are spatially distributed systems that propagate signals without damping. Examples include re propagating through a forest, the Belousov-Zhabotinsky reaction, and cardiac tissue. (1) Excitable media generate waves which synchronize cardiac muscle contraction with each heartbeat. Spa-tiotemporal patterns formed by excitation waves distinguish healthy heart tissues from diseased ones. (3) Discrete Greenberg-Hastings Cellular-Automaton (CA) (1) and the continuous FitzHugh-Nagumo (FHN) model (7 are two methods used to simulate cardiac wave propagation. However, previous observations have shown that these models are not accurately predictive of experimental results as a function of time. We hypothesize that cardiac simulations deviate from the experimental data at a rate that depends on the complexity of the experimental data's initial conditions (I.C.).
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Page 66 McGill Science Undergraduate Research Journal - msurj.com
Comparison of Complexity and
Predictability of a Cellular Automaton Model
in Excitable Media Cardiac Wave Propagation
Compared with a FitzHugh-Nagumo Model
Yujing Zou1,2, Gil Bub2
Abstract
Background: Excitable media are spatially distributed systems that propagate signals without damping. Ex-
amples include re propagating through a forest, the Belousov-Zhabotinsky reaction, and cardiac tissue. (1)
Excitable media generate waves which synchronize cardiac muscle contraction with each heartbeat. Spa-
tiotemporal patterns formed by excitation waves distinguish healthy heart tissues from diseased ones. (3)
Discrete Greenberg-Hastings Cellular- Automaton (CA) (1) and the continuous FitzHugh- Nagumo (FHN)
model (7 are two methods used to simulate cardiac wave propagation. However, previous observations
have shown that these models are not accurately predictive of experimental results as a function of time.
We hypothesize that cardiac simulations deviate from the experimental data at a rate that depends on the
complexity of the experimental data’s initial conditions (I.C.).
Methods: To test this hypothesis, we investigated two types of propagating waves with di erent complex-
ities: a planar (i.e. simple) and a spiral wave (i.e. complex). With the same I.C., we rst compared simulation
results of a Greenberg-Hastings Cellular Automaton (GH-CA) model to that a FitzHugh-Nagumo (FHN) con-
tinuous model which we used as a surrogate for experimental data. We then used median- ltered real-time
cardiac tissue experimental data to initialize the GH-CA model and observe the divergence of wave propa-
gation in the simulation and the experiment.
Results and Conclusion: The alignment between the CA model of a planar wave and the FHN model remains
constant, while the degree of overlap between the CA and FHN models decreases for a spiral wave as a func-
tion of time. CA simulation initialized by a planar wave real-time cardiac tissue data propagates like the ex-
perimental data, however, this is not the case for the spiral wave experimental data. We were able to con rm
our hypothesis that the divergence between the two models are due to initial condition (I.C.) complexity.
Discussion: We discuss a promising strategy to represent a GH-CA model as a Convolutional Neural Network
(CNN) to enhance predictability of the model when an initial condition is given by the experimental data
with higher level of complexity.
Introduction
An excitable media system can be viewed as a group of coupled individual
elements where each element can pass information to its neighbors with
various neighborhood size. A signal over a certain threshold initiates a
wave of activity moving across the excitable medium. (2) ey are spatially
distributed systems that propagate signals without damping. An excitable
media is characterized by its threshold of excitability, which is a certain
level of excitation to be reached before the system can generate travelling
waves whose shape and speed remain unchanged through the medium.
Examples of travelling waves in excitable media include re propagating
through a forest, the Belousov-Zhabotinsky (BZ) reaction, and propagat-
ing waves for means of communication within and across nerves as well as
generating contraction (8) in cardiac tissue. (1) More speci cally, the heart
supports propagating waves in a variety of di erent geometrical patterns
including plane waves, spirals, and multiple spirals. As a physical system
passing signals by di usion, this travelling wave is a result of propagating
electrical activity in cardiac muscle involving sodium and potassium ions
moving to neighboring cells. (2)
e wave dynamics and the resultant spatiotemporal patterns are essential
to the heart’s function. (3) Changes in the spatial patterns of these waves
can cause potentially deadly arrhythmias, therefore, spatiotemporal pat-
terns formed by excitation waves can distinguish healthy heart tissues
from diseased ones. In the context of this paper, we de ne planar waves as
travelling waves emanating from a central pacemaking source that acts to
Research Article.
1Department of Math, McGill
University, Montreal, QC, Canada
2Department of Physiology, McGill
University, Montreal, QC, Canada
synchronize contraction during a healthy heartbeat. In contrast, aberrant
re-entrant waves, which have a characteristic spiral geometry, re-excite the
tissue rapidly and underlie potentially deadly tachycardias and brillation.
(3) We de ne them as spiral waves. erefore, a planar wave’s initial con-
dition is deemed to have less complexity than that of a spiral wave. We use
cardiac monolayers which are thin sheets of heart muscle tissue grown in
a petri dish to examine the dynamics of these propagating waves. ese
cultured cardiac cells can form connections and generate excitation waves
which propagate out from an initiating point (a target) or a re-entrant cir-
cuit (a circuit of electricity where an impulse re-enters and a region of
the heart is repetitively excited) with a spiral shape. us, studying these
waves from the cardiac monolayers can help improve our understanding
of cardiac arrhythmias. (2)
In this paper, we focus on computational methods to study these excitable
waves. 0%)$,'3'* $'&&(&/,2/(3#1/3#.* ;<7=* and $#.3%.(#()* 5%3>?(-42@/2
-(1#*;5?@=*1#"'&) are both well-known systems for simulating cardiac
wave propagation. In a cellular automaton model, each cell has a nite
number of states. ese states are updated based on the states of their
neighbors and their own previous state. It is an extremely powerful meth-
od for studying the dynamics of an excitable media due to its simple rules
which underpin the nature of connected cells, even while the processes
driving the physical system can be rather complicated (Fig. 1, rst row).
