Introduction

Humans and many vertebrates rely on stabilizing gaze on regions of interest in the world to achieve accurate perception. This fixation strategy, combined with the continuous eye, head, and body movements that accompany natural behavior, generates highly structured patterns of motion on the retina that can, in principle, be exploited to control locomotion. A particularly simple and influential case arises when we move approximately in the same direction as we are looking: in this situation the focus of expansion (FOE), the point in the optic flow from which motion vectors appear to radiate, directly specifies the heading direction (Gibson, 1950). The idea that the visual system uses the FOE to recover heading has subsequently dominated both theoretical and empirical work, including the interpretation of neural activity in primate medial superior temporal (MST) area, where neurons exhibit FOE-like tuning (Bremmer et al., 2017; Britten, 2008; Duffy and Wurtz, 1991; Kaminiarz et al., 2014), as well as psychophysical findings showing that humans can make highly accurate heading judgements from optic flow (Li and Warren, 2002; Li and Warren, 2000; Warren and Hannon, 1990; Warren and Hannon, 1988). Building on these observations, computational models have proposed population codes and neural network mechanisms (Beintema et al., 2004; Berg and Beintema, 2000; Heeger and Jepson, 1992; Lappe and Rauschecker, 1993; Perrone and Stone, 1994) that operate on radial flow structure to recover the translation direction within this FOE-centric framework.

However, this FOE-centric view faces several fundamental challenges. In natural environments, observers rarely maintain fixation on their heading direction; instead, they typically look toward behaviorally relevant areas, sometimes located eccentrically in the visual field (Matthis et al., 2018). Under combined eye–head movements, the retinal FOE no longer coincides with the true heading direction, creating a computational problem for relying on the FOE for heading estimation (Royden et al., 1994, 1992), because in this case the retinal flow contains both translational and rotational components. To preserve FOE-based control of navigation, two main strategies have been proposed to decompose the flow field: retinal approaches that attempt to detect and compensate for the rotational component directly from the retinal flow itself (e.g., (Cutting et al., 1992; Heeger and Jepson, 1992; Lappe and Rauschecker, 1993; Longuet-Higgins and Prazdny, 1980)), and extra-retinal approaches that rely on eye movement signals to factor out the rotation (Lappe, 1998; Royden et al., 1994).

Despite these challenges, a functional role of rotational flow components has been largely neglected in both theoretical and experimental work on heading perception. Although some models have included rotation sensitivity or extra-retinal signals to account for eye-movement compensation (Beintema and van den Berg, 1998; Koenderink and Van Doorn, 1981; Lappe, 1998), the curl component of optic flow has typically been treated as something to be removed rather than as a potential source of heading information in its own right. A few exceptions include studies that have considered motion-parallax cues, implying rotation, to determine instantaneous heading (Li and Warren, 2000) or to estimate gaze direction (Cutting, 1986).

Recent evidence suggests that this oversight may be consequential. In real walking scenarios, the FOE expressed in a head-centred reference frame is too variable to be used reliably for heading recovery, whereas the magnitude of retinal curl in the fovea can specify the body trajectory relative to gaze (Matthis et al., 2022). Psychophysical studies have also shown that heading estimates exhibit systematic biases that FOE-based mechanisms cannot fully account for under simulated rotation — i.e. when the retinal flow is indistinguishable from that produced by actual eye or head rotations but without accompanying vestibular or proprioceptive signals (Banks et al., 1996; van den Berg and Brenner, 1994). This bias suggests that rotation is not merely treated as noise, but instead provides information used in heading perception. In addition, neurophysiological work has identified that most MSTd neurons are sensitive to spiral motion (Graziano et al., 1994), consistent with a continuum of motion pattern sensitivity (Layton and Browning, 2014) rather than a decomposition into separate channels (e.g. expansion–contraction vs. rotation components). This continuum is consistent with the amount of rotational components in the retinal image as a function of gaze eccentricity, making gaze-stabilisation or fixation strategies critical for understanding the use of retinal flow patterns (Angelaki and Hess, 2005; Calow and Lappe, 2008; Glennerster et al., 2001).

In this work, we provide the empirical evidence that curl can serve as a control variable during locomotion. By relying on gaze stabilization, which generates a retinal curl component, we reveal a robust heading bias that emerges from prolonged exposure to this underlying curl. The direction of this bias differs from the previously reported simulated-rotation bias (see supplementary videos S1 and S2): when observers viewed flow consistent with linear walking while fixating an eccentric ground target, perceived heading shifts opposite to gaze (left fixation leads to rightward heading, and vice versa). The magnitude of this bias matches the curl expected by gaze-centred flow geometry, and cancelling curl eliminates the bias. A simple controller based on mean image curl reproduces heading behaviour across path conditions. Finally, we present a neurally plausible implementation of this controller illustrating how parietal circuits can transform local curl near gaze into global heading estimates via recurrent dynamics and competition between sensory evidence and priors—without explicit FOE extraction. This reframes retinal curl from a nuisance component to a functional control signal for heading.

