**Many biologically important motions are described by periodic trajectories. Radial frequency (RF) trajectories are one example, in which the motion of a difference of Gaussians (DOG) target moves along a path described by a sinusoidal deviation of the radius from a perfect circle (Or, Thabet, Wilkinson, & Wilson, 2011). Here we explore the hypothesis that visual processing of RF trajectories involves global spatio-temporal processes that are disrupted by motion discontinuity. To test this hypothesis, RF trajectories were used that interspersed smooth, continuous motion with three or four discontinuous jumps to other portions of the trajectory. These jumps were arranged so that the entire trajectory was traversed in the same amount of time as in the continuous motion control condition. The motion discontinuities increased thresholds by a factor of approximately 2.1 relative to continuous motion. This result provides support for global spatio-temporal processing of RF motion trajectories. Comparison with previous results suggests that motion discontinuities erase memory for earlier parts of the trajectory, thereby causing thresholds to be based on only the final segment viewed. Finally, it is shown that RF trajectories obey the 1/3 power law characteristic of biological motion.**

^{2}. All experiments were conducted using VPixx software running on an Apple iMac computer (Apple, Cupertino, CA).

*R*relative to its center by: The space constant

*σ*was set to 7.1 arc min, making the peak spatial frequency 2.17 cpd and the bandwidth 1.79 octaves at half amplitude. The DOG mean luminance was identical to that of the screen, and the contrast was 100%.

*r*is the radius of the trajectory as a function of velocity

*v*and time

*t*. Note that

*r*and

*r*refer to the radius of the RF trajectory relative to the center of the screen, while

_{0}*R*refers to the radius of the target only. Amplitude

*A*of the deviation from circularity was varied to determine the threshold for RF trajectory discrimination, with

*A*= 0.0 corresponding to a circle of radius

*r*. Finally, the radial frequency of the trajectory is given by an integer value of

_{0}*ω*, and the phase by

*ϕ*. Two radial frequencies were used in these experiments:

*ω*= 3 cycles, and

*ω*= 4 cycles, and these will be referred to as RF3 and RF4 trajectories respectively. The base radius

*r*

_{0}= 1.0° in all but one experiment (documented in the following material). Examples are depicted in Figure 1, where the arrows show that motion was always in clockwise direction. The velocity

*v*was either 6.28°/s (1.0 s transit time around the trajectory) or 12.56°/s (0.5 s transit time). Note that these are average velocities (strictly speeds), as speed varied slightly for RF trajectories and only remained constant for the circular pattern. Previous work has shown that keeping the speed strictly constant (by varying the angular velocity appropriately) does not affect threshold measurements (Or et al., 2011).

**Figure 1**

**Figure 1**

**Figure 2**

**Figure 2**

*F*(2, 10) = 25.75,

*p*< 0.0002. Subsequent

*t*tests with Bonferroni correction showed that thresholds for both Jump conditions were significantly greater than the Continuous trajectory threshold, but there was no significant difference between the Max Jump and Min Jump conditions.

**Figure 3**

**Figure 3**

*F*(2, 8) = 19.59,

*p*< 0.0008. Pairwise

*t*tests with Bonferroni correction again revealed that both jump conditions produced significantly higher thresholds than the continuous condition, but there was no statistically significant difference between Max Jump and Min Jump.

**Figure 4**

**Figure 4**

*F*(2, 8) = 13.36,

*p*< 0.003. Pairwise

*t*tests showed that thresholds for the Min Jump condition were significantly greater than those for either the continuous or Max Jump conditions,

*t*(8) = 5.02,

*p*< 0.001;

*t*(8) = 3.56,

*p*< 0.008 respectively. However, there was no significant difference between the Max Jump and continuous conditions,

*t*(8) = 1.44,

*p*= 0.19. Thus, only the Min Jump condition significantly elevated thresholds under these suprathreshold conditions, although there was an insignificant trend in the same direction for the Max Jump condition.

