教案神经网络1

1、Neural OscillationsA nonlinear dynamic approachLogistic Differential EquationKNrNN1NNKK/2NtKLogistic Differential EquationKNrNN1NNKK/2Kd/dt N = - d/dN r N2 (N/3k -1/2)= - d/dN V(N)V(N)03K/2NK/2What is a bifurcation?A qualitative change in the solution !Max & Moritz, W. BuschSaddle Node Bifurcati

2、on (1-dim)2xbxPrototypical example:xx b*xA saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points (or equilibria) of a dynamical system collide and annihilate each other.Transcritical BifurcatoinPrototypical example:2xbxxxx b*xA transcrit

3、ical bifurcation is a particular kind of local bifurcation, meaning that it is characterized by an equilibrium having an eigenvalue whose real part passes through zero.Pitchfork BifurcationPrototypical example:3xbxxxx b*x A pitchfork bifurcation is a particular type of local bifurcation. Pitchfork b

4、ifurcations have two types - supercritical or subcritical. Supercritical SubcriticalSubcritical3xrxxHopf-Bifurcation)()(2222yxyyxyyxxyxxPrototypical example:*xBifurcations ? Catastrophes ?Use simple model from classical mechanicsParticle (m=1) in 1D-potential V(x,)Potential changes very slowly with

5、time (slow time scale represented by )Equations of motion: Stability of fix points:xVppx0, 0exxxVpLoss-of-Equilibrium (Saddle-Node Bifurcation)Loss-of-equilibriumor catastrophe at = cPotential,LyapunovfunctionTranscritical BifurcationEquilibrium branches exchange stability at =cOther possibility:sup

6、ercitical pitchfork bifurcationPotential, Lyapunov functionSubcritical BifurcationConstruct the dynamical System ! d/dt x = - d/dx V(x)V(x) = x2 (b - x2)Dynamics of Two Dimensional Systems1.Find the fixed points in the phase space!2.Linearize the system about the fixed points!3.Determine the eigenva

7、lues of the Jacobian.Limit Cycles and Hopf Bifurcation A vector-field interpretationydxdy 2Let the functions F and G have continuous first partial derivatives in a domain D of the xy-plane. A closed trajectory of the system must necessarily enclose at least one critical (equilibrium) point. If it en

8、closes only one critical point, the critical point cannot be a saddle point.Theorem),(yxFdtdx),(yxGdtdyGraphical Interpretationydtdxxdtdy Graphical InterpretationxyyxSpecific Case of Theorem Find solutions for the following systemDo both functions have continuous first order partial derivatives?)()(

9、2222yxyyxyxxyxyxSpecific Case of Theorem Critical point of the system is (0,0) Eigenvalues are found by the corresponding linear systemwhich turn out to be .yxyx1111i1What does this tell us? Origin is an unstable spiral point for both the linear system and the nonlinear system. Therefore, any soluti

10、on that starts near the origin in the phase plane will spiral away from the origin.22yxxyxdtdx22yxyyxdtdyTrajectories of the SystemdtdxdtdyForming a system out of and yields the trajectories shown. Using Polar CoordinatesUsing x = r cos() y = r sin() r 2 = x 2 + y 222yxxyxdtdx22yxyyxdtdyGoes to:21rr

11、dtdrCritical points ( r = 0 , r = 1 )Thus, a circle is formed at r = 1and a point at r = 0. Stability of Period SolutionsOrbital StabilitySemi-stableUnstableExample of StabilityGiven the Previous Equation:21rrdtdrIf r 1, Then, dr/dt 0 (meaning the solution moves inward)If 0 r 0 (meaning the solution

12、s movies outward)Hopf Bifurcation22yxxyxdtdx22yxyyxdtdy2rrdtdrIntroducing the new parameter ( )Converting to polar form as in previous slide yields: Critical Points are now: r = 0 and r = If you notice, these solutions are extremely similar to those of the previous example y2 = xr = 0r = Hopf Bifurc

13、ationAs the parameter increases through the value zero, the previously asymptotically stable critical point at the origin loses its stability, and simultaneously a new asymptotically stable solution (the limit cycle) emerges. Thus, = 0 is a bifurcation point. This type of bifurcation is called Hopf

14、bifurcation, in honor of the Austrian mathematician Eberhard Hopf who rigorously treated these types of problems in a 1942 paper.Hopf Bifurcation Theorem),(),(yxgyyxfx)0 , 0(),(),(),(),(yxgyxgyxfyxfDFyxyxs.eigenvalue its be )( ),(Let 21,at seigenvalueimaginary purely ofpair a are that theySupposec0|

15、)(Re(2, 1cdd.at stable asymptotic is original theandc0), 0 , 0(), 0 , 0(gfHopf Bifurcation Theorem. withincreases size whosecycle,limit stablea by surrounded unstable, is original the, focus; stable is original the, thatsuch some are therepoint; nbifurcatioa is 2112ccccFitzhugh-Nagumo Model Fitzhugh

16、- Nagumo equations:)(33bWaVdtdWIWVVdtdV08. 08 . 0, 7 . 0ba-3-2-10123-2-1.5-1-0.500.511.52VWAnalysis of Fitzhugh-Nagumo System (1) Jacobian:-b -VA 1 )1 (2)(33bWaVdtdWIWVVdtdV08. 08 . 0, 7 . 0baibVbV42. 05 . 0: )0) 1()1(A of esEigen valu2, 1222The fixed point is:Stable Spiral Response of the resting s

17、ystem (I=0) to a current pulse:Analysis of Fitzhugh-Nagumo System (2)-3-2-10123-2-1.5-1-0.500.511.52WV Response of the resting system (I=0) to a current pulse:Analysis of Fitzhugh-Nagumo System (3)-3-2-10123-2-1.5-1-0.500.511.52WVThreshold is due to fast sodium gating (V nullcline)Hyperpolarization

18、and its termination is due to sodium/potassium channel Response to a steady current :Analysis of Fitzhugh-Nagumo System (4)-b -VA 1 )1 (2)(33bWaVdtdWIWVVdtdV08. 08 . 0, 7 . 0baibVbV32. 041. 0: )0) 1()1(A of esEigen valu2, 1222The fixed point is:Unstable spiral Response of the resting system (I=1) to a steady current:Analysis of Fitzhugh-Nagumo System (