Dynamics of Bursting Spike Renormalization
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Transcript of Dynamics of Bursting Spike Renormalization
Reprint/Preprint Download at: http://www.math.unl.edu/~bdeng1
Bo DengDepartment of Mathematics
University of Nebraska – Lincoln
Outline of Talk Bursting Spike Phenomenon Bifurcation of Bursting Spikes Definition of Renormalization Dynamics of Renormalization
Phenomenon of Bursting Spikes
Rinzel & Wang (1997)Neurosciences
Food Chains Phenomenon of Bursting Spikes
1
1 11 2
2 22
(1 ) : ( , )
( ) : ( , , )
( ) : ( , )
yx x x xf x y
x
x zy y y yg x y z
x y
yz z z zh y z
y
Dimensionless Model:
Bifurcation of Spikes
),( IVgdtdI
IIdtdVC
VRIVdt
dIL
L
LEL
1-d Return Map at = 0
g (V, I) = 0
1-d map
2 time scale system: with ideal situation at = 0.
IIL
V
Bifurcation of Spikes
),( IVgdtdI
IIdtdVC
VRIVdt
dIL
L
LEL
c0
IIL
V
Bifurcation of Spikes
),( IVgdtdI
IIdtdVC
VRIVdt
dIL
L
LEL
c0
f1
10
c0
IIL
V
Homoclinic Orbit at = 0
Food Chains
Phenomenon of Bursting Spikes
Bifurcation of Spikes
),( IVgdtdI
IIdtdVC
VRIVdt
dIL
L
LEL
Def: System is isospiking of n spikes if for every c0 < x0 <=1, thereare exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1].
0 c0 1
f1
Def of Isospike c0
IIL
V
Bifurcation of Spikes
),( IVgdtdI
IIdtdVC
VRIVdt
dIL
L
LEL
c0
Def: System is isospiking of n spikes if for every c0 < x0 <=1, thereare exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1].
c0
IIL
V
Bifurcation of Spikes
),( IVgdtdI
IIdtdVC
VRIVdt
dIL
L
LEL
c0
Isospike of 3 spikes c0
IIL
V
Def: System is isospiking of n spikes if for every c0 < x0 <=1, thereare exactly n points x1, x2, … xn in [0, c0) and xn+1 returns to [c0,1].
Bifurcation of Spikes
Bifurcation of Spikes
1st
2nd3rd
4th
5th
6th
Spike Reset
C/L
Numeric
Silent Phase
Renormalization
Feigenbaum’s Renormalization Theory (1978)• Period-doubling bifurcation for f(x)=x(1-x)• Let n = the 2n-period-doubling bifurcation
parameters, n 0
_
• A renormalization can be defined at each n , referred to as Feigenbaum’s renormalization.• It has a hyperbolic fixed point with eigenvalue
((n+1) - n )/((n+2) - (n+1)) 4.6692016…
which is a universal constant, called the
Feigenbaum number.
Feigenbaum
Renormalization
f
Def of R
Renormalization
f
f 2
Renormalization
f
f 2
00
1 cfc
02
0
1 cfc
Renormalization
f
f 2
00
1 cfc
02
0
1 cfc
R
Renormalization
f
f 2
00
1 cfc
02
0
1 cfc
R
dxxfffYY Y |)(| |)0(||||| with ,:1
0RR
C-1/C0
R( f )1
10
C-1
IIL
Vc0
2 familiesRenormalization
0 c0 1
f
1
0 1
1
0 1
=id
1
11
10
,0
,
x
xxx
0 c0 1
f
1
e-K/
Renormalization
R[0]=0
Y
universalconstant 1
0 1
1
W = { }
Renormalization
R[0]=0
R[]=
0 1
R1
0 1
1
Renormalization
R[0]=0
R[]=
R[n]= n
0 1
R1
0 1
1
Renormalization
R[0]=0
R[]=
R[n]= n
1 is an eigenvalue of DR[0]
20
2
)1/(00
|||| 2
34
-1
|||| ||)(1][][||
RR
20
2
)1/(00
|||| 2
34
-1
|||| ||)(1][][||
RR
0 1
R1
0 1
1
0 1
R1
0 1
1
Renormalization
R[0]=0
R[]=
R[n]= n
1 is an eigenvalue of DR[0]Lemma
1lim1
12
nn
nn
n
q
p
nqn
qnpqn
n
lim
Renormalization
R[0]=0
R[]=
R[n]= n
1 is an eigenvalue of DR[0] - Lemma &
Theorem 1:
q
p
nqn
qnpqn
n
lim
= id
W
U=
Invariant
Invariant
Fixed Point
Eigenvalue:
Renormalizationsuperchaos
YY :R
W
id
Renormalization
YY :R
Theorem 2: R has fixed points whose stable spectrum contains 0 < < 1 in W For any >1 there exists a fixed point repelling at rate and normal to W
0 1
Fixed Points= { }
1
Let W = X0 U X1 with
W
X0
X1
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.
Renormalization
0 1
X0 = { }
1
YY :R
Theorem 2: R has fixed points whose stable spectrum contains 0 < < 1 in W For any >1 there exists a fixed point repelling at rate and normal to W
0 1
X1 = { }
1
id
0 1
X0 = { }
1
Theorem 2: R has fixed points whose stable spectrum contains 0 < < 1 in W For any >1 there exists a fixed point repelling at rate and normal to W
W
X0
X1
id
Every n-dimensional dynamical system can be conjugate embedded into X0 in infinitely many ways.
1 ,: nDRDf n
)()(
s.t ,: ,:
xxf
YDDDf
R
Renormalization
…
slope =
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), …
y0
y1
y2
(x0)
YY :R
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.
W
X0
X1
id
Renormalization
The conjugacy preserves f ’s Lyapunov number L if L <
YY :R
Theorem 2: R has fixed points whose stable spectrum contains 0 < < 1 in W For any >1 there exists a fixed point repelling at rate and normal to W
Every n-dimensional dynamical system can be conjugate embedded into X0 in infinitely many ways.
1 ,: nDRDf n
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.
W
X0
X1
id
Renormalization
f
The conjugacy preserves f ’s Lyapunov number L if L <
Rmk: Neuronal families f through
100 XXf YY :R
Theorem 2: R has fixed points whose stable spectrum contains 0 < < 1 in W For any >1 there exists a fixed point repelling at rate and normal to W
Every n-dimensional dynamical system can be conjugate embedded into X0 in infinitely many ways.
1 ,: nDRDf n
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.
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