Methods

Participants

We tested 12 participants (five self-identified men and seven self-identified women), aged between 24 and 59 years (mean 30 SD: 9), all with normal or corrected-to-normal vision. Except for one, all participants were naïve to the aims of the study and volunteered to take part. The study forms part of an ongoing research program approved by the Ethics Committee of the University of Barcelona and conducted in accordance with the principles of the Declaration of Helsinki.

Displays and conditions

Participant motion was simulated as a translation parallel to the ground plane at a sustained walking speed of approximately 1 m/s. This motion incorporated characteristic bounce and swing components derived from a single gait profile. This profile was recorded once by an independent individual wearing an HMD tracker while walking along the predefined experimental paths in a virtual environment, and then was applied to all participants to ensure stimulus consistency. Each trial lasted between 11 and 12 s.

The ground plane consisted of a 50 × 50 m surface mapped with a naturalistic texture (see Fig. 1A) generated from simplex noise patterns whose spatial power spectrum followed a 1/f ² distribution. The temporal frequency of these patterns generated by simulated self-motion is consistent with the statistics of natural videos as described by (Dong and Atick, 1995) also following a 1/f-type temporal power spectra (exponent of 2). The textures were created in real time using OpenGL shaders within a custom Python program, designed for computational efficiency and allowing online manipulation of the texture in specific experimental conditions (see Flow manipulation conditions below).

Figure 1.

The experiment was run on an Intel i7-based workstation (i7-9700F, Intel, Santa Clara, CA, USA) equipped with an NVIDIA GeForce RTX 2060 SUPER GPU. Images were rendered at 120 Hz with a resolution of 1920 × 1080 pixels and displayed monocularly via a PROPixx projector (VPIxx Technologies, Saint-Bruno, QC, Canada) onto a back-projection screen (2.03 m × 1.16 m), viewed from a distance of 1.0 m, resulting in a visual field of approximately 91º.

To verify fixation, eye movements were recorded with a Pupil Labs Core (Berlin, Germany) eye tracker operating at 200 Hz. Trials were discarded if the median distance between gaze position and the fixation point exceeded 3º.

Path conditions

Participant trajectories could be either straight (length of about 6.5 m) or curved to the left or right (see Fig. 1B). The curved paths resulted in a final heading of ±45° relative to the initial heading (0°) in world coordinates. The three path types (straight, left, and right) were interleaved on a trial-by-trial basis.

Eccentricity conditions

In all trials, a fixation point (yellow dot in Figure 1A) was presented on the ground and remained fixed in world coordinates. Relative to the participant’s initial position (x = 0, z = 0) and heading (0°), the fixation point could appear at one of five lateral positions or eccentricities, x = {−4, −2, 0, 2, 4} m, and was initially located 20 m ahead (see colored dots in Fig. 1B). As simulated self-motion progressed, the fixation point appeared to approach the observer, necessitating a gradual increase in gaze angle. Participants were not explicitly instructed to use specific eye or head movements to maintain fixation; instead, they were permitted to engage the head-eye system naturally to track the target. Under perfect fixation, the expected head/eye rotation rate increased for the largest eccentricity from about 0.4 º/s to 0.8°/s (see Figure S2 in the SI). The eccentricity condition was randomized across trials.

Flow (curl) manipulation conditions

When fixating a stationary eccentric point while translating straight ahead, the retinal image contains an expected rotational component— referred to as curl—around the fovea. This effect is illustrated by the flow lines in Fig. 1A, which shows a snapshot of the retinal image consistent with walking straight ahead while fixating a point located to the left (positive curl). Figures 1C–E display the mean curl distributions for fixation points at varying eccentricities: leftward positions (−4 m, −2 m) produced positive curl, the center (0 m) resulted in near-zero curl, and rightward positions (2 m, 4 m) produced negative curl.

To compute the curl shown in Figs. 1C–E, we simulated the experimental trials and generated the corresponding videos (downsampled to 800 × 452 pixels at 30 Hz). Optic flow was then computed using the Farnebäck algorithm (calcOpticalFlowFarneback, OpenCV), which gives us an estimate of image motion for every pixel (x, y) at each frame t. The curl was extracted for each frame sequence from the image 2D flow:

where u and v are the horizontal and vertical flow components respectively (in pixels per frame). We computed the mean image curl for each frame. These values approximate the mean retinal curl expected under accurate fixation and define the unaltered curl condition.

In some conditions, this expected curl was counteracted (cancelled curl condition) by adding an equal and opposite rotational component centered on the fixation point. Figure 1C,E also show the corresponding curl histograms (lighter colors), which are centered at zero—matching the distribution obtained when fixating straight ahead in the direction of motion (mean zero curl, Figure 1D). In an additional set of trials, the imposed counter-rotation was doubled to produce an overcancelled condition, in which the curl was reversed beyond neutralization (histograms not shown in Figure 1).

