understanding an unf olded border -collision bifurcation in p aced...

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

Cardiac Dynamics

Stimulus

APDAPDAPD

BCL BCL BCL

DI DI

time

timest

imvo

ltag

e

BCL = APD + DI

stim

str

ength

1:1 Behavior

APD

BCL BCL BCL

time

time

volt

age

APD APD

Stimulus

2:2 Behavior(Alternans)Stimulus

stim

str

ength

APD

BCL

DI

time

time

volt

age

APD APD

BCL BCL BCL BCL

APD APD APD

APDAPDAPD

BCL

time

stim

volt

age

APDAPDAPD

BCL BCL BCL

time

Transition

1:1 (slow pacing)2:2 (fast pacing)

BCL BCL BCL BCL

APD APD APD

APDAPDAPD

BCL

time

stim

volt

age

APDAPDAPD

BCL BCL BCL

time

Transition

1:1 (slow pacing)2:2 (fast pacing)

BCL BCL BCL BCL

APD APD APD

2:2 (alternans) linked to ventricular arrhythmias and sudden cardiac death

Bifurcation Diagram

BCL

AP

D

Bbif

1:12:2

AP

D

BCL

Sun, Amellal, Glass,

and Billette (1995)

Supercritical Period-Doubling Bifurcation

Border-Collision Bifurcation

Nolasco and Dahlen

(1968)

BCL

AP

D

AP

D

BCL

AP

D

BCL

Difficult to Distinguish with Discrete Data Points

AP

D

BCL

Investigate 1:1 Regime

Alternate Pacing vary BCL by ! in 1:1 regime

stim

APDsDI

time

time

volt

age

APDlAPDAPDs APDl APDs

BCL + ! BCL - ! BCL + ! BCL - ! BCL + !

Alternate Pacing st

im

APDsDI

time

time

volt

age



APDl -APDs

2 !Gain =

Alternate Pacing SimulationsA

PD

BCL BCL

alternatenon-alternate

Smooth Border-Collision

! = 20 ms ! = 20 ms

APDl -APDs

2 !Gain =

Alternate Pacing SimulationsA

PD

BCL BCL

alternatenon-alternate


BCL

Gai

n

Bbif BCLBbif

Gai

n

Alternate Pacing Trends


BCL

Gai

n

Bbif BCLBbif

Difficult to Distinguish with Discrete Data Points

Alternate Pacing st

im

APDsDI

time

time

volt

age



APDl -APDs

2 !Gain =

Vary ! SizeA

PD

BCL BCL


BCLfixed BCLfixed

!

Gai

n

!

Alternate Pacing TrendsA

PD

BCL BCL


BCLfixed BCLfixed

!

Gai

n

!

Alternate Pacing Trends

Different Trends with Discrete Data Points


Experimental Setup

Voltage-to-period

Stimulator

Microelectrode

BCL

Tissue chamberCardiac Muscle

Oxygenatedsolution

Amplifier

Alternate Pacing Protocol

BCL

BorderCollisionBifurcation

SkewedTentMap!

a"=

b(Zhusubaliyevetal.

Bifu

rcat

ions

and

Cha

osin

Pie

cew

ise-

Sm

ooth

Dyn

amic

alS

yste

ms(2003))

xn+

1=

!

ax

n+

µx

n#

0

bxn

+µ

xn

>0

steadystate

!µ;alternate

!µ±

!

!0.0

2!

0.0

10

0.0

10.0

20.0

3

!0.0

5

!0.0

4

!0.0

3

!0.0

2

!0.0

10

0.0

1

0.0

2

0.0

3

µ

x

ste

ady s

tate

altern

ate

bord

er


SkewedTentMap!

a"=

b(Zhusubaliyevetal.

Bifu

rcat

ions

and

Cha

osin

Pie

cew

ise-

Sm

ooth

Dyn

amic

alS

yste

ms(2003))

xn+

1=

!

ax

n+

µx

n#

0

bxn

+µ

xn

>0

steadystate

!µ;alternate

!µ±

!

!0.0

2!

0.0

10

0.0

10.0

20.0

3

!0.0

5

!0.0

4

!0.0

3

!0.0

2

!0.0

10

0.0

1

0.0

2

0.0

3

µ

xste

ady s

tate

altern

ate

bord

er


SkewedTentMap!

a"=

b(Zhusubaliyevetal.

Bifu

rcat

ions

and

Cha

osin

Pie

cew

ise-

Sm

ooth

Dyn

amic

alS

yste

ms(2003))

xn+

1=

!

ax

n+

µx

n#

0

bxn

+µ

xn

>0

steadystate

!µ;alternate

!µ±

!

!0.0

2!

0.0

10

0.0

10.0

20.0

3

!0.0

5

!0.0

4

!0.0

3

!0.0

2

!0.0

10

0.0

1

0.0

2

0.0

3

µ

x

ste

ady s

tate

altern

ate

bord

er


SkewedTentMap!

a"=

b(Zhusubaliyevetal.

Bifu

rcat

ions

and

Cha

osin

Pie

cew

ise-

Sm

ooth

Dyn

amic

alS

yste

ms(2003))

xn+

1=

!

ax

n+

µx

n#

0

bxn

+µ

xn

>0

steadystate

!µ;alternate

!µ±

!

!0.0

2!

