time-delayed feedback control of complex nonlinear systems eckehard schöll institut für...
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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
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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
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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
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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)
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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
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PublishedOctober 2007
Scope has considerably widened
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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
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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)
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b complex
(1 - )
Beyond Odd Number LimitationBeyond Odd Number Limitation
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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
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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
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Experimental realizationExperimental realization
||
Schikora, Hövel, Wünsche, Schöll, Henneberger, PRL 97, 213902 (2006)
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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
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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)
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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
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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)
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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)
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))()(()(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
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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
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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)
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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::
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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
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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)
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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
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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
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Local delayed feedback control: enhance or suppress synchronization
● Moderately synchronized system (o)
System 1
1:1 synchronization index
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Delayed coupling, no self-feedback + noise
Dahlem,Hiller, Panchuk,Schöll, IJBC in print, 2008
inducesantiphaseoscillations
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Sustained oscillations induced by delayed coupling
excitability parameter a=1.3
a=1.05
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Regime of oscillations
excitability parameter a=1.3
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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
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Spreading depolarization wave(cortical spreading depression SD)
● migraine aura (visual halluzinations)● stroke
Examples:
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Migraine aura: neurological precursor(spatio-temporal pattern on visual cortex)
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Migraine aura: visual halluzinations
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Migraine aura: visual halluzinations
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Migraine aura: visual halluzinations
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Migraine aura: visual halluzinations
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Migraine aura: visual halluzinations
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Migraine aura: visual halluzinations
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Measured cortical spreading depression
Visual cortex
3 mm/ min
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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)
_
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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
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Boundary of propagation of traveling excitation pulses (SD)
excitable:traveling pulses
non-excitable: transient
Propagation verlocitypulse
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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)
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Non-local feedback: suppression of CSD
uu
vvuv
vu
Tissue at risk
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Non-local feedback:shift of propagation boundary
K=+/-0.2
pulse width x
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Time-delayed feedback: suppression of SD
uu vu
uv vv
Tissue at risk
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Time-delayed feedback:shift of propagation boundary
uu vu
vv vu
K=+/-0.2
pulse width t
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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
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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
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PublishedOctober 2007