Procedure

Participants continuously reported their perceived direction of self-motion while maintaining fixation on the yellow dot. The response was given using a rotative encoder as input device, operated like a steering wheel, to align its orientation with the perceived heading direction. The device had an angular resolution of 0.3°. At the beginning of each session, participants completed a five-point calibration procedure for the eye tracker. Each session comprised 45 trials (3 trajectory types × 5 fixation positions × 3 flow manipulation conditions), and each participant completed a total of 10 sessions.

Computation of perceived path

To reconstruct the participant’s estimated path from the angular response, we treated the reported heading angle θ_t (radians) as specifying the instantaneous lateral slope of the perceived trajectory at frame t. For each time step, we computed the incremental simulated forward displacement

and converted the response angle into a lateral increment via

The estimated lateral position was then obtained by forward integration starting from the initial position x₀:

This yields the sample-wise reconstructed lateral trajectory x_t consistent with the participant’s angular responses.

Real-Time Control Model (The controller)

We interpret the reported path as the output of a curl–driven feedback mechanism inspired by point attractor models used previously in steering control (Fajen and Warren, 2007; Wilkie and Wann, 2003a). Let θ_t denote the heading direction (direction of forward velocity in world coordinates) and ψ_t the gaze direction. We consider their difference

as a instantaneous heading error. The mean optic–flow curl is taken as a sensory measurement of this discrepancy, so that , with k_c being a curl–to–angle scale factor. This is motivated by the fact that curl increases whenever there is a discrepancy between θ and ψ. The observer controls heading by applying a yaw rate input :

so that positive curl induces a leftward turn (increasing θ) and negative curl induces a rightward turn. If gaze is held fixed (e.g. steady fixation), and therefore

This is a simple first-order linear differential equation, and the heading error decays exponentially with a correction time constant 1/K_p:

Thus, heading is continuously steered toward gaze until curl vanishes. In our reconstruction we integrate this controlled heading forward in time to generate the perceived trajectory implied by the observed curl (see video S3 in the SI).

Trajectory fits

To test if perceived trajectories can be accounted for by the curl signal, we used the controllerdefined in Equation 5 which we integrated forward with an explicit Euler step to obtain the estimated heading θ. With sampling interval Δt, constant forward speed (v), and initial state (θ₀, x₀, z₀) taken from the first sample of each trial, the discrete update is

This generates a predicted 2D path from the observed mean curl sequence. We then estimate K_p and forward velocity (v) as free parameters by minimizing a time-warped path discrepancy (J) using Dynamic Time Warping (DTW)(Giorgino, 2009) on the (z, x) sequences between the predicted and perceived paths:

where

is the observed path.

We fix K_c (Equation 5) to 1, absorbing the unknown curl-to-angle scale into K_p.

We used two fitting approaches: (1) independent or separate fits, in which parameters were fitted independently for each combination of heading, gaze eccentricity, and retinal flow manipulation and (2) join fits, in which a single set of parameters is used across all heading and gaze eccentricity conditions but they were different for each flow manipulation. Each approach was tested with two parameters (K_p, and the forward velocity, v). The motivation for including forward velocity in the fit was to compensate for variations in the timing of perceived heading responses. For the separate fits, the number of parameters was 30 (for the 2-parameter approach, K_p and v). In contrast, the joined approach used only 2 parameters for all heading and eccentricity conditions.

In order to compare the different models, we employed information criteria adapted for distance-based model comparison. For each model, we computed a surrogate log-likelihood based on the normalized alignment cost:

where D represents the total DTW distance and n the number of observations, with D/n representing the average alignment cost per observation. This normalization ensures appropriate scaling for information criteria computation. We then computed the Akaike Information Criterion as:

where k represents the number of parameters.

Results

All the data and code for the analysis are available at osf link. First, we checked the quality of fixation. The median trial-based deviation from the fixation dot ranged from 1.02^∘ to 1.55^∘ across subjects. On average, the gaze-to-fixation-dot distance exceeded 3^∘ in only 0.46% of the trials which were removed from further analysis.

Figure 2 shows the instant perceived heading in the different gaze and paths conditions in which retinal flow was not altered. The thin traces show this pattern for individual observers, and the thick coloured traces show the group means. Figure 2 illustrates a clear and systematic lateral bias in perceived heading as a function of the inital gaze direction. When observer-simulated motion is straight but fixated an eccentric point on the ground (central vertical panels, blue and green lines), their reported instantaneous heading consistently shifted away from the fixation point: leftward fixation led to rightward heading reports, and rightward fixation produced the opposite pattern. The perceived final lateral displacement is significantly different from 0 (straight ahead, t(11)=5.172, p<0.001) and the magnitude of the bias was not different between sides (F(1,35)<1, p=0.4). This effect was present both when gaze was displaced by 2 m (top row) and 4 m (bottom row), and the effect scales with the lateral distance of the fixation point from the trajectory path (the final lateral distance between 2 m and 4 m condition was marginally significant, F(1,35)=3.49, p=0.07). As we shall see below, the magnitude of the bias is well accounted for by the present mean curl.