0.0

10

0.0

10.0

20.0

3

!0.0

5

!0.0

4

!0.0

3

!0.0

2

!0.0

10

0.0

1

0.0

2

0.0

3

µ

x

ste

ady s

tate

altern

ate

bord

er


SkewedTentMap!

a"=

b(Zhusubaliyevetal.

Bifu

rcat

ions

and

Cha

osin

Pie

cew

ise-

Sm

ooth

Dyn

amic

alS

yste

ms(2003))

xn+

1=

!

ax

n+

µx

n#

0

bxn

+µ

xn

>0

steadystate

!µ;alternate

!µ±

!

!0.0

2!

0.0

10

0.0

10.0

20.0

3

!0.0

5

!0.0

4

!0.0

3

!0.0

2

!0.0

10

0.0

1

0.0

2

0.0

3

µ

x

ste

ady s

tate

altern

ate

bord

er

20 s 20 s20 s20 s

BCL + !1 BCL + !2

BCL + !3

BCL + !4

BCL

BCL - !1BCL - !2

BCL - !3

BCL - !4

120 s 20 sBorderCollisionBifurcation

SkewedTentMap!

a"=

b(Zhusubaliyevetal.

Bifu

rcat

ions

and

Cha

osin

Pie

cew

ise-

Sm

ooth

Dyn

amic

alS

yste

ms(2003))

xn+

1=

!

ax

n+

µx

n#

0

bxn

+µ

xn

>0

steadystate

!µ;alternate

!µ±

!

!0.0

2!

0.0

10

0.0

10.0

20.0

3

!0.0

5

!0.0

4

!0.0

3

!0.0

2

!0.0

10

0.0

1

0.0

2

0.0

3

µ

x

ste

ady s

tate

altern

ate

bord

er

Pac

ing R

ate

Smooth

Gai

n

!

Gai

nExperimental Trends in One Frog

0 10 200.5

1.0

1.5

2.0

0 10 200.5

1.0

1.5

2.0(a) (b)

Smooth

Gai

n

!

Gai

nExperimental Trends in Two Frogs

0 10 200.5

1.0

1.5

2.0

0 10 200.5

1.0

1.5

2.0(a) (b)

!

0 10 200.5

1.0

1.5

2.0

0 10 200.5

1.0

1.5

2.0(a) (b)

Border-Collision

Experiment Trends # of trials

Behavior# of frogs

4 Smooth 3

4Border-

Collision2

3 Flat 3

1 Combo 1

Gai

n

!

Smooth

!

Border-Collision

Gai

n

Combination G

ain

BCL (ms)800 840

B0

(ms)

0.5

1.0

1.5

2.0 (a) (b)

(d)

20 ms15 ms10 ms

5 ms

180 200 220

D (ms)

600

630

660

AP

D (

ms)

0 10 201.0

1.5

2.0

2.5

0 10 20

0.7

0.9

1.1(c)

Single Trial in One Frog

Gai

n

Smooth Trend Close to Bifurcation

800 840

B0

(ms)

0.5

1.0

1.5

2.0 (a) (b)

(d)

20 ms15 ms10 ms

5 ms

180 200 220

D (ms)

600

630

660

AP

D (

ms)

0 10 201.0

1.5

2.0

2.5

0 10 20

0.7

0.9

1.1(c)

800 840

B0

(ms)

0.5

1.0

1.5

2.0 (a) (b)

(d)

20 ms15 ms10 ms

5 ms

180 200 220

D (ms)

600

630

660

AP

D (

ms)

0 10 201.0

1.5

2.0

2.5

0 10 20

0.7

0.9

1.1(c)

BCL

Single Trial in One Frog

Gai

n

Border-Collision Trend Far from Bifurcation

800 840

B0

(ms)

0.5

1.0

1.5

2.0 (a) (b)

(d)

20 ms15 ms10 ms

5 ms

180 200 220

D (ms)

600

630

660

AP

D (

ms)

0 10 201.0

1.5

2.0

2.5

0 10 20

0.7

0.9

1.1(c)

800 840

B0

(ms)

0.5

1.0

1.5

2.0 (a) (b)

(d)

20 ms15 ms10 ms

5 ms

180 200 220

D (ms)

600

630

660

AP

D (

ms)

0 10 201.0

1.5

2.0

2.5

0 10 20

0.7

0.9

1.1(c)

Unfold Border-Collision Bifurcation

where Dn=BCL APDn! !

3

0 10 20

!

0.5

1

1.5

2

"

0 10 20

!

0.5

1

1.5

2(a) (b)

FIG. 2: Results displaying two di!erent trends in " vs. !

as revealed by alternate pacing for two di!erent frogs. Thetrend is consistent with (a) a smooth period-doubling bifur-cation (B0 = 300 ms, alternans observed at B0 = 275 ms sothat 275 ms < Bbif < 300 ms) and (b) a border-collision bi-furcation (B0 = 700 ms, alternans observed at B0 = 675 msso that 675 ms < Bbif < 700 ms). The error bards representthe statistical and systematic error in measuring APD; theirvariation is dominated by systematic errors.

that ! shows a decreasing trend as ! increases for B0

closest to Bbif ; a typical example is shown in Fig. 2(a),which agrees with a smooth bifurcation [recall Fig. 1(e)].However, four other trials from two frogs demonstrate anincreasing trend in ! as ! increases; a typical example isshown in Fig. 2(b), which agrees with a border-collisionbifurcation [recall Fig. 1(f)]. In three other trials fromthree frogs, there is no significant variations in ! for dif-ferent !’s and therefore these trials cannot be classifiedinto either category. These results call into question theassumption that the transition is either a smooth or aborder-collision bifurcation; rather, it appears that bothbehaviors are present to some degree.

