TIME-DELAYED FEEDBACK CONTROL OF COMPLEX NONLINEAR SYSTEMS Eckehard Schöll Institut für Theoretische Physik and Sfb 555 “Complex Nonlinear Processes” Technische Universität Berlin Germany Net-Works 2008 Pamplona

Outline Time-delayed feedback control  Introduction: Time-delayed feedback control of nonlinear systems   control of deterministic states   control of noise-induced oscillations   application: lasers, semiconductor nanostructures Neural systems: control of coherence of neurons and synchronization of coupled neurons  Neural systems: control of coherence of neurons and synchronization of coupled neurons  delay-coupled neurons  delayed self-feedback  delay-coupled neurons  delayed self-feedback Control of excitation pulses in spatio-temporal systems:  Control of excitation pulses in spatio-temporal systems: migraine, stroke migraine, stroke  non-local instantaneous feedback  non-local instantaneous feedback  time-delayed feedback  time-delayed feedback

Why is delay interesting in dynamics?  Delay increases the dimension of a differential equation to infinity: delay t generates infinitely many eigenmodes  Delay has been studied in classical control theory and mechanical engineering for a long time and mechanical engineering for a long time Simple equation produces very complex behavior  Simple equation produces very complex behavior

Delay is ubiquitous  mechanical systems: inertia  electronic systems: capacitive effects (t=RC) latency time due to processing latency time due to processing biological systems: cell cycle time  biological systems: cell cycle time biological clocks biological clocks  neural networks: delayed coupling, delayed feedback  neural networks: delayed coupling, delayed feedback optical systems: signal transmission times  optical systems: signal transmission times travelling waves + reflections travelling waves + reflections  laser coupled to external cavity (Fabry-Perot)  multisection laser  semiconductor optical amplifier (SOA)

Time delayed feedback control methods  Originally invented for controlling chaos (Pyragas 1992): stabilize unstable periodic orbits embedded in a chaotic attractor  More general: stabilization of unstable periodic or stationary states in nonlinear dynamic systems stationary states in nonlinear dynamic systems Application to spatio-temporal patterns:  Application to spatio-temporal patterns:  Partial differential equations  Partial differential equations Delay can induce or suppress instabilities  Delay can induce or suppress instabilities  deterministic delay differential equations  deterministic delay differential equations  stochastic delay differential equations  stochastic delay differential equations

Published October 2007 Scope has considerably widened

Time-delayed feedback control of deterministic systems  Time-delayed feedback (Pyragas 1992): Stabilisation of unstable periodic orbits or unstable fixed points or space-time patterns Time-delay autosynchronisation (TDAS) Extended time-delay autosynchronisation (ETDAS) (Socolar et al 1994) deterministic chaos  =T Many other schemes

Time-delayed feedback control of deterministic systems stability is measured by Floquet exponent L: dx ~ exp(Lt) or Floquet multiplier m=exp(LT)

b complex (1 -  ) Beyond Odd Number Limitation

Example of all-optical time-delayed feedback control in semiconductor laser Optical feedback: | Stabilisation of fixed point: Schikora, Hövel, Wünsche, Schöll, Henneberger, PRL 97, (2006) Laser: excitable unit, may be coupled to others to form network motif

Stabilization of cw emission: Domain of control of unstable fixed point above Hopf bifurcation | Schikora, Hövel, Wünsche, Schöll, Henneberger, PRL 97, (2006) Generic model: phase sensitive coupling Generic model: phase sensitive coupling  =0.5T 0  =0.9T 0   

Experimental realization | Schikora, Hövel, Wünsche, Schöll, Henneberger, PRL 97, (2006)

Control of spatio-temporal patterns: semiconductor nanostructure semiconductor nanostructure Without control: Examples: Chemical reaction-diffusion systems Electrochemical systems Semiconductor nanostructures Hodgkin-Huxley neural models | a(x,t): activator variable u(t): inhibitor variable f(a,u): bistable kinetic function D(a): transverse diffusion coefficient Global coupling: Ratio of timescales:  R DBRT ● Global coupling due to Kirchhoff equation: I Control parameters:  = RC, U 0

Chaotic breathing pattern u 9.1 u min, u max  = 9.1: above period doubling cascade Spatially inhomogeneous chaotic oscillations J. Unkelbach, A.Amann, W. Just, E. Schöll: PRE 68, (2003)

