Extended Data Fig. 9: Low-dimensional latent dynamics in networks performing the delayed match-to-sample (DMS) task.

(a) Circuit diagram representing latent dynamics for a minimal network trained on the DMS task (Eq. (51)). The network was of rank two, so that the latent dynamics were described by two internal variables κ₁ and κ₂. Input A acts as a modulator on the recurrent interactions between the two internal variables. (b) Dynamical landscape for the autonomous latent dynamics in the κ₁ − κ₂ plane (ie. the m⁽¹⁾-m⁽²⁾ plane). Colored lines depict trajectories corresponding to the 4 types of trials in the task (see Sup. Fig. S6 for details of trajectories). Background color and white lines encode the speed and direction of the dynamics in absence of inputs. (c) Two 2d projections of the seven-dimensional connectivity space, with colors indicating the two subpopulations and lines corresponding to linear regressions for each of them on the right panel. (d) Effective circuit diagrams in absence of inputs (left), and when input A (middle) or input B (right) are present (see Supplementary Note 2.4). Filled circles denote positive coupling, open circles negative coupling. Input A in particular induces a negative feedback from κ₂ to κ₁. (e) Distribution of neural gains for each populations (pop. 1: n = 3050, pop. 2: n = 1046), in the three situations described above. The gain of population 1 (green) is specifically modulated by input A. (f) Dynamical landscapes in the 3 situations described above (see Methods). Filled and empty circles indicate respectively stable and unstable fixed points. The negative feedback induced by input A causes a limit cycle to appear in the latent dynamics.