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304 Adaptive Resonance TheoryNote the similarity between Eq. (8.19) and Eq. (6.13) from Section 6.1.3.Both equations describe a competitive layer with on-center off-surround interactions.In Section 6.1.3, we showed how the choice of the functional form ofg(x) influenced the evolution of the activities of the units on the layer. 5 Withoutrepeating the analysis, we assume that the values of the various parameters inEq. (8.19), and the functional form of g(x) have been chosen to enhance theactivity of the single F 2 node with the largest net-input value from F\ accordingto Eq. (8.16). The activities of all other nodes are suppressed to zero. Theoutput of this winning node is given a value of one. We can therefore expressthe output values of F 2 nodes asf 1 T, =max{T A ,}VA:uj = f(x 2] ) = { *• \ (8.20)[ 0 otherwiseWe need to clarify one final point concerning Figure 8.4. The processingelement in that figure appears to violate our standard of a single output per node:The node sends an output of g(x 2 j) to the F 2 units, and an output of f(x2j)to the FI units. We can reconcile this discrepancy by allowing Figure 8.4to represent a composite structure. We can arrange for unit Vj to have thesingle output value x 2 j. This output can then be sent to two other processingelements; one that gives an output of g(x2j), and one that gives an output off(x 2j ). By assuming the existence of these intermediate nodes, or interneurons,we can avoid violating the single-output standard. The node in Figure 8.4 thenrepresents a composite of the Vj nodes and the two intermediate nodes.Top-Down LTM Traces. The equations that describe the top-down LTM traces(weights on connections from F 2 units to F\ units) should be somewhat familiarfrom the study of Chapter 6:^j = (-ztj + h(xu))f(x 2 j) (8.21)Since f(x 2 j} is nonzero for only one value of j (one F 2 node, Vj), Eq. (8.21) isnonzero only for connections leading down from that winning unit. If the jthF 2 node is active and the zth FI node is also active, then Zj, = — z,j + 1 and £,;_/asymptotically approaches one. If the jth F 2 node is active and the zth F, nodeis not active, then Zjj = —z.-,j and z t j decays toward zero. We can summarizethe behavior of z,, as follows:{—Zij + 1 Vj active and v, active—Zij Vj active and ?;,- inactive (8.22)0 Vj and v\ both inactiveRecall from Eq. (8.15) that, if F> is active, then v, can be active only ifit is receiving an input, /,, from below and a sufficiently large net input, Vj,5 Caulion! The function t;(.r) in this section is the analog of f(j-) in Section 6.1.3. In Section 6.1,3,g(.r) was used to mean j-~'/(.r). In this section,