Entire Trajectories Research Articles

Trajectories for the wave function of the universe from a simple detector model

Inspired by Mott's (1929) analysis of particle tracks in a cloud chamber, we consider a simple model for quantum cosmology which includes, in the total Hamiltonian, model detectors registering whether or not the system, at any stage in its entire history, passes through a series of regions in configuration space. We thus derive a variety of well-defined formulas for the probabilities for trajectories associated with the solutions to the Wheeler-DeWitt equation. The probability distribution is peaked about classical trajectories in configuration space. The ``measured'' wave functions still satisfy the Wheeler-DeWitt equation, except for small corrections due to the disturbance of the measuring device. With modified boundary conditions, the measurement amplitudes essentially agree with an earlier result of Hartle derived on rather different grounds. In the special case where the system is a collection of harmonic oscillators, the interpretation of the results is aided by the introduction of ``timeless'' coherent states -- eigenstates of the Hamiltonian which are concentrated about entire classical trajectories.

Morphology of single olivocerebellar axons labeled with biotinylated dextran amine in the rat

The morphology of olivocerebellar (OC) axons originating from the inferior olive (IO) was investigated in the rat by reconstructing the entire trajectories of single axons that had been labeled with biotinylated dextran amine. Virtually all of the OC axons entered the cerebellum through the inferior cerebellar peduncle (ICP) contralateral to the IO, with a few exceptions. Although most OC projection was contralateral, a few axons projected bilaterally by crossing the midline within the cerebellum. Collaterals of OC axons could be classified into thick branches and thin collaterals. Thick branches of each OC axon (6.1 +/- 3.7/OC axon, mean +/- SD for n = 16 axons) terminated as climbing fibers (CFs) on single Purkinje cells (PCs) in a one-to-one relationship. Besides terminal arborization around PC thick dendrites, CFs had terminals that surrounded a PC soma, fine branchlets that extended transversely in the molecular layer, and thin retrograde collaterals that re-entered the PC and granular layers. Innervation of a single PC by two CFs originating from the same axon was seen, although infrequently. Concerning thin collaterals, about half of the OC axons had one or only a few collaterals terminating in the white matter of the ICP, most had 1 to 6 collaterals terminating in a single cerebellar nucleus, and all had 3 to 16 collaterals that terminated mainly in the granular layer, but occasionally in the cerebellar white matter and the PC layer. Some swellings of thin collaterals touched somata of presumed Golgi cells and PCs. No OC axons terminated solely in the ICP, cerebellar nucleus or granular layer without giving rise to CFs.

A vector-integration-to-endpoint model for performance of viapoint movements

Viapoint (VP) movements are movements to a desired point that are constrained to pass through an intermediate point. Studies have shown that VP movements possess properties, such as smooth curvature around the VP, that are not explicable by treating VP movements as strict concatenations of simpler point-to-point (PTP) movements. Such properties have led some theorists to propose whole-trajectory optimization models, which imply that the entire trajectory is precomputed before movement initiation. This paper reports new experiments conducted to systematically compare VP with PTP trajectories. Analyses revealed a statistically significant early directional deviation in VP movements but no associated curvature change. An explanation of this effect is offered by extending the vector-integration-to-endpoint (VITE) model (Bullock, D., & Grossberg, S. (1988a). Neural dynamics of planned arm movements: Emergent invariants and speed-accuracy properties during trajectory formation. Psychological Review, 95, 49–90; Bullock, D., & Grossberg, S. (1988b). The VITE model: A neural command circuit for generating arm and articulator trajectories. In J.A.S. Kelso, A.J. Mandell & M.F. Schlesinger (Eds.), Dynamic patterns in complex systems (pp. 305–326). Singapore: World Scientific Publishers.), which postulates that voluntary movement trajectories emerge as internal gating signals control the integration of continuously computed vector commands based on the evolving, perceptible difference between desired and actual position variables. The model explains the observed trajectories of VP and PTP movements as emergent properties of a dynamical system that does not precompute entire trajectories before movement initiation. The new model includes a working memory and a stage sensitive to time-to-contact information. These cooperate to control serial performance. The structural and functional relationships proposed in the model are consistent with available data on forebrain physiology and anatomy.

Signal transmission in the parallel fiber-Purkinje cell system visualized by high-resolution imaging.

We investigated the synaptic transmission in the parallel fiber-Purkinje cell system at high spatio-temporal resolution by using voltage-sensitive dyes and an imaging system. In rat cerebellar slices, cut in the frontal plane or in a plane of the cerebellar surface, local electrical stimulation induced volleys of action potentials in the parallel fibers; subsequent postsynaptic responses from Purkinje cells were observed along the volleys' entire trajectories. Furthermore, the formation of an ordered spatial gradient in parallel fiber conduction velocity across the depth of the molecular layer during postnatal development was observed. In preparations of adult, but not of immature rats, the conduction velocity of parallel fibers in the deep molecular layer was faster than in its more superficial regions. Our observations demonstrate that parallel fibers can mediate Purkinje cell excitation effectively and over considerable distances in a well-organized spatio-temporal manner, thus supporting the classical view of the physiological role assigned to the parallel fibers.

A primal‐dual approach to comparative dynamics with time‐dependent parameters in variational calculus

AbstractBy changing the viewpoint of a variational problem so that the time‐dependent parameters are the choice variables, this paper presents a dual way of establishing the effects that perturbations in such parameters have on the entire optimal trajectories of the original choice variables. The beauty of this approach is that it provides a qualitative characterization of the effects that perturbations in time‐dependent parameters have on the entire optimal trajectories in a symmetric negative semidefinite matrix. Such a characterization is especially important in economics, since nearly all refutable implications in economics can be summarized by such matrices. Furthermore, a one‐line proof of a generalized dynamic envelope theorem is achieved via this methodology.

Venusian saltation

Estimates of the trajectories of saltating particles on Venus show the level of saltation on Ve low when compared to either Earth or Mars. Particles in saltation on Venus obtain maximum heights of only 1 cm over a wide range in particle size and surface wind speeds. Their path lenghts are only a few centimeters at the wind speed of 1 and 2 m/sec. The entire saltation process and particle trajectories are insensitive to changes in surface pressure over the range from 70 to 100 bars and to changes in surface temperature over the range from 600 to 900°K. Secondly, the net rate of surface material transport due to saltation on Venus is small when compared to Earth or Mars. This result is due to the dense Venusian atmosphere. It is estimated that approximately 10 times more surface materials is transported by saltation on Earth than on Venus for dynamically similar conditions. And approximately 250 times more material is moved by the saltation process on Mars than on Venus, again for dynamically similar conditions. Both these estimates apply over a wide range of particle diameter, from 0.01 to 7 mm. Thirdly, the ripple wavelenghts may be small, such that thay may not be detected by the high-resolution radar images of the surface of Venus.