It has been debated whether kinematic features, such as the number of peaks or decomposed submovements in a velocity profile, indicate the number of discrete motor impulses or result from a continuous control process. The debate is particularly relevant for tasks involving target perturbation, which can alter movement kinematics. To simulate such tasks, finite-horizon models require two preset movement durations to compute two control policies before and after the perturbation. Another model employs infinite- and finite-horizon formulations to determine, respectively, movement durations and control policies, which are updated every time step. We adopted an infinite-horizon optimal feedback control model that, unlike previous approaches, does not preset movement durations or use multiple control policies. It contains both control-dependent and independent noises in system dynamics, state-dependent and independent noises in sensory feedbacks, and different delays and noise levels for visual and proprioceptive feedbacks. We analytically derived an optimal solution that can be applied continuously to move an effector toward a target regardless of whether, when, or where the target jumps. This single policy produces different numbers of peaks and “submovements” in velocity profiles for different conditions and trials. Movements that are slower or perturbed later appear to have more submovements. The model is also consistent with the observation that subjects can perform the perturbation task even without detecting the target jump or seeing their hands during reaching. Finally, because the model incorporates Weber's law via a state representation relative to the target, it explains why initial and terminal visual feedback are, respectively, less and more effective in improving end-point accuracy. Our work suggests that the number of peaks or submovements in a velocity profile does not necessarily reflect the number of motor impulses and that the difference between initial and terminal feedback does not necessarily imply a transition between open- and closed-loop strategies.

Modeling stereo transparency with physiologically plausible mechanisms is challenging because in such frameworks, large receptive fields mix up overlapping disparities, whereas small receptive fields can reliably compute only small disparities. It seems necessary to combine information across scales. A coarse-to-fine disparity energy model, with both position- and phase-shift receptive fields, has already been proposed. However, because each scale decodes only one disparity for each location and uses the decoded disparity to select cells at the next scale, this model cannot represent overlapping surfaces at different depths. We have extended the model to solve stereo transparency. First, we introduce multiplicative connections from cells at one scale to the next to implement coarse-to-fine computation. The connection is the strongest when the presynaptic cell’s preferred disparity matches the postsynaptic cell’s position-shift parameter, encouraging the next scale to encode residual disparities with the more reliable phase-shift mechanism. This modification not only eliminates the artificial decoding and selection steps of the original model but also enables maintenance of complete population responses throughout the coarse-to-fine process. Second, because of this modification, explicit decoding is no longer necessary but rather is for visualization only. We use a simple threshold criterion to decode multiple disparities from population energy responses instead of a single disparity in the original model. We demonstrate our model using simulations on a variety of transparent and nontransparent stereograms. The model also reproduces psychophysically observed disparity interactions (averaging, thickening, attraction, and repulsion) as the depth separation between two overlapping planes varies.