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

An application of optimal control on neuronal dynamics using koopman operator Putian He Contact: putianhe@gmail.com

Control system – last time

ሶ ሶ Fitzhugh-Nagumo (excitable cell) Limit cycle attractor 𝑤 = −𝑥 − 𝑤 𝑤 − 1 𝑤 − 𝑏 + 𝐽 𝑥 = 𝜗(𝑤 − 𝛿𝑥) “In general, finding isochrons is a daunting mathematical task” - Eugene M. Izhikevich [1] Fixed points attractor [1] “ Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting”

ሶ ሶ ሶ ሶ Dynamical system 𝒗 𝑢 = ℒ𝒗 𝑢 + 𝑂(𝒗(𝑢)) 𝒗 𝐶. 𝐷𝑡 1. Linear system ℒ 𝒗 𝑢 = ℒ𝒗 𝑢 ∞ ∞ ℒ𝜔 𝑙 = λ 𝑙 𝜔 𝑙 ℒ𝑣 𝑢 = ℒ ෍ 𝑐 𝑙 𝜔 𝑙 = ෍ λ 𝑙 𝑐 𝑙 𝜔 𝑙 𝑙=0 𝑙=0 𝑂? 𝒗 ? 𝐶. 𝐷𝑡 2. Non-linear system 𝒉(𝒗) = 𝒧𝒉(𝒗) 𝒗 𝑢 = 𝑂(𝒗(𝑢)) Koopman theory State space (Lifting) Hilbert space of observable functions g 𝒧: 𝑙𝑝𝑝𝑞𝑛𝑏𝑜 𝑝𝑞𝑓𝑠𝑏𝑢𝑝𝑠 (𝑚𝑗𝑜𝑓𝑏𝑠) 𝒧𝜒 𝑙 = λ 𝑙 𝜒 𝑙 𝑕 1 (𝒗) ∞ ∞ 𝑕 2 ⋮ (𝒗) 𝒧𝒉 𝒗 = 𝒧 = 𝒧 ෍ 𝒘 𝑙 𝜒 𝑙 (𝒗) = ෍ λ 𝑙 𝒘 𝑙 𝜒 𝑙 (𝒗) 𝑕 𝑂 (𝒗) 𝑙=0 𝑙=0

Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics - A. Mauroy, I. Mezic, and J. Moehlis Real 𝜇 : Observable: 𝑔 𝑤, 𝑥 = 𝑤 − 𝑤 ∗ + (𝑥 − 𝑥 ∗ ) Linear Jacobian Nonlinearity Koopman Eigenfunction with slowest rate – dominant pole (new coordinate) Laplace average was used to calculate j-th koopman eigenfunction

ሶ ሶ ሶ Absolute value of Koopman eigenfunction around a fixed point r with Complex eigenvalue Fitzhugh-Nagumo 𝑤 = −𝑥 − 𝑤 𝑤 − 1 𝑤 − 𝑏 + 𝐽 𝑥 = 𝜗(𝑤 − 𝛿𝑥) Global linearization around basin of attractions of stable fixed point r with real eigenvalue 𝑠 = 𝜏𝑠 Code from Prof. Alexandre Mauroy and also available from https://sites.google.com/site/alexmauroy/

ሶ 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 = 𝜏𝑠 + 𝜗 𝜖𝑠 𝜖𝒀 ∙ 𝐺 𝒀 + 𝜗𝐻 𝒀, 𝑢 𝜖𝒀 ∙ 𝐻 𝒀, 𝑢 𝑒𝑠 𝑒𝑢 = 𝜏𝑠 + 𝜗 𝜖𝑠 𝜖𝑣 𝑣 𝑢 𝑠 = 𝜏𝑠 + 𝜗𝑎(𝑣 𝑢 ) 𝑎: 𝜖𝑠 𝑠 𝜖𝑣 evaluated at the stable manifold of the fixed points attractors 𝑠 𝑣

ሶ ሶ ሶ ሶ ሶ Linear quadratic regulator Quadratic cost Regulator Linear ∞ 𝑠 = 𝜏𝑠 + መ 𝑎 𝑣 𝑢 (𝒔 𝑈 𝑅 𝒔 + 𝒗 𝑈 𝑆𝒗 ) dt 𝐾 = න 𝑣 = −𝐿𝑠 መ −∞ 𝑎 𝑗𝑡 𝑏 𝑚𝑗𝑜𝑓𝑏𝑠𝑗𝑨𝑏𝑢𝑗𝑝𝑜 𝑝𝑔 𝑎 𝑠 = 𝜏𝑠 − መ 𝑎𝐿 𝑠 ∞ 𝑠 = (𝜏 − መ 𝑎𝐿) 𝑠 𝐾 = න (𝑠𝑅 𝑠 + 𝑣𝑆𝑣 ) dt 𝑠 = 𝜏 𝑜𝑓𝑥 𝑠 −∞ 𝑠 𝑣 𝑅 = 100, 𝑆 = 0.001 𝐿 = 231.6625 𝜏 𝑜𝑓𝑥 = -0.6411 𝜏 𝑝𝑚𝑒 = −0.1933 𝑠 = −0.1933𝑠 + 0.001933 𝑣 𝑢 Faster actuation and silencing of neurons?

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 ”

ሶ Control system on excitable neurons Voltage measurement State estimation 𝑣 Neuronal plant Offline Linearization 𝑜𝑓𝑥 𝑑𝑝𝑝𝑠𝑒𝑗𝑜𝑏𝑢𝑓𝑡 𝜏 :real 𝜏 :complex 𝑠 𝑠 Linear quadratic regulator ∞ 𝐾 = න (𝑠𝑅 𝑠 + 𝑣𝑆𝑣 ) dt (Build up more functions) Neuronal system −∞ 𝑣 = −𝐿𝑠 𝑣 = 𝜖𝑠 𝑠 𝑠 = 𝜏𝑠 + 𝜗𝑎(𝑣 𝑢 ) 𝜖𝑣 𝑎

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