the presence of the second term, which is responsible for the feedback from higher layers. These top-down signals help neurons in lower layers to decide on their response to the presented stimuli. Following the general recipe it is convenient to introduce a Lagrangian function for the -th hidden layer, which depends on the activities of all the neurons in that layer. The activation functions in that layer can be defined as partial derivatives of the Lagrangian With these definitions the energy (Lyapunov) function is given by If the Lagrangian functions, or equivalently the activation functions, are chosen in such a way that the Hessians for each layer are positive semi-definite and the overall energy is bounded from below, this system is guaranteed to converge to a fixed point attractor state. The temporal derivative of this energy function is given by Thus, the hierarchical layered network is indeed an attractor network with the global energy function. This network is described by a hierarchical set of synaptic weights that can be learned for each specific problem. See also Associative memory (disambiguation) Autoassociative memory Boltzmann machine – like a Hopfield net but uses annealed Gibbs sampling instead of gradient descent Dynamical systems model of cognition Ising model Hebbian theory References External links Hopfield Network Javascript The Travelling Salesman Problem – Hopfield Neural Network JAVA Applet Neural Lab Graphical Interface – Hopfield Neural Network graphical interface (Python & gtk) Neural network architectures