synchronization detection of biological cam plants …...w,x,y, are measurable g(z,t) is known...
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Synchronization Detection of
Biological CAM Plants Using
Instantaneous Lyapunov Exponent
Yusuke Totoki, Akira Goto,
Haruo Suemitsu, Takami Matsuo
Oita univ, Oita, Japan
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Outline
◆Background and Objectives
◆Decay Rate and Synchronization Measures
Instantaneous Lyapunov Exponent(ILE)
Synchronization Mesures
◆Numerical Example
System with finite escape time
Biological system
◆Conclusion
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Background and Objectives
Information about both stability properties and transient responses
of nonlinear differential equations is important for practical
engineering systems design.
Decay estimates of solutions of dynamical systems provide
qualitative descriptions for the asymptotic behavior of the
solutions of systems and have been mathematically studied by
many researchers
The Lyapunov exponent gives a measure of the mean
decay/divergence rates of the flows of nonlinear systems.
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Background and Objectives
The finite-time Lyapunov exponent (Goldhirsch et al, 1987)
It dose not require the limit of infinite time interval.
it requires the knowledge of system parameters.
A computational method to estimate the Lyapunov exponents
(Zeng et al.1991)
That use limited experimental data.
It is obtained from the calculation of a difference equation derived by
discrete-time series data
The instantaneous Lyapunov exponent (Shin et al.1998)
It is defined the a function of time in order to calculate it for each time,
which includes a differential operator instead of the division of time.
The Lyapunov exponent needs an infinite time interval of flows and
the Jacobian matrix of system dynamics
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Background and Objectives
We propose an instantaneous decay rate that is a kind of
generalized Lyapunov exponent and call the Instantaneous
Lyapunov Exponent (ILE) with respect to a decay function.
1. We propose two types ILE’s
using the continuous-time series data of the nonlinear flow.
using the linearized flow.
2. We propose two synchronization measures of continuous-time
signals using the ILE.
3. The proposed ILE’s are applied to the synchronization
detection of the endogenous circadian photosynthesis
oscillations of plants performing CAM and its adaptive
estimator.
compare two
ILE’s
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Decay Rate and Synchronization Measures
◆Instantaneous Lyapunov Exponent
0lim,0
0,
,||||
exp||||
log1
)(
,||||
exp||||
log1
)(
tt
xxxwhere
t
tx
t
tx
tt
t
tx
t
tx
tt
t
T
l
n
Based on
nonlinear flow
Based on
linearized flow
Comparison Function
To avoid unboundedness when 0tx
If λ(t) < 0, then the decay of x(t) is faster than the comparison function.
If a signal converges to zero as t tends to zero, then the ILE’s approach zero
from the negative side.
tx 0t
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Decay Rate and Synchronization Measures
If the linearized flow cannot be obtained measurement data directly
The ILE is sensitive to dynamical noise
►Adaptive differential filter (Nomura et.al,2008)
xxtxx
xxtxxkx
ˆˆ,ˆˆ
ˆsgnˆˆˆˆ
is given by tx
dxxtxt
ˆˆˆ0
Introduce the moving average of ILE
lnciTtN
t c
N
tc ,
1 1
0
T:sampling time
N:Number of data
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Decay Rate and Synchronization Measures
Another type Instantaneous Lyapunov Exponent(Shin et.al)
It is defined as the derivative of the logarithm of the divergence rate
We call Shin’s instantaneous Lyapunov exponent the differential type ILE.
The relation between proposed ILE and differential type ILE
The ILE defined in this paper becomes the time average of
flow theof nature deforminglocally the todue
ellipsoid theof axis principalth -i oflength The :
0log
tP
P
tP
dt
dt
i
i
ii
t
ii dllt
t0
1
ti
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Decay Rate and Synchronization Measures
Synchronization measures
iTtN
t
tx
tx
tt
n
N
in
n
1
0
2
1
1
log1
||1||
||||log
||1||
||||log
1)(
2
1
2
1
2 tx
tx
tx
tx
ttn
Moving average of
assigning the comparison
function as another signal.
