1. 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

2. 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

3. Introduction
Experiments
Models ???

4. Introduction Typical electrical responses Light
Ozone (sprayed for 2 minutes)

5. Black Box models Linear estimator Generalized least-square estimator
{A,B,F,C,D} are rational polynomials
Linear estimator

6. Black Box models Nonlinear Hammerstein-Wiener model structure
System output
Cost function
This cost function is minimized using optimization

7. Black Box models

8. 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

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

10. ABC Approximate Bayesian Computation 𝑝(𝜃|𝑦)≈1(∆(𝑦, 𝑥) ≤ 𝜀)𝑓(x|𝜃)π(θ)
posterior
Likelihood
Prior

11. 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

12. 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

13. 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

14. ABC

15. 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

16. Future work
Model needs to be extended to capture the variability seen among different electrical responses.
More models are required to represent other stimuli.

17. 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.