However, previous results have demonstrated that neither of the CA and
FHN models show consistently accurate predictions of excited wave be-
haviors that align with experimental results (Fig. 2, second row) a er a
Keywords.
Cellular automaton,
FitzHugh-Nagumo model,
excitable media, cardiac dynamics,
convolutional neural network
Email Correspondence
yujing.zou@mail.mcgill.ca
Submitted: 01/15/20
Page 67
Volume 15 | Issue 1 | April 2020
Comparison of Complexity and
Predictability of a Cellular Automaton Model
in Excitable Media Cardiac Wave Propagation
Compared with a FitzHugh-Nagumo Model
Yujing Zou1,2, Gil Bub2
Abstract
Background: Excitable media are spatially distributed systems that propagate signals without damping. Ex-
amples include re propagating through a forest, the Belousov-Zhabotinsky reaction, and cardiac tissue. (1)
Excitable media generate waves which synchronize cardiac muscle contraction with each heartbeat. Spa-
tiotemporal patterns formed by excitation waves distinguish healthy heart tissues from diseased ones. (3)
Discrete Greenberg-Hastings Cellular- Automaton (CA) (1) and the continuous FitzHugh- Nagumo (FHN)
model (7 are two methods used to simulate cardiac wave propagation. However, previous observations
have shown that these models are not accurately predictive of experimental results as a function of time.
We hypothesize that cardiac simulations deviate from the experimental data at a rate that depends on the
complexity of the experimental data’s initial conditions (I.C.).
Methods: To test this hypothesis, we investigated two types of propagating waves with di erent complex-
ities: a planar (i.e. simple) and a spiral wave (i.e. complex). With the same I.C., we rst compared simulation
results of a Greenberg-Hastings Cellular Automaton (GH-CA) model to that a FitzHugh-Nagumo (FHN) con-
tinuous model which we used as a surrogate for experimental data. We then used median- ltered real-time
cardiac tissue experimental data to initialize the GH-CA model and observe the divergence of wave propa-
gation in the simulation and the experiment.
Results and Conclusion: The alignment between the CA model of a planar wave and the FHN model remains
constant, while the degree of overlap between the CA and FHN models decreases for a spiral wave as a func-
tion of time. CA simulation initialized by a planar wave real-time cardiac tissue data propagates like the ex-
perimental data, however, this is not the case for the spiral wave experimental data. We were able to con rm
our hypothesis that the divergence between the two models are due to initial condition (I.C.) complexity.
Discussion: We discuss a promising strategy to represent a GH-CA model as a Convolutional Neural Network
(CNN) to enhance predictability of the model when an initial condition is given by the experimental data
with higher level of complexity.
Introduction
An excitable media system can be viewed as a group of coupled individual
elements where each element can pass information to its neighbors with
various neighborhood size. A signal over a certain threshold initiates a
wave of activity moving across the excitable medium. (2) ey are spatially
distributed systems that propagate signals without damping. An excitable
media is characterized by its threshold of excitability, which is a certain
level of excitation to be reached before the system can generate travelling
waves whose shape and speed remain unchanged through the medium.
Examples of travelling waves in excitable media include re propagating
through a forest, the Belousov-Zhabotinsky (BZ) reaction, and propagat-
ing waves for means of communication within and across nerves as well as
generating contraction (8) in cardiac tissue. (1) More speci cally, the heart
supports propagating waves in a variety of di erent geometrical patterns
including plane waves, spirals, and multiple spirals. As a physical system
passing signals by di usion, this travelling wave is a result of propagating
electrical activity in cardiac muscle involving sodium and potassium ions
moving to neighboring cells. (2)
e wave dynamics and the resultant spatiotemporal patterns are essential
to the heart’s function. (3) Changes in the spatial patterns of these waves
can cause potentially deadly arrhythmias, therefore, spatiotemporal pat-
terns formed by excitation waves can distinguish healthy heart tissues
from diseased ones. In the context of this paper, we de ne planar waves as
travelling waves emanating from a central pacemaking source that acts to
Research Article.
1Department of Math, McGill
University, Montreal, QC, Canada
2Department of Physiology, McGill
University, Montreal, QC, Canada
synchronize contraction during a healthy heartbeat. In contrast, aberrant
re-entrant waves, which have a characteristic spiral geometry, re-excite the
tissue rapidly and underlie potentially deadly tachycardias and brillation.
(3) We de ne them as spiral waves. erefore, a planar wave’s initial con-
dition is deemed to have less complexity than that of a spiral wave. We use
cardiac monolayers which are thin sheets of heart muscle tissue grown in
a petri dish to examine the dynamics of these propagating waves. ese
cultured cardiac cells can form connections and generate excitation waves
which propagate out from an initiating point (a target) or a re-entrant cir-
cuit (a circuit of electricity where an impulse re-enters and a region of
the heart is repetitively excited) with a spiral shape. us, studying these
waves from the cardiac monolayers can help improve our understanding
of cardiac arrhythmias. (2)
In this paper, we focus on computational methods to study these excitable
waves. 0%)$,'3'* $'&&(&/,2/(3#1/3#.* ;<7=* and $#.3%.(#()* 5%3>?(-42@/2
-(1#*;5?@=*1#"'&) are both well-known systems for simulating cardiac
wave propagation. In a cellular automaton model, each cell has a nite
number of states. ese states are updated based on the states of their
neighbors and their own previous state. It is an extremely powerful meth-
od for studying the dynamics of an excitable media due to its simple rules
which underpin the nature of connected cells, even while the processes
driving the physical system can be rather complicated (Fig. 1, rst row).
However, previous results have demonstrated that neither of the CA and
FHN models show consistently accurate predictions of excited wave be-
haviors that align with experimental results (Fig. 2, second row) a er a
Keywords.
