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

Outline of Talk Bursting Spike Phenomenon Bifurcation of Bursting Spikes Definition of Renormalization Dynamics of Renormalization

Phenomenon of Bursting Spikes Rinzel & Wang (1997) Neurosciences

Phenomenon of Bursting Spikes Food Chains Dimensionless Model:

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

Bifurcation of Spikes c0 V I IL

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

Phenomenon of Bursting Spikes Food Chains

Bifurcation of Spikes c0 Def of Isospike V 1 f I IL 0 c0 1 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].

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].

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].

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

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

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.

Renormalization f Def of R

Renormalization f f 2

Renormalization f f 2

Renormalization f R f 2

Renormalization f R f 2 R

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

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

Renormalization Y R[y0]=y0 0 1 1 W = { } universal constant 1

R[ym]=ym / (1-m) R m / (1-m) R[y0]=y0 Renormalization ym /(1-m) ym 1 m 0 1 ym R 1 m 1-m ym /(1-m) m / (1-m)

R[y1/(n+1) ]= y1/n R m / (1-m) R[y0]=y0 R[ym]=ym/(1-m) Renormalization 0 1 ym R 1 m 1-m ym/(1-m) m / (1-m)

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] 0 1 ym R 1 m 1-m ym/(1-m) m / (1-m)

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 0 1 ym R 1 m 1-m ym/(1-m) m / (1-m)

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 &

superchaos Eigenvalue: Invariant Fixed Point Invariant Renormalization U={ym} Invariant chaos y0 = id Fixed Point W Invariant

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 0 1 Fixed Points= { } 1 chaos id r < 1 W

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 0 1 X0 = { } 1 chaos id r < 1 X1 chaotic X0 0 1 X1 = { } 1 W

0 1 X0 = { } 1

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

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

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

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.