2. 2
Introduction
The
electrical
activity
of
an
animal’s
heart
helps
determine
its
heartbeat.
The
diffusion
of
charged
ions
across
a
cardiomyocyte
membrane
allows
the
cell
to
contract,
in
turn
pumping
blood
through
the
body.
When
this
electrical
behavior
becomes
disrupted,
the
heartbeat
becomes
irregular,
causing
a
myriad
of
heart
conditions
defined
as
cardiac
arrhythmias.
To
analyze
the
various
component
interactions
that
contribute
to
the
beating
of
the
heart,
such
as
the
voltage
level
of
the
cell
and
the
blocking
of
voltage-‐
gated
channels
on
the
membrane
which
prevents
charged
ions
from
coming
in
and
out
of
the
cell,
the
stability
of
equilibriums
of
the
ordinary
nonlinear
differential
equations
are
determined
to
understand
the
pattern
of
depolarization
and
repolarization.
The
patterns
of
voltage
(v(t))
and
channel
blockage
(h(t))
are
compared
to
each
other
to
analyze
their
patterns
as
the
heart
contracts
and
relaxes,
and
the
Action
Potential
Duration,
the
time
that
the
voltage
is
high
enough
for
the
cells
to
contract,
is
measured
for
different
heart
rates.
If
these
components
become
disrupted
or
irregular,
the
risk
of
succumbing
to
cardiac
arrest
or
stroke
increase.
3.1 Analytical Work
To
determine
equilibrium
points
of
the
system,
the
ODE’s
are
set
equal
to
zero
to
find
the
nullclines
of
the
system.
The
ODE’s
are
defined
below,
with
predefined
variables
tabulated
in
Table
1.
The
equilibrium
points
can
be
found
in
Table
1.2
1 𝑣′(𝑡) = −𝑘𝑣 𝑣 − 𝑎 𝑣 − 1 − 𝑣ℎ + 𝑠𝑡𝑖𝑚𝑢𝑙𝑢𝑠
2 ℎ′(𝑡) = (𝜖! +
𝜇!ℎ
𝑣 + 𝜇!
)(−ℎ − 𝑘𝑣 𝑣 − 𝑎 − 1 )
Table
1
k 8
a 0.15
ϵ0 0.002
u1 0.2
u2 0.3
3. 3
Table
1.2
v(t) h(t)
0 0
0.15 0
1 0
Equilibrium
points
are
located
where
vertical
nullclines
and
horizontal
nullclines
intersect.
Three
nullclines
are
located
along
the
v-‐axis,
as
seen
in
Table
1.2,
the
first
being
located
at
the
origin,
(v,h)=(0,0).
This
resting
state
equilibrium
represents
when
the
heart
is
fully
relaxed
and
the
cells
are
negatively
charged,
around
-‐90
mV
in
humans.
Biologically,
this
resting
state
should
be
stable
because
after
the
heart
contracts
fully,
it
should
relax
to
its
non-‐contracted
state
in
order
to
pump
the
same
amount
of
blood
again.
If
this
equilibrium
were
unstable,
the
cells
would
start
at
rest
but
after
contracting
would
never
return
to
the
resting
state,
preventing
the
full
capacity
of
blood
to
be
pumped
around
the
body,
making
the
oxygenation
of
blood
slower,
a
condition
called
diastolic
dysfunction.
To
prove
that
the
resting
state
is
a
stable
equilibrium,
the
eigenvalues
of
the
Jacobian
matrix
would
both
have
to
be
negative.
The
Jacobian
is
used
to
create
a
linear
approximation
of
the
nonlinear
system.
Negative
eigenvalues
would
cause
the
general
solution
to
decay
to
the
origin.
The
eigenvalues
for
this
system
are
found
to
be
(λ1,
λ2)=(-‐1.2,-‐0.002).
Both
of
these
values
are
negative,
proving
that
the
origin
is
a
stable
equilibrium.
Figure
1
shows
the
nullclines
of
the
system.
The
vertical
nullclines
correspond
to
v’(t)=0,
and
the
horizontal
nullclines
correspond
to
h’(t)=0.
4. 4
Figure
1
The
vector
plot
of
Figure
1
describes
the
movement
of
the
system
between
the
nullclines
as
time
progresses.
It
is
evident
that
the
origin
is
a
stable
equilibrium
based
on
the
direction
progression
of
the
vectors.
If
a
stimulus
larger
than
the
second
equilibrium
point,
which
is
around
(v,h)=(0.15,0),
then
the
trajectory
would
move
towards
the
final
equilibrium
point
around
(v,h)=(1,0).
The
second
equilibrium
point
represents
the
voltage
threshold
required
to
allow
the
cardiomyocytes
to
contract,
around
-‐70
mV
in
humans.
Naturally,
this
threshold
is
surpassed
as
positively
charged
sodium
and
calcium
ions
diffuse
through
the
cell’s
permeable
membrane
from
a
neighboring
cell.
Once
the
cell
reaches
a
voltage
around
-‐70
mV,
the
sodium
voltage-‐gated
channels
open
up,
allowing
for
an
abundance
of
sodium
to
diffuse
into
the
cell.
