TIME-DELAYED FEEDBACK CONTROL OF COMPLEX NONLINEAR SYSTEMS

Eckehard Schöll

Institut für Theoretische Physikand

Sfb 555 “Complex Nonlinear Processes”Technische Universität Berlin

Germany

http://www.itp.tu-berlin.de/schoell

Net-Works 2008 Pamplona 10.6.2008

OutlineOutline

Introduction: Time-delayed feedback controlTime-delayed feedback control of nonlinear systems

control of deterministic statescontrol of noise-induced oscillations application: lasers, semiconductor nanostructures

Neural systems:Neural systems: control of coherence of neurons and control of coherence of neurons and synchronization of coupled neuronssynchronization of coupled neurons

delay-coupled neuronsdelay-coupled neurons delayed self-feedbackdelayed self-feedback

Control of excitation pulses in Control of excitation pulses in spatio-temporal systemsspatio-temporal systems:: migraine, stroke migraine, stroke non-local instantaneous feedbacknon-local instantaneous feedback time-delayed feedback time-delayed feedback

Why is delay interesting in dynamics?Why is delay interesting in dynamics?

Delay increases the dimension of a differential equation to infinity:

delay generates infinitely many eigenmodes

Delay has been studied in Delay has been studied in classical control theoryclassical control theory and and mechanical engineeringmechanical engineering for a long time for a long time

Simple equation produces very Simple equation produces very complexcomplex behavior behavior

Delay is ubiquitousDelay is ubiquitous

mechanical systems: inertia

electronic systems: electronic systems: capacitive effects capacitive effects ((=RC)=RC) latency time latency time due to processingdue to processing

biological systems: biological systems: cell cycle timecell cycle time biological clocksbiological clocks

neural networks: neural networks: delayed coupling, delayed feedbackdelayed coupling, delayed feedback

optical systems: optical systems: signal transmission timessignal transmission times travelling waves + reflectionstravelling waves + reflections

laser coupled to external cavity (Fabry-laser coupled to external cavity (Fabry-Perot)Perot)multisection lasermultisection lasersemiconductor optical amplifier (SOA)semiconductor optical amplifier (SOA)

Time delayed feedback control methodsTime delayed feedback control methods

Originally invented for controlling chaos (Pyragas 1992): stabilize unstable periodic orbits embedded in a chaotic attractor

More general: More general: stabilization of stabilization of unstable periodic or unstable periodic or stationary statesstationary states in nonlinear dynamic systems in nonlinear dynamic systems

Application to Application to spatio-temporal patterns:spatio-temporal patterns: Partial differential equationsPartial differential equations

Delay can Delay can induce or suppressinduce or suppress instabilities instabilities deterministic delay differential equationsdeterministic delay differential equationsstochastic delay differential equationsstochastic delay differential equations

PublishedOctober 2007

Scope has considerably widened

Time-delayed feedback control Time-delayed feedback control of deterministic systemsof deterministic systems

Time-delayed feedback (Pyragas 1992):Time-delayed feedback (Pyragas 1992):

Stabilisation of unstable periodic orbits Stabilisation of unstable periodic orbits or unstable fixed points or space-time patterns or unstable fixed points or space-time patterns

Time-delay autosynchronisation(TDAS)

Extended time-delay autosynchronisation(ETDAS) (Socolar et al 1994)

)()1((0

txtxRK

)}()({ txtxK

deterministic chaosdeterministic chaosdeterministic chaosdeterministic chaos

=T=T=T=T

Many other schemes

Time-delayed feedback control of deterministic systemsTime-delayed feedback control of deterministic systems

stability is measured byFloquet exponent : x ~ exp(t)or Floquet multiplier =exp(T)

b complex

(1 - )

Beyond Odd Number LimitationBeyond Odd Number Limitation

Example of all-optical time-delayed Example of all-optical time-delayed feedback control in semiconductor laserfeedback control in semiconductor laser

Optical feedback:Optical feedback:

latencynlatency

ni

nin

n

nib

ntt

tEetEeRKetE

,,

)()()(

00

10

||

Stabilisationof fixed point:Schikora, Hövel, Wünsche, Schöll, Henneberger, PRL 97, 213902 (2006)

Laser: excitable unit, may be coupled to others to form network motif

Stabilization of cw emission:Stabilization of cw emission:Domain of control of unstable fixed pointDomain of control of unstable fixed point

above Hopf bifurcation above Hopf bifurcation

||

Schikora, Hövel, Wünsche, Schöll, Henneberger , PRL 97, 213902 (2006)

Generic model:

phase sensitive coupling

Generic model:

phase sensitive coupling

=0.5T0 =0.9T0

Experimental realizationExperimental realization

||

Schikora, Hövel, Wünsche, Schöll, Henneberger, PRL 97, 213902 (2006)

