states that in order to calculate the updated state of the -th neuron the network compares two energies: the energy of the network with the -th neuron in the ON state and the energy of the network with the -th neuron in the OFF state, given the states of the remaining neuron. The updated state of the -th neuron selects the state that has the lowest of the two energies. In the limiting case when the non-linear energy function is quadratic these equations reduce to the familiar energy function and the update rule for the classical binary Hopfield Network. The memory storage capacity of these networks can be calculated for random binary patterns. For the power energy function the maximal number of memories that can be stored and retrieved from this network without errors is given byFor an exponential energy function the memory storage capacity is exponential in the number of feature neurons Continuous variables Modern Hopfield networks or dense associative memories can be best understood in continuous variables and continuous time. Consider the network architecture, shown in Fig.1, and the equations for neuron's states evolutionwhere the currents of the feature neurons are denoted by , and the currents of the memory neurons are denoted by ( stands for hidden neurons). There are no synaptic connections among the feature neurons or the memory neurons. A matrix denotes the strength of synapses from a feature neuron to the memory neuron . The synapses are assumed to be symmetric, so that the same value characterizes a different physical synapse from the memory neuron to the feature neuron . The outputs of the memory neurons and the feature neurons are denoted by and , which are non-linear functions of the corresponding currents. In general these outputs can depend on the currents of all the neurons in that layer so that and . It is convenient to define these activation functions as derivatives of the Lagrangian functions for the two groups of neuronsThis way the specific form of the equations for neuron's states is completely defined once the Lagrangian functions are specified. Finally, the time constants for the two groups of neurons are denoted by and , is the input current to the network that can be driven by the presented data. General systems of non-linear differential equations can have many complicated behaviors that can depend on the choice of the non-linearities and the