Since
sodium
has
a
voltage
around
67
mV,
the
cell
depolarizes
dramatically
as
these
ion
enter,
driving
the
voltage
to
about
20
mV.
Biologically,
the
final
equilibrium
point,
where
the
cell
is
fully
depolarized
to
20
mV,
represents
the
voltage
at
which
the
heart
is
fully
contracted.
As
the
h
value
increases,
as
is
natural
with
heart
compression,
the
trajectory
flows
counter-‐clockwise
back
towards
the
origin,
or
resting
state.
This
is
represented
by
the
following
repolarization
steps.
As
the
cell
depolarizes,
the
voltage-‐gated
sodium
channels
close,
and
the
voltage-‐gated
potassium
channels
open,
allowing
potassium
to
diffuse
to
the
outside
of
the
cell
more
quickly.
The
cell
reaches
a
voltage
of
about
5mV,
5. 5
and
the
calcium
voltage-‐gated
channels
open,
allowing
calcium,
with
a
voltage
around
123
mV,
to
diffuse
into
the
cell.
This
stabilizes
the
voltage
of
the
cell
until
the
calcium
channels
suddenly
close,
blocking
calcium
from
entering
the
cell.
The
potassium
channels
remain
open,
however,
and
the
cell’s
voltage
repolarizes
to
-‐90
mV
again,
as
visualized
by
the
vector
plot
flowing
counterclockwise
back
towards
the
initial
equilibrium
point,
or
the
steady
state.
Once
this
voltage
is
reached,
the
potassium
channels
close
again,
and
potassium
seeps
through
the
semipermeable
membrane
just
like
at
the
beginning
of
the
cycle.
This
flow
represents
the
depolarization
and
repolarization
of
the
cardiomyocytes.
The
initial
movement
of
the
trajectory
with
a
stimulus
bigger
than
0.15,
shown
in
green,
can
be
seen
in
Figure
2.
Figure
2
Evident
in
these
nullcline
plots,
once
the
voltage
of
the
cell
passes
the
threshold,
as
time
continues,
the
voltage
and
channel
blocking
return
to
the
steady
state
at
which
the
heart
is
relaxed.
6. 6
3.2 Numerical Simulations
Plotting
v(t)
and
h(t)
versus
time
provides
a
visual
representation
of
the
voltage
change
and
voltage-‐
gated
channel
blockage
as
the
heart
beats
over
a
time
period.
The
duration
of
the
plot
in
Figure
3
is
a
time
of
500
and
a
stimulation
period
of
100.
The
stimulation
period
represents
the
time
the
heart
requires
to
undergo
a
full
cycle
of
stimulation,
contraction,
and
relaxation,
or
a
full
heart
beat.
The
voltage
reaches
a
maximum
of
1,
which
aligns
with
the
nullcline
plot
analysis.
Figure
3
Figure
4,
the
plot
of
v(t)
versus
h(t),
establishes
the
relationship
between
the
voltage
of
the
cell
and
the
voltage-‐gated
channels
opening
and
closing.
When
the
cell
is
initially
stimulated,
the
closure
of
the
channels
grows
exponentially,
and
maximizes
at
a
value
of
2.
As
the
voltage
reaches
1,
and
the
cells
are
fully
contracted,
more
channels
gradually
open.
These
points
align
with
the
sodium,
potassium,
and
calcium
voltage-‐gated
channels
described
in
3.1.
The
maximum
values
of
h
and
t
in
Figure
4
align
with
the
maximum
values
of
Figure
1.
7. 7
Figure
4
The
Action
Potential
Duration
represents
the
amount
of
time
between
when
the
voltage
passes
the
stimulus
threshold
initially
and
when
it
gets
passed
again
on
the
way
back
to
the
steady
state.
The
equation
for
APD
can
be
found
below.
𝐴𝑃𝐷 = 𝑡!"## !!!"#!!"# !"#$! !"#$ − 𝑡!"## !!!"#!!"# !"#$! !"
These
values
were
calculated
and
tabulated
in
Table
2
below.
Table
2
t
=
duration T
=
stimulation
period APD
1000 50 16.8000
1000 60 17.8000
1000 70 18.6000
1000 80 19.4000
1000 90 19.8000
1000 100 20.4000
As
expected,
the
general
trend
is
that
APD
values
increase
as
the
period
of
the
heart
rate
increases.
Essentially,
as
the
heart
rate
slows,
the
time
it
takes
the
voltage
to
return
to
the
threshold
value
increases.
APD
and
T
are
positively
correlated,
as
seen
in
Figure
5.
8. 8
Figure
5
Biologically,
larger
animals,
such
as
humans,
require
a
high
enough
APD
in
order
to
survive.
This
means
that
if
the
human’s
heart
rate
is
too
fast,
there
are
many
risks
that
could
put
their
life
in
danger.
This
condition
is
called
Atrial
Flutter,
and
symptoms
include
heart
palpitations,
shortness
of
breath,
chest
pain,
dizziness,
and
fainting.