Cellular automaton,
FitzHugh-Nagumo model,
excitable media, cardiac dynamics,
convolutional neural network
Email Correspondence
yujing.zou@mail.mcgill.ca
Submitted: 01/15/20
few seconds. Consequently, it is crucial to determine the roots of the dis-
crepancies seen between the CA model and experimental result in order to
generate an accurate model of the dynamics. erefore, we hypothesized
that cardiac simulations deviate from the experimental data as a function
of the complexity of initial conditions (I.C.) of the experimental data.
e cardiac membrane potential function of the FHN model is contin-
uous. It is relatively simple and not computationally expensive so it was
used to produce surrogate data as our “ground truth” for wave propaga-
tions. Here, we provide a direct comparison between Greenberg-Hastings
Cellular Automaton and FHN model simulations for two types of excited
waves with di erential complexity, a planar wave I.C. for the simple type
and a spiral wave I.C. for the complex type. Our results were able to con-
rm our hypothesis that when the same I.C. was given, the two models
diverge earlier in simulation time steps when the I.C. is more complex.
Methods
To study what gives rise to the divergence between a cellular automaton
model and experimental results over time, we investigated two types of
propagating waves with di erent complexity: a planar and a spiral wave.
We rst built a discrete GH-CA model of size 100 by 100 cells. en we
used di usively coupled FHN equations to generate surrogate data sets as
our “ground truth experimental data” since its (fast and slow) variables
together re ect the cardiac membrane potential, which is continuous. We
then compared the CA and FHN models with the same initial condition
using algorithms to quantify wave behavior.
Greenberg-Hastings Cellular Automaton Model
e GH-CA model follows a simple set of rules to represent the complex
physiological processes that result in electrical impulse generation, con-
duction, and propagation. It does so by representing electrical activity
propagation by cardiac action potentials on a discrete lattice of points in
space (i.e. representing the volume of the myocardium) as a form of infor-
mation transmission. e cellular automaton model is made up of discrete
integer numbers where each number represents its own state. States in a
CA model are categorized as being at rest, excited, or refractory. Import-
ant parameters used in the CA rules for governing wave propagation are
the following: the threshold (T) where 0<A<1, the excitatory state (E), the
refractory state (R), the resting state (de ned as 0), size of the cardiac tis-
sue (N) which is the number of cells in a row of a 2D CA square array, and
the neighborhood size (r) which determines the number of neighboring
cells that a ect the current cell state in next time step. We de ne a ‘cell’
state as a discrete integer that represents the state of the cell at position
(%BC) in the 2D square matrix. For the wave to progress, the state of each
cell must be updated during each generation based on the simple rules we
de ne in our GH-CA model. Let (C;3= be a cell state at a certain time step
(or generation). If 1 (C;3=*E+R, then (C;3D1)= (C;3=*+1. In simulations
using a neighborhood with square boundaries (a Moore neighborhood),
we saw unrealistic sharp edges (Fig.2) in our CA simulations. is is due
to the condition for a resting cell (state=0) to become the rst state of an
excited cell (state=1). For square boundaries, when
a resting state becomes 1. We therefore adapted a method where we cre-
ated a new coordinate system initially developed by !"#*'3*/&E*(1) into our
GH-CA model. is algorithm made the edges of our waves signi cantly
smoother (Fig. 2). Speci cally, we re-de ned the original coordinate of any
cell from (x0, y0), where x0 and y0 are integer numbers, to (x0 + ε x, y0 + ε y),
where ε x and ε y are uniformly distributed decimal values between -0.5
and 0.5. We also assigned a random weight Sj to each cell in a 2D GH-CA
matrix where Sj is uniformly distributed between 0.5 and 1.5. Here, new
coordinates and random weights are assigned to each cell every new time
step. en we compute the distance between a resting cell to all its Moore
neighboring cells 0FCB%G. A resting cell becomes excited if
is randomization process of the new coordinate system successfully
eliminated unwanted edges in our GH-CA simulation with the Moore
neighborhood counting method is used (Fig. 2).
FitzHugh-Nagumo Equations Model
e FHN model (5) is popular for simulating excitable media because of
its analytical tractability (7), relative simplicity, and ease of geometrical
analysis. e basic form of the FHN model has two coupled, non-linear
ordinary di erential equations. One of these depicts the fast evolution of
the neuronal membrane voltage while the other equation represents the
slower recovery (refractory) action of sodium channel de-inactivation and
potassium channel deactivation. For simulating a travelling wave, a spatial
di usion term (i.e. a second derivative in spatial coordinates) is needed for
the rst equation to model an action potential propagation process, which
turns the FHN model into a coupled-di usive partial di erential equa-
tion. (7) e electrical propagation properties in an excitable media like
nerve bers are analogous to that of myocardium. Since the model tracks
Figure 1. Greenberg-Hastings Cellular Automaton (GH-CA)
wave ( rst row) propagation simulations generated at various
parameter values can accurately represent experimental data.
The GH-CA algorithm is detailed in section 2.1. The rst row
shows Cellular Automaton simulations of wave propagation:
where the rst sub gure shows a planar wave and the sec-
ond and the third sub gure show a spiral wave with various
refractory states where the spiral source is to the very left of
the panel. In the second row: the rst sub gure is a snapshot
of a spiral wave during tachycardia, the second and the third
pictures show an example output of wave segmentation and
tracking using an automated wave tracking software in cul-
tured cardiac monolayers called ‘Cco nn’. (4)
Figure 2. The e ec t of integrating the new coordinate system
into our GH-CA model. We could see in the left column; edge
wave front appears in the CA simulation whereas the wave-
fronts become much smoother when our new coordinate
algorithm is adopted.
Page 68 McGill Science Undergraduate Research Journal - msurj.com
the membrane voltage continuously and is easily controllable compared
to real-time experimental cardiac tissue data, we chose to use the FHN
model to create our surrogate cardiac tissue wave propagation dataset as
our ‘ground truth’ to be compared with our GH-CA simulation.
