(A) Schematics of the mean-field dynamics after learning. The qualitative difference between the dynamics with or without the constraint is the feedback from to m (dotted curve). (B–E) Average response before (red) and after (blue) learning with different and . As the case with the constraint, the oscillation can be generated by the strong positive feedback and slow negative feedback in the m dynamics, and the synchronous oscillations in and m are determined by the signs of and . When and have the same signs, a positive feedback loop through further boosts the oscillation in the m dynamics (B,E), while the opposite signs of and suppress the oscillation (C,D). Synchrony oscillation between and m requires >0 (B,D), and >0 further increase overall oscillation as well as average rates (B). On the other hand, <0 diminishes the oscillation while decreasing average rates (D). None of these cases can reproduce the experimental observations, synchronous oscillations in and m with a decrease in the average rates. Thus, additional changes as the feedforward synaptic plasticity are still required without the constraint on the pre-synaptic dependence. The simulation of the mean-field dynamics is the same as in Figure 3—figure supplement 1 except wR = −0.1, a = −6.5, b = 5.5, t0 = 400 ms, t1 = 40 ms, =±0.1 and =±1.