Control of spatio-temporal patterns:Control of spatio-temporal patterns: semiconductor nanostructuresemiconductor nanostructure

Without control:Without control:

Examples: Chemical reaction-diffusion systemsChemical reaction-diffusion systemsElectrochemical systemsElectrochemical systemsSemiconductor nanostructuresSemiconductor nanostructuresHodgkin-Huxley neural modelsHodgkin-Huxley neural models

rJuUdt

tdu

x

aaD

xuaf

t

txa

0

1)(

)(),(),(

||

a(x,t): activator variableu(t): inhibitor variable f(a,u): bistable kinetic function D(a): transverse diffusion coefficient

Global coupling:Ratio of timescales:

L

dxuajL

J0

),(1

R

DBRTI

I totU 0

C

U

● Global coupling due to Kirchhoff equation:

jdxUURdt

dUC 0

1 I

Control parameters: = RC, U0

Chaotic breathing pattern

j

u

9.1

u min , u m

ax

= 9.1: above period doubling cascade

Spatially inhomogeneous chaotic oscillations

J. Unkelbach, A.Amann, W. Just, E. Schöll: PRE 68, 026204 (2003)J. Unkelbach, A.Amann, W. Just, E. Schöll: PRE 68, 026204 (2003)

Stabilisation of unstable period-1 orbit

u min , u m

ax

●Period doubling bifurcations generate a family of unstable periodic orbits (UPOs)

● Period-1 orbit:

Breathing oscillationsBreathing oscillations

Resonant tunneling diodeResonant tunneling diodea(x,t): electron concentrationa(x,t): electron concentration in quantum well in quantum well u(t): voltage across diodeu(t): voltage across diode

tracking

Time-delayed feedback control Time-delayed feedback control of noise-induced oscillations of noise-induced oscillations

Stabilisation of UPOStabilisation of UPO

noise-inducednoise-inducedoscillationsoscillations

noise-inducednoise-inducedoscillationsoscillations

??

no deterministic orbits!no deterministic orbits!

)}()({ txtxK )}()({ txtxK

deterministic chaosdeterministic chaosdeterministic chaosdeterministic chaos

=T=T=T=T

K. Pyragas, Phys. Lett. A 170, 421 (1992)K. Pyragas, Phys. Lett. A 170, 421 (1992) N. Janson, A. Balanov, E. Schöll, PRL 93 (2004)N. Janson, A. Balanov, E. Schöll, PRL 93 (2004)

Time-delayed feedback control of injection laser with Fabry-Perot resonator

Suppression of noise-induced relaxations oscillations in semiconductor lasers

||

Lang-Kobayashi model:Power spectral densityof optical intensity

Suppression of noisefor 0.5TRO

Flunkert and Schöll,PRE 76, 066202 (2007)

))()(()(1)(

),()(),(),(

0

tutuKtDrJuUdt

tdu

txDx

aaD

xuaf

t

txa

u

a

Feedback control of noise-induced space-time patterns in the DBRT nanostructure

G. Stegemann, A. Balanov, E. Schöll, PRE 73, 016203 (2006)G. Stegemann, A. Balanov, E. Schöll, PRE 73, 016203 (2006)

=4, K=0.4 Du = 0.1, Da = 10-4

Enhancement of temporal regularity:correlation time vs. noise amplitude

vs. feedback gain

=7: increase=7: increase=5: decrease=5: decrease

G. Stegemann, A. Balanov, E. Schöll, PRE 73, 016203 (2006)G. Stegemann, A. Balanov, E. Schöll, PRE 73, 016203 (2006)

Large effect for small noise intensity

Du = 0.1, Da = 10-4

Coherence resonanceCoherence resonance

0

)( dsstcor – – normalized normalized autocorrelation functionautocorrelation function

Correlation time:Correlation time:

Simplified FitzHugh-Nagumo (FHN) system: excitable neuron Simplified FitzHugh-Nagumo (FHN) system: excitable neuron

Excitable SystemExcitable SystemExcitable SystemExcitable System

a=1.1a=1.1=0.01=0.01

Gang, Ditzinger, Ning, Haken, PRL 71, 807 (1993)Gang, Ditzinger, Ning, Haken, PRL 71, 807 (1993)Pikovsky, Kurths, PRL 78, 775 (1997)Pikovsky, Kurths, PRL 78, 775 (1997)

Example of coherence resonance: neuronExample of coherence resonance: neuron

Simulation from Simulation from S.-G. LeeS.-G. Lee, A., A. Neiman Neiman, , S. KimS. Kim, ,

PREPRE 57, 3292 57, 3292 ( (19981998).).

