Table showing examples of where patterns may converge in multiple trials in the Hopfield network and in the network after Gram-Schmidt orthogonalization when two very similar patterns are inscribed in a network of size N = 100.

We first inscribe 4 patterns (ξ(1) to ξ(4)) in the network, then choose ξ(ν) randomly from one of these 4 to make a fifth pattern ξ(5) which is similar (98% similarity) to it, and differs on the sites marked as ‘Differences’. We now store this fifth pattern and check whether all the 5 patterns are attractors. The inscribed patterns other than the chosen ξ(ν) remain attractors even as p changes from 4 to 5. We are interested in the situation with the two similar patterns, and whether ξ(ν) and ξ(5) are attractors (✓) or not (×), when p = 5. When ξ(5) is presented to the network, three possibilities arise: (i) ξ(5) falls within the basin of attraction of ξ(ν) and converges there (Case 1), (ii) ξ(5) is an attractor, and ξ(ν) which was previously stable now falls within the basin of attraction of ξ(5) (Case 2), or (iii) both ξ(5) and ξ(ν) converge to a third pattern which is not any of the inscribed patterns and has 99% overlap with both ξ(5) and ξ(ν) (Case 3). We have used examples from different trials to illustrate each of these cases.