Understanding an Unfolded
Border-Collision Bifurcation in
Paced Cardiac Tissue
Carolyn M. Berger, Xiaopeng Zhao, David G. Schaeffer,
Salim F. Idriss, Ned Rouse, David Hall
and Daniel J. Gauthier
Duke University
NSF PHY-0243584 & PHY-0549259 NIH 1RO1-HL-72831
Outline
Bifurcation in Cardiac Tissue
Technique to Uncover Bifurcation Type
Experimental Results
New Model
Stimulus
voltage stim
Cardiac Dynamics
BCL
BCL
BCL
time
APD DI APD DI APD
time
BCL = APD + DI
stim strength
voltage
1:1 Behavior
Stimulus
BCL
BCL
BCL
time
APD
APD
APD
time
voltage
stim strength
2:2 Behavior(Alternans)
Stimulus
BCL
BCL
BCL
BCL
BCL
time
APD
DI
APD
APD
APD APD
APD
time
Transition
voltage
stim
2:2 (fast pacing)
1:1 (slow pacing)
BCL BCL BCL BCL BCL
APD
APD
APD
APDAPD
APD
time
BCL
APD
BCL
APD
BCL
APD
time
Transition
voltage
stim
2:2 (fast pacing)
1:1 (slow pacing)
BCL BCL BCL BCL BCL
APD
APD
APD
APDAPD
APD
time
BCL
APD
BCL
APD
BCL
APD
time
2:2 (alternans) linked to ventricular arrhythmias
and sudden cardiac death
Bifurcation Diagram
1:1
APD
2:2
Bbif
BCL
APD
Supercritical Period-Doubling Bifurcation
APD
Border-Collision Bifurcation
Nolasco and Dahlen
(1968)
BCL
Sun, Amellal, Glass,
and Billette (1995)
BCL
APD
BCL
APD
BCL
APD
Difficult to Distinguish with Discrete Data Points
BCL
APD
Investigate
1:1 Regime
BCL
Alternate Pacing
voltage
stim
vary BCL by ! in 1:1 regime
BCL + ! BCL - ! BCL + ! BCL - ! BCL + !
time
APDs
DI l
APD
APDs
APD
APD
APDs
l
time
Alternate Pacing
voltage
stim
Gain =
APDl -APDs
2!
BCL + ! BCL - ! BCL + ! BCL - ! BCL + !
time
APDs
DI l
APD
APDs
APD
APD
APDs
l
time
Alternate Pacing Simulations
Smooth
Border-Collision
! = 20 ms
APD
! = 20 ms
non-alternate
alternate
BCL
Gain =
BCL
APDl -APDs
2!
Alternate Pacing Simulations
Border-Collision
APD
Smooth
non-alternate
alternate
Bbif
BCL
Gain
Gain
BCL
BCL
Bbif
BCL
Alternate Pacing Trends
Difficult to Distinguish with Discrete Data Points
Border-Collision
Gain
Smooth
Bbif
BCL
Bbif
BCL
Alternate Pacing Trends
Difficult to Distinguish with Discrete Data Points
Border-Collision
Gain
Smooth
Bbif
BCL
Bbif
BCL
Alternate Pacing
voltage
stim
Gain =
APDl -APDs
2!
BCL + ! BCL - ! BCL + ! BCL - ! BCL + !
time
APDs
DI l
APD
APDs
APD
APD
APDs
l
time
Vary ! Size
Border-Collision
APD
Smooth
BCL
BCLfixed
BCL
BCLfixed
Alternate Pacing Trends
Border-Collision
APD
Smooth
BCL
BCLfixed
Gain
BCLfixed
BCL
!
!
Alternate Pacing Trends
Border-Collision
APD
Smooth
BCL
BCLfixed
Gain
BCLfixed
BCL
!
!
Alternate Pacing Trends
Different Trends with Discrete Data Points
Border-Collision
Gain
Smooth
!
!
Experimental Setup
BCL
Voltage-to-period
Stimulator
Amplifier
Microelectrode
Oxygenated
solution
Cardiac Muscle
Tissue chamber
BCL
120 s
BCL + !1 BCL + !