Where we used 6;+=H2+;/2+=;I2+=
Where we used a=0.25, b=0.001, g=0.003, and D=0.05. e function 6;+=*
is a third order polynomial that describes the fast evolution of the cardiac
membrane voltage, whereas the slower recovery variable , provides neg-
ative feedback. We used the ode45 solver from Matlab 2018b to solve the
above equations (Fig. 3).
Fraction Method
To compare our GH-CA simulation with our FHN surrogate data, we de-
ne a measure called the Fraction Method which we used as a rst pass
to test our hypothesis that GH-CA simulation deviates from our surro-
gate FHN data at a rate depending on the initial condition complexity.
e same initial condition representing a simple (planar) and a complex
(spiral) wave were given to both the CA-GH and the FHN model. We com-
puted a ratio of (Eq. 4) for each iteration of the CA and FHN
simulations for both the planar and spiral waves. is algorithm can track
wave propagation dynamics regardless of wave travelling direction. A lim-
itation of the Fraction Method has to do with the observation that it does
not track the direction of the wave. In other words, the result of the same
wave travelling from the le and the right of the 2D CA matrix is the same.
However, this method still provides a general sense of wave dynamics re-
gardless of the travelling direction of the wave.
To give the same IC to both models, we initially created a spiral wave from
the FHN model. At a certain iteration step of interest in the FHN simula-
tion, we rst identi ed which CA-equivalent state this cell is in, and then
directly converted the FHN values into discrete values that corresponded
to the state values in a CA model based on the parameter value of / from
the FHN model. Let the FHN value be γ. If 0≤γ<a, then this cell received
an equivalent resting state of 0. If γ>a and γ<1, then we assumed this cell
to be at an excited state; since we would not be sure which discrete state the
cell is in, we gave the all FHN values belonging to this range a value of 1
as its excitatory CA state. e rest of the values from the FHN model is as-
sumed to be in a refractory state; similarly, as we do not know the speci c
state the cell is in, we gave all cells in this FHN value range a state of E+1,
the rst possible refractory state. is was the initial condition retrieved
from our FHN simulation converted into discrete values that our GH-CA
model could accept (Fig. 5).
Overlap Method
Due to the di erence in travelling wave velocity between both models giv-
en the same initial condition, the CA simulation always travels seemingly
faster than that of the FHN model. erefore, though the Fraction Method
portrays the dynamics for both models, it cannot track how much two
simulation with the same initial condition align as a function of time. We
then quantify how much the two models overlap in order to test whether
di erences in initial conditions lead to discrepancies between the GH-CA
model and experimental data. us, to directly compare the two models,
we designed a technique to quantify the degree of overlap between the CA
simulation and our ‘surrogate’ FHN data when both models were given
the same initial condition (I.C.) by calculating a ratio of (Eq.
5) at each time step. We call this ratio the given I.C.’s complexity score.
An “overlapping cell” is de ned as when a cell is at the same state in both
models (i.e. excited, refractory or at rest). is algorithm generates either
a spiral or a planar wave in the FHN model, then initializes the GH-CA
model with the same initial condition. Due to FHN model’s slower wave
propagation velocity, we let it iterate 200 steps before starting the GH-CA
simulation. We then found the time point of maximal overlap between the
two models by determining the ratio in Eq.5 while excluding resting cells
with a state of 0. We then iterated the FHN model for another 200 steps
and waited for the GH-CA model to achieve the largest match between the
two models, then compute the ratio. e same process was repeated for 4
times at ve di erent threshold values of the GH-CA model (Fig. 8).
Using Experimental data as an Initial Condition for the Cellular
Automaton Model
We rst obtain experimental real-time cardiac activity data (i.e. movies
capturing cell’s activity frame by frame) that depict a planar wave and a
spiral wave from our microscope built by !"#*'3*/&E*published in their @/2
3(,'*J4#3#.%$) paper. (3) With an algorithm written in Python, we con-
Figure 3. Simulation results of FHN model of a planar wave
(left) and a spiral wave (right)
Figure 4. relating the discrete states and parameters of the
GH-CA model (i.e. E, R, resting, and T ) to an action potential
simulated by the FHN model. The GH-CA model simulates
cardiac wave propagation whose underlying process is an
action potential in discrete states, while the FHN model does
so with continuous membrane voltage values. This gure was
generated by the Simulink toolbox of Matlab 2018b to simu-
late a cardiac cell action potential.
Figure 5. FHN simulation spiral wave propagation (left) where
2D values at a certain instance (middle) were retrieved as an
I.C. for the CA, which were then converted into discrete val-
ues a GH-CA model can accept (right). This 2D discrete IC was
then fed into the GH-CA model which then was let run and
created a spiral wave propagation in the GH-CA simulation
(right).
Page 69
Volume 15 | Issue 1 | April 2020
the membrane voltage continuously and is easily controllable compared
to real-time experimental cardiac tissue data, we chose to use the FHN
model to create our surrogate cardiac tissue wave propagation dataset as
our ‘ground truth’ to be compared with our GH-CA simulation.
Where we used 6;+=H2+;/2+=;I2+=
Where we used a=0.25, b=0.001, g=0.003, and D=0.05. e function 6;+=*
is a third order polynomial that describes the fast evolution of the cardiac
membrane voltage, whereas the slower recovery variable , provides neg-
ative feedback. We used the ode45 solver from Matlab 2018b to solve the
above equations (Fig. 3).
Fraction Method
To compare our GH-CA simulation with our FHN surrogate data, we de-
ne a measure called the Fraction Method which we used as a rst pass
to test our hypothesis that GH-CA simulation deviates from our surro-
gate FHN data at a rate depending on the initial condition complexity.
e same initial condition representing a simple (planar) and a complex
(spiral) wave were given to both the CA-GH and the FHN model. We com-
puted a ratio of (Eq. 4) for each iteration of the CA and FHN
simulations for both the planar and spiral waves. is algorithm can track
wave propagation dynamics regardless of wave travelling direction. A lim-
itation of the Fraction Method has to do with the observation that it does
not track the direction of the wave. In other words, the result of the same
wave travelling from the le and the right of the 2D CA matrix is the same.
