1. Dynamics of Bursting Spike Renormalization
Bo Deng
Department of Mathematics
University of Nebraska – Lincoln
Reprint/Preprint Download at:

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
Food Chains
Dimensionless Model:

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

6. Bifurcation of Spikes c0 V I IL

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

8. Phenomenon of Bursting Spikes
Food Chains

9. Bifurcation of Spikes c0
Def of Isospike V 1 f I IL c
Def: System is isospiking of 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 isospiking of 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 isospiking of 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. 1/2 1/3 1/n 1 2 3 n Bifurcation of Spikes Isospike Distribution 1/x…
1/2
1/3
1/n 1 2 3 n
# of Spikes
1/x
Isospike Distribution

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

14. 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))  …
which is a universal constant, called the
Feigenbaum number.

15. Renormalization f
Def of R

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

21. 2 families fm f0 m m e-K/m ym y0=id m 1-m Renormalization 1 1 0 c0 1 ym 1 m
y0=id
1-m

22. Renormalization Y
R[y0]=y0 1
W = { }
universal constant 1

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

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

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

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

27. R[y0]=y0 R[ym]=ym/(1-m) R[y1/(n+1) ]= y1/n 1 is an eigenvalue
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 Eigenvalue: Invariant Fixed Point Invariant Renormalization
U={ym}
Invariant
chaos
y0 = id
Fixed Point W
Invariant

29. Fixed Points= { } 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
l > 1
l > 1
l = 1 ym
Fixed Points= { } 1
chaos id
r < 1 W

30. chaotic Let W = X0 U X1 with Every point in X1 goes to a fixed point
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
Let W = X0 U X1 with
Every point in X1 goes to a fixed
point
X0 is a 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
X0 = { } 1
chaos id
r < 1 X1
chaotic X0
X1 = { } 1 W

31.
X0 = { } 1

32. chaotic … For each orbit { x0 , x1= f (x0), x2= f (x1), …} in [0,1],
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
slope = l …
Let W = X0 U X1 with y0
q (x0)
Every point in X1 goes to a fixed
point
X0 is a 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. y1
l > 1 y2
l = 1 ym
Every n-dimensional dynamical
system
can be conjugate embedded into
X0 in infinitely many ways.
chaos
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. chaotic The conjugacy preserves f ’s Lyapunov number L if L < l
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
Let W = X0 U X1 with
Every point in X1 goes to a fixed
point
X0 is a 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
Every n-dimensional dynamical
system
can be conjugate embedded into
X0 in infinitely many ways.
chaos id X1
r < 1
The conjugacy preserves f ’s
Lyapunov number L if L < l
chaotic X0 W

34. fm chaotic The conjugacy preserves f ’s Every n-dimensional dynamical
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
Let W = X0 U X1 with
Every point in X1 goes to a fixed
point
X0 is a 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
Every n-dimensional dynamical
system
can be conjugate embedded into
X0 in infinitely many ways.
chaos id X1
r < 1
The conjugacy preserves f ’s
Lyapunov number L if L < l
chaotic X0 W
Rmk: Neuronal families fm through

35. Summary 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.