2
ax
µ xn ≤ 0
n+
txn+1
Map
→
a
=
"
b
µ; =alternate → µ ± δ
µ xn > 0 Dynamical Systems (200
n +in Piecewise-Smooth
l. Bifurcations andbx
Chaos
!
→ µ;steady
alternate
→ µx±
δ 0
ax
n+µ
n ≤
state
Map=→ a "= b
xt n+1
alternate
bxn +in Piecewise-Smooth
µ xn > 0 Dynamical Systems (200
l..03Bifurcations
and Chaos
border
steady state
!
alternate
.02
→ µ; alternate
→ µx±
δ 0
borderaxn + µ
≤
n
t n+1
Map=→ a "= b
.01
x
µ xn > 0 Dynamical Systems (200
l..030Bifurcations andbx
Chaos
n +in Piecewise-Smooth
steady state
!
.01
alternate
.02
→ µ; alternate
→ µx±
δ 0
≤
borderaxn + µ
n
.02
.01
t n+1
Map=→ a "= b
x
µ xn > 0 Dynamical Systems (200
n +in Piecewise-Smooth
l..030Bifurcations andbx
Chaos
steady
state
!
.04
.01
alternate
.02
→ µ; alternate
→ µx±
δ 0
≤
borderaxn + µ
n
.05
.02
.01
x
= !0.01 0 0.01 0.02 0.03
n+1!0.02
bxn +µµ xn > 0
.03
0
!
!
r Collision Bifurcation
axn + µ xn ≤ 0
t+1
Map
= → a "= b
µ
x
>
0
l. Bifurcations andbx
Chaos
in+
Piecewise-Smooth
Dynamical
Systems (200
n
n
r Collision Bifurcation
ifurcations and ChaosPacing
in Piecewise-Smooth
Dynamical
Rate
Alternate Pacing Protocol
20 s
r Collision Bifurcation
BCL - !1
20 s
r Collision Bifurcation
BCL - !2
BCL - !3
20 s
20 s
20 s
0.01
.01
.05
→ µ; alternate
→
border
µ µ±δ
0
BCL - !4
.02
.04
.01
BCL
!0.02alternate
!0.01
steady state
0.02
0.03
r Collision Bifurcation
BCL + !3
BCL + !4
Experimental Trends in One Frog
(a)
'Gain
2.0
Smooth
2.0
1.5
1.5
1.0
1.0
10
D
Gain
0.5
0
!
20
0.5
0
(b)
10
D
20
Experimental Trends in Two Frogs
1.5
(a)
1.5
2.0
(a)
Border-Collision
2.0
(b)
1.5
1.5
1.0
1.0
0.5
200
0.5
010
(b)
'
'Gain
2.0
Smooth
2.0
1.0
0.5
0
0.5
010
D
10
20
D
D
10
20
D
Gain
1.0
!
!
20
Experiment Trends
Behavior
# of
frogs
4
Smooth
3
Gain
# of
trials
Smooth
!
2
3
1
Flat
Combo
3
1
Border-Collision
Gain
4
BorderCollision
!
20 ms
15 ms
10 ms
5 ms
Gain
'
2.0
1.5
(a)
1.0
0.5
800
2.5
840
B0 (ms)
BCL
(ms)
APD (ms)
Combination
660
630
600
18
0.5
22
(c) 2.0
(d)
B0 (ms)
D (ms)
2.0
'
1.5
1.5
1.0
0
600
200
800 Trial
840 in One
180 Frog
Single
Gain
'
2.5
A
1.0
1.1
0.9
20 ms
15 ms
10 ms
5 ms
(a)
1.0
0.5
10
D
20
0.7
0 800
10840
BBCL
(ms) D
0
2.5
Smooth Trend Close to Bifurcation
2.0
(c)
20
AP
600
Single
200Trial
220in One Frog
180
D (ms)
(d) 2.0
20 ms
15 ms
10 ms
5 ms
1.5
0.9
'
Gain
1.1
(a)
1.0
0.7
0.5
0
10
D
20
800
B0 (ms)
840
2.5
Border-Collision Trend Far from Bifurcation (c)
2.0
results.
