What is so scary about a perfectly uniform What is so scary about a perfectly uniform heartbeat?heartbeat?

P G Vaidya

Mathematical Modelling Unit

National Institute of Advanced Studies

Bangalore 560 012

International Biomedical Modeling School and WorkshopInternational Biomedical Modeling School and WorkshopNCBS, 2 March 2008NCBS, 2 March 2008

An ECG

0 1000 2000 3000 4000 5000 6000 7000 80005

0

5

105.54027

3.20009

X2ppp

8000130 ppp

Data Supplied by Dr. Pradhan of NIMHANS, Bangalore, Healthy Male, Age 36

Can

Hea

rt A

ttac

k b

e P

red

icte

d?

Phase distribution of the RR intervals of a healthy Phase distribution of the RR intervals of a healthy subject subject

600 650 700 750 800 850600

650

700

750

800

850

Gaps 1

Gaps

1 0.5 0 0.5 11

0.5

0

0.5

1

sin s sin ss

cos s cos ss

3D state space picture of the ECG data using “Embedding”

Can

Hea

rt A

ttac

k b

e P

red

icte

d?

X5low Y5hi X5hi( )

Can we synchronize them by giving

the same push to both ?

Position and Velocity of the Swing

0 20 40 60 80 1001

0.5

0

0.5

1

POSITIONn

VELOCITYn

n

State Space and Phase of Swing

1 0.5 0 0.5 10.2

0.1

0

0.1

0.2

VELOCITYn

POSITION n

Portrait of a nonlinear autonomous system Portrait of a nonlinear autonomous system

With the solitary exception of a trajectory starting at the origin, all others approach a limit cycle of unit radius : 

2 1 0 1 21

0

10.65993

0.64332

Vs

1.508451.00029 Xs

What is the phase outside of the limit cycle?

"2.jpg"

Effect of Driving ForceEffect of Driving Force

d

dtX Y X 1 X

2 Y

2

n

An t Tn

d

dtY Y 1 X

2 Y

2 X

EFFECT of a single PULSE

"1.jpg"

Change of Phase due to a Single PulseChange of Phase due to a Single Pulse

F atansin

cos

Change in Phase due to single Pulse

The graph and the equation shows that there is a fixed point at 0, which is stable and a fixed point at pi which is unstable. Therefore, multiple periodic pulses would lead to synchronization

0 2 4 6 81

0

11

1

F ( )

60

Example of ECG DataExample of ECG Data

Results from the Simulation EquationsResults from the Simulation Equations--11

State Space with an Apparently Simple Limit CycleState Space with an Apparently Simple Limit CycleX8 Y8 Z8( )

From a “normal” person’s data an approximate autonomous nonlinear equation was derived. (Experimental Chaos, Florence 2004)

Two Stable and Two Unstable Fixed PointsTwo Stable and Two Unstable Fixed Points

For this case, the before and after phase was determined by running several cases with different initial conditions. It can be seen from the map that now there are two stable and unstable fixed points.

0 1 2 3 4 5 6 72

0

2

4

6

8

Starting Phase Shift

Pha

se S

hift

aft

er P

ulse

6.0362478933603

0.255168660946595

after mm

beforemm

6.03624789336030 beforemm

Change in Phase due to single Pulse

0 2 4 6 81

0

1

F 0

Two Stable Rest Points Two Stable Rest Points

GRAPHIC REPRESENTATION OF POSSIBLE PROBLEM

•

•

••

S1

S2

U1U2

Multiple pulsesMultiple pulses

.

n 1 f n2 T

pn

2 T

pn

UNIFORM CASE

0 5 10 15 200

1

2

3

4

54.398229715026

0

HUk 0

HUk 30

HUk 70

200 k

ConsequencesConsequences

Thus it is clear that it is possible that if two cells are nearly synchronized, chances are that they will continue to get closer. An occasional pulse which puts them on the wrong side would not cause lasting damage.

But once they are farther, the chances of their drifting further increase.

Pulses at a random interval from one another

0 20 40 60 80 1004

2

0

2

4

6

HRk 0

HRk 30

HRk 70

k

All drift, but together !

PCEPCEPHASE COHERNCE of an ENSEMBLEPHASE COHERNCE of an ENSEMBLE

. PCE calculated in a manner similar to TSC (pl see poster)

PCE=1 if all cells are in phase PCE=0 if phases are uniformly distributed across the circle

PCE of RANDOM forcing vs. UNIFORM forcing

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

RANDOMk

UNIFORMk

k

A plausible Explanation of theA plausible Explanation of the desirability of RR variabilitydesirability of RR variability

It does seem that Uniform Interval forcing, in most cases, could lead to more than one cluster of cells,

each going at their own phases, well separated from each other A variable rate of pulsing, if with well distributed phases

Could reduce this possibility

Some theoretical justification based on Markov Modeling

Special thanks are due to

Professor V. Kannan, Univ. Hyderabad, Professor R. Narasimha, JNC, Professor Ivanov of Boston University and Dr. Kanters, University of Copenhagen and others