Visual systems exploit temporal continuity principles to achieve stable spatial perception, manifested as the serial dependence and central tendency effects. These effects are posited to reflect a smoothing process whereby past and present information integrates over time to decrease noise and stabilize perception. Meanwhile, the basic spatial coordinate—Cartesian versus polar—that scaffolds the integration process in two-dimensional continuous space remains unknown. The spatial coordinates are largely related to the allocentric and egocentric reference frames and presumably correspond with early and late processing stages in spatial perception. Here, four experiments consistently demonstrate that Cartesian outperforms polar coordinates in characterizing the serial bias—serial dependence and central tendency effect—in two-dimensional continuous spatial perception. The superiority of Cartesian coordinates is robust, independent of task environment (online and offline task), experimental length (short and long blocks), spatial context (shape of visual mask), and response modality (keyboard and mouse). Taken together, the visual system relies on the Cartesian coordinates for spatiotemporal integration to facilitate stable representation of external information, supporting the involvement of allocentric reference frame and top-down modulation in spatial perception over long time intervals.

*. The second task was the blind spot task, in which participants fixed their eyesight on a black square while a red dot was moving from the center to left. They were asked to press keys as soon as they could not see the dot (at the blind spot). The length that the dot traveled through in pixel was denoted by parameter*

**a***. Because the blind spot position is constant among people (*

**s***, about 13.5°), we could calculate the viewing distance, by:*

**θ****error-ori**, responded orientation minus stimulus orientation) and relative orientation of previous trial (

**relative-ori**, stimulus orientation in previous trial minus stimulus orientation in current trial). To quantify the serial dependence effect, we used the well-established Gaussian derivative (DoG) function (e.g., see Fischer & Whitney, 2014) to fit the data, given by:

*x*is

**relative-ori**,

*y*is

**error-ori**,

*a*is the amplitude of the curve peak,

*w*scales the curve width,

*c*is the constant \(\sqrt 2 /{{\rm{e}}}^{ - 0.5}\) scaling the curve to make

*a*parameter equal to the peak amplitude, and

*b*describes the vertical shifts along the

*y*axis. We combined all participants’ data into aggregated data for model fitting. Note that we normalized response error (

**error-ori**) within participants to reduce variability among participants, and thus the dependent variable would be the z-score of

**error-ori**.

*x*and

*y*axes, with each location being represented by a pair of numerical coordinates. Polar coordinates consist of ρ (

**distance**) and φ (

**angle**) axes, characterizing a location by its distance from a reference point and the angle from a reference direction. The former is a grid-like coordinate system and the latter is a radial-like one. Notably, the two coordinate systems share the same original point, that is, the center of the screen.

*x*axis,

**error-x,**and projection of error-loc on the

*y*axis,

**error-y**, respectively. The projection of the relative-loc onto the

*x*and

*y*axes resulted in

**relative-x**and

**relative-y**(Figure 1C). For polar coordinates, the projections onto the ρ axis were

**error-ρ**and

**relative-φ**, and the projections onto the φ axis were

**error-ρ**and

**relative-φ (**Figure 1F).

*x*,

*y*, and ρ axes are in noncircular forms, and the variable on the φ axis is circular ranging from –180° to 180°. These two kinds of variables exhibit different relationships between “

**relative**” and “

**error**.” For the circular variable on the φ axis, it is in DoG form (Manassi et al., 2018); the circular variable for orientation in previous studies (Fischer & Whitney, 2014). For the noncircular variables on the

*x*,

*y*, and ρ axes, the relationships are in a linear form as an approximation of the DoG function whereby independent variables are restricted to a range of small values, corresponding with the limited space (e.g., Motala, Zhang, & Alais, 2020).

**error**” and “

**relative**” for the

*x*,

*y*, ρ, and φ axes separately. Specifically, the DoG function for φ was defined in Equation 1. For variables on the

*x*,

*y*, and ρ axes, a linear fitting function relates

*x*(“

**relative**”) to

*y*(“

**error**”), by:

*x*,

*y*, and ρ axes. For each participant, the same trials in the serial dependence effect analysis were chosen so that the two effects were comparable in later analysis. We characterized the relationship between response error and current stimulus location to quantify the central tendency effect. Previous studies on magnitude estimation discovered the linear relationship between response error and current stimulus (Petzschner et al., 2015), but we found that the raw data pattern was more complicated than the linear relationship, suggesting an alternative model. Therefore, we proposed a model called the “complex central tendency model (complex CT)” to capture the relationship between the current stimulus location (

*x*) and the response error (

*y*) with parameters a to e:

*y*is the response error,

*x*

_{1}is relative,

*x*

_{2}is the current stimulus location, and

*a*to

*f*are free parameters. If the compound model outperformed both the complex CT model and the serial dependence model (SD), it would support the coexistence of two effects in one trial. Because the central tendency effect only exists for noncircular variables such as the

*x*,

*y*, and ρ axes, we proposed the compound model as a combination of linear serial dependence effect and complex central tendency effect for the

*x*,

*y*, and ρ axes.

*x*and

*y*axes to get the AICc of Cartesian coordinates. We added the AICc of the best model among five candidates for the ρ axis and the best model among two candidates for φ axis (constant/DoG serial dependence) to get the AICc of the polar coordinates. We then compared the AICc of the two coordinates.