Figure 2.

Note that when both simulated motion and gaze are straight ahead (orange traces), there is essentially no bias: path is perceived straight. This is consistent with curl staying near zero throughout the trial, so the controller receives no systematic steering signal (see Figure 1D). In this case, heading and gaze remain aligned and perceived trajectory stays close to the physical straight path.

Although the bias is more easily interpretable in the straight ahead condition, systematic effects due to gaze direction are present in the curved paths as well. Perceived curvature was systematically underestimated when gaze was directed into the direction of the physical curve: for leftward curves this underestimation is most evident when gaze is to the left (blue traces), and for rightward curves when gaze is to the right (green traces). This follows directly from the curl dynamics: as the observer-simulated motion proceeds along the curved path, the instantaneous mean curl decreases, because the heading and gaze directions become progressively aligned (i.e. ϕ(t) → 0). Under the proportional controller we have defined above, the yaw command scales with this error, ω(t) = K_pϕ(t), hence reduced curl implies reduced corrective turning. The model, therefore, predicts a progressive flattening of the perceived path as curl dies out over time. This is exactly the pattern observed empirically, and it is symmetric for left and right curves. By contrast, when gaze is directed opposite to the direction of curvature, ϕ(t) does not collapse toward zero, curl remains sustained, and therefore no underestimation of curvature is expected or observed. This pattern is symmetric for left and right curves too.

Effects of flow manipulation

Figure 3 shows the reported instant heading for the different flow manipulations. We also show the unaltered condition (yellow-green) again for the sake of comparison. Importantly, when curl is cancelled (cyan), these biases essentially disappear – the perceived path remains much closer to the physical trajectory across gaze directions. Note that the central row (straightahead simulated motion) provides the clearest demonstration: the unaltered curl shows the previously reported lateral bias, but when curl is cancelled (counteracted) the perceived trajectory becomes near-veridical. This confirms that the mean curl signal itself – not gaze eccentricity per se – is the functional driver of the bias. In the over-cancelled condition (purple), the pattern reverses, with biases in the opposite direction. To maintain the full factorial design (Eccentricity × Flow Manipulation × Heading), we included trials where gaze was centered and straight ahead. Although no natural curl is generated at this zero-eccentricity position, we purposely introduced curl at rotation speeds corresponding to the ‘cancelled’ and ‘over-cancelled’ conditions (simulating random leftward or rightward gaze). As shown in the central panel of Figure 3, this introduced flow induced the predicted biases despite the absence of gaze eccentricity, confirming that the retinal flow pattern, rather than the physical orientation of the eyes, is the primary driver of the perceived heading shift.

Figure 3.

The last column in Figure 3 shows the average 2D displacement (in meters) required for the observed paths to align with the physical ones. This measure gives an idea of the similarity between the reported paths and the physical exposed trajectories in the experiment. In general, the difference between perceived and physical paths is smaller when the path is straight (central panel of last column in Figure 3). Consistently with the reported paths in the different flow manipulation conditions, the displacement is smaller in the cancelled flow curl (color-coded) for all headings (different rows). This resulted in a significant quadratic effect of Flow manipulation, β = 0.09, SE = 0.023, t(526) = 3.997, p < .001, indicating that mean deviation was lower in the cancelled condition compared with the unaltered and overcancelled conditions.

Fitting the controler

To assess how well the curl-based controller accounts for reported perceived headings, we fitted the model as described in the Methods. The fit incorporates the mean image curl computed via Equation 1 from the experimental videos, assuming perfect gaze stabilization on the fixation point—a reasonable assumption, given the high quality of fixation recorded. Fits were performed for both individual participants (see Figures S4 to S15 in the SI) and the aggregate data (see Figure 4 and Figure S3 in the SI). As shown in Figure 4, the model with separate parameters captures the average perceived trajectories with remarkable accuracy. The observed instant headings (thin lines) are overlaid with the separate fits per condition (thick red lines). In these separate fits, the mean 2D deviation per step was minimal, ranging from 0.066 m in the cancelled flow condition to 0.080 m in the unaltered condition (see table S1 in the SI). Statistical analysis via ANOVA on the individual fits confirmed that while separate fits yielded a significantly smaller loss than the join approach (F (1, 55) = 206.27, p < 0.001), flow manipulation itself had only a marginal effect on fit quality (F (2, 55) = 2.69, p = 0.077). Interestingly, the cancelled condition showed a slightly smaller loss than the other two (Quadratic contrast t(55) = 2.356, p = 0.022). Although the join fit (using only two parameters for all heading and eccentricity conditions) naturally results in a larger 2D deviation, it effectively captures the core qualitative trends across the entire conditions: a) The model successfully tracks the near-straight perceived trajectories in the central row when curl is cancelled; b) The join model accurately reproduces the systematic underestimation of curvature when gaze eccentricity aligns with the direction of the curve and, c) Even with constrained parameters, the model accounts for the reversal of bias directions in the over-cancelled condition (purple lines) across different gaze eccentricities. Despite the higher deviation inherent in compromising parameters across conditions, the join model is statistically superior as indicated by its lower AIC measures (see Table S1). This suggests that a unified curl-based mechanism provides a parsimonious and robust explanation for the instant heading perception across diverse gaze and flow conditions.

