The method of constant stimuli combined with a temporal two-interval forced-choice (2IFC) procedure was used to estimate detection thresholds for RF trajectories of 2–5 cycles. To investigate the effect of rotational speed on detection, we conducted three separate conditions using polar coordinate angular speeds v of 180°/s, 360°/s, and 720°/s, equivalent to tangential speeds of 3.14 deg/s, 6.28 deg/s, and 12.57 deg/s, respectively. Each stimulus was composed of 57 frames displaying the dot at instantaneous points sampled at identical angular intervals of 6.43° around an RF trajectory. The last of these frames displayed the dot at the starting point, which was always directly to the right of the fixation point. The last frame was thus identical to the first frame, ensuring that the trajectory motion appeared to complete one revolution. The 57 frames were presented in rapid succession without an inter-frame interval such that the dot appeared to move around the RF trajectory. Each frame lasted for 35.7, 17.9, and 8.9 ms, respectively, for the three speed conditions, corresponding to stimulus durations of 2.0, 1.0, and 0.5 s, respectively. Note that the dot speeds in the three conditions differed only by changing the duration of a frame, while the number of instantaneous points sampled around the RF trajectory (i.e., the number of frames) was always the same (which was 57). Importantly, these settings ensured that all peaks and troughs of an RF trajectory were accurately represented, and observers reported that they experienced smooth trajectory motion.
While the angular speeds are constant in the three conditions, the Cartesian (i.e., instantaneous linear) speed varies as the dot's polar location
r changes when it moves around the RF trajectory. To examine such speed variation as a potential cue for detection, one additional experimental condition was introduced in which the Cartesian speed
v′ was held constant at 6.28 deg/s by continuously varying the angular speed
v over the course of revolution, using the following equation:
Using
Equation 3, the dot's instantaneous locations were sampled at unequal angular intervals around an RF trajectory to achieve a constant Cartesian speed (but a varying angular speed). Each frame still remained for 17.9 ms, identical to that in the corresponding (6.28 deg/s) constant angular speed condition. Given a constant Cartesian speed, the total number of frames required to complete one revolution inevitably increased as a function of the radial modulation amplitude
A, as the length of the trajectory became longer. Fortunately, such increase was minor: the total number of frames ranged from 57 to 59 at the range of
A we used, and a maximum 2-frame difference resulted in a change in stimulus duration by 36 ms only (cf. the average stimulus duration was 1.0 s). Importantly, pilot experiments show that observers could not reliably discriminate between the durations of two circular trajectories of such a small time difference.
At the start of each experimental run, observers were informed of the speed and radial frequency to be used. The phase angle was set to 90°. The only parameter varied within a run was the radial amplitude. A centrally presented fixation cross remained on the screen over the course of a trial, and observers were required to maintain fixation throughout. Following central fixation for 500 ms at the beginning of each trial, observers were presented with two intervals of trajectory motion, temporally separated by a 1.5-s period of mean luminance (
Supplementary Figure 1). One interval contained the target trajectory with radial deformations, and the other interval always contained a circular trajectory matched in mean radius and speed. The order of presentation of the two intervals was randomized from trial to trial. Observers were instructed to report by button press which of the two intervals contained deformations from a circular trajectory. No feedback was provided.
Sixteen conditions (4 radial frequencies (2–5 cycles) × 4 speeds) were tested in separate blocks. For each condition, at least 5 levels of radial amplitude (40 trials/level) were tested to estimate a threshold. All seven observers participated in the 12 conditions using constant angular speeds of 3.14 and 6.28 deg/s and constant Cartesian speed of 6.28 deg/s. The order of these 12 conditions was randomized for each observer. Data collection was split into several sessions spanning a few days. Five of the same seven observers participated in the four conditions using constant angular speed of 12.57 deg/s.
For each observer and condition, data collected in different sessions were combined for analysis. The proportion of correct responses for each radial amplitude was then calculated, and a psychometric function was derived by fitting a Quick (
1974) or Weibull (
1951) function to the data using maximum likelihood estimation. The point of 75% correct response on the curve was chosen as the threshold amplitude. An estimate of the standard deviation (
SD) of the threshold amplitude was computed using a bootstrapping procedure.