Symmetries in stimulus statistics shape the form of visual motion estimators

Abstract

The estimation of visual motion has long been studied as a paradigmatic neural computation, and multiple models have been advanced to explain behavioral and neural responses to motion signals. A broad class of models, originating with the Reichardt correlator model, proposes that animals estimate motion by computing a temporal cross-correlation of light intensities from two neighboring points in visual space. These models provide a good description of experimental data in specific contexts but cannot explain motion percepts in stimuli lacking pairwise correlations. Here, we develop a theoretical formalism that can accommodate diverse stimuli and behavioral goals. To achieve this, we treat motion estimation as a problem of Bayesian inference. Pairwise models emerge as one component of the generalized strategy for motion estimation. However, correlation functions beyond second order enable more accurate motion estimation. Prior expectations that are asymmetric with respect to bright and dark contrast use correlations of both even and odd orders, and we show that psychophysical experiments using visual stimuli with symmetric probability distributions for contrast cannot reveal whether the subject uses odd-order correlators for motion estimation. This result highlights a gap in previous experiments, which have largely relied on symmetric contrast distributions. Our theoretical treatment provides a natural interpretation of many visual motion percepts, indicates that motion estimation should be revisited using a broader class of stimuli, demonstrates how correlation-based motion estimation is related to stimulus statistics, and provides multiple experimentally testable predictions.

Fig. 1.

Fundamentals of motion estimation. (A) The task of the organism is to estimate its rotational velocity from the unknown contrast pattern C(θ, t). (B) The simplest Reichardt correlator uses time delays τ to estimate motion. The mirror symmetry makes the model sensitive to motion in either direction. (C) The generalized Reichardt hypothesis allows arbitrary filters {f₁^(ij), f₂^(ij), f₃^(ij), f₄^(ij)} before the multiplicative nonlinearity as well as filters {f₅^(ij), f₆^(ij)} postnonlinearity.

Fig. 2.

The utility of odd-ordered estimators. (A) Schematic showing two- and three-point correlators to detect rightward motion. (B) When motion is rightward, the two-point correlator returns a positive response to light or dark spots (small black ovals). The three-point correlator gives a positive response to light spots (large black oval) and a negative response to dark spots (large red oval). This difference yields an ambiguity between rightward moving light spots and leftward moving dark spots. (C) Motion signals also arise randomly. This coincidence is less likely for dark spots, so the estimator should interpret negative three-point output as motion to the right.

Fig. 3.

The structure of multipoint correlators. (A–C) Left, probability of a dark signal to be 0.4; Center, 0.5; and Right, 0.6. (A) To demonstrate the asymmetry, we show an example stimulus and cartoon of the contrast prior. (B) The Reichardt correlator is insensitive to P_dark. (C) The third-order correlator varies strongly with P_dark. The sign of the kernel inverts as P_dark crosses 0.5, indicating the switch of confidence (Fig. 2). (D) Structure of the Reichardt and third-order correlator, fixing the third, second, or first coordinate.