Nonlinear phase coupling functions: a numerical study Michael - PowerPoint PPT Presentation

Nonlinear phase coupling functions: 
 a numerical study Michael Rosenblum and Arkady Pikovsky Institute of Physics and Astronomy, Potsdam University, Germany URL: www.stat.physik.uni-potsdam.de/~mros Trieste , 9 May 2019

Contents of the talk 1. An introduction: - phase of a limit cycle oscillator 
 - models of phase dynamics 
 - linear and nonlinear coupling functions 2. Nonlinear coupling functions: a numerical approach 3. A simple case: forced Stuart-Landau oscillator 4. A less simple case: - forced Rayleigh oscillator 
 - forced Rössler oscillator 5. Conclusions � 2

Phase dynamics: brief summary Consider general N -dimensional 
 x 1 self-sustained oscillator x = G(x) , x = ( x 1 , x 2 , . . . , x N ) ˙ with a stable limit cycle x T x 2 Phase is defined from the condition · φ = ω = 2 π / T and can be introduced in two steps : 1. phase on the limit cycle 2. phase in the basin of attraction of the limit cycle � 3

Phase on the limit cycle We start with some (arbitrary) zero point, x ( t 0 ) → φ ( x ( t 0 )) = 0 φ = 2 π t − t 0 and define phase as T A remark: phase can be defined either on interval or 
 [0,2 π ) on the real line � 4

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Phase reduction Perturbation technique for weak coupling, Malkin 1956, Kuramoto 1984 · The forced system x = G ( x ) + ε p ( x , t ) coupling strength, small parameter ε ⌧ | λ − | Negative Lyapunov exponent (determines the stability of the limit cycle) � 6

Phase reduction Perturbation technique for weak coupling, Malkin 1956, Kuramoto 1984 · The forced system x = G ( x ) + ε p ( x , t ) coupling strength, small parameter in the first approximation in one writes ε φ ( x ) = ∂ φ x = ∂ φ · · ∂ x [ G ( x ) + ε p ( x , t ) ] ∂ x = ω + ∂ φ ∂ x ε p ( x , t ) ≈ ω + ∂ φ ε p ( x T , t ) ∂ x x T where we 1. use the phase definition for the unperturbed system 2. we compute the r . h . s . on the cycle � 7

Phase reduction The forced system · x = G ( x ) + ε p ( x , t ) φ ( x ) = ω + ∂ φ · ε p ( x T , t ) ∂ x x T Let force be periodic we characterise it by its phase ψ , · ψ = ν Then p ( x T , t ) → p ( ψ , t ) Points on the limit cycle are in a one-to-one correspondence to , 
 φ i.e. we obtain a closed equation for the phase: x T = x T ( φ ) · φ = ω + ε Q ( φ , ψ ) � 8

Phase reduction: the coupling function · The forced system x = G ( x ) + ε p ( x , t ) coupling function · Phase equation φ = ω + ε Q ( φ , ψ ) Q ( φ , ψ ) = ∂ φ where p ( x T ( φ ), ψ ) ∂ x x T � 9

Phase reduction: the Winfree form · The forced system x = G ( x ) + ε p ( x , t ) coupling function · Phase equation φ = ω + ε Q ( φ , ψ ) Q ( φ , ψ ) = ∂ φ where p ( x T ( φ ), ψ ) ∂ x x T Notice : if the forcing is scalar, then p ( x , ψ ) = p ( ψ ) Q ( φ , ψ ) = Z ( φ ) p ( ψ ) Phase Sensitivity Curve, or 
 Phase Response Curve (PRC) · Thus, we have φ = ω + ε Q ( φ , ψ ) = ω + ε Z ( φ ) p ( ψ ) the Winfree form � 10

Phase reduction: the Kuramoto-Daido form · The forced system x = G ( x ) + ε p ( x , t ) coupling function · Phase equation φ = ω + ε Q ( φ , ψ ) ∥ ε Q ∥≪ ω If norm the phase equation can be averaged, keeping the resonance terms ω / ν ≈ m / n If then averaging yields · φ = ω + ε h ( n φ − m ψ ) the Kuramoto-Daido form � 11