The remaining experimental trial provides the mostcompelling evidence for the coexistence of both types ofbehaviors. Specifically, we see a smooth behavior whenB0 !Bbif " 25 ms, but a border-collision behavior when25 ms " B0 ! Bbif , as shown in Figs. 3(c) and (d), re-spectively. Figure 3(b) is a plot of steady-state APDvs. DI, where DI = B0 ! APD; the significance of Fig-ure 3(b) will be discussed in the limitations paragraph.Figure 3(a) shows ! vs. B0 for di"erent values of !,where it is seen that these curves cross one another. Thiscrossing is not consistent with either a smooth or border-collision period-doubling bifurcation. The presence ofsuch a crossing is one of our most revealing experimentalresults.

Our experimental observations can be explained witha 1-D mathematical model of the form

APDn+1 = f(Dn) (3)

where n is the beat number and Dn = B0!APDn is thediastolic interval. First, suppose that f is a piecewiselinear function of the form

APDn+1 = A0 + " (Dn ! Dth) + # |(Dn ! Dth)| , (4)

800 840

B0

(ms)

0.5

1

1.5

2

"

(a) (b)

(d)

20 ms15 ms10 ms5 ms

180 200 220

DI (ms)

600

630

660

AP

D (

ms)

0 10 20

!

1

1.5

2

2.5

"

0 10 20

!

0.7

0.9

1.1(c)

FIG. 3: Experimental evidence consistent with both a smoothand border-collision bifurcation in a single trial. (a) The trendin " vs. B0 for four di!erent values of ! (legend). We observealternans at B0 = 750 ms so that 750 ms < Bbif < 775 ms.(b) The corresponding dynamic restitution curve (see textfor details). Some of the data from this trial is re-plotted in(c) and (d) as " vs. ! for B0 = 775 ms and B0 = 800 ms,respectively. The behavior in (c) for B0 = 775 ms is consistentwith a smooth bifurcation and in (d) for B0 = 800 ms isconsistent with a border-collision bifurcation.

800 840B

0 (ms)

0.5

1

1.5

2

"

(c) (d)

20 ms15 ms10 ms5 ms

200 250DI (ms)

600

630

660

AP

D (

ms)

0 10 20

!

1

1.5

2

2.5

"

0 10 20

!

0.2

0.4

0.6

(a) (b)

FIG. 4: Theoretically predicted behavior of the unfoldedborder-collision bifurcation [map (5)]. (a) The trend in "vs. B0 for four di!erent values of ! (legend). Alternans oc-curs at Bbif = 750 ms. Some of the data from this simulationis re-plotted in (c) and (d) as " vs. ! for B0 = 775 ms andB0 = 800 ms, respectively. The trend in " vs. ! is consistentwith a smooth bifurcation in panel (c) and a border-collisionbifurcation in panel (d). Panel (b) shows the dynamic resti-tution curve as an illustration of the steady-state behavior forthe range of B0’s in (a).


where Dn=BCL APDn! !

3

0 10 20

!

0.5

1

1.5

2

"

0 10 20

!

0.5

1

1.5

2(a) (b)







APDn+1 = f(Dn) (3)



800 840

B0

(ms)

0.5

1

1.5

2

"

(a) (b)

(d)

20 ms15 ms10 ms5 ms

180 200 220

DI (ms)

600

630

660

AP

D (

ms)

0 10 20

!

1

1.5

2

2.5

"

0 10 20

!

0.7

0.9

1.1(c)


800 840B

0 (ms)

0.5

1

1.5

2

"

(c) (d)

20 ms15 ms10 ms5 ms

200 250DI (ms)

600

630

660

AP

D (

ms)

0 10 20

!

1

1.5

2

2.5

"

0 10 20

!

0.2

0.4

0.6

(a) (b)


3

FIG. 2: Typical experimental results displaying two di!erenttrends in " vs. ! as revealed by alternate pacing for twodi!erent frogs. The trend is consistent with (a) a smoothperiod-doubling bifurcation (data collected at B0 = 300 ms,alternans observed at B0 = 275 ms so that 275 ms < Bbif <

300 ms) and (b) a border-collision bifurcation (data collectedat B0 = 700 ms, alternans observed at B0 = 675 ms so that675 ms < Bbif < 700 ms). For all experimental data in thispaper, the error bars represent the statistical and systematicerror in the measurement of APD. Variations in the error barsare due to the systematic error.

However, four other trials from two frogs demonstrate anincreasing trend in ! as ! increases; a typical example isshown in Fig. 2(b), which agrees with a border-collisionbifurcation [recall Fig. 1(f)]. In three other trials fromthree frogs, there is no significant variations in ! for dif-ferent !’s and therefore these trials cannot be classifiedinto either category. These results call into question theassumption that the transition is either a smooth or aborder-collision bifurcation; rather, it appears that bothbehaviors are present to some degree.

