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 circuit's flow field. We propose that single-trial fitting can be reframed as inferring the low-dimensional control parameters that reshape a shared circuit's 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. We validate the model in a teacher-student setting, where it recovers the ground-truth dynamics and bifurcation structure from activity alone. Applied to mouse and monkey motor cortex, the model infers control spaces with interpretable axes that track behavioral engagement in mouse and task structure in monkey. A generative extension reproduces the structure of trial-to-trial variability and partially transfers across sessions and subjects. 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.