An application of optimal control on neuronal dynamics using koopman - - PowerPoint PPT Presentation

An application of optimal control on neuronal dynamics using koopman

• perator

Putian He

Contact: putianhe@gmail.com

Control system – last time

Fitzhugh-Nagumo (excitable cell)

ሶ 𝑤 = −𝑥 − 𝑤 𝑤 − 1 𝑤 − 𝑏 + 𝐽 ሶ 𝑥 = 𝜗(𝑤 − 𝛿𝑥)

[1] “Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting”

Limit cycle attractor Fixed points attractor “In general, finding isochrons is a daunting mathematical task” - Eugene M. Izhikevich [1]

Dynamical system

ሶ 𝒗 𝑢 = 𝑂(𝒗(𝑢)) ሶ 𝒗 𝑢 = ℒ𝒗 𝑢

• 1. Linear system
• 2. Non-linear system

ℒ𝜔𝑙 = λ 𝑙𝜔𝑙

ℒ𝑣 𝑢 = ℒ ෍

𝑙=0 ∞

𝑐𝑙𝜔𝑙 = ෍

𝑙=0 ∞

λ 𝑙𝑐𝑙𝜔𝑙

ሶ 𝒗 𝑢 = ℒ𝒗 𝑢 + 𝑂(𝒗(𝑢))

ሶ 𝒉(𝒗) = 𝒧𝒉(𝒗) 𝒧𝜒𝑙 = λ 𝑙𝜒𝑙

𝒧𝒉 𝒗 = 𝒧 𝑕1(𝒗) 𝑕2

⋮(𝒗)

𝑕𝑂(𝒗) = 𝒧 ෍

𝑙=0 ∞

𝒘𝑙𝜒𝑙(𝒗) = ෍

𝑙=0 ∞

λ 𝑙𝒘𝑙𝜒𝑙(𝒗) Koopman theory

𝒧: 𝑙𝑝𝑝𝑞𝑛𝑏𝑜 𝑝𝑞𝑓𝑠𝑏𝑢𝑝𝑠 (𝑚𝑗𝑜𝑓𝑏𝑠) 𝒗 𝐶. 𝐷𝑡 ℒ 𝑂? 𝐶. 𝐷𝑡 𝒗 ?

(Lifting) State space Hilbert space of

• bservable functions g

Linear Jacobian Nonlinearity Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics - A. Mauroy,

• I. Mezic, and J. Moehlis

Koopman Eigenfunction with slowest rate – dominant pole (new coordinate) Laplace average was used to calculate j-th koopman eigenfunction Real 𝜇 : Observable: 𝑔 𝑤, 𝑥 = 𝑤 − 𝑤∗ + (𝑥 − 𝑥∗ )

Absolute value of Koopman eigenfunction around a fixed point

Code from Prof. Alexandre Mauroy and also available from https://sites.google.com/site/alexmauroy/

r with Complex eigenvalue r with real eigenvalue ሶ 𝑤 = −𝑥 − 𝑤 𝑤 − 1 𝑤 − 𝑏 + 𝐽 ሶ 𝑥 = 𝜗(𝑤 − 𝛿𝑥) ሶ 𝑠 = 𝜏𝑠 Fitzhugh-Nagumo Global linearization around basin of attractions of stable fixed point

Isostable response curve – a function that transforming external control input to subspaces spanned by eigenfunctions observerbles of koopman operator

𝑒𝒀 𝑒𝑢 = 𝐺(𝒀) + 𝜗𝐻(𝒀, 𝑢) 𝑒𝑠 𝑒𝑢 = 𝜖𝑠 𝜖𝒀 ∙ 𝑒𝒀 𝑒𝑢 = 𝜖𝑠 𝜖𝒀 ∙ 𝐺 𝒀 + 𝜗𝐻 𝒀, 𝑢 = 𝜏𝑠 + 𝜗 𝜖𝑠 𝜖𝒀 ∙ 𝐻 𝒀, 𝑢 𝑒𝑠 𝑒𝑢 = 𝜏𝑠 + 𝜗 𝜖𝑠 𝜖𝑣 𝑣 𝑢 ሶ 𝑠 = 𝜏𝑠 + 𝜗𝑎(𝑣 𝑢 ) 𝑠