The remaining experimental trial provides the mostcompelling evidence for the coexistence of both types ofbehaviors. Specifically, we see a smooth behavior whenB0 !Bbif " 25 ms, but a border-collision behavior when25 ms " B0!Bbif , as shown in Figs. 3(c) and (d), respec-tively. Figure 3(a) shows ! vs. B0 for di"erent values of!, where it is seen that these curves cross one another.This crossing is not consistent with either a smooth orborder-collision period-doubling bifurcation. The pres-ence of such a crossing is one of our most revealing ex-perimental results.


APDn+1 = f(Dn) (3)



where A0, Dth, ", and # are constants. The derivativeof f is discontinuous when Dn = Dth, where APDn =A0. This map exhibits a border-collision period-doublingbifurcation under the condition: !1 < "+# < 1 < "!#

800 840

B0

(ms)

0.5

1

1.5

2

!

(a) (b)

(d)

20 ms15 ms10 ms5 ms

180 200 220

DI (ms)

600

630

660

AP

D (

ms)

0 10 20

"

1

1.5

2

2.5

!

0 10 20

"

0.7

0.9

1.1(c)

FIG. 3: Experimental evidence consistent with both a smoothand border-collision bifurcation in a single trial. (a) The trendin " vs. B0 for four di!erent values of ! (legend). We observealternans at B0 = 750 ms so that 750 ms < Bbif < 775 ms.(b) The corresponding dynamic restitution curve (see textfor details). The " vs. ! for some of the data from thistrial is re-plotted in (c) and (d). The behavior in (c) forB0 = 775 ms is consistent with a smooth bifurcation andin (d) for B0 = 800 ms is consistent with a border-collisionbifurcation.

800 840B

0 (ms)

0

1

2

!

(c) (d)

20 ms15 ms10 ms5 ms

200 250DI (ms)

620

630

640

AP

D (

ms)

0 10 20

"

1

1.5

2

2.5

!

0 10 20

"

0.2

0.4

0.6

(a) (b)

FIG. 4: Theoretically predicted behavior of the unfoldedborder-collision bifurcation [map (5)]. (a) " as a functionof the bifurcation parameter B0 for various values of the per-turbation size !. Some of the data from this simulation isre-plotted in (c) and (d) as " vs. ! for B0 = 775 ms andB0 = 800 ms, respectively. The trend in " vs. ! is consistentwith a smooth bifurcation in panel (c) and a border-collisionbifurcation in panel (d). Panel (b) shows the dynamic resti-tution curve as an illustration of the steady state behavior forthe range of B0’s in (a).


where Dn=BCL APDn!

3

0 10 20

!

0.5

1

1.5

2

"

0 10 20

!

0.5

1

1.5

2(a) (b)







APDn+1 = f(Dn) (3)



800 840

B0

(ms)

0.5

1

1.5

2

"

(a) (b)

(d)

20 ms15 ms10 ms5 ms

180 200 220

DI (ms)

600

630

660

AP

D (

ms)

0 10 20

!

1

1.5

2

2.5

"

0 10 20

!

0.7

0.9

1.1(c)


800 840B

0 (ms)

0.5

1

1.5

2

"

(c) (d)

20 ms15 ms10 ms5 ms

200 250DI (ms)

600

630

660

AP

D (

ms)

0 10 20

!

1

1.5

2

2.5

"

0 10 20

!

0.2

0.4

0.6

(a) (b)


3






APDn+1 = f(Dn) (3)




800 840

B0

(ms)

0.5

1

1.5

2

!

(a) (b)

(d)

20 ms15 ms10 ms5 ms

180 200 220

DI (ms)

600

630

660

AP

D (

ms)

0 10 20

"

1

1.5

2

2.5

!

0 10 20

"

0.7

0.9

1.1(c)


800 840B

0 (ms)

0

1

2

!

(c) (d)

20 ms15 ms10 ms5 ms

200 250DI (ms)

620

630

640

AP

D (

ms)

0 10 20

"

1

1.5

2

2.5

!

0 10 20

"

0.2

0.4

0.6

(a) (b)

FIG. 4: Theoretically predicted behavior of the unfoldedborder-collision bifurcation [map (5)]. (a) " as a functionof the bifurcation parameter B0 for various values of the per-turbation size !. Some of the data from this simulation isre-plotted in (c) and (d) as " vs. ! for B0 = 775 ms andB0 = 800 ms, respectively. The trend in " vs. ! is consistentwith a smooth bifurcation in panel (c) and a border-collisionbifurcation in panel (d). Panel (b) shows the dynamic resti-tution curve as an illustration of the steady state behavior forthe range of B0’s in (a).Recall BCL is the bifurcation parameter

!

3






APDn+1 = f(Dn) (3)




800 840

B0

(ms)

0.5

1

1.5

2

!

(a) (b)

(d)

20 ms15 ms10 ms5 ms

180 200 220

DI (ms)

600

630

660

AP

D (

ms)

0 10 20

"

1

1.5

2

2.5

!

0 10 20

"

0.7

0.9

1.1(c)


800 840B

0 (ms)

0

1

2

!

(c) (d)

20 ms15 ms10 ms5 ms

200 250DI (ms)

620

630

640

AP

D (

ms)

0 10 20

"

1

1.5

2

2.5

!