Figure 4.

Neural Network Model of Gaze-Contingent Heading Bias

We present a neural model that provides a neurophysiologically plausible implementation of the controller. The model demonstrates that the bias emerges from the interaction between gaze-modulated visual flow processing and a weak straight-ahead prior in parietal heading representation, implemented through standard cortical connectivity. After presenting the model, we show different dynamic aspects of the model and reproduce the bias for the straight ahead trajectories at different trajectories. The python source code of the model is available at osf link

Local Motion Encoding

Visual motion is encoded by direction-selective units analogous to primate MT cortex, with 8 directional preferences:

The response of each direction channel at image-plane position p_i = (x_i, y_i) is given by rectified cosine tuning (Simoncelli and Heeger, 1998):

where v_i is the local optical flow vector. In our implementation, optic flow v_i is computed, as before, using the Farnebäck algorithm as a substitute for early visual processing (V1/MT complex). This provides the motion signals that the visual system would extract through V1 to MT processing. The model itself begins with the input to the 8-sector MT-like directional encoding (Equation 10).

Curl Computation (MSTd)

To quantify the rotational (curl) component of optic flow around the current gaze position, we project local flow vectors onto the tangential direction of the circle centered at gaze. For each sampled location p_i, we first compute its position relative to the gaze point g:

This vector points from the gaze location to the sample point. The direction orthogonal to this vector corresponds to the local tangential direction of rotation around gaze. We obtain this unit tangential vector by rotating r_i by 90° and normalizing:

Finally, the contribution of the optic flow at that point to local rotational motion is given by the projection of the flow vector v_i onto this tangential direction:

Positive values indicate counterclockwise motion around gaze, and negative values indicate clockwise motion. The mean curl is computed as follows:

where S represents samples satisfying ‖r_i‖ > r_min after trimming outliers. In our implementation, N_total = 400 samples are drawn within a gaze-centered region.

Ring Attractor Dynamics: Parietal Heading Representation

Heading direction θ is represented by a ring attractor mechanism consistent with previous studies (Zhang, 1996), with neural activity x(ϕ, t) at preferred heading ϕ evolving as:

We implement this dynamics by using a discrete ring of N = 181 units with preferred headings ϕ_j uniformly spaced between [−ϕ_max, ϕ_max]. The activity dynamics then follow:

The recurrent connectivity follows a standard Mexican hat profile (Ben-Yishai et al., 1995):

where d(ϕ_j, ϕ_k) is the circular distance between preferred headings. A_e and A_i were set to 1.9 and 1.0 respectively.

External Inputs

The external input I_ext(ϕ, t) consists of two components:

The gaze-modulated inhibitory input is:

where ϕ_g is the gaze direction in heading coordinates, K_p is the inhibitory gain, and σ_κ controls the width of the gaze-centered modulation, determining how broadly the inhibition spreads around the gaze direction ϕ_g

The straight-ahead prior is:

I₀ is the prior strength and σ_p sets the width of the straight-ahead prior, with smaller values producing a sharper preference for ϕ = 0

Bias generation as Sensory–Prior Competition

Within this framework, the systematic bias opposite to gaze emerges from the interaction between the gaze-centered inhibitory drive and a weak straight-ahead prior within a stabilizing recurrent Mexican-hat network. A strong prior (I₀ ≳ 0.25) keeps the heading estimate near straight-ahead, while a weak prior I₀ ≈ 0.03 allows sensory evidence to shift the attractor state. The symmetric recurrent dynamics maintain a coherent and stable activity bump. For leftward gaze (ϕ_g < 0) and forward motion Gaze-centered inhibition creates activity suppression at ϕ_g; 2) With weak prior, activity bump shifts toward opposite side and 3) Recurrent dynamics translate inhibition into bump displacement.

The heading direction is decoded using population vector readout:

where N is the number of ring units (N = 181). Unlike some heading perception models (Beintema et al., 2004; Lappe and Rauschecker, 1993), we do not apply sigmoid nonlinearities to neural activities prior to decoding but a linear readout. Gaze-contingent bias emerges even with linear decoding, suggesting it is a fundamental property of the network dynamics.

Relation to the controller (phenomenological model)

The neural implementation provides a mechanistic basis for the phenomenological relationship by demonstrating how rotational flow is converted into a change in perceived heading. In the neural model, K_p is not a single fixed parameter, but rather a dynamic property of the recurrent network. The shift in the activity bump (representing ) is driven by the spatial asymmetry of the inputs. When the gaze-located inhibition suppresses the sensory evidence at the fixation point, it causes the stable bump of activity to “drift” toward the uninhibited regions of the sensory input. The speed of this drift is directly proportional to the strength of the inhibitory gain (K_p in Equation 17) and the magnitude of the local curl signal (ω). Thus, the network naturally performs the integration required by the controller: it transforms localized, gaze-contingent inhibition into a global heading shift that matches the relationship observed in participants.

