What is so scary about a perfectly uniform heartbeat? P G Vaidya Mathematical Modelling Unit...
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Transcript of What is so scary about a perfectly uniform heartbeat? P G Vaidya Mathematical Modelling Unit...
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