Algorithms for estimating the states and parameters of neural mass models for epilepsy

Transcription

1 Algorithms for estimating the states and parameters of neural mass models for epilepsy Michelle S. Chong Department of Automatic Control, Lund University Joint work with Romain Postoyan (CNRS, Nancy, France), Dragan Nesic (University of Melbourne-UoM), Levin Kuhlmann (UoM)

2 Outline } Motivation } Neural mass models } Background and overview of our recent results } Supervisory observer } The big picture 1/17

3 Motivation EEG measurement Estimation algorithm Seizure / No Seizure? 2/17

4 Motivation EEG measurement Model-based estimation algorithm Seizure / No Seizure? 2/17

5 Motivation EEG measurement Model-based estimation algorithm Seizure / No Seizure? Which mathematical model to use? 2/17

6 Motivation EEG measurement Model-based estimation algorithm Seizure / No Seizure? Which mathematical model to use? Neural mass models 2/17

7 Neural mass models

8 } Average behaviour of neuronal populations: excitatory and inhibitory interneurons } Reproduce EEG patterns related to seizures } A class of neural mass models convenient for seizure prediction } A network of neural mass models 3/17 Credits: istockphoto/guido Vrola and

9 Input Output/ Measurement: EEG 4/17 Cortical column Credits: istockphoto/guido Vrola and

10 Neural mass models } See Deco et. al. (2008) for an overview. Model by Stam et. al. (1999) Model by Jansen and Rit (1995) Model by Wendling et. al. (2005) 5/17

11 u Neural mass model (System) ẋ = Ax + G( ) (Hx)+ (u, y, ) y = Cx y 6/17

12 Background and overview

13 Parameter and state estimation - Background Estimation Algorithm ˆxˆ } Rich literature: adaptive control and system identification } Stochastic and deterministic methods 7/17

14 Parameter and state estimation - Background Estimation Algorithm ˆxˆ } Rich literature: adaptive control and system identification } Stochastic and deterministic methods } Specific to classes of systems 7/17

15 Parameter and state estimation - Background Estimation Algorithm ˆxˆ } Rich literature: adaptive control and system identification } Stochastic and deterministic methods } Specific to classes of systems } Our objective : a multi-model framework for estimation inspired by works in supervisory control (Vu & Liberzon; Hespanha, Morse, et al) 7/17

16 Overview of our estimation algorithms } Supervisory observer - Static sampling policy - Dynamic sampling policy - DIviding RECTangles (DIRECT) sampling policy? is unknown & constant Estimation Algorithm ˆxˆ } An adaptive observer 8/17

17 Overview of our estimation algorithms } Supervisory observer - Static sampling policy - Dynamic sampling policy - DIviding RECTangles (DIRECT) sampling policy? is unknown & constant Estimation Algorithm ˆxˆ } An adaptive observer See book chapter in Recent Advances in Predicting and Preventing Epileptic Seizures, pp D. Freestone, L. Kuhlmann, M. Chong, D. Nesic, D.B. Grayden, P. Aram, R. Postoyan, M.J. Cook 8/17

18 Overview of our estimation algorithms } Supervisory observer - Static sampling policy - Dynamic sampling policy - DIviding RECTangles (DIRECT) sampling policy? is unknown & constant Estimation Algorithm ˆxˆ } An adaptive observer See IEEE Transactions of Automatic Control (2015) M. Chong, D. Nesic, R. Postoyan, L. Kuhlmann 8/17

19 Overview of our estimation algorithms } Supervisory observer - Static sampling policy - Dynamic sampling policy - DIviding RECTangles (DIRECT) sampling policy? is unknown & constant Estimation Algorithm ˆxˆ } An adaptive observer To appear IEEE Conference on Decision and Control 2017 M. Chong, R. Postoyan, D. Nesic Available on arxiv. Journal version to come. 8/17

20 Supervisory observer

21 Let Multiple-model architecture Θ 9/17

22 Let Multiple-model architecture Θ 9/17

23 Let Multiple-model architecture Θ State observer 1 State observer 2 State observer N 9/17

24 Multiple-model architecture Let Θ State observer 1 State observer 2 State observer N ˆx 1 ˆx 2 ˆx N Some scheme to provide ˆx, ˆ ˆx ˆ 9/17

25 Supervisory observer u System y Multiobserver θ 1 : State observer θ 2 : State observer θ N : State observer ˆx 1 ˆx 2 ˆx N Monitoring signals Supervisor Selection criterion, σ(t) Estimator ˆx(t) := ˆx (t) 10/17

