One Circuit, Many Flow Fields: Mechanistic Models of Single-Trial Neural Dynamics

A single neural circuit can exhibit qualitatively different dynamics across trials: one circuit, many flow fields. Standard models treat this trial-to-trial variability as noise around a fixed dynamical system or as discrete switches between regimes, yet neither captures how continuous internal-state variables, such as arousal or engagement, can gradually deform the circuits flow field. We propose that singletrial fitting can be reframed as inferring the low-dimensional control parameters that reshape a shared circuits flow field. We realize this with a low-rank recurrent network in which trial-specific static input biases act as bifurcation parameters: constant within a trial, they deform the flow field without directly driving activity over time. In a teacher-student setting, the model recovers the underlying dynamical system and its bifurcation structure from activity alone. Applied to large-scale recordings of mouse motor cortex during a delayed movement task, the model identifies a disengagement axis that separates engaged from disengaged trials and, when perturbed in silico, causally shifts the flow field between engaged and disengaged regimes. A generative extension reproduces the distribution of single-trial activity, and the inferred latent structure partially transfers across sessions and animals, suggesting shared low-dimensional structure across motor-cortical circuits. Together, these results reframe a methodological problem of fitting single-trial activity as a scientific opportunity: reading off the control parameters of the underlying dynamics, and connecting data-driven inference of neural dynamics to mechanistic theories of how a single circuit reuses its dynamics for flexible behavior.