Without
treatment,
Atrial
Flutter
can
cause
the
heart
to
be
significantly
inefficient
at
pumping
blood
around
the
body,
leading
to
risks
like
blood
clots,
which
has
high
risks
of
heart
attack
or
stroke.
The
inefficiency
on
the
heart
to
pump
blood
around
the
body
can
be
explained
through
the
pattern
of
voltage-‐gated
channel
blocking
as
the
heart
beats.
With
a
time
duration
of
t=1000,
various
stimulation
periods
are
evaluated
to
determine
the
heart’s
ability
to
return
to
steady
state
before
the
next
stimulation
period
begins.
These
values
are
calculated
and
tabulated
in
Table
3
below.
9. 9
Table
3
t=duration T=stimulation
period Steady
State
h
1000 50 0.0439
1000 60 0.0342
1000 70 0.0278
1000 80 0.0233
1000 90 0.0200
1000 100 0.0174
As
T
increases,
the
steady
state
value
decreases,
meaning
that
as
the
heart
rate
slows,
it
has
an
easier
time
returning
to
its
steady
state.
Stimulation
period
and
steady
state
are
negatively
correlated
and
are
visually
represented
in
Figure
6
below.
Figure
6
The
trend
of
correlation
between
stimulation
period
and
voltage-‐gated
channel
blocking
suggests
that
as
the
heart-‐rate
increases,
not
all
of
the
channels
return
to
steady
state
by
the
time
the
next
heart
beat
begins.
Based
on
this
data,
the
conclusion
is
that
the
cardiomyocytes
don’t
fully
repolarize
before
depolarizing
again.
Biologically,
the
inability
of
the
heart
to
return
to
the
steady
state
value
means
that
10. 10
the
heart
does
not
fully
relax
to
-‐90
mV
before
contracting
again.
As
mentioned
earlier,
when
the
heart
does
not
fully
contract,
less
blood
enters
the
heart
atria
and
less
blood
is
pumped
around
the
body,
characteristic
of
diastolic
dysfunction.
A
high
resting
heart
rate
could
easily
increase
the
risk
of
heart
attack
and
stroke
in
humans.
Conclusion
Cardiac
arrhythmias
categorize
numerous
heart
conditions
that
concern
electrical
malfunctions
within
cardiomyocytes.
The
mathematical
model
of
the
nonlinear
system
describes
the
heart’s
relaxed
steady
state
as
a
stable
fixed
point,
so
as
time
progresses,
the
heart
typically
returns
to
this
state
between
contractions
if
performing
correctly,
confirmed
in
the
nullcline
and
Jacobian
analyses
in
3.1.
The
balance
of
cardiomyocyte
voltage
and
heart
rate
is
disrupted,
however,
as
the
heart
rate
increases,
which
can
be
seen
in
the
analysis
of
APD
and
voltage-‐gated
channel
blocking
in
3.2.
If
the
heart
beats
too
quickly
too
often,
less
blood
can
be
pumped
around
the
body
at
a
time,
increasing
risk
for
blood
clots,
and
decreasing
the
speed
at
which
the
blood
becomes
oxygenated.
The
data
collected
representing
the
electrical
activity
of
the
heart
mathematically
attributes
dysfunction
of
this
activity
to
heart
complications.
11. 11
Appendix
system_template.m
function [vprime] = system_template(t,v)
% Import global parameters
global e_0 k a mu1 mu2 T;
% Set stimulation by periodicity
if (mod (t,T) >= 10.0) && (mod (t,T) <= 13.0)
stim = 0.25;
else
stim = 0.0;
end
% ODEs
vprime(1,1) = stim +(- k.*v(1).*(v(1)-a)).*(v(1)-1)-v(1).*v(2);
vprime(2,1) = (e_0+((mu1.*v(2)./(v(1)+mu2)))).*(-v(2) - k.*v(1).*(v(1) - a -1));
scrpit_template.m
% Set Parameters
close all
global e_0 k a mu1 mu2 T;
% Set parameter values %
e_0 = 0.002;
k = 8;
a = 0.15;
mu1 = 0.2;
mu2 = 0.3;
% Set period T and simulation duration dur
12. 12
T = 50;
dur = 1000;
% Run the simulation and plot the data
[t,v] = ode45 ('system_template',0:0.2:dur,[0,0]); % This runs the simulation
%3.2.1
figure
plot(t,v)
title('v and h vs. t');
xlabel('t');
ylabel('v(t),h(t)')
legend('v(t)','h(t)');
figure
plot(v(:,1),v(:,2))
title('h vs. v');
xlabel('v');
ylabel('h')
% Calculate APDs and minimum h values for each beat
grt=find(v(:,1)>0.1)
%3.2.2
t(4656)-t(4554) %T=100
%3.2.3
t(4653)-t(4554) %T=90
t(4951)-t(4854) %T=80
t(4697)-t(4604) %T=70
t(4943)-t(4854) %T=60
t(4888)-t(4804) %T=50
% Plot APDs and minimum h's vs beat and each other