Stabilisation of unstable period-1 orbit u min, u max ● Period doubling bifurcations generate a family of unstable periodic orbits (UPOs) ● Period-1 orbit: Breathing oscillations Resonant tunneling diode a(x,t): electron concentration in quantum well in quantum well u(t): voltage across diode tracking

Time-delayed feedback control of noise-induced oscillations Stabilisation of UPO noise-inducedoscillations ? no deterministic orbits! deterministic chaos  =T K. Pyragas, Phys. Lett. A 170, 421 (1992) 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 density of optical intensity Suppression of noise for  0.5  T RO Flunkert and Schöll, PRE 76, (2007)

Feedback control of noise-induced space- time patterns in the DBRT nanostructure G. Stegemann, A. Balanov, E. Schöll, PRE 73, (2006)  =4, K=0.4  D u = 0.1, D a = 10 -4

Enhancement of temporal regularity: correlation time vs. noise amplitude vs. feedback gain  =7: increase  =5: decrease G. Stegemann, A. Balanov, E. Schöll, PRE 73, (2006) Large effect for small noise intensity D u = 0.1, D a = 10 -4

Correlation time vs. delay time Real parts of eigenvalues Period: inverse imaginary parts of eigenvalues  Control of time- scales: basic period of oscillations Control of coherence: optimum 

Coherence resonance  – normalized autocorrelation function autocorrelation function Correlation time: Correlation time: Simplified FitzHugh-Nagumo (FHN) system: excitable neuron Excitable System a=1.1  =0.01 Gang, Ditzinger, Ning, Haken, PRL 71, 807 (1993) Pikovsky, Kurths, PRL 78, 775 (1997)

Example of coherence resonance: neuron Simulation from S.-G. Lee, A. Neiman, S. Kim, PRE 57, 3292 (1998). S.-G. Lee, A. Neiman, S. Kim, PRE 57, 3292 (1998). Time series of the membrane potential for various noise intensity:

FitzHugh-Nagumo model with delay Janson, Balanov, Schöll, PRL 93, (2004) Excitability a=1: excitability threshold u activator (membrane voltage) v inhibitor (recovery variable) e time-scale ratio

Coherence vs.  and K D=0.09 D=0.09; K=0.2 Numerics: Balanov, Janson, Schöll, Physica D 199, 1 (2004) Analytics: Prager, Lerch, Schimansky-Geier, Schöll, J. Phys.A 40, (2007)

2 coupled FitzHugh-Nagumo systems: coupled neuron model as network motif ● 2 non-identical stochastic oscillators: diffusive coupling frequencies tuned by    D 1,    D 2 B. Hauschildt, N. Janson, A. Balanov, E. Schöll, PRE 74, (2006) a= 1.05,  1 =0.005,  2 = 0.1, D 2 =0.09 : coherence resonance as function of D 1

Time series for various noise intensities ● C= 0.07

Stochastic synchronization ● Frequency synchronization : mean interspike intervals (ISI) ● Phase synchronization: 1:1 synchronization index (Rosenblum et al 2001) o X + + weakly synchronized o moderately synchronized x strongly synchronized

Local delayed feedback control: enhance or suppress synchronization ● Moderately synchronized system (o) System 1 1:1 synchronization index

Can local delayed feedback de-synchronize 2 coupled neurons? ● Weakly synchronized system (+) ● Strongly synchronized system (x)

Delayed coupling, no self-feedback + noise Dahlem, Hiller, Panchuk, Schöll, IJBC in print, 2008 induces antiphase oscillations

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

Regimes of synchronized oscillation modes

Different modes of oscillations excitability parameter a=1.3 anti-phase in-phase oscillator death bursting

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

Measured cortical spreading depression Visual cortex 3 mm/ min

FitzHugh-Nagumo (FHN) system with activator diffusion u activator (membrane voltage) v inhibitor (recovery variable) D u diffusion coefficient e time-scale ratio of inhibitor and activator variables b 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 b and e

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 al Chaos (2008)

Non-local feedback: suppression of CSD uu vv uv vu Tissue at risk

Non-local feedback: shift of propagation boundary K=+/-0.2 pulse width Dx

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 Dt

Conclusions   Delayed feedback control of excitable systems   Control 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 feedback   Modulation by varying delay time   Delay-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 Collaborators Andreas Amann Alexander Balanov Bernold Fiedler Natalia Janson Wolfram Just Sylvia Schikora Hans-Jürgen Wünsche Markus Dahlem Postdoc

Published October 2007