Our experimental observations can be explained wit
Unfold
Border-Collision
Bifurcation
a 1-D mathematical model of the form
AP Dn+1 = f (Dn )
(3
! where Dn=BCL ! APDn
where n is the beat number and Dn = B0 − AP Dn is th
diastolic interval. First, suppose that f is a piecewis
linear function of the form
AP Dn+1 = A0 + α (Dn − Dth ) + β |(Dn − Dth )| , (4
results.
a 1-D mathematical model of the form
Our experimental observations can be explained wit
Unfold
Border-Collision
Bifurcation
AP Dn+1
= fform
(Dn )
a 1-D mathematical model
of the
where n is the beat
=
B
−
AP
D
is
n
0
n
APnumber
Dn+1 =and
f (DD
)
(3
n
diastolic interval. First, suppose that f is a piecew
where
D
!
n=BCL ! APDn
linear
of number
the formand Dn = B0 − AP Dn is th
where
n function
is the beat
diastolic interval. First, suppose that f is a piecewis
APfunction
Dn+1 =of
A0the
+ αform
(Dn − Dth ) + β |(Dn − Dth )| ,
linear
where A0 , Dth , α, and β are constants. The derivat
AP Dn+1 = A0 + α (Dn − Dth ) + β |(Dn − Dth )| , (4
of f is discontinuous when Dn = Dth , where AP Dn
A0 . This map exhibits a border-collision period-doubl
bifurcation under the condition: −1 < α + β < 1 < α
results.
a 1-D mathematical model of the form
Our experimental observations can be explained wit
Unfold
Border-Collision
Bifurcation
AP Dn+1
= fform
(Dn )
a 1-D mathematical model
of the
where n is the beat
=
B
−
AP
D
is
n
0
n
APnumber
Dn+1 =and
f (DD
)
(3
n
diastolic interval. First, suppose that f is a piecew
where
D
!
n=BCL ! APDn
linear
of number
the formand Dn = B0 − AP Dn is th
where
n function
is the beat
diastolic interval. First, suppose that f is a piecewis
APfunction
Dn+1 =of
A0the
+ αform
(Dn − Dth ) + β |(Dn − Dth )| ,
linear
where A0 , Dth , α, and β are constants. The derivat
AP Dn+1 = A0 + α (Dn − Dth ) + β |(Dn − Dth )| , (4
of f isRecall
discontinuous
Dn =parameter
Dth , where AP Dn
BCL is thewhen
bifurcation
A0 . This map exhibits a border-collision period-doubl
bifurcation under the condition: −1 < α + β < 1 < α
where n is the beat number and Dn = B0 − AP Dn is t
diastolic
interval.
First, suppose Bifurcation
that f is a piecew
Unfold
Border-Collision
linear function of the form
AP Dn+1 = A0 + α (Dn − Dth ) + β |(Dn − Dth )| , (
where A0 , Dth , α, and β are constants. The derivat
parameters
of !
f is
discontinuous when Dn = Dth , where AP Dn
A0 . This map exhibits a border-collision period-doubli
bifurcation under the condition: −1 < α + β < 1 < α −
where n is the beat number and Dn = B0 − AP Dn is t
diastolic
interval.
First, suppose Bifurcation
that f is a piecew
Unfold
Border-Collision
linear function of the form
AP Dn+1 = A0 + α (Dn − Dth ) + β |(Dn − Dth )| , (
where A0 , Dth , α, and β are constants. The derivat
parameters
of !
f is
discontinuous when Dn = Dth , where AP Dn
A0 . This map exhibits a border-collision period-doubli
bifurcation under the condition: −1 < α + β < 1 < α −
where n is the beat number and Dn = B0 − AP Dn is t
diastolic
interval.
First, suppose Bifurcation
that f is a piecew
Unfold
Border-Collision
linear function of the form
AP Dn+1 = A0 + α (Dn − Dth ) + β |(Dn − Dth )| , (
APD
APD
where A0 , Dth , α, and β are constants. The derivat
parameters
of !
f is
discontinuous when
D
=
D
,
where
AP
D
n
th
n
2
2
andexhibits
−1 < αa border-collision
− β! < 1. Now,period-doubli
let us replac
A0 . This map
2 + Dmay
2 , wher
cates
that
current
ionic
models
place
|(D
−
D
)|
n
th
(D
−
D
)
in
map
(4)
with
bifurcation under the condition: n−1 < th
α + β < 1s < nee
α−
to include
slightly smoothed border-co
where Ds is a small
parameter
so that
Modifications to current models may in
! whe
parameter that becomes activated
(D
AP
=
A
+
α
(D
−
D
)
+
β
0
n
th
2
2 Dn+1
state space is crossed. For example, anp
Dn − Dth ) + Ds .