26 Supervisory observer u System y Assumptions : 1. System : Uniformly bounded solutions. 2. Multiobserver: Observers are robust w.r.t parameter mismatch. Multiobserver θ 1 : State observer θ 2 : State observer θ N : State observer ˆx 1 ˆx 2 ˆx N Monitoring signals Supervisor Selection criterion, σ(t) Estimator ˆx(t) := ˆx (t) 3. Supervisor : - Monitoring signals µ i (t) := Z t 0 e (t s) y(s) ŷ i (s) 2 ds satisfy a PE condition, i.e. 4. The parameter set Θ is sampled appropriately. 10/17

27 Theorem (Convergence guaranteed!) See Chong et. al. (2015) IEEE Transactions of Automatic Control Let Θ Suppose all assumptions hold. For all Δ x > 0 and any margins υ!x,υ!p > 0, there exist T > 0 and sufficiently large N * > 0 such that for any N N * p (t) (t) apple p, 8t T, lim sup t!1 x (t) (t) apple x for any initial conditions x(0), ˆx i (0) B Δx. 10/17

28 Theorem (Convergence guaranteed!) See Chong et. al. (2015) IEEE Transactions of Automatic Control Let Θ Suppose all assumptions hold. For all Δ x > 0 and any margins υ!x,υ!p > 0, there exist T > 0 and sufficiently large N * > 0 such that for any N N * needs large N! p (t) (t) apple p, 8t T, lim sup t!1 x (t) (t) apple x for any initial conditions x(0), ˆx i (0) B Δx. 10/17

29 Supervisory observer u System Multiobserver θ 1 : State observer ˆx 1 θ 2 : State observer ˆx 2 θ N : State observer ˆx N y Supervisor Monitoring signals Selection criterion, σ(t) Estimator ˆx(t) := ˆx (t) Recall the monitoring signals µ i (t) := Z t 0 e (t s) y(s) ŷ i (s) 2 ds and the selection criterion (t) := arg min µ i (t) i2{1,...,n} 12/17

30 Supervisory observer u System Multiobserver θ 1 : State observer ˆx 1 θ 2 : State observer ˆx 2 θ N : State observer ˆx N y Supervisor Monitoring signals Selection criterion, σ(t) Estimator ˆx(t) := ˆx (t) Recall the monitoring signals µ i (t) := µ(,t):= Z t 0 Z t 0 e (t s) y(s) ŷ i (s) 2 ds or e (t s) y(,s) ŷ(,s) 2 ds and the selection criterion (t) := arg min µ i (t) i2{1,...,n} 12/17

31 DIviding RECTangles (DIRECT) } compact is a Lipschitz function i.e. there exists L > 0 such that µ(a) µ(b) L a b 13/17

32 DIviding RECTangles (DIRECT) } compact is a Lipschitz function i.e. there exists L > 0 such that µ(a) µ(b) L a b 13/17

33 DIviding RECTangles (DIRECT) } compact is a Lipschitz function i.e. there exists L > 0 such that µ(a) µ(b) L a b 13/17

34 DIviding RECTangles (DIRECT) } compact is a Lipschitz function 13/17

35 DIviding RECTangles (DIRECT) } compact is a Lipschitz function 13/17

36 DIviding RECTangles (DIRECT) } compact is a Lipschitz function 13/17

37 DIviding RECTangles (DIRECT) } compact is a Lipschitz function 13/17

38 DIviding RECTangles (DIRECT) } compact is a Lipschitz function 13/17

39 DIviding RECTangles (DIRECT) } compact is a Lipschitz function 13/17

40 DIviding RECTangles (DIRECT) } compact is a Lipschitz function Now, identify the potentially optimal boxes: There exists L>0 such that 1. µ(p i ) Ld i apple µ(p j ) Ld j 2. µ(p i ) Ld i apple ˆµ k 1 ˆµ k 1, > 0. 13/17

41 DIviding RECTangles (DIRECT) } compact is a Lipschitz function Now, identify the potentially optimal boxes: There exists L>0 such that 1. µ(p i ) Ld i apple µ(p j ) Ld j 2. µ(p i ) Ld i apple ˆµ k 1 ˆµ k 1, > 0. 13/17

42 DIviding RECTangles (DIRECT) } compact is a Lipschitz function Further divide each of the potentially optimal boxes according to the dividing procedure before. 13/17

43 DIviding RECTangles (DIRECT) } compact is a Lipschitz function Further divide each of the potentially optimal boxes according to the dividing procedure before. 13/17

44 DIviding RECTangles (DIRECT) } compact is a Lipschitz function Further divide each of the potentially optimal boxes according to the dividing procedure before. 13/17