Time series of the membrane potentialTime series of the membrane potential for for various noise intensityvarious noise intensity::

FitzHugh-Nagumo model with delay FitzHugh-Nagumo model with delay

)()(3

3

tDyyKaxy

yx

xx

)()(3

3

tDyyKaxy

yx

xx

12.4;1.1;01.0 a 12.4;1.1;01.0 a

Janson, Balanov, Schöll, PRL 93, 010601 (2004)

Excitabilitya=1: excitabilitythreshold

u activator (membrane voltage) v inhibitor (recovery variable) time-scale ratio

Coherence vs. Coherence vs. and K and K

D=0.09D=0.09D=0.09D=0.09

D=0.09; K=0.2D=0.09; K=0.2D=0.09; K=0.2D=0.09; K=0.2

Numerics: Balanov, Janson, Schöll, Physica D 199, 1 (2004)Analytics: Prager, Lerch, Schimansky-Geier, Schöll, J. Phys.A 40, 11045 (2007)

2 coupled FitzHugh-Nagumo systems:coupled neuron model as network motif

● 2 non-identical stochastic oscillators: diffusive coupling

frequencies tuned by D1 , D2

B. Hauschildt, N. Janson, A. Balanov, E. Schöll, PRE 74, 051906 (2006)

a= 1.05, 1=0.005, 2= 0.1, D2=0.09 : coherence resonance as function of D1

Stochastic synchronization

● Frequency synchronization : mean interspike intervals (ISI)

● Phase synchronization: 1:1 synchronization index

(Rosenblum et al 2001)

oX+

+ weakly synchronizedo moderately synchronizedx strongly synchronized

Local delayed feedback control: enhance or suppress synchronization

● Moderately synchronized system (o)

System 1

1:1 synchronization index

Delayed coupling, no self-feedback + noise

Dahlem,Hiller, Panchuk,Schöll, IJBC in print, 2008

inducesantiphaseoscillations

Sustained oscillations induced by delayed coupling

excitability parameter a=1.3

a=1.05

Regime of oscillations

excitability parameter a=1.3

Delayed coupling and delayed self-feedback

excitability parameter a=1.3,oscillatory regime,C=K=0.5

Average phase synchronization time:

Schöll, Hiller,Hövel, Dahlem,2008

Spreading depolarization wave(cortical spreading depression SD)

● migraine aura (visual halluzinations)● stroke

Examples:

Migraine aura: neurological precursor(spatio-temporal pattern on visual cortex)

Migraine aura: visual halluzinations

Migraine aura: visual halluzinations

Migraine aura: visual halluzinations

Migraine aura: visual halluzinations

Migraine aura: visual halluzinations

Migraine aura: visual halluzinations

Measured cortical spreading depression

Visual cortex

3 mm/ min

FitzHugh-Nagumo (FHN) system with FitzHugh-Nagumo (FHN) system with activator diffusionactivator diffusion

u activator (membrane voltage) v inhibitor (recovery variable)Du diffusion coefficient time-scale ratio of inhibitor and activator variables excitability parameter

Dahlem, Schneider, Schöll, Chaos (2008)

_

Transient excitation: tissue at risk (TAR)pulses die out after some distance

Dahlem, Schneider, Schöll, J. Theor. Biol. 251, 202 (2008)

different values of and

Boundary of propagation of traveling excitation pulses (SD)

excitable:traveling pulses

non-excitable: transient

Propagation verlocitypulse

FHN system with feedback

Non-local, time-delayed feedback:

Instantaneous long-range feedback:

Time-delayed local feedback:

(electrophysiological activity)

(neurovascular coupling)

Dahlem et alChaos (2008)

Non-local feedback: suppression of CSD

uu

vvuv

vu

Tissue at risk

Non-local feedback:shift of propagation boundary

K=+/-0.2

pulse width x

Time-delayed feedback: suppression of SD

uu vu

uv vv

Tissue at risk

Time-delayed feedback:shift of propagation boundary

uu vu

vv vu

K=+/-0.2

pulse width t

Conclusions

Delayed feedback control of excitable systemsControl of coherence and spectral properties

Stabilization of chaotic deterministic patterns

2 coupled neurons as network motif FitzHugh-Nagumo system: suppression or enhancement of

stochastic synchronization by local delayed feedbackModulation by varying delay timeDelay-coupled neurons:

delay-induced antiphase oscillations of tunable frequency delayed self-feedback: synchronization of oscillation modes

Failure of feedback as mechanism of spreading depression

non-local or time-delayed feedback suppresses propagation of excitation pulses for suitably chosen spatial connections or

time delays

Students

● Roland Aust● Thomas Dahms● Valentin Flunkert● Birte Hauschildt● Gerald Hiller● Johanne Hizanidis● Philipp Hövel● Niels Majer● Felix Schneider

CollaboratorsAndreas AmannAlexander BalanovBernold FiedlerNatalia JansonWolfram JustSylvia SchikoraHans-Jürgen Wünsche

Markus Dahlem

Postdoc

PublishedOctober 2007