Neural simulations

To demonstrate the neurophysiological feasibility of the curl-based controller, we conducted a series of neural network simulations using a custom Python implementation of the recurrent ring network, with subsequent 3D trajectory reconstruction. The model implementation read the video sequences of the experimental straight paths conditions, assuming again perfect gaze stabilization on the fixation point across all trials. As commented above, we computed the optic flow for each frame, before proceeding with each step of the neural model, from the local motion computation to the linear decoding of the ring activity.

We explored the parameter space to identify the drivers of the heading bias, varying the prior strength (I₀ ∈ [0.03, 0.24]), the prior width (σ_p ∈ [0.18, 0.25]), the spatial extent of gaze inhibition (σ_k ∈ [0.02, 0.04, 0.08, 0.12]) and the inhibitory gain (K_p ∈ [0.4, 0.8]). Table S2 in the SI shows the range for the different parameters used in the simulations.

Simulation results

The network dynamics provide a clear mechanistic explanation for the systematic heading bias observed during eccentric gaze. In Figure 5, we observe the network components for a trial in which the gaze direction is 4 m to the left during forward translation. Figure 5 shows the recurrent connectivity following a standard Mexican hat profile, characterized by local excitation and broader surround inhibition. The activity bump does not settle at the objective straight-ahead (0^∘) but stabilizes at a positive (rightward) heading offset, as illustrated in Figure 5. This displacement is driven by the competition shown in the mechanism panel (Figure 5): the gaze-centered inhibitory drive (red dashed line) suppresses the leftward portion of the ring, effectively ‘pushing’ the neural activity bump (black line) away from the gaze location and to the right of the straight-ahead prior (blue dotted line). The Phase Portrait (Figure 5) confirms this as a stable state; the drift dynamics converge toward a stable fixed point where the curve crosses zero at a rightward offset. Together, these results demonstrate how localized inhibition, within a parietal ring attractor, transforms gaze-contingent sensory signals into a shifted global heading estimate. For results across a range of eccentricities during forward translation, see Figure S16 in the SI. The presence of temporal oscillations in the activity bump when gaze is aligned with the direction of travel (Figure S16 Row B) or at low eccentricities (rows A and C) represents a state of high competition within the attractor. As seen in Figure S16 B, when the gaze-centered inhibition —the sensory pull— overlaps directly with the straight-ahead prior —the internal push— it creates a central suppression that counteracts the recurrent excitation of the network. This conflict results in a concave activity profile and rhythmic fluctuations in bump intensity as the network continuously attempts to re-stabilize the representation against the central inhibitory zone. Notably, these oscillations persist when gaze is less eccentric (e.g., at −2 and 2 m in rows A and C respectively), as the proximity of the inhibitory signal to the prior creates a similar destabilizing interaction. Despite these internal dynamics, the decoded heading remains accurate due to the symmetry of the competing forces, suggesting that while representation coherence may decrease, the population vector readout remains robust to central sensory-prior conflict.

Figure 5.

From the neural model heading estimates (, in pixels) we reconstructed the 3D paths using a pinhole camera model (see Heading Estimation and 3D Path Reconstruction section in SI) that are shown in Figure S17 of the SI. This figure demonstrates that the observed instant headings pattern shown in Figure 2 is best captured by a network configuration featuring a weak straight-ahead prior (I₀ ≈ 0.03) and moderate inhibitory gains (K_p between 0.4 and 0.8). Within this parameter space, the localized, gaze-contingent inhibition effectively shifts the population activity bump, reproducing the characteristic lateral spread seen in the behavioral data. A low prior value (I₀ = 0.03 in the simulation grid) is essential. Higher prior strengths (e.g., I₀ = 0.24) overly anchor the heading estimate to the center (0 m), failing to produce the significant lateral spread observed in the human data. K_p = 0.4 closely matches the tighter trajectory fan seen in the top central panel of the empirical data, while K_p = 0.8 captures the wider lateral deviations observed in the bottom central panel, where the bias is more pronounced. Concerning the inhibition width (σ_k), values between 0.08 and 0.12 provide the most realistic structural match. Very low values (σ_k = 0.02, 0.04) produce negligible bias even at high gain, as the localized inhibition does not sufficiently overlap with the heading activity bump to drive a shift. The prior width value of σ_p = 0.18 appears to provide a more sensitive range for capturing the gaze-contingent shift compared to the wider σ_p = 0.25.

