Review 1.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
1.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 4.Additive activation models 衣，=-4x,+∑S,(y)ni+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). Cx=R+s,m,+
4.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 5.Additive bivalent models x+1=∑S,Oy5)m:+1 y=∑S,(x)m,+1 Lyapunov Functions Cannot find a lyapunov function,nothing follows; Can find a lyapunov function,stability holds
5.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 A dynamics system is stable,ifL≤O asymptotically stable,if <O Monotonicity of a lyapunov function is a sufficient not necessary condition for stability and asymptotic stability
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
Review Bivalent BAM theorem. Every matrix is bidirectionally stable for synchronous or asynchronous state changes. Synchronous:update an entire field of neurons at a time. ● Simple asynchronous:only one neuron makes a state- change decision. Subset asynchronous:one subset of neurons per field makes state-change decisions at a time
Bivalent BAM theorem. Every matrix is bidirectionally stable for synchronous or asynchronous state changes. • Synchronous:update an entire field of neurons at a time. • Simple asynchronous:only one neuron makes a statechange decision. • Subset asynchronous:one subset of neurons per field makes state-change decisions at a time. Review