However, this method still provides a general sense of wave dynamics re-
gardless of the travelling direction of the wave.
To give the same IC to both models, we initially created a spiral wave from
the FHN model. At a certain iteration step of interest in the FHN simula-
tion, we rst identi ed which CA-equivalent state this cell is in, and then
directly converted the FHN values into discrete values that corresponded
to the state values in a CA model based on the parameter value of / from
the FHN model. Let the FHN value be γ. If 0≤γ<a, then this cell received
an equivalent resting state of 0. If γ>a and γ<1, then we assumed this cell
to be at an excited state; since we would not be sure which discrete state the
cell is in, we gave the all FHN values belonging to this range a value of 1
as its excitatory CA state. e rest of the values from the FHN model is as-
sumed to be in a refractory state; similarly, as we do not know the speci c
state the cell is in, we gave all cells in this FHN value range a state of E+1,
the rst possible refractory state. is was the initial condition retrieved
from our FHN simulation converted into discrete values that our GH-CA
model could accept (Fig. 5).
Overlap Method
Due to the di erence in travelling wave velocity between both models giv-
en the same initial condition, the CA simulation always travels seemingly
faster than that of the FHN model. erefore, though the Fraction Method
portrays the dynamics for both models, it cannot track how much two
simulation with the same initial condition align as a function of time. We
then quantify how much the two models overlap in order to test whether
di erences in initial conditions lead to discrepancies between the GH-CA
model and experimental data. us, to directly compare the two models,
we designed a technique to quantify the degree of overlap between the CA
simulation and our ‘surrogate’ FHN data when both models were given
the same initial condition (I.C.) by calculating a ratio of (Eq.
5) at each time step. We call this ratio the given I.C.’s complexity score.
An “overlapping cell” is de ned as when a cell is at the same state in both
models (i.e. excited, refractory or at rest). is algorithm generates either
a spiral or a planar wave in the FHN model, then initializes the GH-CA
model with the same initial condition. Due to FHN model’s slower wave
propagation velocity, we let it iterate 200 steps before starting the GH-CA
simulation. We then found the time point of maximal overlap between the
two models by determining the ratio in Eq.5 while excluding resting cells
with a state of 0. We then iterated the FHN model for another 200 steps
and waited for the GH-CA model to achieve the largest match between the
two models, then compute the ratio. e same process was repeated for 4
times at ve di erent threshold values of the GH-CA model (Fig. 8).
Using Experimental data as an Initial Condition for the Cellular
Automaton Model
We rst obtain experimental real-time cardiac activity data (i.e. movies
capturing cell’s activity frame by frame) that depict a planar wave and a
spiral wave from our microscope built by !"#*'3*/&E*published in their @/2
3(,'*J4#3#.%$) paper. (3) With an algorithm written in Python, we con-
Figure 3. Simulation results of FHN model of a planar wave
(left) and a spiral wave (right)
Figure 4. relating the discrete states and parameters of the
GH-CA model (i.e. E, R, resting, and T ) to an action potential
simulated by the FHN model. The GH-CA model simulates
cardiac wave propagation whose underlying process is an
action potential in discrete states, while the FHN model does
so with continuous membrane voltage values. This gure was
generated by the Simulink toolbox of Matlab 2018b to simu-
late a cardiac cell action potential.
Figure 5. FHN simulation spiral wave propagation (left) where
2D values at a certain instance (middle) were retrieved as an
I.C. for the CA, which were then converted into discrete val-
ues a GH-CA model can accept (right). This 2D discrete IC was
then fed into the GH-CA model which then was let run and
created a spiral wave propagation in the GH-CA simulation
(right).
verted the experimental data into readable 128 x 128 sized matrix whose
further manipulation was performed in Matlab. We then removed the
noise from a chosen arbitrary frame using a median lter. A er this l-
tering, this frame’s pixel values become integers. For the planar wave (Fig.
5), this chosen frame was fed into the GH-CA model directly as an initial
condition. We arbitrarily let the excitatory state level (E) be 7, refractory
state level (R) be 1 and the threshold value be 0.3. For the spiral wave (Fig.
6), instead of median- ltering one frame as the initial condition, all frames
from the recording were ltered.
Results
Following the Fraction Method, for a simple planar wave, we observed a
rather constant ratio a er the FHN model is caught up with the
GH-CA model at about the 60th GH-CA iteration (Fig. 7). A planar wave
(i.e. simple wave) in the cellular automaton model is always better aligned
with its FitzHugh-Nagumo model of the same initial condition than a spi-
ral wave. We show in Fig.7 that the ratio from the FHN and CA simula-
tions becomes the same at 0.05 as iteration step continues to 100, which
indicates the two models are behaving the same with a planar wave initial
condition. In contrast, when applying the Fraction Method with a spiral
wave I.C.to both the FHN and GH-CA models, we saw a strong deviation
between the ratio between excited and unexcited cells as iteration steps
increase for both models. Since a CA wave travels faster than an FHN wave
because of its state’s discreteness, the CA was iterated for 100 steps and the
FHN was iterated for 872 steps, which achieve the same ‘cell distance’. Evi-
dently in Fig.8, the fraction method traced the FHN simulation dynamics
of a spiral wave until the wave disappears at the end of its simulation at
the edges of the 2D FHN matrix, which was why its excited to unexcited
cells ratio dropped dramatically to 0 at its 780 iteration step. However, it
the FHN spiral wave behaviors is still captured by the fraction method
precisely. In contrast, the GH-CA simulation whose IC was given by CA
at a certain instant never completely le the edges of the 2D GH-CA ma-
trix. erefore, its excited to unexcited cells ratio was signi cantly higher
than that of the FHN simulation around 50 GH-CA iterations. However,
regardless of whether the spiral wave exits the 2D array or not, a noticeable
discrepancy between the excited to unexcited cell count ratio can be seen
between the GH-CA (red) and the FHN (blue) models.