𝑣

𝑠 𝑎: 𝜖𝑠

𝜖𝑣 evaluated at the stable manifold of the fixed points attractors

Linear quadratic regulator

ሶ 𝑠 = 𝜏𝑠 + መ 𝑎 𝑣 𝑢 Linear መ 𝑎 𝑗𝑡 𝑏 𝑚𝑗𝑜𝑓𝑏𝑠𝑗𝑨𝑏𝑢𝑗𝑝𝑜 𝑝𝑔 𝑎 ሶ 𝑠 = −0.1933𝑠 + 0.001933 𝑣 𝑢 𝑠

𝑣

𝑣 = −𝐿𝑠 Regulator ሶ 𝑠 = (𝜏 − መ 𝑎𝐿) 𝑠 ሶ 𝑠 = 𝜏𝑠 − መ 𝑎𝐿 𝑠 ሶ 𝑠 = 𝜏𝑜𝑓𝑥 𝑠 𝜏𝑜𝑓𝑥 = -0.6411 𝜏𝑝𝑚𝑒 = −0.1933 Faster actuation and silencing of neurons? 𝐾 = න

−∞ ∞

(𝒔𝑈𝑅 𝒔 + 𝒗𝑈𝑆𝒗 ) dt Quadratic cost 𝐾 = න

−∞ ∞

(𝑠𝑅 𝑠 + 𝑣𝑆𝑣 ) dt 𝐿 = 231.6625 𝑅 = 100, 𝑆 = 0.001

State estimation using unscented kalman filter for excitable cell

UKF Code available from paper “Nonlinear dynamical system identification from unscented and indirect measurements - Henning U. Voss etc”

Neuronal system

Control system on excitable neurons

Offline Linearization 𝑠 𝑠 Voltage measurement State estimation ሶ 𝑠 = 𝜏𝑠 + 𝜗𝑎(𝑣 𝑢 ) 𝜏:real 𝜏:complex 𝑠

𝑣 = 𝜖𝑠

𝜖𝑣 𝑎 Neuronal plant 𝑣 = −𝐿𝑠 𝐾 = න

−∞ ∞

(𝑠𝑅 𝑠 + 𝑣𝑆𝑣 ) dt Linear quadratic regulator 𝑣 𝑜𝑓𝑥 𝑑𝑝𝑝𝑠𝑒𝑗𝑜𝑏𝑢𝑓𝑡 (Build up more functions)

Summary

Future works

• Fine tuning parameters in models
• Calculating koopman eigenfunctions of unstable fixed point and apply

control for stabilizing

• Data-driven eigenfunctions discovery for high dimensional neural
• networks. Dynamic mode decomposition. Koopman neural networks?

Backup slides

A digression - what do fluid dynamicists say about nonlinear optimal control?

Other applications of Koopman operator

Van der Pol’s triode – relaxation oscillator

https://en.wikipedia.org/wiki/Van_der_Pol_oscillator

Bistable switch

https://mcb.berkeley.edu/courses/mcb137/exercises/

Koopman operator

http://homepages.laas.fr/henrion/ecc15/mezic-workshop-ecc15.pdf By: Igor Mezić http://homepages.laas.fr/henrion/ecc15/mauroy1-workshop-ecc15.pdf by Alexandre Mauroy Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control, Steven L. Brunton , Bingni W. Brunton, Joshua L. Proctor, J. Nathan Kutz

My works outlines

• 1. State estimation previously were done on

Hodgkin-Huxley dynamics by [3] using kalman filter framework, however their control methods were not mathematically given. I am trying to fill that gap with optimal control algorithms on nonlinear neuronal dynamics.

• 2. Apply global linearization of Fitzhugh-Nagumo

using Koopman operator which was developed theoretically by [1] A. Maurov, I. Mezic, J Moehlis,

• r to an alternative data-driven koopman

eigenfunction discovery methods by fluid dynamicists Kaiser, E; Kutz, N; Brunton, S [2]

[4] http://www.isn.ucsd.edu/classes/beng260/2017/abstracts/2017_Group10.pdf - Putian He [1] Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics -Mauroy, A.; Mezić, I.; Moehlis, J. [3] Tracking and control of neuronal Hodgkin-Huxley dynamics - Ghanim Ullah and Steven J. Schiff [2] Data-driven discovery of Koopman eigenfunctions for control - Kaiser, Eurika; Kutz, J. Nathan; Brunton, Steven L.

[4]