These results suggest that the interplay between sensory-driven inhibition and a stabilizing straight-ahead prior within a standard Mexican-hat recurrent architecture is sufficient to account for the gaze-contingent biases observed in human heading perception

Discussion

Retinal Curl as a Functional Control Variable revealed by a bias

Our findings provide empirical evidence that retinal curl has a functional role in the perceived heading. Manipulating the retinal flow to which the participants were exposed affects the perceived heading in predictable ways. This main finding suggests that curl could be used as a control signal rather than being a “nuisance” component of the optic flow that must be filtered out. While previous work has emphasized the decomposition of flow into translational and rotational components to recover heading (Beintema et al., 2004; Heeger and Jepson, 1992; Longuet-Higgins and Prazdny, 1980), we show that the visual system exploits the curl generated by gaze stabilization to perceive heading. This is consistent with recent findings in real-world walking, where retinal curl magnitude in the fovea specifies body trajectory relative to gaze (Matthis et al., 2022). In this sense, curl alone combined with extra-retinal information (e.g. rotation of eye, or head or both relative to the body) provides heading information.

Importantly, the functional nature of the curl signal is demonstrated by our curl cancellation condition. When the naturally occurring curl was counteracted, the systematic heading bias disappeared. This confirms that the bias is not an artifact of gaze eccentricity itself, but a direct consequence of the underlying flow geometry due to sustained gaze stabilization. In addition, the flow manipulation reveals a direct link between the amount of curl and perceived heading: not only did the bias opposite to fixation disappear when the curl was cancelled or nullified but also the bias re-appeared towards fixation (opposite to the bias in the unaltered flow condition) when the curl was over-cancelled (see paths in the middle row in Figure 3). This manipulation led to counter-intuitive findings like the reported path was closer to the physical/experimental one in the cancelled condition (last column in Figure 3)

Perceiving a curved heading when exposed to a physically straight path is not new. Several studies have reported heading errors consistent with this misperception in the past (e.g. (Royden et al., 1994, 1992; van den Berg and Brenner, 1994)). A mathematical analysis based on image point positions had been put forward to explain the perceived curvature when exposed to straight paths (Royden, 1994). However, there are large differences between the bias we report and theirs. First, the biases reported in the previous cited studies appear in simulated rotation conditions and when the simulated rotation rate is larger than 1º/s. When simulated rotation rate was smaller than 1º/s heading was accurate and no different than conditions with actual eye/head rotation (Warren and Hannon, 1990). None of these two conditions apply in our study. Our retinal rotation was contingent with actual gaze rotation and the rotation rate was always less than 1º/s (see figure S2 of the SI). Moreover, the stimulus duration in all these studies was smaller than 2 or 3 seconds, while our stimulus duration was larger than 10 seconds. Our biases take some time to build up. Looking at the time course of the heading responses in our experiment, the biases show up after 3 m or 5 s (see Figure 2). Finally, most of previous studies examined heading based on a binary response after the last frame, while we used a continuous response. We believe that these differences lead to the most important difference between our heading bias and the previously reported ones: the bias direction. While our bias is in the direction opposite to fixation, the biases in previous studies were consistent with the direction of fixation caused by the simulated rotation. Since, it was sustained gaze stabilization that allowed us to reveal this bias, our findings are more consistent with time varying signal account of the use of optic flow (Burlingham and Heeger, 2020) rather than the mediation of instant flow (Cutting et al., 1992; Li and Cheng, 2011) to recover heading. We encourage the reader to experience this bias firsthand: when walking in an open space while maintaining a sustained gaze on a lateral landmark, one may notice a drift in their trajectory.

Active Steering vs. Heading Recovery

An important distinction in navigation research lies between the passive perception of heading (estimating where am I going?) and the active control of steering (adjusting where do I want to go?) (Goodridge et al., 2023; Powell et al., 2024; Tuhkanen et al., 2021; Wilkie and Wann, 2003a, 2003b, 2002). While traditional models posit that accurate heading recovery is a prerequisite for steering or controlling navigation and, given that, our experimental task requires heading, our results support the view that explicit heading estimation is not strictly necessary for control (Wilkie and Wann, 2003a). Our curl-based controller reinforces this distinction: it can successfully maintain a trajectory by simply nulling the error signal derived from retinal curl, without ever needing to calculate the instantaneous focus of expansion or resolve the ambiguity of the heading point. We reproduced the results of Wilkie and Wann (2003a) (experiment 3) of active steering towards 6 targets at different eccentricities (see Figure S17 and table S3 in the SI). We introduced the heading/gaze discrepancy in these eccentricities and by nullifying the curl the controller could find solutions very close to the empirical steering paths reported in Wilkie and Wann (2003a). This suggests that the “bias” observed in our perceived headings may reflect the operation of a control law optimized for action rather than a failure of a perceptual system designed for passive estimation.