45 DIviding RECTangles (DIRECT) } compact is a Lipschitz function 13/17

46 DIviding RECTangles (DIRECT) } compact is a Lipschitz function 13/17

47 DIviding RECTangles (DIRECT) } compact is a Lipschitz function 13/17

48 DIviding RECTangles (DIRECT) } compact is a Lipschitz function 13/17

49 DIviding RECTangles (DIRECT) } compact is a Lipschitz function 13/17

50 DIviding RECTangles (DIRECT) } compact is a Lipschitz function 13/17

51 Combining DIRECT with the supervisory observer u System y Multiobserver θ 1 : State observer θ 2 : State observer θ N : State observer ˆx 1 ˆx 2 ˆx N Monitoring signals Supervisor Selection criterion, σ(t) Estimator ˆx(t) := ˆx (t) 14/17

52 Combining DIRECT with the supervisory observer u System y At k=0: Multiobserver θ 1 : State observer θ 2 : θ N : Supervisor ˆx 1 ˆx(t) := Monitoring signals Selection criterion, σ(t) Estimator ˆx (t) 14/17

53 Combining DIRECT with the supervisory observer At k=1: u System y Multiobserver θ 1 : State observer θ 2 : State observer θ N : State observer ˆx 1 ˆx 2 Monitoring signals Supervisor Selection criterion, σ(t) Estimator ˆx(t) := ˆx (t) 14/17

54 Combining DIRECT with the supervisory observer At k=2: u System y Multiobserver θ 1 : State observer θ 2 : State observer θ N : State observer ˆx 1 ˆx 2 Monitoring signals Supervisor Selection criterion, σ(t) Estimator ˆx(t) := ˆx (t) 14/17

55 Theorem: Convergence is guaranteed! K :α(0) = 0, strictly increasing α(r) as r. } Given any Δ x,δ u > 0, accuracy margins d,η > 0, there exist a class K function γ!x, a large enough T > 0 s.t. for any sampling interval T d T, there exist a class K function υ!p and a constant T * > 0 s.t. p (t) (t) apple p (d )+, 8t T lim sup x (t) (t) apple x (d )+ t!1 for all initial conditions x(0) B Δx and inputs u bounded by satisfying the PE condition Δ u 15/17

56 Simulation example: Jansen and Rit 95 neural mass model } States: mean membrane potential of each population } Parameters ( p 1, p 2 ): synaptic gain of excitatory and inhibitory populations 16/17

57 Simulation example: Jansen and Rit 95 neural mass model 28 t [0,10) s } States: mean membrane potential of each population p 2 } Parameters ( p 1, p 2 ): 23 synaptic gain of excitatory and inhibitory populations p /17

58 Simulation example: Jansen and Rit 95 neural mass model 28 t t [10,20) [0,10) s } States: mean membrane potential of each population p 2 } Parameters ( p 1, p 2 ): 23 synaptic gain of excitatory and inhibitory populations p 1 16/17

59 Simulation example: Jansen and Rit 95 neural mass model 28 t t [10,20) [20,30) [0,10) s } States: mean membrane potential of each population p 2 } Parameters ( p 1, p 2 ): 23 synaptic gain of excitatory and inhibitory populations p 1 16/17

60 Simulation example: Jansen and Rit 95 neural mass model 28 t t [10,20) [20,30) [30,40) [0,10) s } States: mean membrane potential of each population p 2 } Parameters ( p 1, p 2 ): 23 synaptic gain of excitatory and inhibitory populations p 1 16/17

61 Simulation example: Jansen and Rit 95 neural mass model 28 t t [40,50) [10,20) [20,30) [30,40) [0,10) s } States: mean membrane potential of each population p 2 } Parameters ( p 1, p 2 ): 23 synaptic gain of excitatory and inhibitory populations p 1 16/17

62 Simulation example: Jansen and Rit 95 neural mass model 28 t t [40,50) [50,60) [10,20) [20,30) [30,40) [0,10) s } States: mean membrane potential of each population p 2 } Parameters ( p 1, p 2 ): 23 synaptic gain of excitatory and inhibitory populations p 1 16/17

63 Simulation example: Jansen and Rit 95 neural mass model Parameter error State error } States: mean membrane potential of each population } Parameters ( p 1, p 2 ): synaptic gain of excitatory and inhibitory populations ˆp p Time, t (s) Normalised state error Time, t (s) 16/17

64 Summary } A framework for state and parameter estimation adapted for neural mass models. } Convergence of estimates is guaranteed. } Computationally tractable. } Future work includes application to EEG data } Big picture: 17/17