Desynchronization of oscillatory networks by intermittent delayed feedback control
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1 Desynchronization of oscillatory networks by intermittent delayed feedback control Center for Physical Sciences and Technology A. Gostauto 11, LT Vilnius LITHUANIA The 5th International Conference on Chaotic Modeling and Simulation, CHAOS 2012
2 Outline Motivation Algorithm scheme Landau-Stuart oscillators desynchronisation Hodgkin-Huxley neurons desynchronisation Conclusions
3 Motivation Possible application for the deep brain stimulation Previously proposed desynchronisation methods P.A. Tass, Biol. Cybern., 89:81 88, 2003 K. Pyragas, O.V. Popovich and P.A. Tass, Europhys. Lett., 80:40002, 2007 N. Tukhlina, M. Rosenblum, A. Pikovsky, and J. Kurths, Phys. Rev. E, 75:011918, Drawbacks It uses more than one electrode Feedback is not protected from stimulation signal direct impact
4 Motivation Possible application for the deep brain stimulation Previously proposed desynchronisation methods P.A. Tass, Biol. Cybern., 89:81 88, 2003 K. Pyragas, O.V. Popovich and P.A. Tass, Europhys. Lett., 80:40002, 2007 N. Tukhlina, M. Rosenblum, A. Pikovsky, and J. Kurths, Phys. Rev. E, 75:011918, Drawbacks It uses more than one electrode Feedback is not protected from stimulation signal direct impact PID controller Mean field of one subpopulation effects other subpopulation system
5 Algorithm scheme Stage I In the first stage, we measure and memorize the output of the controlfree system.
6 Algorithm scheme Stage I In the first stage, we measure and memorize the output of the controlfree system. Stage II In the second stage, we apply the feedback control using the memorized signal. Both stages take equal amount of time τ.
7 Desynchronisation of Landau-Stuart oscillators (1) oscillator coupling Complex variable: Effect of the averaged field: The feedback coefficient P is complex measure. System synchronisation is defined by order parameter: control
8 Desynchronisation of Landau-Stuart oscillators (1) oscillator coupling Complex variable: Effect of the averaged field: The feedback coefficient P is complex measure. System synchronisation is defined by order parameter: control
9 Desynchronisation of Landau-Stuart oscillators (1) oscillator coupling control Complex variable: Effect of the averaged field: The feedback coefficient P is complex measure. synchronised state System synchronisation is defined by order parameter: desynchronised state
10 Desynchronisation of Landau-Stuart oscillators (1) oscillator coupling control Complex variable: Effect of the averaged field: The feedback coefficient P is complex measure. synchronised state System synchronisation is defined by order parameter: Object is to reset r to 0 desynchronised state
11 Equation for order parameter Equations for oscillators Equation for the order parameter r
12 Equation for order parameter Equations for oscillators Equation for the order parameter r Equations for oscillators phases (Kuramoto model)
13 Equation for order parameter Equations for oscillators Equation for the order parameter r Equations for oscillators phases (Kuramoto model) Assumptions: 1. All oscillators have the same radius. 2. The number of oscillators is infinite i.e. continuous case. Ott-Antonsen ansatz infinite size coupled oscillators behave low dimensional dynamics 3. The intrinsic oscillators frequencies are distributed by the Lorentzian (with central frequency and width ). Edward Ott and Thomas M. Antonsen, Chaos, 18:037113, 2008
14 Desynchronisation of Landau-Stuart oscillators (2)
15 Desynchronisation of Landau-Stuart oscillators (2) Solutions of Landau-Stuart equations and reduced equation for order parameter: Parameters: T=2 K=0.5 P= i τ=0.4
16 Desynchronisation of Landau-Stuart oscillators (2) Solutions of Landau-Stuart equations and reduced equation for order parameter: Parameters: T=2 K=0.5 P= i τ=0.4 Which P and τ desynchronise?
17 Desynchronisation stability zones Linearization reduces initial problem to unstable fixed point stabilisation: (3)
18 Desynchronisation stability zones Linearization reduces initial problem to unstable fixed point stabilisation: (3) Zeroth point stability can be estimated studying one registrationstimulation period. Stable when
19 Desynchronisation stability zones Linearization reduces initial problem to unstable fixed point stabilisation: (3) Zeroth point stability can be estimated studying one registrationstimulation period. Stable when Color code shows oder parameter absolute value calculated from integration of original problem. According linear analysis the order parameter relax to zero between black lines.
20 Synaptically coupled HodgkinHuxley(HH) neurons Realistic neuron model: (4) Standart HH model - neurons membrane potential - regulate neurons frequency - synaptic current- makes synchronisation Coupling Control
21 Synaptically coupled HodgkinHuxley(HH) neurons Desyncronisation parameter is defined as ratio between variance of mean field when stimulation is on and free system: - smaller is better Depicted in colors. Solution of HH equations. Green- sync state (withouth algorithm), red- algorithm is on, blue line represent stimulation current.
22 Conclusions Separation of the registration and stimulation stages in time allow implement algorithm with one electrode and avoid stimulation direct influence to feedback signal; Analytical estimations and numerical simulations confirm that the algorithm desynchronise globally coupled Landau-Stuart oscillators and synaptically coupled Hodgkin-Huxley neurons. Acknowledgments This research was funded by the European Social Fund under the Global Grant measure (grant No. VP1-3.1-SMM-07-K )
23 The end
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