sum is positive. Thus, is pulled by j towards its value when , the contribution of j in the weighted sum is negative. Then again, is pushed by j towards its value Thus, the values of neurons i and j will converge if the weight between them is positive. Similarly, they will diverge if the weight is negative. Working principles of discrete and continuous Hopfield networks Bruck shed light on the behavior of a neuron in the discrete Hopfield network when proving its convergence in his paper in 1990. A subsequent paper further investigated the behavior of any neuron in both discrete-time and continuous-time Hopfield networks when the corresponding energy function is minimized during an optimization process. Bruck shows that neuron j changes its state if and only if it further decreases the following biased pseudo-cut. The discrete Hopfield network minimizes the following biased pseudo-cut for the synaptic weight matrix of the Hopfield net. where and represents the set of neurons which are −1 and +1, respectively, at time . For further details, see the recent paper. The discrete-time Hopfield Network always minimizes exactly the following pseudo-cut The continuous-time Hopfield network always minimizes an upper bound to the following weighted cut where is a zero-centered sigmoid function. The complex Hopfield network, on the other hand, generally tends to minimize the so-called shadow-cut of the complex weight matrix of the net. Energy Hopfield nets have a scalar value associated with each state of the network, referred to as the "energy", E, of the network, where: This quantity is called "energy" because it either decreases or stays the same upon network units being updated. Furthermore, under repeated updating the network will eventually converge to a state which is a local minimum in the energy function (which is considered to be a Lyapunov function). Thus, if a state is a local minimum in the energy function it is a stable state for the network. Note that this energy function belongs to a general class of models in physics under the name of Ising models; these in turn are a special case of Markov networks, since the associated probability measure, the Gibbs measure, has the Markov property. Hopfield network in optimization Hopfield and Tank presented the Hopfield network application in solving the classical traveling-salesman problem in 1985. Since then, the Hopfield network has been widely used for optimization. The idea of using the Hopfield