Fig. 1. Method demonstration with adaptation of a two-node network.

a The functional dependency of the synthesis terms f₁ and f₂ for g₁ and g₂ are evaluated by a small feed-forward NN (dashed box). f₁ and f₂ (shaded blue and green circles) can depend on all the three variables: g₁, g₂ and the input signal I. Time evolution of the dynamic system corresponds to recurrent iteration of the NN block. The output g(t) = (g₁(t), g₂(t)) is compared with the target value to define the Loss function for training. b We require g₁ to be adaptive to the input change (pink line) and set its target time-course values as indicated by the blue dotted line. No constraints on g₂ are imposed. Blue and green solid lines are the time evolution of g₁ and g₂, respectively, after training. c f₁ (blue dotted line) and f₂ (green dotted line) as functions of I, g₁ and g₂, after training. The three panels show their dependence on I, g₁ or g₂ with the other two variables fixed. In all three subpanels, the horizontal and vertical axis both ranges from 0 to 1. d The regulation network drawn from the information of (c). For example, the first panel of (c) indicates that f₁ increases with I, implying I activates g₁.