This does not necessarily mean that the information used by the participants in Wilkie and Wann (2003a) is the curl signal. They argued that steering can be achieved through a mechanism that can combine different information in addition to retinal flow, like visual direction or perceived target location (Rushton et al., 1998). This is also a possibility in our experiment, that heading responses were based on the perceived location of our experimental fixation point. However, the different flow manipulation conditions (cancelled and over-cancelled) were able to completely change the perceived instant heading. While visual direction is undoubtedly a cue, our results indicate that retinal flow dynamics can override it. In our experiments, even when the fixation point remained eccentric (providing a stronger visual direction cue), cancelling the retinal curl eliminated the bias or induced in the other direction (over-cancelling condition). This runs counter the possibility of participants using the visual direction of the fixation point. The large field of view in our experiment (>90º) could also have contributed to exploit retinal flow to a larger extent. This suggests that behaviors previously attributed to visual direction strategies may instead be driven by the minimization of retinal curl, offering a parsimonious account of steering control that unifies fixation and flow processing (Matthis et al., 2022).

Re-evaluating the Focus of Expansion (FOE)

Overall, our results suggest, at least in simulated walking, that when rotation in the retinal flow is real and active, the system treats curl as a signal to be regulated rather than noise to be subtracted. Historically, heading perception theory has been dominated by the extraction and use of the Focus of Expansion (FOE) (Gibson, 1950; Warren et al., 1988; Warren and Hannon, 1990). However, the relevance of the FOE in natural, active vision has been increasingly questioned (Matthis et al., 2022; Muller et al., 2023). In the presence of eye movements, the retinal FOE is rarely aligned with the heading, creating a complex computational problem, known as the rotational problem (Cutting et al., 1992; Koenderink and Van Doorn, 1981; Longuet-Higgins and Prazdny, 1980; Royden et al., 1994).

Although our results support the view that explicit FOE extraction may not be required for controlling navigation or locomotion, this does not mean that FOE is render useless. Instead, the visual system may rely on different flow field structures and FOE, or divergence can be one of them (Matthis et al., 2022). For example, in more complex scenes, pseudo-FOE can lead to heading biases (Layton and Fajen, 2016) due to relative motion between objects in the background. FOE can also be more relevant in high-speed locomotion, in which momentary changes of gaze direction due to rotation of head or eyes, maybe less frequent than in walking, leading to a less disrupted head-based flow (Muller et al., 2023). This shift away from FOE-based heading is further supported by recent computational models (Layton and Browning, 2014). The continuum of observed cells (from radial to circular) in MSTd would simultaneously code curvature for trajectory estimation and heading across the neural population, with curvature encoded through the spirality of the most active cell and heading through the visuotopic location of its receptive field center.

Neural Implementation of Curl-Based Steering

We propose a relatively simple neural model that can reproduce the bias when exposed to straight ahead video scene, providing a biologically plausible bridge between the phenomenological model, the controller, and neural implementation. This neural circuit model demonstrates that gaze-contingent heading biases can emerge from fundamental principles of sensory integration in cortical networks. Therefore, standard center–surround (Mexican-hat) recurrent connectivity is sufficient to generate the observed bias. The inherent push–pull dynamics transform localized inhibition into a global shift of the activity bump. The key insight is that weak straight-ahead priors, combined with standard cortical connectivity with gaze-centered inhibition, are sufficient to explain the observed psychophysical biases. The model integrates direction-selective motion processing—analogous to area MSTd, where neurons are known to be sensitive to spiral and curl patterns (Duffy and Wurtz, 1991; Graziano et al., 1994) —with a parietal ring-attractor network. Spiral tuning also appears in VIP Schaafsma and Duysens (1996) and 7a Read and Siegel (1997), to which MSTd projects Born and Bradley (2005).

Conclusion

We conclude that the rotational component of optic flow (curl), generated during gaze stabilization, is an actively used signal to control heading. It acts as a navigational cue rather than noise, as evidenced by the elimination of steering biases when curl is experimentally cancelled. Our findings challenge the necessity of explicitly extracting the Focus of Expansion for online control of locomotion. The observed behaviors are supported by a neural model based on established properties of motion processing areas. The interaction between sensory flow inputs and internal priors within a recurrent network suffices to explain the gaze-contingent biases observed in our experimental data.

Data availability

All the data and code for the analysis are available at https://osf.io/b37rg/overview?view_only=35b748a234b7459a9e9df1b41a2d4a44

Additional files

Supplementary Information.

Video S1. Retinal flow pattern during forward translation with fixation on a target (white dot) located to the left of the heading. For optimal viewing, the display should be centered relative to the observer with a field of view exceeding 50º (e.g., a 60 cm screen viewed from a distance of 60 cm).

Video S2. The same as video S1, retinal flow pattern during forward translation and fixating a target (white dot) located to the left of the current path, but, the rotational component of the flow has been counteracted, so there is very little curl around the fovea. For optimal viewing, the display should be centered relative to the observer with a field of view exceeding 50º (e.g., a 60 cm screen viewed from a distance of 60 cm).

Video S3. Controller illustration: how heading is corrected to match gaze by applying equation 5 in the main text.

Additional information

Funding

MEC | Agencia Estatal de Investigación (AEI) (MICIU/AEI/10.13039/501100011)

• Konytessa I Zorpala