We quanti ed the degree of overlap between the CA model simulation
using the !"#$%&'()#*+,-. We were able to generate the degree of overlap
of a planar and a spiral wave compared between the FHN and CA mod-
els. We directly compared the CA and FHN models for ve di erent CA
thresholds (color-coded) shown in Fig. 9. We can see 1) the cellular au-
tomaton model’s planar waves are better aligned with the FHN models ini-
tiated by the same I.C. than the spiral waves, and 2) the t with the spirals
decreases as the number of iterations increase, while the plane waves do
not decrease as much.
Figure 5. For the planar wave, the left image is the raw data
from our experiment converted to be a matrix appearing to
be noisy; the right image is the left image that has been me-
dian ltered, we can see (at the arrow) a very clear front wave
compared to the left image.
(a) (b) (c)
Figure 6. For a spiral wave, (a) a chosen starting frame of the raw
data from our experiment converted to be a matrix appearing to
be noisy; (b) is image (a) after it has been median ltered; since
all frames of the raw data have been ltered, (c) and (d) represent
the spiral wave at two later time points from (b), we can clearly
see a wave spiraling into the center of the plate in the experiment.
(d)
Figure 8. Fraction method performed to a complex spiral
wave initial condition given to the FHN (blue, bottom line)
and GH-CA (red, top line) models for 100 GH-CA iterations
and 872 FHN iterations which travel the same cell equivalent
distance.
Figure 7. Fraction method performed to a simple planar wave
initial condition given to the FHN (blue) and GH-CA (orange)
models for 100 GH-CA and FHN iterations.
Figure 9. Overlapping to non-overlapping cells ratio between
GH-CA and FHN models by capturing the maximal overlap af-
ter allowing the FHN simulation to run 200 iterations ahead of
the GH-CA one and let the latter to catch up.
Page 70 McGill Science Undergraduate Research Journal - msurj.com
Furthermore, our method of initializing the CA with one frame of medi-
an- ltered real-time experimental cardiac tissue data was successful in
reproducing the behavior of the raw data for the planar wave (Fig. 10), but
not for the spiral wave. As for the spiral wave, its initial condition evolves
to be a planar wave rather than spiraling into the center.
Discussion
Discussion of results and possible improvements
We used the FitzHugh-Nagumo model to generate data sets of simple and
complex propagating waves in excitable media. We evaluated whether a
cellular automaton model can reproduce these wave dynamics using two
algorithmic tests, a 6,/$3%#.*1'34#"and an #+',&/:*1'34#"K. For future
work, since we chose our 2D matrix cell size to be 100 for both the GH-CA
and the FHN models, we could try varying the size of our simulation to see
if it changes our results. For all results generated other than experimental
result simulation, we relied on the condition that the GH-CA and the FHN
models were initiated by the same I.C.. is was retrieved from the FHN
voltage values at a certain instance and converted into discrete cell states
that the GH-CA model can accept as an IC too (see Methods). We strictly
used the parameter a from the FHN model to set the ranges for the excited
and refractory cell states. In addition, we assigned a 1 for all FHN values in
the “excited range” and an E+1 for all that’s in the “refractory range”. Both
parameter assignments increase the probability that the discrete values
we converted the FHN values to may not be entirely representative of the
original FHN simulation, which can a ect the consequent GH-CA simula-
tion results and the comparison results between the two model.
Proposed strategies of integrating Machine Learning techniques into
our GH-CA model to lessen the deviation of its simulation from
experimental data
Our future strategy to resolve the challenge of spiral wave experimental
data initial condition not being able to induce spirals is inspired by Gilpin’s
(6) recent paper <'&&(&/,* /(3#1/3/* /)* $#.+#&(3%#./&* .'(,/&* .'3L#,M).
Gilpin (6) showed that /.N Cellular Automaton may readily be represent-
ed using a convolutional neural network (CNN) with a network-in-net-
work architecture. ey built a CNN that can learn the dynamical rules
for arbitrary CA when given videos of the CA as training data. ey
trained ensembles of networks on randomly sampled CA and showed
that CA with simpler rule tables produce trained networks with hierar-
chical structure and layer specialization, while more complex CA tend to
produce shallower representations. is is analogous to our results. A er
they con rmed that arbitrary cellular automaton may be represented by
convolutional perceptrons with nite layers and units, automated train-
ing of neural networks on time series of cellular automaton images was
carried out. By training ensembles of convolutional neural networks on
random images and random CA rulesets. More speci cally, they de ned a
CA as an explicit mapping between an input pixel and an output pixel. For
a neighborhood size of 1, there are 512 possible neighbor state groupings
(i.e. a center cell has 29=512 neighboring state possibilities for 3-by-3-pixel
groups in a binary image) for an input pixel. erefore, their training data
is an ensemble of randomly generated binary images where each image
contains an equal number of black and white pixels on average. For each
training input pixel, its output pixel (‘label’) is produced by the input pixel
undergoing a randomly selected CA Conway Game of Life rule set. Subse-
quently, this CA map is applied to an ensemble of random binary images
(the training data), in order to produce a new output binary image set (the
training labels). ey used a su cient number of images and training data
batches (500 images) to ensure that the training data contains at least one
instance of each rule.
We may adapt this technique to train a Convolutional Neural Network
(CNN) representing our CA with experimental median- ltered spiral
wave data. Complex wave propagation such as a spiral wave presents dy-
namical rules to be learnt by the CNN like Conways’ Game of Life. e
input training data would be our experimental data at time with frame n,
its output label data would be the experimental data at time 3+1 of frame
.+1. Iteratively, the next training image would be frame .+1 whose output
label is frame .+2. An initial step would be to train the CNN representing
our CA on synthetic wave data generated by the FHN model where its con-
tinuous values are converted into equivalent discrete states of CA values.
is procedure is promising to adeptly train a CNN for leaning spiral wave
propagation with an ability surpassing that of a GH-CA model.
