Symmetry constraints on feedback-driven orientation tuning dynamics in translation-invariant models of primate V1

Orientation selectivity in primate V1 is shaped by feedforward drive, intracortical recurrence, and corticocortical feedback. Yet a basic conceptual question remains: what aspects of orientation tuning can (or cannot) be altered by a recurrent/feedback loop that is rotationally symmetric in cortical space? We develop a minimal translation-invariant linear field model in which stimulus drive and recurrent interactions are analyzed in the spatial Fourier domain (k, {phi}). Our central analytic result is an exact symmetry constraint: for any temporally filtered yet isotropic recurrent coupling W0(k, {omega}), the response at fixed (k, {omega}) is multiplied by an angle-independent complex gain. Consequently, any normalized orientation metric computed at fixed k--including circular variance, orientation selectivity index (OSI), and bandwidth--is invariant under isotropic recurrence. Isotropic recurrence can nevertheless rescale gain and select a spatial band via its k-dependence. We then show that adding a weak anisotropic component, especially when endowed with a slower timescale than a fast isotropic pathway, produces delayed sharpening and modest preferred-angle drifts. A perturbative calculation predicts linear scaling of tuning changes with anisotropy strength, and a simple phasor-sum formula explains{phi} *(t) drift when feedforward and feedback anisotropies compete. We provide stability criteria for the temporally filtered (auxiliary-variable) implementation and verify them numerically. Finally, we identify concrete regimes in which isotropic recurrence can alter measured tuning in practice, including finite-k readouts with k-dependent feedforward anisotropy and nonlinear (tuned) divisive-normalization readouts. Together these results sharpen a null hypothesis for perturbation experiments and delineate the minimal feature-specific ingredients required to explain dynamic tuning in macaque V1. Author summaryNeurons in the primary visual cortex (V1) respond more strongly to some stimulus orientations than others. This "orientation selectivity" is often attributed to feedforward input from the thalamus, but it is also shaped by recurrent and feedback connections within and between cortical areas. A common intuition is that if feedback is not itself tuned for orientation--for example, if it is spatially symmetric and acts like a global gain control--then it should change response strength without changing tuning. Here we turn that intuition into an exact mathematical statement in a simple but widely used approximation: locally translation-invariant, small-signal (linearized) dynamics analyzed in the spatial Fourier domain. We prove that any rotationally symmetric recurrent coupling cannot change normalized tuning metrics at a fixed spatial frequency, even if it has rich temporal filtering. We then show what is minimally required to change tuning: a weak orientation-biased component, especially if it has a slower timescale than an untuned pathway. This two-timescale mechanism produces delayed sharpening and small drifts in preferred orientation, features reported in macaque V1. We also explain why tuning can still appear to change under untuned feedback in real experiments, for example when measurements combine multiple spatial scales or when normalization pools are orientation selective.