model [8] was used recently to capture
(5)
BCL[19]
We refer
to mapwhich
(5) ascould
an unfolding
of
BCL
dynamics
contribute
to the
of map (4), which
Simulation
20 ms (a) 2.0 660
15 ms
10 ms
1.5
5 ms
630
'Gain
'
APD (ms)
2.0
1.5
1.0
B
(ms)
0
BCL
800
200
840
(b)
840250
BBCL
(ms)
D
(ms)
0
2.5
(c)Close to0.6
Smooth Trend
Bifurcation
2.0
'
'
(a)
0.5 600
800
2.0
20 ms
15 ms
10 ms
5 ms
1.0
0.5
2.5
Experiment
0.4
(c) (d)
Simulation
Experiment
20 ms (a) 2.0 660
15 ms
10 ms
1.5
5 ms
630
'Gain
'
APD (ms)
2.0
1.5
1.0
20 ms
15 ms
10 ms
5 ms
(a)
(b)
1.0
0.5
0.5 600
800
B
(ms)
0
BCL
800
200
840
840250
BBCL
(ms)
D
(ms)
0
'
'
2.5
2.5 Border-Collision
(c)Close
Smooth Trend
Trend
Far
to0.6
from
Bifurcation
Bifurcation (c) (d)
2.0
2.0
0.4
Conclusions
Bifurcation to Alternans exhibits BOTH
smooth and border-collision-like features
Far from bifurcation
Insensitive to !
Close to bifurcation
Sensitive to !
Unfolded Border-Collision Bifurcation
So What...
Fundamentally still Smooth Bifurcation
Importance: Connection to other
dynamical processes occurring in
cardiac cells
Identify the “border”
Main Players
K+
Voltage
Ca2+
Na+
K+
time
Dubin, D., Ion Adventures in the Heartland. (2003)
Main Players
K+
Voltage
Ca2+
Na+
K+
time
Main Players
K+
Voltage
Ca2+
Na+
K+
time
Main Players
K+
Voltage
Ca2+
Na+
K+
time
Calcium Effects Plateau
dV
= IN a + IK + ICa + Istim
Cm
dt
K+
Voltage
Ca2+
Na+
K+
time
Calcium Effects Plateau
dV
= IN a + IK + ICa + Istim
Cm
dt
K+
Voltage
Ca2+
Na+
Chudin, E. J. et al. Biophys. J. 77, 2930 (1999)
K+
time
Calcium’s Role
Calcium responsible for contraction
Stores of Calcium in the cell get “stuffed”
and then release
Study Store or Intracellular Space
Karin R. Sipido, Understanding Cardiac Alternans:
The Answer Lies in the Ca2+ Store, Circulation Res., 94: 570-572
2004.
Experimental Setup
Camera
LED
Microelectrode
Stimulation
Cardiac Tissue
Pacing Scheme
Pace for 150 s at constant BCL
Alternate pace for 20 s (! = 20 ms)
Repeat with a new BCL
Calcium Waves
High
Ca2+
Low
Ca2+
Stimulus
BCL = 1000 ms
Alternate Pacing
Digital Number
4400
4300
4200
4100
4000
3900
6000
6500
7000
7500
time (ms)
8000
8500
Voltage (mV)
Perturbative Pacing
-0.1
-0.12
-0.14
-0.16
-0.18
6000
6500
7000
7500
time (ms)
8000
8500
Calcium
0.18
0.16
DN
Calcium Amplitude
0.2
0.14
0.12
time
0.1
500
600
700
800
900
1000
BCL (ms)
350
APD (ms)
voltage
Action Potential
300
250
time
200
500
600
700
800
BCL (ms)
900
1000
Conclusion
Previous result: APD relatively insensitive to
perturbations in BCL
Initial Result: Calcium more sensitive to
perturbations
Hypothesis: Calcium instability in cardiac
cells drives electrical instability
steady-state behavior with respect to the diastolic [Ca2!]
2!