Conclusion
We compared the dynamics of two models of cardiac propagation: a dis-
crete Greenberg-Hastings Cellular Automaton (GH-CA) model and a
continuous FitzHugh-Nagumo (FHN) model for excitable propagation.
Our results were able to con rm our hypothesis that the divergence be-
tween the two models are due to initial condition (IC) complexity (i.e.
the more complex the I.C. is, the faster the GH-CA simulation results
deviate from the FHN model results where we treat our FHN simulation
as our ground-truth surrogate experimental data). is was shown by the
alignment between the CA model of a planar wave (i.e. simple) and the
FHN model remaining constant, while the degree of overlap between the
CA and FHN models decreasing for a spiral wave (i.e. complex) at higher
time steps. Once a median- ltered frame of real-time cardiac tissue exper-
imental data was fed to the CA model as an initial condition, the planar
wave simulation propagates forward as the experimental data shows, but
the spiral wave CA simulation did not successfully spiral in. is result
further identi es the issue that a simpler I.C. such as a planar wave can be
modelled more accurately by CA than a complex I.C. such as a spiral wave.
Acknowledgements
is project was a ful llment for the course PHGY 461 D1/D2: Experi-
mental Physiology at McGill University. e author would like to thank
McGill M.Sc. candidate Miguel Romero Sepulvida for generously provid-
ing experimental data for simulation results of this paper and Harvard
NSF-Simons Research Fellow Dr. William Gilpin for fruitful discussions
and further collaboration on Convolutional Neural Network as Cellular
Automata model for our cardiac experimental data. Especially, the author
would like to sincerely thank her supervisor Dr. Gil Bub from McGill’s De-
partment of Physiology for continuously o ering his patience, intellectual
guidance, stimulating discussions and support.
References
1. Bub G, Shrier A, Glass L. Global organization of dynamics in oscillato-
ry heterogeneous excitable media [Internet]. Physical review letters. U.S.
National Library of Medicine; 2005 [cited 2020Jan15]. Available from:
https://www.ncbi.nlm.nih.gov/pubmed/15698236
2. Bub G. Contents [Internet]. Optical Mapping of Pacemaker Interactions.
[cited 2020Mar7]. Available from: https://www.physiol.ox.ac.uk/~gb1/
cnd/bub/thesis.html#tth_sEc1.3
Figure 10. After a median- ltered initial condition of the ex-
perimental result of a planar wave (a) have been fed into a
GH-CA model, (b) – (e) shows this planar wave’s wave propa-
gation evolution, successfully reproduced the raw data.
(a) (b) (c) (d) (e)
Page 71
Volume 15 | Issue 1 | April 2020
Furthermore, our method of initializing the CA with one frame of medi-
an- ltered real-time experimental cardiac tissue data was successful in
reproducing the behavior of the raw data for the planar wave (Fig. 10), but
not for the spiral wave. As for the spiral wave, its initial condition evolves
to be a planar wave rather than spiraling into the center.
Discussion
Discussion of results and possible improvements
We used the FitzHugh-Nagumo model to generate data sets of simple and
complex propagating waves in excitable media. We evaluated whether a
cellular automaton model can reproduce these wave dynamics using two
algorithmic tests, a 6,/$3%#.*1'34#"and an #+',&/:*1'34#"K. For future
work, since we chose our 2D matrix cell size to be 100 for both the GH-CA
and the FHN models, we could try varying the size of our simulation to see
if it changes our results. For all results generated other than experimental
result simulation, we relied on the condition that the GH-CA and the FHN
models were initiated by the same I.C.. is was retrieved from the FHN
voltage values at a certain instance and converted into discrete cell states
that the GH-CA model can accept as an IC too (see Methods). We strictly
used the parameter a from the FHN model to set the ranges for the excited
and refractory cell states. In addition, we assigned a 1 for all FHN values in
the “excited range” and an E+1 for all that’s in the “refractory range”. Both
parameter assignments increase the probability that the discrete values
we converted the FHN values to may not be entirely representative of the
original FHN simulation, which can a ect the consequent GH-CA simula-
tion results and the comparison results between the two model.
Proposed strategies of integrating Machine Learning techniques into
our GH-CA model to lessen the deviation of its simulation from
experimental data
Our future strategy to resolve the challenge of spiral wave experimental
data initial condition not being able to induce spirals is inspired by Gilpin’s
(6) recent paper <'&&(&/,* /(3#1/3/* /)* $#.+#&(3%#./&* .'(,/&* .'3L#,M).
Gilpin (6) showed that /.N Cellular Automaton may readily be represent-
ed using a convolutional neural network (CNN) with a network-in-net-
work architecture. ey built a CNN that can learn the dynamical rules
for arbitrary CA when given videos of the CA as training data. ey
trained ensembles of networks on randomly sampled CA and showed
that CA with simpler rule tables produce trained networks with hierar-
chical structure and layer specialization, while more complex CA tend to
produce shallower representations. is is analogous to our results. A er
they con rmed that arbitrary cellular automaton may be represented by
convolutional perceptrons with nite layers and units, automated train-
ing of neural networks on time series of cellular automaton images was
carried out. By training ensembles of convolutional neural networks on
random images and random CA rulesets. More speci cally, they de ned a
CA as an explicit mapping between an input pixel and an output pixel. For
a neighborhood size of 1, there are 512 possible neighbor state groupings
(i.e. a center cell has 29=512 neighboring state possibilities for 3-by-3-pixel
groups in a binary image) for an input pixel. erefore, their training data
is an ensemble of randomly generated binary images where each image
contains an equal number of black and white pixels on average. For each
training input pixel, its output pixel (‘label’) is produced by the input pixel
undergoing a randomly selected CA Conway Game of Life rule set. Subse-
quently, this CA map is applied to an ensemble of random binary images
(the training data), in order to produce a new output binary image set (the
training labels). ey used a su cient number of images and training data
batches (500 images) to ensure that the training data contains at least one
instance of each rule.