0 10 20

"

0.2

0.4

0.6

(a) (b)



parameters!

3






APDn+1 = f(Dn) (3)




800 840

B0

(ms)

0.5

1

1.5

2

!

(a) (b)

(d)

20 ms15 ms10 ms5 ms

180 200 220

DI (ms)

600

630

660

AP

D (

ms)

0 10 20

"

1

1.5

2

2.5

!

0 10 20

"

0.7

0.9

1.1(c)


800 840B

0 (ms)

0

1

2

!

(c) (d)

20 ms15 ms10 ms5 ms

200 250DI (ms)

620

630

640

AP

D (

ms)

0 10 20

"

1

1.5

2

2.5

!

0 10 20

"

0.2

0.4

0.6

(a) (b)



parameters!!

3






APDn+1 = f(Dn) (3)




800 840

B0

(ms)

0.5

1

1.5

2

!

(a) (b)

(d)

20 ms15 ms10 ms5 ms

180 200 220

DI (ms)

600

630

660

AP

D (

ms)

0 10 20

"

1

1.5

2

2.5

!

0 10 20

"

0.7

0.9

1.1(c)


800 840B

0 (ms)

0

1

2

!

(c) (d)

20 ms15 ms10 ms5 ms

200 250DI (ms)

620

630

640

AP

D (

ms)

0 10 20

"

1

1.5

2

2.5

!

0 10 20

"

0.2

0.4

0.6

(a) (b)



parameters! 4

and !1 < !2 ! "2 < 1. Now, let us replace |(Dn ! Dth)|in map (4) with

!

(Dn ! Dth)2 + D2s , where Ds is a small

parameter so that

APDn+1 = A0 + ! (Dn !Dth) + "!

(Dn ! Dth)2 + D2s .

(5)We refer to map (5) as an unfolding [19] of map (4), whichreduces to map (4) when Ds = 0. For any Ds "= 0, the un-folded map (5) is smooth and exhibits what is technicallya smooth period-doubling bifurcation. Nevertheless, thedynamics of map (4) and map (5) exhibit no significantdi!erences except when B ! Bbif is less than or on theorder of Ds.

We simulate the alternate pacing protocol withmap (5) using the parameters ! = 0.79, " = !0.69,A0 = 621, Dth = 164, and Ds = 10. Figure 4(a) shows" vs. B0 for di!erent values of #. These curves cross oneanother in the region 775 ms < B0 < 800 ms as shownin Fig. 4(a), similar to our experimental results shown inFig. 3(a). In Fig. 4(c), " vs. # for B0 = 775 ms displaysa trend consistent with a smooth bifurcation [compara-ble to Fig. 3(c)]. On the other hand, in Fig. 4(d), "vs. # for B0 = 800 ms shows a trend consistent with aborder-collision bifurcation, also comparable to our ex-perimental results shown in Fig. 3(d). To summarize,the period-doubling bifurcation of map (5) is smooth,but the e!ects of this smoothness can be seen only in anarrow range of B0, when B0!Bbif is on the order of Ds,exactly the behavior observed experimentally. Further-more, in di!erent experimental trials, di!erent values ofthe parameters, including Ds, are needed to fit to the ex-perimental data and therefore the smooth behavior is notalways easy to capture. Thus, our experimental resultspresented in Figure 2 are elucidated.

Since the unfolded border collision bifurcation modelfits the data obtained through alternate pacing, this indi-

cates that current ionic models may need to be adjustedto include slightly smoothed border-collision behavior.Modifications to current models may involve tracking aparameter that becomes activated when a threshold instate space is crossed. For example, a piecewise smoothmodel [8] was used recently to capture calcium cyclingdynamics which could contribute to the mechanism thatgives rise the unfolded border collision bifurcation behav-ior.

We note that map (5) faces some limitations. For ex-ample, it is known that the dynamic restitution behaviorof cardiac tissue, how steady state values of APD de-pend on DI, cannot be adequately described by a simple1-D map [20]. Not surprisingly, although map (5) fits thegain under alternate pacing rather well, it cannot captureall the restitution phenomena. For example, comparingfig. 3(b) to fig. 4(b), we see that the agreement betweenthe two dynamic restitution curve is poor. Nevertheless,map (5) suggests that existing models should be modifiedto include non-smooth features.

Finally, our results suggest that the proposed clinicaluse of alternate pacing [15] may not be successful. In ap-plying alternate pacing to a map that exhibits a smoothbifurcation, one expects large " for slower pacing rates.As a result, in the clinic, the propensity for alternanscould be revealed using pacing rates that are slow enoughto avoid inducing a life-threatening arrhythmia. How-ever, we find that " remains small until the pacing ratesare decreased to a value very close to the bifurcation,greatly diminishing diagnostic value of such a procedure.

We gratefully acknowledge the financial support of theNSF under grant PHY-0243584 and PHY-0549259 andthe NIH under grant 1R01-HL-72831.

[1] S. H. Strogatz, Nonlinear Dynamics and Chaos (Wester-view Press, Cambridge, 1994), Ch. 3.

[2] Z. T. Zhusubaliyev and E. Mosekilde, Bifurcations andChaos in Piecewise-smooth Dynamical Systems (WorldScientific Publishing Co., Singapore, 2003).

