Figure 2
From: Constructing neural networks with pre-specified dynamics
Example graphs G (upper left of each panel), matrix \(\mathrm {\textbf{G}}\) (upper right of each panel, with columns showing stimulus, source node and target node), and their associated graph D (lower left of each panel). (a) Graph D forming a cycle of 3 nodes. If a neuron is active in one node and not active in the other node, we arrive at an inconsistency. (b) Inconsistency in (a) is resolved by adding node \(v_{4}\) with same targets as \(v_{1}\) (node \(v_{1}\) is expanded). The associated graph D has no cycles. (c) A graph with 3 nodes and 3 stimuli (dashes stand for unassigned target nodes). Graph D has no cycles because it is not possible to flow from a green arc to a cyan arc. (d) Green and cyan arcs in graph D are superimposed, putting them in the same traversable set. Hence, there is a cycle. (e) Green and cyan arcs in graph D are superimposed but they have opposite directions. These deltas are linked, but arcs labelled with one of these deltas must be inverted. (f) Same graph D as in (e), but cyan arcs have been inverted. All arcs belong to the same traversable set, but no cycles are present. Graph G is consistent. (g) Arcs green and cyan are not directly superimposed but they are linked: defining \(\delta _{1,2}\) or \(\delta _{2,3}\) also defines \(\delta _{1,3}\). (h) Composed superposition appears any time two nodes are reachable by more than one path (regardless of arcs direction).