We may adapt this technique to train a Convolutional Neural Network
(CNN) representing our CA with experimental median- ltered spiral
wave data. Complex wave propagation such as a spiral wave presents dy-
namical rules to be learnt by the CNN like Conways’ Game of Life. e
input training data would be our experimental data at time with frame n,
its output label data would be the experimental data at time 3+1 of frame
.+1. Iteratively, the next training image would be frame .+1 whose output
label is frame .+2. An initial step would be to train the CNN representing
our CA on synthetic wave data generated by the FHN model where its con-
tinuous values are converted into equivalent discrete states of CA values.
is procedure is promising to adeptly train a CNN for leaning spiral wave
propagation with an ability surpassing that of a GH-CA model.
Conclusion
We compared the dynamics of two models of cardiac propagation: a dis-
crete Greenberg-Hastings Cellular Automaton (GH-CA) model and a
continuous FitzHugh-Nagumo (FHN) model for excitable propagation.
Our results were able to con rm our hypothesis that the divergence be-
tween the two models are due to initial condition (IC) complexity (i.e.
the more complex the I.C. is, the faster the GH-CA simulation results
deviate from the FHN model results where we treat our FHN simulation
as our ground-truth surrogate experimental data). is was shown by the
alignment between the CA model of a planar wave (i.e. simple) and the
FHN model remaining constant, while the degree of overlap between the
CA and FHN models decreasing for a spiral wave (i.e. complex) at higher
time steps. Once a median- ltered frame of real-time cardiac tissue exper-
imental data was fed to the CA model as an initial condition, the planar
wave simulation propagates forward as the experimental data shows, but
the spiral wave CA simulation did not successfully spiral in. is result
further identi es the issue that a simpler I.C. such as a planar wave can be
modelled more accurately by CA than a complex I.C. such as a spiral wave.
Acknowledgements
is project was a ful llment for the course PHGY 461 D1/D2: Experi-
mental Physiology at McGill University. e author would like to thank
McGill M.Sc. candidate Miguel Romero Sepulvida for generously provid-
ing experimental data for simulation results of this paper and Harvard
NSF-Simons Research Fellow Dr. William Gilpin for fruitful discussions
and further collaboration on Convolutional Neural Network as Cellular
Automata model for our cardiac experimental data. Especially, the author
would like to sincerely thank her supervisor Dr. Gil Bub from McGill’s De-
partment of Physiology for continuously o ering his patience, intellectual
guidance, stimulating discussions and support.
References
1. Bub G, Shrier A, Glass L. Global organization of dynamics in oscillato-
ry heterogeneous excitable media [Internet]. Physical review letters. U.S.
National Library of Medicine; 2005 [cited 2020Jan15]. Available from:
https://www.ncbi.nlm.nih.gov/pubmed/15698236
2. Bub G. Contents [Internet]. Optical Mapping of Pacemaker Interactions.
[cited 2020Mar7]. Available from: https://www.physiol.ox.ac.uk/~gb1/
cnd/bub/thesis.html#tth_sEc1.3
Figure 10. After a median- ltered initial condition of the ex-
perimental result of a planar wave (a) have been fed into a
GH-CA model, (b) – (e) shows this planar wave’s wave propa-
gation evolution, successfully reproduced the raw data.
(a) (b) (c) (d) (e)
3. Burton RAB, Klimas A, Ambrosi CM, Tomek J, Corbett A, Entche-
va E, et al. Optical control of excitation waves in cardiac tissue [Inter-
net]. Nature photonics. U.S. National Library of Medicine; 2015 [cited
2020Jan15]. Available from: https://www.ncbi.nlm.nih.gov/pmc/articles/
PMC4821438/
4. Cco nn: Automated Wave Tracking in Cultured Cardiac ... [Internet].
[cited 2020Jan15]. Available from: https://www.cell.com/biophysj/pdf/
S0006-3495(16)30816-5.pdf
5. FitzHugh R. Impulses and Physiological States in eoretical Models of
Nerve Membrane [Internet]. Biophysical Journal. Cell Press; 2009 [cited
2020Jan15]. Available from: https://www.sciencedirect.com/science/arti-
cle/pii/S0006349561869026
6. Gilpin, William. Cellular automata as convolutional neural networks
[Internet]. arXiv.org. 2019 [cited 2020Jan15]. Available from: https://arxiv.
org/abs/1809.02942
7. Murray JD. Mathematical Biology - I. An Introduction: James D. Mur-
ray [Internet]. Springer. Springer-Verlag New York; [cited 2020Jan15].
Available from: https://www.springer.com/gp/book/9780387952239
8. Pálsson E, Cox EC. Origin and evolution of circular waves and spirals
in Dictyostelium discoideum territories [Internet]. Proceedings of the Na-
tional Academy of Sciences of the United States of America. U.S. National
Library of Medicine; 1996 [cited 2020Mar7]. Available from: https://www.
ncbi.nlm.nih.gov/pmc/articles/PMC40047/
ResearchGate has not been able to resolve any citations for this publication.
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Origin and evolution of circular waves and spirals in Dictyostelium discoideum territories
  • E Pálsson
  • E C Cox
Pálsson E, Cox EC. Origin and evolution of circular waves and spirals in Dictyostelium discoideum territories [Internet]. Proceedings of the National Academy of Sciences of the United States of America. U.S. National Library of Medicine; 1996 [cited 2020Mar7]. Available from: https://www. ncbi.nlm.nih.gov/pmc/articles/PMC40047/