[3] A. Karma, Chaos 4, 461 (1994).[4] J. M. Pastore, S. D. Girouard, K. R. Laurita, F. G. Akar,

and D. S. Rosenbaum, Circulation 99, 1385 (1999).[5] D. S. Rosenbaum, L. E. Jackson, J. M. Smith, H. Garan,

J. N. Ruskin, and R. J. Cohen, New Engl. J. Med. 330,235 (1994).

[6] T. Thom et al., Circulation 113, e85 (2006).[7] J. W. M. Bassani, W. Yuan, and D. M. Bers, Am. J.

Physiol.-Cell Ph. 268, C1313 (1995).[8] Y. Shiferaw, D. Sato, and A. Karma. Phys. Rev. E 71,

021903 (2005).[9] G. M. Hall, S. Bahar, and D. J. Gauthier, Phys. Rev.

Lett. 82, 2995 (1999).[10] J. B. Nolasco and R. W. Dahlen, Appl. Physiol. 25, 191

(1968).[11] J. Sun, F. Amellal, L. Glass, and J. Billette, J. Theor.

Biol. 173, 79 (1995).[12] D. S. Chen, H. O. Wang, and W. Chin, Proc. 1998 IEEE

Int’l Symp. Circuits and Systems (CA), 3, 635 (1998).[13] M. A. Hassouneh and E. H. Abed, Int. J. Bifurcat. Chaos

14, 3303 (2004).[14] J. Heldstab, H. Thomas, T. Geisel, and G. Randons, Z.

Phys. B 50, 141 (1983).[15] A. Karma and Y. Shiferaw, Heart Rhythm 1, S290,

(2004).[16] X. Zhao, D. G. Schae!er, C. M. Berger, and D.

J. Gauthier, to appear in Nonlinear Dynam. (2006).http://arxiv.org/abs/nlin.CD/0609009

[17] X. Zhao and D. G. Schae!er, to appear in Nonlinear Dy-nam. (2006). http://arxiv.org/abs/math.DS/0609106

[18] All procedures are approved by the Duke University In-stitutional Animal Care and Use Committee (DIACUC).

[19] M. Golubitsky and D. Schae!er, Singularities and Groupsin Bifurcation Theory, (Springer, Berlin, 1985).

[20] S.S. Kalb et al.. J. Cardiovas. Electr. 15, 698, (2004).

4

and !1 < !2 ! "2 < 1. Now, let us replace |(Dn ! Dth)|in map (4) with

!

(Dn ! Dth)2 + D2s , where Ds is a small

parameter so that

APDn+1 = A0 + ! (Dn !Dth) + "!

(Dn ! Dth)2 + D2s .

(5)We refer to map (5) as an unfolding [19] of map (4), whichreduces to map (4) when Ds = 0. For any Ds "= 0, the un-folded map (5) is smooth and exhibits what is technicallya smooth period-doubling bifurcation. Nevertheless, thedynamics of map (4) and map (5) exhibit no significantdi!erences except when B ! Bbif is less than or on theorder of Ds.

We simulate the alternate pacing protocol withmap (5) using the parameters ! = 0.79, " = !0.69,A0 = 621, Dth = 164, and Ds = 10. Figure 4(a) shows" vs. B0 for di!erent values of #. These curves cross oneanother in the region 775 ms < B0 < 800 ms as shownin Fig. 4(a), similar to our experimental results shown inFig. 3(a). In Fig. 4(c), " vs. # for B0 = 775 ms displaysa trend consistent with a smooth bifurcation [compara-ble to Fig. 3(c)]. On the other hand, in Fig. 4(d), "vs. # for B0 = 800 ms shows a trend consistent with aborder-collision bifurcation, also comparable to our ex-perimental results shown in Fig. 3(d). To summarize,the period-doubling bifurcation of map (5) is smooth,but the e!ects of this smoothness can be seen only in anarrow range of B0, when B0!Bbif is on the order of Ds,exactly the behavior observed experimentally. Further-more, in di!erent experimental trials, di!erent values ofthe parameters, including Ds, are needed to fit to the ex-perimental data and therefore the smooth behavior is notalways easy to capture. Thus, our experimental resultspresented in Figure 2 are elucidated.

Since the unfolded border collision bifurcation modelfits the data obtained through alternate pacing, this indi-

cates that current ionic models may need to be adjustedto include slightly smoothed border-collision behavior.Modifications to current models may involve tracking aparameter that becomes activated when a threshold instate space is crossed. For example, a piecewise smoothmodel [8] was used recently to capture calcium cyclingdynamics which could contribute to the mechanism thatgives rise the unfolded border collision bifurcation behav-ior.

We note that map (5) faces some limitations. For ex-ample, it is known that the dynamic restitution behaviorof cardiac tissue, how steady state values of APD de-pend on DI, cannot be adequately described by a simple1-D map [20]. Not surprisingly, although map (5) fits thegain under alternate pacing rather well, it cannot captureall the restitution phenomena. For example, comparingfig. 3(b) to fig. 4(b), we see that the agreement betweenthe two dynamic restitution curve is poor. Nevertheless,map (5) suggests that existing models should be modifiedto include non-smooth features.

