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Hopfield Network

The Hopfield Network can be thought of as a network automaton with a complete graph. The nodes (or neurons) are binary units, and the activity rule is a simple threshold rule, where the weighted inputs to a node are summed and compared to a threshold value. The weights are defined in the adjacency matrix.

Netomaton already comes with a Hopfield Net implementation. To use it, we must first train the network, by giving it a set of patterns:

from netomaton import *

zero = [
    0, 1, 1, 1, 0,
    1, 0, 0, 0, 1,
    1, 0, 0, 0, 1,
    1, 0, 0, 0, 1,
    1, 0, 0, 0, 1,
    0, 1, 1, 1, 0]
one = [
    0, 1, 1, 0, 0,
    0, 0, 1, 0, 0,
    0, 0, 1, 0, 0,
    0, 0, 1, 0, 0,
    0, 0, 1, 0, 0,
    0, 0, 1, 0, 0]
two = [
    1, 1, 1, 0, 0,
    0, 0, 0, 1, 0,
    0, 0, 0, 1, 0,
    0, 1, 1, 0, 0,
    1, 0, 0, 0, 0,
    1, 1, 1, 1, 1]
# replace the zeroes with -1 to make these vectors bipolar instead of binary
one = [-1 if x == 0 else x for x in one]
two = [-1 if x == 0 else x for x in two]
zero = [-1 if x == 0 else x for x in zero]

hopfield_net = HopfieldNet(n=30)

hopfield_net.train([zero, one, two])

As is seen above, we must instantiate an instance of a HopfieldNet, specifying the number of nodes in the network. Then, we must call train, providing a list of training examples.

Using the Hopfield Net involves providing a potentially incomplete pattern, and evolving the network for a pre-specified number of timesteps. The network state should settle into a pattern that resembles those seen during training. It acts like a content-addressable (associative) memory.

half_two = [
        0, 0, 0, 0, 0,
        0, 0, 0, 0, 0,
        0, 0, 0, 0, 0,
        0, 1, 1, 0, 0,
        1, 0, 0, 0, 0,
        1, 1, 1, 1, 1]
half_two = [-1 if x == 0 else x for x in half_two]

initial_conditions = half_two

trajectory = evolve(initial_conditions=initial_conditions, 
                    network=hopfield_net.network,
                    timesteps=hopfield_net.num_nodes * 7, 
                    activity_rule=hopfield_net.activity_rule)

# view the time evolution of the Hopfield net as it completes the given pattern
activities = get_activities_over_time_as_list(trajectory)
animate_activities(activities[::hopfield_net.num_nodes], shape=(6, 5), interval=150)

The full source code for this example can be found here.

See the following resources for some more information on Hopfield nets:

J. J. Hopfield, "Neural networks and physical systems with emergent collective computational abilities", Proceedings of the National Academy of Sciences of the USA, vol. 79 no. 8 pp. 2554–2558, April 1982.