Phase reduction: beyond first approximation Thus, for weak coupling, i.e. in the first approximation one obtains · φ = ω + ε Q ( φ , ψ ) Let us denote this explicitly: · φ = ω + ε Q 1 ( φ , ψ ) order of approximation � 12

Phase reduction: beyond first approximation Thus, for weak coupling, i.e. in the first approximation we obtained · φ = ω + ε Q ( φ , ψ ) Let us denote this explicitly: · φ = ω + ε Q 1 ( φ , ψ ) order of approximation Generally, one expects · φ = ω + ε Q 1 + ε 2 Q 2 + ε 3 Q 3 + … = ω + Q ( φ , ψ ) … but it is unknown, how to compute Q 2 , Q 3 , … � 13

Phase reduction: problems 1. Even computation of is difficult if the isochrons are Q 1 not known analytically 2. Power series representation remains a conjecture; there no algorithms for computation of Q 2 , Q 3 , … � 14

Phase reduction: problems 1. Even computation of is a problem if the isochrons are Q 1 not known analytically 2. Power series representation remains a conjecture; there no algorithms for computation of Q 2 , Q 3 , … … and approaches 1. Extension to the case of strong coupling with account of deviations from the limit cycle: a number of attempts, see e.g. recent review 6 B. Monga et al, Biol. Cybern., 2019 2. We suggest a numerical approach � 15

The simplest model: the Stuart-Landau system · A = ( μ + i η ) A − (1 + i α ) | A | 2 A + ε p ( ψ ), ψ = ν t A = Re i θ In polar coordinates, with : · R = μ R − R 3 + ε p ( ψ ) ⋅ cos θ (*) · θ = η − α R 2 − ε p ( ψ ) ⋅ sin θ / R Isochrons are known analytically: φ = θ − α ln( R / R 0 ) with R 0 = μ Derivation with account of (*) yields φ = ω − α cos θ + sin θ · ε p ( ψ ) with ω = η − αμ R � 16

The Stuart-Landau system: PRC and coupling functions φ = ω − α cos θ + sin θ · ε p ( ψ ) with ω = η − αμ R For weak force R ≈ R 0 = well-known results μ , θ ≈ φ Z ( φ ) = − μ − 1/2 ( α cos φ + sin φ ) PRC: Linear coupling function for harmonic forcing : p ( ψ ) = cos ψ Q 1 ( φ , ψ ) = − μ − 1/2 ( α cos φ + sin φ )cos ψ Averaging for yields the Kuramoto-Daido function ν ≈ ω Q 1 h ( φ − ψ ) = − 0.5 μ − 1/2 [ α cos( φ − ψ ) + sin( φ − ψ )] � 17

Computing nonlinear coupling functions · φ = ω + ε Q 1 + ε 2 Q 2 + ε 3 Q 3 + … = ω + Q ( φ , ψ ) Q ( φ , ψ ) = ε Q 1 + Q nlin known from the theory shall be obtained numerically We simulate the forced Stuart-Landau system to obtain φ ( t ), · φ ( t ) φ r = · · φ − ω − ε Q 1 and fit the rest term by a function of φ , ψ Practically: we use kernel density estimation on an 100x100 grid � 18

Fitting the coupling function we fit the equation · φ r = Q nlin ( φ , ψ ) We use kernel density estimation on an grid n × n K ( x , y ) = exp [ 2 π (cos x + sin y ) ] n with kernel We start with time series φ r , k , φ k , ψ k and for each point on the equidistant grid compute φ , ψ ∑ k · φ r , k K ( φ − φ k , ψ − ψ k ) Q ( φ , ψ ) = ∑ k K ( φ − φ k , ψ − ψ k ) 2 π 2 π Notice: the fitting works in the absence of locking! 0 0 � 19 2 π 0 2 π 0