time-dependent activity . The temporal evolution has a time constant , which in general can be different for every neuron. This network has a global energy function where the first two terms represent the Legendre transform of the Lagrangian function with respect to the neurons' currents . The temporal derivative of this energy function can be computed on the dynamical trajectories leading to (see for details) The last inequality sign holds provided that the matrix (or its symmetric part) is positive semi-definite. If, in addition to this, the energy function is bounded from below the non-linear dynamical equations are guaranteed to converge to a fixed point attractor state. The advantage of formulating this network in terms of the Lagrangian functions is that it makes it possible to easily experiment with different choices of the activation functions and different architectural arrangements of neurons. For all those flexible choices the conditions of convergence are determined by the properties of the matrix and the existence of the lower bound on the energy function. Hierarchical associative memory network The neurons can be organized in layers so that every neuron in a given layer has the same activation function and the same dynamic time scale. If we assume that there are no horizontal connections between the neurons within the layer (lateral connections) and there are no skip-layer connections, the general fully connected network (), () reduces to the architecture shown in Fig.4. It has layers of recurrently connected neurons with the states described by continuous variables and the activation functions , index enumerates the layers of the network, and index enumerates individual neurons in that layer. The activation functions can depend on the activities of all the neurons in the layer. Every layer can have a different number of neurons . These neurons are recurrently connected with the neurons in the preceding and the subsequent layers. The matrices of weights that connect neurons in layers and are denoted by (the order of the upper indices for weights is the same as the order of the lower indices, in the example above this means that the index enumerates neurons in the layer , and index enumerates neurons in the layer ). The feedforward weights and the feedback weights are equal. The dynamical equations for the neurons' states can be written as with boundary conditions The main difference of these equations from the conventional feedforward networks is