Two signals synchronize strongly when
χ: the tolerance of the
synchronization error
0tn
To detect the synchronization as a negative number of
measures, we introduce another synchronization measures
tn
If 0tn txtxtx 212 11
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Numerical Example
texxx 2
When ,
When , is the finite escape time 120 x
120 x
20
0log
x
xt
0lim
txt
t
ttx
t
1,10524.1,1
10,1.0,2
0
0
System with finite escape time
Initial condition
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Numerical Example
The ILE’s when 10 tx
Dynamical response when 10524.10 tx The ILE’s when 10524.10 tx
Dynamical response when 10 tx
Blue line :
Green line: l
n
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Numerical Example
The ILE’s when 10 txDynamical response when 10 tx
Dynamical response with noise N(0,0.01) The ILE’s
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x y
z
tL
tT
3u
2u
1u
w
Numerical Example
tCext
cytoplasm
vacuole :internal CO2 concentration
:malate concentration in the cytoplasm
:malate concentration in the vacuole
:a variable that describes the ordering of the
lipid molecules in the tonoplast membrane
w
xy
z
yTzgz
uy
uux
uuw
,
1
21
32
The minimal CAM model (Blasius et.al)
Biological System (CAM plant)
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x y
z
tL
tT
3u
2u
1u
w
Numerical Example
tCext
cytoplasm
vacuole
1
1
3
2
1
exp
ww
w
LtL
Lc
wtLw
wtCcu
xx
wu
z
ycxu
k
KR
extj
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Numerical Example
Dynamical response
2238.0T2246.0T
Limit cycle Asymptotically stable
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Numerical Example
2246.0when
signalanothor and between
T
wn
2246.0when
signalanothor and between
T
yn
Based on x (fast system) Based on y (slow system)
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Conclusion
We proposed two ILE’s that are the measures to estimate the decay
rates of flows of nonlinear systems by assigning a comparison
function and can apply a stable system whose decay rate is slower
than an exponential function.
Future work
The physical meaning of the ILE.
Control theory using the ILE.
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Biological System (CAM plant)
Numerical Example
CAM (Crassulacean Acid Metabolism )
CAM is a special mode of photosynthesis providing a mechanism for
plants to concentrate CO2 and economize water use.
Blasius et.al. used throughout continuous time differential equations.
CAM plant
daytime
vacuole
chloroplast
CO2
night
malate malate
pore
To other organ starch
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The CAM model consists of
Input variables :
State variables :
Small time constants :
Constant parameters :
Nonlinear function :
Numerical Example
Problem
Estimate the tonoplast order
extCLT ,,
zyxw ,,, ,
,,,,, 1wLccc kRj
Tzg ,
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Assumption
w,x,y, are measurable
g(z,T) is known
Numerical Example
yTzgz
uy
uux
uuw
,
1
21
32
yTzgz
z
ycxy
,
yxw ,,
Tzg ,
The model can be reduced to the second order dynamical
system
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Numerical Example
yykyTzgz
z
ycxy
ˆ,ˆ1
ˆ
ˆ
ˆˆ
observer-type dynamic estimator
Parameter and Initial condition
2.00,56.00,62.00,4.00
1.0,5.0,1.0,5.1,35.0,001.0
1,1,5.5,2246.0,2238.0,1,1
1
zyxw
RLw
cccTLC
K
RJext
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Numerical Example
yn ofyl of
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Numerical Example
2246.0when
ˆ valueestimate and of response
T
zz
2246.0 when ˆ and between Tzzn
2246.0 when ˆ and between Tzzn
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1.0
2
0
2
t
exxx t
10517.12
8097.12
2
1
0
0
2
0
0
t
tt
e
eexxx
Numerical Example
System with finite escape time
Initial condition
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Compare to Lyapunov spectrum
The ILE’s when 10 txLyapunov spectrum
(using Sunday ChaosTimes
AIHARA Electrical Engineering Co.Ltd )
System with finite escape time at 10 tx
Blue line :
Green line: l
n