synchronization detection of biological cam plants …...w,x,y, are measurable g(z,t) is known...

Synchronization Detection of

Biological CAM Plants Using

Instantaneous Lyapunov Exponent

Yusuke Totoki, Akira Goto,

Haruo Suemitsu, Takami Matsuo

Oita univ, Oita, Japan

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

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.


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


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

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


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


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


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

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


Initial condition

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

Numerical Example

The ILE’s when 10 txDynamical response when 10 tx

Dynamical response with noise N(0,0.01) The ILE’s

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)

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

Numerical Example

Dynamical response

2238.0T2246.0T

Limit cycle Asymptotically stable

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)

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.

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

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 ,

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

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

Numerical Example

yn ofyl of

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

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


Initial condition

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

synchronization detection of biological cam plants …...w,x,y, are measurable g(z,t) is known...

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