Dynamics of Bursting Spike Renormalization

1. Dynamics of Bursting Spike Renormalization Bo Deng Department of Mathematics University of Nebraska – Lincoln Reprint/Preprint Download at: http://www.math.unl.edu/~bdeng1

2. Outline of Talk • Bursting Spike Phenomenon • Bifurcation of Bursting Spikes • Definition of Renormalization • Dynamics of Renormalization

3. Phenomenon of Bursting Spikes Rinzel & Wang (1997) Neurosciences

4. Phenomenon of Bursting Spikes Dimensionless Model: Food Chains

5. 1-d map Bifurcation of Spikes 2 time scale system: 0 < e << 1, with ideal situation at e= 0. 1-d Return Map at e = 0 V g (V, I) = 0 I IL

6. Bifurcation of Spikes c0 V I IL

7. Bifurcation of Spikes c0 1 f 0 c0 1 Homoclinic Orbit at e= 0 V I IL

8. Phenomenon of Bursting Spikes Food Chains

9. Def of Isospike Bifurcation of Spikes c0 V 1 f I IL 0 c01 Def: System is isospikingof n spikes if for every c0< x0 <=1, there are exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1].

10. Bifurcation of Spikes c0 V I IL c0 Def: System is isospikingof n spikes if for every c0< x0 <=1, there are exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1].

11. Bifurcation of Spikes c0 Isospike of 3 spikes V I IL c0 Def: System is isospikingof n spikes if for every c0< x0 <=1, there are exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1].

12. Bifurcation of Spikes n 3 # of Spikes 2 1 … 1/n … 1/3 1/2 1 0 Isospike Distribution 1/x

13. Numeric Bifurcation of Spikes Silent Phase Spike Reset 6th 5th 4th 3rd 2nd 1st m = C/L

14. Feigenbaum Renormalization Feigenbaum’s Renormalization Theory (1978) • Period-doubling bifurcation for • fl(x)=lx(1-x) • Let ln = the 2n-period-doubling bifurcation • parameters, ln  l0_ • A renormalization can be defined at each ln , • referred to as Feigenbaum’s renormalization. • It has a hyperbolic fixed point with eigenvalue • (l(n+1) - ln )/(l(n+2) - l(n+1))  4.6692016… • which is a universal constant, called the • Feigenbaum number.

15. Def of R Renormalization f

16. Renormalization f f 2

17. Renormalization f f 2

18. Renormalization f R f 2

19. Renormalization f R f 2 R

20. C-1 1 R( f ) 0 C-1/C0 1 c0 V I IL

21. 2 families m Renormalization 1 1 fm f0 m 0 m e-K/m 0 c01 0 c01 1 1 ym y0=id m 0 m 01 1-m 01

22. universalconstant 1 Renormalization 1 W = { } 01 Y • R[y0]=y0

23. Renormalization 1 1 R m / (1-m) ym /(1-m) ym m 01 01 1-m • R[y0]=y0 • R[ym]=ym / (1-m)

24. Renormalization 1 1 R m / (1-m) ym/(1-m) ym m 01 01 1-m • R[y0]=y0 • R[ym]=ym/(1-m) • R[y1/(n+1)]= y1/n

25. Renormalization 1 1 R m / (1-m) ym/(1-m) ym m 01 01 1-m • R[y0]=y0 • R[ym]=ym/(1-m) • R[y1/(n+1)]= y1/n • 1 is an eigenvalue of DR[y0]

26. Renormalization 1 1 R m / (1-m) ym/(1-m) ym m 01 01 1-m • R[y0]=y0 • R[ym]=ym/(1-m) • R[y1/(n+1)]= y1/n • 1 is an eigenvalue of DR[y0] • l- Lemma

27. Renormalization Theorem 1: • R[y0]=y0 • R[ym]=ym/(1-m) • R[y1/(n+1)]= y1/n • 1 is an eigenvalue of DR[y0] • l- Lemma &

28. superchaos Renormalization Eigenvalue: l = 1 U={ym} Invariant y0 = id Fixed Point W Invariant

29. Theorem 2: • R has fixed points whose stable • spectrum contains 0 < r < 1 in W • For any l >1 there exists a fixed point • repelling at rate l and normal to W Renormalization 1 Fixed Points= { } 01 l > 1 l > 1 l = 1 ym id r < 1 W

30. Theorem 2: • R has fixed points whose stable • spectrum contains 0 < r < 1 in W • For any l >1 there exists a fixed point • repelling at rate l and normal to W Renormalization 1 1 X1 = { } X0 = { } 01 01 • Let W = X0U X1 with Every point in X1 goes to a fixed point X0isa chaotic set: (1) dense set of periodic orbits; (2) every point is connected to any other point; (3) sensitive dependence on initial conditions; (4) dense orbits. l > 1 l > 1 l = 1 ym id r < 1 X1 chaotic X0 W

31. 1 X0 = { } 01

32. Theorem 2: • R has fixed points whose stable • spectrum contains 0 < r < 1 in W • For any l >1 there exists a fixed point • repelling at rate l and normal to W Renormalization y0 q (x0) y1 y2 … Every n-dimensional dynamical system can be conjugate embedded into X0in infinitely many ways. slope = l • Let W = X0U X1 with Every point in X1 goes to a fixed point X0isa chaotic set: (1) dense set of periodic orbits; (2) every point is connected to any other point; (3) sensitive dependence on initial conditions; (4) dense orbits. l > 1 l = 1 ym For each orbit { x0 , x1= f (x0), x2= f (x1), …} in [0,1], let y0 = S(x0), y1 = R-1S(x1), y2 = R-2S(x2), … id r < 1 X1 chaotic X0 W

33. Theorem 2: • R has fixed points whose stable • spectrum contains 0 < r < 1 in W • For any l >1 there exists a fixed point • repelling at rate l and normal to W Renormalization Every n-dimensional dynamical system can be conjugate embedded into X0in infinitely many ways. The conjugacy preserves f ’s Lyapunov number L if L < l • Let W = X0U X1 with Every point in X1 goes to a fixed point X0isa chaotic set: (1) dense set of periodic orbits; (2) every point is connected to any other point; (3) sensitive dependence on initial conditions; (4) dense orbits. l > 1 l = 1 ym id X1 r < 1 chaotic X0 W

34. Theorem 2: • R has fixed points whose stable • spectrum contains 0 < r < 1 in W • For any l >1 there exists a fixed point • repelling at rate l and normal to W Renormalization Every n-dimensional dynamical system can be conjugate embedded into X0in infinitely many ways. Rmk: Neuronal families fm through The conjugacy preserves f ’s Lyapunov number L if L < l • Let W = X0U X1 with Every point in X1 goes to a fixed point X0isa chaotic set: (1) dense set of periodic orbits; (2) every point is connected to any other point; (3) sensitive dependence on initial conditions; (4) dense orbits. l > 1 fm l = 1 ym id X1 r < 1 chaotic X0 W

35. Zero is the origin of everything. • One is a universal constant. • Infinity is the number of copies every dynamical • system can be found inside a chaotic square. • It can be taught to undergraduate students who • have learned separable spaces. Summary