Two insights about the rate of convergence s First,the individual energies decrease nontrivially.The BAM system does not creep arbitrary slowly down the Lyapunov or "energy surface toward the nearest local minimum.The system takes definite hops into the basin of attraction of the fixed point. Second,a synchronous BAM tends to converge faster than an asynchronous BAM.In another word, asynchronous updating should take more iterations to converge. 12
12 Two insights about the rate of convergence First,the individual energies decrease nontrivially. The BAM system does not creep arbitrary slowly down the Lyapunov or “energy” surface toward the nearest local minimum. The system takes definite hops into the basin of attraction of the fixed point. Second,a synchronous BAM tends to converge faster than an asynchronous BAM. In another word, asynchronous updating should take more iterations to converge
Review Neuronal Dynamical Systems We describe the neuronal dynamical systems by first- order differential or difference equations that govern the time evolution of the neuronal activations or membrane potentials. 13
13 Neuronal Dynamical Systems We describe the neuronal dynamical systems by firstorder differential or difference equations that govern the time evolution of the neuronal activations or membrane potentials. ( , , ) ( , , ) X Y X Y y h F F x g F F = = Review
Review Additive activation models x,=-Ax,+∑S,y,)ni+1, i=1 立，=-A,y,+∑S,(x)m,+J Hopfield circuit: i=1 1.Additive autoassociative model; 2.Strictly increasing bounded signal function (S>0); 3.Synaptic connection matrix is symmetric (M=M C=是+s,,m+1 14
14 Additive activation models = = = − + + = − + + n i j j j i i ij j p j i i i j j ji i y A y S x m J x A x S y n I 1 1 ( ) ( ) Hopfield circuit: 1. Additive autoassociative model; 2. Strictly increasing bounded signal function ; 3. Synaptic connection matrix is symmetric . (S 0) ( ) T M = M = − + + j j j ji i i i i i S x m I R x C x ( ) Review
Review Additive bivalent models x1=S,(0y5)mn+1, y=∑S,(x)m,+1 Lyapunov Functions Cannot find a lyapunov function,nothing follows; Can find a lyapunov function,stability holds. 15
15 Additive bivalent models = + = + + + n i ij j k i i k j p j ji i k j j k i y S x m I x S y m I ( ) ( ) 1 1 Lyapunov Functions Cannot find a lyapunov function,nothing follows; Can find a lyapunov function,stability holds. Review
Review 1- OL Ox A dynamics system is Ox,at stable,ifL≤O asymptotically stable,if L<0 Monotonicity of a lyapunov function is a sufficient not necessary condition for stability and asymptotic stability. 16
16 A dynamics system is stable , if ; asymptotically stable, if . L 0 L 0 Monotonicity of a lyapunov function is a sufficient not necessary condition for stability and asymptotic stability. Review = = n i i i n i i i x x L x x L L t