Finally, our results suggest that the proposed clinicaluse of alternate pacing [15] may not be successful. In ap-plying alternate pacing to a map that exhibits a smoothbifurcation, one expects large " for slower pacing rates.As a result, in the clinic, the propensity for alternanscould be revealed using pacing rates that are slow enoughto avoid inducing a life-threatening arrhythmia. How-ever, we find that " remains small until the pacing ratesare decreased to a value very close to the bifurcation,greatly diminishing diagnostic value of such a procedure.

We gratefully acknowledge the financial support of theNSF under grant PHY-0243584 and PHY-0549259 andthe NIH under grant 1R01-HL-72831.

[1] S. H. Strogatz, Nonlinear Dynamics and Chaos (Wester-view Press, Cambridge, 1994), Ch. 3.

[2] Z. T. Zhusubaliyev and E. Mosekilde, Bifurcations andChaos in Piecewise-smooth Dynamical Systems (WorldScientific Publishing Co., Singapore, 2003).

[3] A. Karma, Chaos 4, 461 (1994).[4] J. M. Pastore, S. D. Girouard, K. R. Laurita, F. G. Akar,

and D. S. Rosenbaum, Circulation 99, 1385 (1999).[5] D. S. Rosenbaum, L. E. Jackson, J. M. Smith, H. Garan,

J. N. Ruskin, and R. J. Cohen, New Engl. J. Med. 330,235 (1994).

[6] T. Thom et al., Circulation 113, e85 (2006).[7] J. W. M. Bassani, W. Yuan, and D. M. Bers, Am. J.

Physiol.-Cell Ph. 268, C1313 (1995).[8] Y. Shiferaw, D. Sato, and A. Karma. Phys. Rev. E 71,

021903 (2005).[9] G. M. Hall, S. Bahar, and D. J. Gauthier, Phys. Rev.

Lett. 82, 2995 (1999).[10] J. B. Nolasco and R. W. Dahlen, Appl. Physiol. 25, 191

(1968).[11] J. Sun, F. Amellal, L. Glass, and J. Billette, J. Theor.

Biol. 173, 79 (1995).[12] D. S. Chen, H. O. Wang, and W. Chin, Proc. 1998 IEEE

Int’l Symp. Circuits and Systems (CA), 3, 635 (1998).[13] M. A. Hassouneh and E. H. Abed, Int. J. Bifurcat. Chaos

14, 3303 (2004).[14] J. Heldstab, H. Thomas, T. Geisel, and G. Randons, Z.

Phys. B 50, 141 (1983).[15] A. Karma and Y. Shiferaw, Heart Rhythm 1, S290,

(2004).[16] X. Zhao, D. G. Schae!er, C. M. Berger, and D.

J. Gauthier, to appear in Nonlinear Dynam. (2006).http://arxiv.org/abs/nlin.CD/0609009

[17] X. Zhao and D. G. Schae!er, to appear in Nonlinear Dy-nam. (2006). http://arxiv.org/abs/math.DS/0609106

[18] All procedures are approved by the Duke University In-stitutional Animal Care and Use Committee (DIACUC).

[19] M. Golubitsky and D. Schae!er, Singularities and Groupsin Bifurcation Theory, (Springer, Berlin, 1985).

[20] S.S. Kalb et al.. J. Cardiovas. Electr. 15, 698, (2004).

AP

D

BCL BCL

AP

D

Simulation Experiment

Gai

n

BCL

800 840

B0

(ms)

0.5

1.0

1.5

2.0 (a) (b)

(d)

20 ms15 ms10 ms

5 ms

180 200 220

D (ms)

600

630

660

AP

D (

ms)

0 10 201.0

1.5

2.0

2.5

0 10 20

0.7

0.9

1.1(c)

800 840B

0(ms)

0.5

1.0

1.5

2.0

(c) (d)

20 ms15 ms10 ms

5 ms

200 250D (ms)

600

630

660

AP

D (

ms)

0 10 201.0

1.5

2.0

2.5

0 10 200.2

0.4

0.6

(a) (b)

BCL

Smooth Trend Close to Bifurcation

Simulation Experiment

Gai

n

BCL

800 840

B0

(ms)

0.5

1.0

1.5

2.0 (a) (b)

(d)

20 ms15 ms10 ms

5 ms

180 200 220

D (ms)

600

630

660

AP

D (

ms)

0 10 201.0

1.5

2.0

2.5

0 10 20

0.7

0.9

1.1(c)

800 840B

0(ms)

0.5

1.0

1.5

2.0

(c) (d)

20 ms15 ms10 ms

5 ms

200 250D (ms)

600

630

660

AP

D (

ms)

0 10 201.0

1.5

2.0

2.5

0 10 200.2

0.4

0.6

(a) (b)

BCL

Smooth Trend Close to Bifurcation Border-Collision Trend Far from Bifurcation

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

Ca2+

Na+

K+

K+

Dubin, D., Ion Adventures in the Heartland. (2003)time

Vo

ltag

e

Main Players

Ca2+

Na+

K+

K+

time

Vo

ltag

e

Ca2+

Na+

K+

K+

time

Vo

ltag

e

Calcium Effects PlateauCm

dV

dt= INa + IK + ICa + Istim (1)

1

Ca2+

Na+

K+

K+

time

Vo

ltag

e

Calcium Effects PlateauCm

dV

dt= INa + IK + ICa + Istim (1)

1

Chudin, E. J. et al. Biophys. J. 77, 2930 (1999)

