Fig. 3: Nonoptimal networks balance periods of stability and instability.
a, A linear subsystem of active neurons can be tuned to encode a continuum of orientations over a fixed interval (heatmap; left). Multiple line attractors can be stitched together at orientations where the active subset simultaneously gains and loses an active neuron (middle), thereby generating a ring attractor (right). b, Without precise tuning, each linear subsystem (shaded region; left) encodes a single unstable or stable fixed point (‘FP’; markers). When stitched together (middle), the set of linear subsystems can stably encode only a finite number of orientations (‘point attractors’; right). c, Top: the dynamics of each linear subsystem are governed by the leading eigenvalue λ of the active submatrix of the connectivity (Fig. 2h). Bottom: in the unstable regime (orange), the bump accelerates away from an unstable fixed point at rate λu > 0; in the stable regime (turquoise), the bump decelerates toward a stable fixed point at rate λs < 0. d, Bump dynamics depend on the fixed-point orientations (square markers), drift rates λ (color map), and angular span of each regime (colored areas). Illustrated without velocity input. e–h, Bump dynamics without velocity input. e, Simplified energy landscape. f, Same as e for different JE. As JE approaches an optimal value, one region of the landscape flattens and fills the entire ring; the other sharpens and shrinks in span. g, Bump dynamics for energy landscapes in f. h, Net drift speed, computed analytically (line) and by simulation (markers). i–l, Bump dynamics with velocity input. i, Small velocities shift the fixed points toward the boundary between stable and unstable regimes, tipping the energy landscape in the direction of the input. At a threshold velocity (equation (5)), the fixed points meet at the boundary, and the bump slides continuously down the landscape. j, Same as i for different JE, given a fixed input velocity. JE affects how quickly the fixed points move through the energy landscape, and, thus, how readily the landscape tips for a given velocity. k, Bump dynamics for energy landscapes in j. l, Threshold velocity (solid curve) and linearity of integration (dashed curves), computed analytically (lines) and by simulation (markers).