**Figure 5**

**Figure 5**

*F*(2, 10) = 5.59,

*p*< 0.024. Subsequent

*t*tests revealed that only the Min Jump condition was significantly larger than the continuous condition,

*t*(10) = 3.32,

*p*< 0.008, although the Max Jump versus continuous condition approached significance,

*t*(10) = 1.97,

*p*= 0.077. Max Jump versus Min Jump thresholds were not significantly different,

*p*(10) = 1.35,

*p*= 0.21. Thus, the same pattern of results for suprathreshold increment thresholds was apparent for both RF3 and RF4 trajectories.

**Figure 6**

**Figure 6**

*r*

_{0}= 1.0°. As a control experiment to determine whether these variables were critical to our results, thresholds for discrimination between circular trajectories and RF4 or RF3 trajectories were repeated under two further conditions. In the Double Radius condition,

*r*

_{0}= 2.0°, and target speed was also doubled to 12.56°/s. In the Speed 2× condition, only the speed was doubled to 12.56°/s, while the radius remained at 1.0°. Results for six subjects tested with RF4 trajectories are graphed in Figure 7. The baseline condition replots the means for these subjects from Figure 3. A two-way ANOVA with Jump condition and Speed as factors showed a significant effect of Jump condition,

*F*(2, 45) = 12.37,

*p*< 0.0001, but no effect of Speed,

*F*(2, 45) = 2.17,

*p*= 0.126. In addition, there was no interaction between conditions,

*F*(4, 45) = 0.274,

*p*= 0.894. Pairwise comparisons of the three Jump conditions revealed that all three comparisons were statistically significant at the

*p*< 0.02 level.

**Figure 7**

**Figure 7**

*F*(2, 36) = 12.23,

*p*< 0.0001, and the Speed condition,

*F*(2, 36) = 5.50,

*p*< 0.01. There was no interaction,

*F*(4, 36) = 1.39,

*p*= 0.258. The effect of jump condition was expected and was consistent with all previous data. However, the significant Speed effect here was not expected given the results with RF4 trajectories in Figure 7 above. Subsequent pairwise

*t*tests showed that the Baseline versus Double Radius and Baseline versus Speed 2× were both significant,

*t*(36) = 2.55,

*p*< 0.02, and

*t*(36) = 3.11,

*p*< 0.005 respectively. However, there was no significant difference between Double Radius and Speed 2×,

*t*(36) = 0.56,

*p*= 0.577. From Figure 8 it seems apparent that these results are due to the smaller disruptions produced by the Max Jump and Min Jump conditions in the Double Radius and Speed 2× conditions relative to baseline.

**Figure 8**

**Figure 8**

*V*to the local radius of trajectory curvature R (Lacquaniti, Terzuolo, & Viviani, 1983). Specifically: where

*k*represents a gain factor. Furthermore, trajectories of a moving dot on a screen perceptually seem most natural and to have the least variation in velocity when this power law is obeyed (Viviani & Stucchi, 1992). As RF trajectories were not specifically designed with this law in mind, it is worth asking whether they are consistent with it. For low amplitudes

*A*in Equation 2, there is little variation from a circle, for which

*R*and

*V*are constant, so these stimuli are certainly consistent with this law. To test relatively large variations in radius of curvature for adherence to this law, analytic formulas for both V and R along an RF trajectory were derived using Symbolic Calculator for OSX (Voxeloid Kft). The result was: where

*θ*is angular velocity. Approximating this by a first order Taylor series in

*θ*near

*θ*= 0 yields: This is exactly equivalent to Equation 3 above with the term in brackets representing the value of the constant

*k*as a function of

*A*and

*ω*. Figure 9 shows plots of Equation 5 for

*A*= 0.5 and three values of

*ω*compared with the exact results in Equation 4. Clearly, RF trajectory motion precisely replicates the biological motion relationship represented in Equation 3 across a wide range of radial frequencies and amplitudes.

**Figure 9**

**Figure 9**

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