*fitlm*function (“statistics and machine learning” toolbox), which uses an iteratively reweighted least squares algorithm. The nonlinear model fitting was conducted with the

*fitnlm*function, which uses the Levenberg–Marquardt nonlinear least squares algorithm. We used the Bayesian model comparison method, using the index AICc (Hurvich & Tsai, 1989) to select the best model among several candidate models, that is, the model with the lowest AICc. We subtracted the best model's AICc from each model's AICc to get the Δ

*AICc*. If the difference in AICc between two models is greater than 10, it suggests strong evidence in favor of the model with lower AICc (Burnham & Anderson, 2004).

^{2}of those fittings formed a null distribution against which the adjusted R

^{2}of raw data's fitting were compared. The

*P*value was taken as the proportion of adjusted R

^{2}in null distribution, which was larger than or equal to the raw data's fitting adjusted R

^{2}.

*fitglme*function and

*nlmefit*function on the offline experiment data (experiments 1 and 4).

*AICc*= 252.4, model comparison against constant model; adjusted R

^{2}= 0.0112,

*P*= 10

^{−4}, permutation test).

*x*and

*y*in the Cartesian coordinates (Figure 1C) and by ρ and φ in the polar coordinates (Figure 1F). We next built five candidate models to characterize the spatial serial bias (Figure 1I): a constant model with constant biases (Constant), a

*SD*, a linear central tendency model (linear CT), a complex central tendency model (complex CT), and a compound model comprising serial dependence and complex central tendency effects (SD + complex CT).

*x*and

*y*axis. Specifically, as shown in Figure 1D, the perception error was positively proportional to the target location shift across consecutive trials (SD model vs. Constant model;

*x*: Δ

*AICc*= 2418.8;

*y*: Δ

*AICc*= 3824.7). Similarly, in the polar coordinates (Figure 1G), the perception error defined at ρ axis also showed a positive linear pattern (SD model vs. constant model; Δ

*AICc*= 1382.8). The serial dependence effect for φ followed a DoG-shaped curve(Δ

*AICc*= 266.6), consistent with previous findings (Bliss et al., 2017; Manassi et al., 2018). Permutation tests further supported significant serial dependence for all the parameters in the two coordinates (

*P*= 10

^{−4}for all axes).

*x*axis and

*y*axis in the Cartesian coordinates and the ρ axis in the polar coordinates revealed a linear relationship between the position error and the shift of target location from the center. Note that the φ axis in the polar coordinates was not applicable to quantify the central tendency effect (right panel, Figure 1H). Given that the complex CT model surpassed both the linear CT model (X: Δ

*AICc*= 1118.5; Y: Δ

*AICc*= 224.8; ρ: Δ

*AICc*= 128.2) and the constant model (X: Δ

*AICc*= 4104.3; Y: Δ

*AICc*= 5290.3; ρ: Δ

*AICc*= 1541.3), we used the complex CT model in the subsequent analyses.

*x*,

*y*, and ρ (Figure 1I) (compound model vs. SD model: Δ

*AICc*= 1951.7 for X; Δ

*AICc*= 1807.8 for Y; Δ

*AICc*= 394.5 for ρ; compound model vs. complex CT model: Δ

*AICc*= 266.2 for X, Δ

*AICc*= 342.2 for Y; Δ

*AICc*= 236.0 for ρ).

*x*and

*y*; polar coordinates: ρ and φ) and the five different models (Constant, SD, linear CT, complex CT, SD + complex CT). Importantly, for all the three noncircular variables (

*x*,

*y*, ρ), the SD + complex CT model outperformed the other four models (denoted by ※), suggesting that spatial perception involves concurrent serial dependence and central tendency effect. Note that because the circular variable φ was not applicable for the central tendency effect, it only applied to the constant and SD models and was denoted in the other three models by ⊘.

*x*,

*y*, ρ, and φ) and added the values belonging to the same coordinates respectively (Cartesian coordinates:

*x*and

*y*; polar coordinates: ρ and φ). As shown in Figure 1J, the Cartesian coordinates performed better than the polar coordinates (Δ

*AICc*= 7959.2). Furthermore, a mixed effect analysis that considers random effect also supports that Cartesian outperforms Polar coordinates (Supplementary Figure S1A). Therefore, stable spatial perception in a 2D continuous space over time is better characterized via Cartesian rather than Polar coordinates.

*AICc*= 2073.1). Therefore, the role of the Cartesian coordinates in mediating serial bias in continuous 2D spatial perception does not rely on strictly controlled environmental settings as well as lengthy tests, but instead emerges rapidly in a natural context.

*AICc*= 2237.8), suggesting that the role of the Cartesian coordinates in serial bias in continuous 2D spatial perception is not due to the spatial context the following mask might bring about. Instead, the function of the Cartesian coordinates persists even in spatial contexts that likely favor the polar coordinate.

*AICc*= 11499.5). A mixed-effect analysis that considers random effect supports the same conclusion (Supplementary Figure S1B). Therefore, the engagement of Cartesian coordinate in stable spatial perception over time is not due to the response modality either.

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