Modelling dynamics of electrical responses in plantsSanmitra Ghosh

Supervisor: Dr Srinandan Dasmahapatra & Dr Koushik Maharatna

Electronics and Software Systems,School of Electronics & Computer Science

University of Southampton

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OutlineIntroductionBlack Box models (System Identification)Modelling plant responses as ODEsCalibration of Models (Parameter Estimation

in ODE using ABC-SMC)ABC-SMC using Gaussian processesFuture workReferences

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IntroductionExperiments

Models ???

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IntroductionTypical electrical responses

Light

Ozone (sprayed for 2 minutes)

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Black Box models

1 11

1 1

( ) ( )( ) ( ) ( ) ( )( ) ( )B q C qA q y t u t e tF q D q

Generalized least-square estimator {A,B,F,C,D} are rational polynomials

1

1

( )( )C qD q

1

1

( )( )B qF q

1

1( )A q

( )e t

( )y t( )u t

Linear estimator

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Black Box models

Linear BlockInput Nonlinearity

Output Nonlinearity

fBF h

( )u t ( )w t ( )x t ( )y t

Nonlinear Hammerstein-Wiener model structure

1

1

( )( ) ( ( ( )))( )pB qy t h f u tF q

2

1

( ( ) ( ))N

N p mt

V y t y t

System output

Cost function

This cost function is minimized using optimization

Black Box models

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Modelling responses as ODEs

Proposed model:

(t) models a latent stimulus (v) is a chosen non-linear function of voltage V(t)

𝑑𝐼𝑑𝑡=−𝜇 𝐼

𝑑𝑉𝑑𝑡 =𝐼+ 𝑓 (𝑣)

2 4 6 8 10

4.05

4.10

4.15

4.20

time

Voltage

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Modelling responses as ODEs

(v) is chosen as Micheles-Menten (sigmoidal) and Fitzhugh-Nagumo (cubic) type non-linear function of v(t) (voltage)

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ABC

𝑝 (𝜃∨𝑦 )≈1( ∆( 𝑦 , 𝑥 ) ≤ 𝜀¿ 𝑓 ( x |𝜃 ) π ( θ )

Approximate Bayesian Computation

Prior

Likelihoodposterior

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ABC

ABC Rejection Sampler (Pickard, 1999)

1. Given , π(θ), (x|) 2. Sample a parameter ∗ from the prior

distribution π().

3. Simulate a dataset x from model (x| ∗) with parameter θ .∗

4. if ∆(, ) ≤ then

5. Accept ∗ otherwise reject.

Note: to generate data x from model (x| ∗) we have to solve the ODE

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ABC

ABC-Sequential Monte Carlo (Toni et al, 1999) …

Limitation: extremely slow due to large number of explicit ODE solving for generating simulated data

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ABC

Data

Gaussian Process

== + noise

𝑑 X̂ ( t )𝑑𝑡  

≤

The Gaussian process trick

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ABC

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ABCPredator-Pray a b

generated 2.10

2.10

estimated (ABC-SMC) 2.10

2.07

estimated (ABC-SMC-GPDist)

2.10

2.06

Fitzhugh-Nagumo a b c

generated 0.20 0.20 3.00

estimated (ABC-SMC) 0.19 0.20 2.97

estimated (ABC-SMC-GPDist)

0.21 0.22 2.62

Mackay-Glass β n γ τ

generated 2.00 9.65 1.00 2.00

estimated (ABC-SMC) 2.07 9.42 1.03 2.01

estimated (ABC-SMC-GPDist)

2.04 9.16 1.00 2.04

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Future workModel needs to be extended to capture the

variability seen among different electrical responses.

More models are required to represent other stimuli.

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References1. J K Pritchard, M T Seielstad, a Perez-Lezaun, and M W Feldman. Population growth of human Y chromosomes: a study of Y chromosome microsatellites. Mol. Biol. Evol., 16(12):1791–8, December 1999.

2. T. Toni, D. Welch, N. Strelkowa, a. Ipsen, and M. P.H Stumpf. Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. J. R. Soc. Interface, 6(31):187–202, February 2009.