Fig. 7
The adaptation dynamics in the \(\Delta Z \times \theta\) plane. Visualization of the two adaptation pathways as two trajectories in the simplified model’s internal state represented by \(\Delta Z\) and \(\theta\). The arrows represent the “downhill” direction for learning: the opposite direction of the loss gradient with respect to \(\Delta Z\) and \(\theta\). The blue and red nullclines represent the loci where \(\dot{\Delta Z}=0\) and \(\dot{\theta }=0\), respectively. Their intersections correspond to fixed points in which learning halts. They intersect at the stable fixed points at \(\Delta Z=\theta =\pm r/\sqrt{2}\), and at the unstable fixed point \(\Delta Z=\theta =0\). Before reversing the rule, the network start with positive \(\Delta Z\) and \(\theta\). Following the reversal of the rule, \(\Delta Z\) flips its sign (yellow circles). Adaptation when \(\alpha\) is large (left panel) leads to reversing the sign of \(\Delta Z\), returning to the positive fixed point, whereas when \(\alpha\) is small (right panel) it is \(\theta\) that changes its sign to match the negative \(\Delta Z\).
