Fig. 8
How initial and new rule strengths determine the adaptation pathway. This figure explores how the network adapts when the rule changes from an initial strength (\(\alpha _{1}\)) to an opposite rule of a different strength (\(-\alpha _{2}\)). The choice of adaptation pathway can be understood as a “race” between the representational module (\(\Delta Z\)) and the relational module (\(\theta\)). Two factors determine the winner. First, as illustrated in panel (a), the ratio of the new rule’s strength to the old one (\(\alpha _{2}/\alpha _{1}\)) determines the “starting position” for the race (here, \(\alpha _2\) is fixed at 1, while \(\alpha _1\) is 2 for the blue trajectory and 0.5 for the red). Second, as shown in panel (b), the strength of the new rule (\(\alpha _{2}\)) influences the relative “speed” of adaptation, changing the direction of the learning process (we used \(\alpha _2=0.5\) for the blue trajectory and \(\alpha _2=2\) for the red, while fixing the starting points by setting \(\alpha _1=\alpha _2\)). (c) The final race outcome for various combinations of \(\alpha _{1}\) and \(\alpha _{2}\). The boundary between the two pathways is determined by the combined strength of the rules. Specifically, the line \(\alpha _{1}\alpha _{2}=1\) (black line) is a good fit for this boundary.
