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**Modelling dynamics of electrical responses in plants**

Sanmitra Ghosh Supervisor: Dr Srinandan Dasmahapatra & Dr Koushik Maharatna Electronics and Software Systems, School of Electronics & Computer Science University of Southampton

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**Outline Introduction Black Box models (System Identification)**

Modelling plant responses as ODEs Calibration of Models (Parameter Estimation in ODE using ABC-SMC) ABC-SMC using Gaussian processes Future work References

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Introduction Experiments Models ???

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**Introduction Typical electrical responses Light**

Ozone (sprayed for 2 minutes)

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**Black Box models Linear estimator Generalized least-square estimator**

{A,B,F,C,D} are rational polynomials Linear estimator

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**Black Box models Nonlinear Hammerstein-Wiener model structure**

System output Cost function This cost function is minimized using optimization

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Black Box models

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**Modelling responses as ODEs**

Proposed model: 𝑑𝐼 𝑑𝑡 =−𝜇𝐼 𝑑𝑉 𝑑𝑡 =𝐼+𝑓(𝑣) 𝑓(v) is a chosen non-linear function of voltage V(t) 𝐼(t) models a latent stimulus Voltage time

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**Modelling responses as ODEs**

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

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**ABC Approximate Bayesian Computation 𝑝(𝜃|𝑦)≈1(∆(𝑦, 𝑥) ≤ 𝜀)𝑓(x|𝜃)π(θ)**

posterior Likelihood Prior

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**ABC ABC Rejection Sampler (Pickard, 1999) Given 𝑦, π(θ), 𝑓(x|𝜃)**

Sample a parameter 𝜃∗ from the prior distribution π(𝜃). Simulate a dataset x from model 𝑓(x| 𝜃∗) with parameter θ∗. if ∆(𝑦, 𝑥) ≤ 𝜀 then 5. Accept 𝜃∗ otherwise reject. Note: to generate data x from model 𝑓(x| 𝜃∗) we have to solve the ODE

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**ABC ABC-Sequential Monte Carlo (Toni et al, 1999) 𝜀 1 ≥… ≥ 𝜀 𝑇**

𝜀 1 ≥… ≥ 𝜀 𝑇 Limitation: extremely slow due to large number of explicit ODE solving for generating simulated data

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**ABC The Gaussian process trick 𝑑𝑋(𝑡) 𝑑𝑡 =𝑓(𝑋 𝑡 ,𝜃) Data**

𝑑𝑋(𝑡) 𝑑𝑡 =𝑓(𝑋 𝑡 ,𝜃) Data 𝑌= X(t) + noise Gaussian Process 𝑑 X(t) 𝑑𝑡 𝑑 X(t) 𝑑𝑡 −𝑓(𝑋 𝑡 ,𝜃) 2 ≤ 𝜀

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ABC

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**ABC Predator-Pray a b generated 2.10 estimated (ABC-SMC) 2.07**

estimated (ABC-SMC-GPDist) 2.06 Fitzhugh-Nagumo a b c generated 0.20 3.00 estimated (ABC-SMC) 0.19 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 estimated (ABC-SMC) 2.07 9.42 1.03 2.01 estimated (ABC-SMC-GPDist) 2.04 9.16

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Future work Model 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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References 1. 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.

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