Why Is Alternans Indeterminate?

Xiaopeng ZhaoMechanical, Aerospace and Biomedical EngineeringUniversity of Tennessee Xzhao9@utk.edu

350 375 400BCLms1.4

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350 375 400BCLms1.4

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Ventricular Fibrillation

Courtesy of Flavio H. Fenton, Cornell University

ECG

Death in the US

stroke

167,000

lung cancer

160,000http://www.wrongdiagnosis.com

sudden cardiac arrest

350,000

Ventricular Fibrillation Leads to SCA

Basic Physiology

Calcium Contraction -- Action Potential Duration-- Diastolic Interval-- Basic Cycle Length

Alternans: period doubling

Slow pacing1:1 response

Rapid pacing2:2 response(alternans)

Model

Fox et al., Am J Physiol, 2002 Shiferaw et al., Biophy, 2003

Alternans in Single Cells

Simulation of the Shiferaw-Fox Model

Alternans in Fibers

Alternans in Fibers

A “fiber” of Two Cells

desynchronized synchronized

Influence of Junctional Diffusion

7 2 2voltage: 5*10 m /s, calcium: 300 m /sD D D.M. Bers, 2001

Spatial Pattern of Alternans

• Existing theory:– Pattern determined by length of fiber;

short fibers: spatially concordant;long fibers: spatially discordant

• Shiferaw-Fox model:– Pattern determined by initial condition;

spatial concordance independent of length of fiber

Voltage/Calcium Coupling

1

1

,

,n n n

n n n

A F D C

C G D C

n nA D B

A Single Cell

C

C

D

D

FJ

G GF

: slope of APD restitution: slope of Ca relation

D

C

FG

Voltage/Calcium Coupling

1 1 2

1 1 1

1 2 2

1/ 2 , 1/ 2 ,

,

,

n n n n n

n n n

n n n

A F D C x F D C x

C x G D C x

C x G D C x

/ 2 / 20

0

D C C

D C

D C

F F FJ G G

G G

Two Coupled Cells

Eigenvalues

D C

D C

F FJ

G G

/ 2 / 20

0

D C C

D C

D C

F F FJ G G

G G

Stability Boundary0C DF G 0C DF G

Single Cell

Coupled Cells

Spatial Synchrony

1 1 2

1 1 1

1 2 2

1/ 2 , 1/ 2 ,

,

,

n n n n n

n n n

n n n

A F D C x F D C x

C x G D C x

C x G D C x

0 1 1 and / 1 1C DG G Eigenvectors:

Calcium Dynamics

Diaz et al., Circ Res, 2002Shiferaw et al, Biophy, 2003

CG

Border-collision Bifurcation0C DF G 0C DF G

Single Cell

Coupled Cells

Simulation of Mapping

350 375 400BCLms1.4

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a single cell two coupled cells

Bifurcation Diagram

Simulation of Mapping

0 25 50 75 100cell number

240

260

280

300

DPAsm

0 25 50 75 100cell number

240

260

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300

DPAsm

0 25 50 75 100cell number

1.4

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aCM

0 25 50 75 100cell number

1.4

1.6

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aCM

Future Work

1

1

,

,

n i j n j n j

n i n i n i

A x w F D x C x

C x G D x C x

Open questions:

stability, synchronization, …experiments, prediction, control, …

Coupled Oscillators

Take-home Message

The spatiotemporal pattern of calcium-driven alternans is indeterminate.

350 375 400BCLms1.4

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350 375 400BCLms1.4

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Acknowledgement

• Code of the Shi-Fox model, courtesy of Yohannes Shiferaw

• Useful discussion with Wanda Krassowska, David Schaeffer, Daniel Gauthier, Daisuke Sato, and Alain Karma

Positive Vm/Ca Coupling

Summary

• In tissue, spatial coupling in APD is strong whereas the coupling in Ca is weak

• Multiple alternans (period-2) solutions coexist if alternans is induced by instability in calcium dynamics

• At a first approximation, the phenomenon occurs through a border-collision bifurcation.