Calcium’s RoleCalcium 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

LED

Cardiac Tissue

Camera

StimulationMicroelectrode

Pacing Scheme

Pace for 150 s at constant BCL

Alternate pace for 20 s (! = 20 ms)

Repeat with a new BCL

BCL = 1000 ms

Calcium Waves

StimulusLow

Ca2+

High

Ca2+

Alternate Pacing

6000 6500 7000 7500 8000 8500

time (ms)

3900

4000

4100

4200

4300

4400

Dig

ital

Nu

mb

er

Perturbative Pacing

6000 6500 7000 7500 8000 8500

time (ms)

-0.18

-0.16

-0.14

-0.12

-0.1

Volt

age

(mV

)

500 600 700 800 900 1000

BCL (ms)

0.1

0.12

0.14

0.16

0.18

0.2C

alci

um

Am

pli

tude

time

DN

Calcium

500 600 700 800 900 1000

BCL (ms)

200

250

300

350

AP

D (

ms)

time

volt

age

Action Potential

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

Conclusion

steady-state behavior with respect to the diastolic [Ca2!]SR

versus Ca2! release relationship. During steady alternans,however, data points clustered with very little variation in2 completely separate areas, where the large [Ca2!]SR

depletion was preceded by a higher diastolic [Ca2!]SR andthe small depletion, by a lower diastolic [Ca2!]SR. Figure2B and 2C also indicates the great sensitivity of these[Ca2!]SR measurements, where a difference of "3% indiastolic [Ca2!]SR signal is quite easily resolved duringalternans (Figure I in the online data supplement).

The upper curve in Figure 2C, connecting large deple-tions during alternans along their transition to regulardepletions has the monotonic positive slope expected fromprior work studying the relationship between SR Ca2!

content and SR Ca2! release.1,2,21,23 That is, increased[Ca2!]SR is associated with increased SR Ca2! release. Theadditional lower limb (connecting regular depletions andsmall depletions during alternans) indicates that eventhough diastolic [Ca2!]SR was higher preceding the smallalternans beat, SR Ca2! release was lower. This most likelyreflects a change in either the trigger for SR Ca2! release(ICa) or responsiveness of the RyRs to the trigger (eg,attributable to RyR refractoriness). Thus, whereas fluctu-ations in diastolic [Ca2!]SR are occurring here in associa-tion with Ca2! alternans, factors different from [Ca2!]SR

limit SR Ca2! release during the small beats duringalternans.

We assessed the maximal SR Ca2! release flux based onthe maximal rate of [Ca2!]SR decline (#d[Ca2!]SR/dt),analogous to the analysis of cytoplasmic Ca2! transients.The duration of release was measured as time to nadir(Figure 2D). Normalized to regular depletions at 1.5 Hz,

during alternans, the release flux was reduced to 54$10%during small depletions (P%0.01, n&26 depletions) andincreased to 159$3% during large depletions (P%0.01,n&25). The duration of release was reduced to 82$7%(P%0.01, n&26) during small depletions and slightly,although significantly, reduced during large depletions.Furthermore, comparing small versus large depletionsduring alternans SR Ca2! release flux and duration ofrelease was less during small depletions (P%0.01).

We also examined whether diastolic [Ca2!]SR was lim-ited by the time between beats during alternans. Whenstimulation was stopped after a small or a large depletion,respectively, differences in diastolic [Ca2!]SR remained.Figure 3A shows that during a pause after the smalldepletion, [Ca2!]SR attained the same higher level as duringdiastole after small depletions in ongoing alternans. Whenthe stimulation was paused after a large depletion (Figure3B), resting [Ca2!]SR stayed at the lower level typical ofdiastolic [Ca2!]SR after the large depletions during alter-nans and therefore lower than after small depletions. Thissuggests that the diastolic [Ca2!]SR during alternans issubstantially controlled by the cellular Ca2! availablerather than being limited mainly by the diastolic interval.

The [Ca2!]SR depletion after a 7-second rest in Figure 3Aand 3B was potentiated, compared with depletions duringalternans and was also similar in magnitude for both cases.This emphasizes that the time between beats can stronglyimpact fractional SR Ca2! release at constant [Ca2!]SR. Thisis consistent with prior work showing that enhancedfractional SR Ca2! release during postrest potentiationover this time scale is attributable to a slow phase of RyRrecovery rather than altered [Ca2!]SR or ICa.24

Figure 2. Ca2! alternans with distinct diastolic[Ca2!]SR fluctuations. A, Stable depletion alter-nans was present at 2.2 Hz, which ceasedwhen the frequency was gradually reduced to1.5 Hz. Bottom, Stimulation frequency. B,Regions a and b in A are shown on anexpanded time scale. C, Relationship betweendiastolic [Ca2!]SR and the following depletionamplitude (individual data points and SD forstable conditions). Open symbols representtransitional depletions between stable alternansand regular depletions. D, Normalized SR Ca2!

release flux and duration of release (measuredas d[Ca2!]SR/dt and time to nadir, respectively)for regular depletions and for the small andlarge alternating depletions. *P%0.01 vs noalternans, §P%0.01 vs small depletions.

742 Circulation Research September 29, 2006

at Duke University--Durham on July 29, 2007 circres.ahajournals.orgDownloaded from

understanding an unf olded border -collision bifurcation in p aced...

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