Hopfield NetworkHopfield network is a special kind of neural network whose response is different from other neural networks. It is calculated by converging iterative process. It has just one layer of neurons relating to the size of the input and output, which must be the same. When such a network recognizes, for example, digits, we present a list of correctly rendered digits to the network. Subsequently, the network can transform a noise input to the relating perfect output. In 1982, John Hopfield introduced an artificial neural network to collect and retrieve memory like the human brain. Here, a neuron is either on or off the situation. The state of a neuron(on +1 or off 0) will be restored, relying on the input it receives from the other neuron. A Hopfield network is at first prepared to store various patterns or memories. Afterward, it is ready to recognize any of the learned patterns by uncovering partial or even some corrupted data about that pattern, i.e., it eventually settles down and restores the closest pattern. Thus, similar to the human brain, the Hopfield model has stability in pattern recognition. A Hopfield network is a singlelayered and recurrent network in which the neurons are entirely connected, i.e., each neuron is associated with other neurons. If there are two neurons i and j, then there is a connectivity weight w_{ij} lies between them which is symmetric w_{ij} = w_{ji }. With zero selfconnectivity, W_{ii} =0 is given below. Here, the given three neurons having values i = 1, 2, 3 with values X_{i}=±1 have connectivity weight W_{ij}. Updating rule:Consider N neurons = 1, … , N with values X_{i = }+1, 1. The update rule is applied to the node i is given by: If h_{i } _{≥ 0} then x_{i} _{→ } 1 otherwise x_{i → } 1 Where h_{i }= is called field at i, with b£ R a bias. Thus, x_{i} _{→} sgn(h_{i}), where the value of sgn(r)=1, if r ≥ 0, and the value of sgn(r)=1, if r < 0. We need to put b_{i}=0 so that it makes no difference in training the network with random patterns. We, therefore, consider h_{i}=. We have two different approaches to update the nodes: Synchronously: In this approach, the update of all the nodes taking place simultaneously at each time. Asynchronously: In this approach, at each point of time, update one node chosen randomly or according to some rule. Asynchronous updating is more biologically realistic. Hopfield Network as a Dynamical system:Consider, K = {1, 1} ^{N} so that each state x £ X is given by x_{i }£ { 1,1 } for 1 ≤ I ≤ N Here, we get 2^{N} possible states or configurations of the network. We can describe a metric on X by using the Hamming distance between any two states: P(x, y) = # {i: xi≠y_{i}} N Here, P is a metric with 0≤H(x,y)≤ N. It is clearly symmetric and reflexive. With any of the asynchronous or synchronous updating rules, we get a discretetime dynamical system. The updating rule up: X → X describes a map. And Up: X → X is trivially continuous. Example:Suppose we have only two neurons: N = 2 There are two nontrivial choices for connectivities: w_{12 }= w_{21 }= 1 w_{12}= w_{21 }= 1 Asynchronous updating: In the first case, there are two attracting fixed points termed as [1,1] and [1,1]. All orbit converges to one of these. For a second, the fixed points are [1,1] and [1,1], and all orbits are joined through one of these. For any fixed point, swapping all the signs gives another fixed point. Synchronous updating: In the first and second cases, although there are fixed points, none can be attracted to nearby points, i.e., they are not attracting fixed points. Some orbits oscillate forever. Energy function evaluation:Hopfield networks have an energy function that diminishes or is unchanged with asynchronous updating. For a given state X ∈ {−1, 1} N of the network and for any set of association weights W_{ij} with W_{ij} = w_{ji} and w_{ii} =0 let, Here, we need to update X_{m} to X'_{m} and denote the new energy by E' and show that. E'E = (X_{m}X'_{m} ) ∑_{i≠m}WmiXi. Using the above equation, if X_{m} = X_{m}' then we have E' = E If X_{m} = 1 and X_{m}' = 1 , then X_{m}  X_{m}' = 2 and hm= ∑_{i}WmiXi ? 0 Thus, E'  E ≤ 0 Similarly if X_{m} =1 and X_{m}'= 1 then X_{m}  X_{m}' = 2 and h_{m}= ∑_{i}WmiXi < 0 Thus, E  E' < 0. Note:
