Intrinsic Coordinate System Research Articles

Calculation of Movement Direction from Firing Activities of Neurons in Intrinsic Co-ordinate Systems Defined by Their Preferred Directions

In this work we carried out computer simulations to compare different coding algorithms (the tensor network theoretical approach of Pellionisz & Llinas, 1979; and the weighted average population coding model of Georgopoulos et al., 1986) that were originally devised to recompute vectors of the external world from firing rate responses of neurons of the central nervous system. Georgopoulos and his colleagues (1986, 1988) observed, in electrophysiological experiments, that certain neurons of the primate motor cortex are selective to the direction of arm movement in a three-dimensional space (directional neurons). The discharge rate of a cell is highest with movement in a certain direction (the cell's "preferred direction") and decreases as a linear function of the cosine of the angle between the direction of movement and the cell's preferred direction. They calculated a population vector to predict the direction of arm movement from neuronal responses as a weighted linear combination of the preferred direction vectors using several sets of weighting coefficients. It was implicitly assumed in this approach that if the brain uses such coding, the calculations are carried out by a further layer of neurons. The tensor network theory also gives an algorithm to calculate an external vector in an intrinsic co-ordinate system whose basis vectors are distinguished vectors assigned to the individual neurons based on results of physiological observations. However, it goes beyond providing a simple mathematical formula to recompute an external vector (Pellionisz & Llinas, 1979). It is a promising theoretical solution for the problem faced by sensorimotor systems; how to transform information about the environment, measured by a diverse set of sensors, into appropriate responses executed by multiple muscles acting in concert. As the weighting coefficients used by Georgopoulos et al. in their calculations differed from those used by Pellionisz & Llinas, we show here how they are related. We compared the exactness and robustness of the two approaches in computer simulations assuming that the firing rate responses of individual neurons would change from trial to trial even when the movement direction is the same. We also allowed that different sets of preferred directions were used in different trials mimicking the case when different movement-related directional neurons would be active from trial to trial. In our computer simulations the outcome of the different algorithms were fairly similar. No experimental protocol can be devised in which the activity of all the possible active motor cortex neurons taking part in coding the direction of movement could be simultaneously monitored. Therefore we provide a mathematical algorithm showing how the effective weights between pairs of directional neurons of the motor cortex and a further layer of directional neurons might reveal which coding algorithm is implemented in the network. These algorithms were simple and gave a good approximation for the movement direction but we relied on the experimental result that the discharge rate of directional motor cortex neurons is a simple linear cosine function of the angle between their preferred direction and the movement direction. For this reason we set aside the vectorial description and devised an alternative population coding solution where this would not be required.

Population coding by simultaneous activities of neurons in intrinsic coordinate systems defined by their receptive field weighting functions

In this paper we address the problem of how external stimuli from the outside world, which can be represented as vectors by a set of numbers, might be expressed, transformed, and possibly reconstructed in neuronal networks of sensory brain areas. Our work is based on the approach of modeling the firing rate responses of simple linear sensory neurons. As characteristic functions, the receptive field weighting profiles can be determined from the firing rate responses of linear neurons to external stimuli in electrophysiological experiments. The responses of such neurons to arbitrary stimuli can be described as a linear function of the dot product of the stimulus vector and a special vector, the receptive field weighting function. The stimulus vector can be reconstructed in the coordinate system whose basis vectors are the receptive field weighting functions of linear neurons as a linear combination of the basis vectors and some weighting coefficients calculated from the firing rates of the neurons. Georgopoulos, Kettner, and Schwartz developed a scheme of population coding to reconstruct a simple three-dimensional vector, the direction of an arm movement from results of electrophysiological observations in primate motor cortex. Daugman introduced an iterative algorithm to compress and reconstruct image vectors in a network of neural elements whose receptive field weighting functions were similar to receptive field weighting functions of orientation-selective neurons in cat visual cortex. Pellionisz and Llinas showed in their tensor network theory how sensorimotor transformations of vectors might take place in neuronal networks. We show how these seemingly different approaches are related and compare their advantages and disadvantages. We adopted the iterative algorithm of Daugman to code image vectors and calculated the weights of neural elements taking place in the reconstruction of the stimuli in different layers of the network. Then we determined what the receptive field profiles of neurons in higher layers look like at different stages of the iteration process and at equilibrium. Finally, we provide theoretical predictions that might be verified in electrophysiological experiments to reveal which population coding scheme is applied by the particular neuronal network in sensory areas of the brain.

Brainstem control of orienting movements: intrinsic coordinate systems and underlying circuitry.

A fundamental issue in the understanding of how the nervous system processes information is the way in which sensory information is used to initiate and guide movements. Recent progress has been made by taking an information processing approach in which information--for example, the spatial location of an object towards which an animal will orient--is tracked through the nervous system from sensory to motor levels. In this approach, neurally encoded information is characterized in terms of its representation within a neural or intrinsic coordinate system or set of neural coding parameters. For example, the retina codes spatial location in terms of the location of activity on the retinal surface, whereas motoneurons code spatial location in terms of the pulling directions of the muscles they activate. In between these two peripheral stages, the information passes through intermediate coordinate systems. These intermediate coordinate systems can be characterized by recording or altering the activity of small groups of neurons while an animal is performing a well-defined sensorimotor task. Spatial location information is used to guide orienting movements, those movements made by the eyes, ears, head, or body which function to center an object of interest in the animal's visual field. The optic tectum and forebrain, their connections to the medial mesencephalic and rhombencephalic brainstem tegmental cell groups, and subsequent connections to brainstem motor nuclei and spinal cord are employed to control fundamental aspects of this behavior. Studies reviewed herein indicate that following the retinotopic coding of spatial location in the retina and tectum, spatial location information appears to enter a different coordinate system at tegmental levels in which spatial aspects of orienting movement are coded in terms of their discrete horizontal and vertical components. This Cartesian coordinate system is an example of an abstract neural coordinate system, in that it is a simple, low-dimensional representation of spatial location which differs greatly from both sensory and motor representations. Also, this Cartesian representation may be common to many orienting movements, yet it appears to differ from the coordinate systems controlling other movement types such as stabilization or phasic movements. This suggests an hypothesis in which coordinate systems, especially at intermediate levels of processing, may be organized according to behavioral task as opposed to being determined by the particular sensory or motor system involved in the behavior. Understanding the evolutionary heritage and computational function of abstract neural coordinate systems, and the relation between different coordinate systems and behavioral tasks may be useful in understanding general aspects of sensory information processing and motor control.

Perception of forearm angles in 3-dimensional space

The purpose of this study was to determine a preferred coordinate system for representation of forearm orientation in 3-dimensional space. In one experiment, the ability of human subjects to perceive angles of the forearm in 3-dimensional space (forearm elevation and yaw--extrinsic coordinate system) was compared to their ability to perceive elbow joint angle (intrinsic coordinate system). While blindfolded, subjects performed an angle reproduction task in which the experimenter first positioned the upper limb in a reference trial. This was followed, after movement of the subject's entire upper limb to a different position, by an attempt to reproduce or match a criterior angle of the reference trial by motion of the forearm in elbow flexion or extension only. Note that matching of the criterion forearm angle in the new upper limb position could not be accomplished by reproducing the entire reference upper limb position, but only by angular motion at the elbow. Matching of all 3 criterion angles was accomplished with about equal accuracy in terms of absolute constant errors and variable errors. Correlation analysis of the perceptual errors showed that forearm elevation and elbow angle perception errors were not biased but that forearm yaw angle matching showed a bias toward elbow angle matching in 7 of 9 subjects. That is errors in forearm yaw perception were attributed to a tendency toward a preferred intrinsic coordinate system for perception of forearm orientation. These results show that subjects can accurately perceive angles in both extrinsic and intrinsic coordinate systems in 3-dimensional space. Thus, these data conflict with previous reports of highly inaccurate perception of elbow joint angles in comparison to perception of forearm elevation. In an attempt to resolve this conflict with previous results, a second experiment was carried out in which perception of forearm elevation and elbow joint angles with the forearm motion constrained to a vertical plane. Results of this experiment showed that during a two-limb elbow angle matching task, four of five subjects exhibited a clear bias toward forearm elevation angle. During a one-limb angle reproduction task only two of five subjects exhibited such a bias. Perception of elevation angles show little bias toward elbow angle matching.(ABSTRACT TRUNCATED AT 400 WORDS)

Is there a preferred coordinate system for perception of hand orientation in three-dimensional space?

The purpose of this experiment was to determine the preferred coordinate system for representation of hand orientation in 3-dimensional space. The ability of human subjects to perceive angles of the hand in 3-dimensional space (elevation, yaw, roll angles-extrinsic coordinate system) was compared to their ability to perceive hand angles relative to the proximal upper limb segments (wrist joint angles, forearm pronation-intrinsic coordinate system). With eyes closed, subjects performed a matching task in which the experimenter positioned the left arm, forearm and hand and the right arm and forearm. Subjects were then told to match an angle in one of the two coordinate systems by moving only the right hand at the wrist or the forearm as in pronation or roll matching. Absolute constant error (ACE), variable error (VE) and normalized variable error (NVE-normalized to tested range of motion) of matching were quantified for each subject for each of the six angles matched. It was hypothesized that matching angles in a preferred coordinate system would be associated with lower ACE, VE and NVE. Overall, ACE and VE were lower for matching hand angles in the intrinsic coordinate system. This suggests that the preferred coordinate system involved specification of hand angles relative to forearm and arm angles (joint angles) rather than the hand angles relative to axes external to the upper limb. However, matching of pronation angles was associated with larger VE and NVE than roll angle matching. There were no significant differences in ACE between pronation and roll matching. In a second experiment subjects with their forearms constrained at different elevations matched hand elevation and wrist flexion angles. Thus, errors in matching the angles in the non-preferred coordinate system were predictable if the subjects were biased toward matching angles in the preferred coordinate system. Trends in the data suggested that subjects preferred matching hand elevation angles but these trends were not consistent within or between subjects. Thus a preferred intrinsic coordinate system for wrist flexion matching was not observed in this experiment. We suggest that matching angles when proximal limb segments are constrained is a simpler task for the subjects (VE lower than in the first experiment) and may bias the matching toward the extrinsic coordinate system. Thus, hand orientation in 3-dimensional space may be perceived as follows: wrist flexion and abduction angles together with forearm elevation and yaw are used to specify hand elevation and yaw; these together with hand roll angle, completely specify the hand angle in 3-dimensional space.

Construction of inner space representation of latticed network circuits by learning

A position-detective sensor with an intrinsic inner coordinate system using a parallel calculation schema is proposed. The schema keeps the calculation time independent of the resolution or the size. Because the inner coordinates are defined as a representation that is suited to the usage of the sensor, they are different from the physical coordinates, which are provided by the arrangement of the detectors. The architecture has a latticed resistive network circuit that can detect moments of input distribution by using its boundary currents. The position of an object to be measured is acquired as an analog value by the circuit. This circuit retains inner coordinates in the resistance distribution. In addition, it is shown that the inner coordinates can be obtained by the adaptive modification of the resistance distribution. Finally, the construction of a position-detective sensor with a non-uniform detector arrangement is simulated and the validity of the method confirmed.

The intrinsic electrophysiological properties of mammalian neurons: insights into central nervous system function.

This article reviews the electroresponsive properties of single neurons in the mammalian central nervous system (CNS). In some of these cells the ionic conductances responsible for their excitability also endow them with autorhythmic electrical oscillatory properties. Chemical or electrical synaptic contacts between these neurons often result in network oscillations. In such networks, autorhythmic neurons may act as true oscillators (as pacemakers) or as resonators (responding preferentially to certain firing frequencies). Oscillations and resonance in the CNS are proposed to have diverse functional roles, such as (i) determining global functional states (for example, sleep-wakefulness or attention), (ii) timing in motor coordination, and (iii) specifying connectivity during development. Also, oscillation, especially in the thalamo-cortical circuits, may be related to certain neurological and psychiatric disorders. This review proposes that the autorhythmic electrical properties of central neurons and their connectivity form the basis for an intrinsic functional coordinate system that provides internal context to sensory input.

A hydrodynamic model developed for the condensate flowing over a sinusoidal fluted tube

Fluted tubes which were designed by Gregorig considering the use of surface tension forces are commonly used in evaporators and condensers for desalination purposes. In this study the surface profile of a fluted tube, which cannot be described by any conventional coordinate system is obtained numerically by using an intrinsic-coordinate system. Navier-Stokes equations are solved after some simplifications for the condensate flowing over the sinusoidal fluted profile to find the volumetric flow rate. The results are found to be in good agreement with the experimental findings.

Path constraints on point-to-point arm movements in three-dimensional space

In this paper data are presented concerning the kinematic and dynamic characteristics of point-to-point arm movements which are inwardly or outwardly directed in three-dimensional space. Elbow and wrist position as well as elbow angle of extension were measured. From these data, other angles were computed trigonometrically and elbow and shoulder torques were calculated. Some of the angles describing arm and forearm motion were found to be linearly related for any given movement. Changes in shoulder and elbow torque were found to be similar to those described for movements restricted to one degree of freedom. Shoulder and elbow motions were not affected when it was required that the orientation of the hand in space remain constant. These observations were taken to indicate that shoulder and elbow motions are tightly coupled for movements in three-dimensional space and that wrist motion has no influence on this coupling. Linear relations between angles express such coupling. They are taken to result from functional constraints and may facilitate the mapping between extrinsic and intrinsic coordinate systems. Some of the observations pertaining to the torque lead to the hypothesis of a further constraint limiting the number of possible trajectories in a point-to-point movement.

The magnetotelluric impedance tensor properties with respect to rotations

A serious limitation to conventional data analysis is that the data refer mainly to elongated bodies. When three‐dimensional distortions are present, quantitative interpretation based only on the off‐diagonal elements of the conventionally rotated impedance tensor is inadequate, because these off‐diagonal elements are insensitive to the tensor trace. The impedance tensor eigenstate formulation proposed in the literature defines a complete set of parameters suitable for recognition of three‐dimensionality. Generally, though, the eigenvalues do not stand for the off‐diagonal elements of an impedance tensor measured in a physical coordinate system. It is shown how the eigenvalues are modified when the relationship between coordinate system rotations and the eigenstate formulation is clarified. A generalization of the conventional analysis results, but the rotation angle obtained is neither unique nor complete To improve the situation, two new analytical rotation angles are proposed. These angles define two complete intrinsic coordinate systems suitable for magnetotelluric data analysis when a general three‐dimensional structure is involved.

Tensorial computer model of gaze—I. Oculomotor activity is expressed in non-orthogonal natural coordinates

The central nervous system expresses its function in natural frames of reference. A most conspicuous feature of such frames is their non-orthogonality. Gaze stabilization and, in particular, the sensorimotor transformations performed by the vestibulo-ocular reflex, are prime examples of such general coordinate transformations between and within multidimensional non-orthogonal frames. Since such operations can be described by tensor formalisms in an abstract manner, this methodology is applied here to develop a tensorial computer model of gaze stabilization. The representation of sensorimotor transformations by a reference-frame independent method obviates the necessity to simplify the intrinsic coordinate systems either by a reduction of the dimensionality or by a presumption of orthogonality. The frames of reference intrinsic to vestibulo-ocular reflex transformation (the vestibular semicircular canals and extraocular muscles) as well as the covariant character of the sensory input and the contravariant character of the motor output are physically obvious. A model built on these intrinsic systems of coordinates first serves to quantitate the degree of non-orthogonality in the extraocular muscle system, and thus to demonstrate both the necessity and the applicability of representing them by a formalism suitable for non-orthogonal systems, such as tensor network theory. The actual non-orthogonality of the gaze-stabilization system can be quantitated on the basis of the difference of covariant and contravariant expressions as follows. Tensor network theory describes sensorimotor transformations by employing a covariant embedding procedure. This, however, yields a covariant intention-type motor vector. If the central nervous system were to transmit these sensory-type components directly to the extraocular muscle motor mechanism, an error-angle would occur since covariants do not physically compose the intended movement. The error in every direction of gaze would be zero only if the extraocular muscle system would constitute an orthogonal set of rotation axes. Otherwise, the error, called refraction angle, is a measure of non-orthogonality. The complexity of the quantitation of non-orthogonality is compounded by the fact that these rotation axes change with the moving eye. Calculation of eye movements, executed both by covariant and contravariant vectors from primary and secondary eye positions, is based on the simplest assumption that the central nervous system establishes the covariant-contravariant transformation in the retinal tangent plane. The results quantitate (1) the significant degree of non-orthogonality in the oculomotor system, (2) the degree of sensitivity of non-orthogonality to the direction of eye movement and to the initial eye position. With the degree of non-orthogonality made explicit, the errors inherent in remanent orthogonal treatments are quantitated. Presenting the differences of covariants and contravariants in the form of eye movement trajectories provides experimentally testable predictions.

Dynamical evolution of angular momentum in damped nuclear reactions: (I). Accumulation of angular momentum by nucleon transfer

The dynamical accumulation of angular momentum in the course of a damped nuclear reaction is studied within the framework of the nucleon exchange transport model. The dinuclear spin distribution is described by the mean values and the covariances of the two prefragment spins and their orbital angular momentum L . Using an intrinsic coordinate system aligned with the fluctuating direction of L , the equations of motion for the spin distribution are derived and discussed. The ultimate transformation to an externally defined reference frame is also discussed. The evolution of other observables and their coupling to the spin variables are included and, by integrating conditional distributions over all impact parameters, results are obtained for differential cross sections corresponding to a specified loss of relative kinetic energy. The characteristic features of the evolution of the spin distribution is discussed in detail. First the stationary solution of the equations of motion is considered and its different appearance in the various relevant coordinate systems is exhibited. The dynamical evolution is discussed in terms of the time-dependent relaxation times associated with the six different intrinsic modes of rotation in the disphere. Due to the relative smallness of the window size the positive modes will dominate (for not too long times), resulting in a predominantly positive correlation between the fragment spin fluctuations. Illustrative applications to cases of experimental interest are made and a critical discussion is given of other models addressing angular momentum in damped nuclear reactions.

Intrinsic Lie group and nuclear collective rotation about intrinsic axes

For any Lie group G, is so called intrinsic Lie group G is introduced, G and G being commutative and anti-isomorphic. It is shown that in the parameter space of a Lie group G, the corresponding intrinsic Lie group G is just the second parameter group. The general relations between the first and second parameter groups are derived in a simple way. For the group SO(3), there is the intrinsic Lie group SO(3). The infinitesimal generators of SO(3) are precisely the components of the angular momentum in the intrinsic coordinate system. Therefore the intrinsic Lie group SO(3) provides an appropriate mathematical formalism for the description of collective rotations of nuclei about their intrinsic axes.

Diffusive transport across corrugated plates and tubes

AbstractThe steady diffusion across a constant thickness corrugated barrier is analyzed through the intrinsic coordinate system. Perturbation solutions are found when the thickness is small compared to the minimum radius of curvature. For a given perimeter length, diffusive transfer is decreased, while for given maximum geometric dimensions, diffusive transfer may be increased by corrugations.

Solutions for Three-Dimensional Over- or Underexpanded Exhaust Plumes

An efficient method for solving the problem of a slightly underor overexpanded supersonic jet exhausting into a subsonic coflowing stream is presented. This paper extends an earlier work on two-dimensional and axisymmetric flows to general three-dimensional flows with specific application to high-aspect-ratio slot jet flows. The viscous problem is rendered parabolic through a unique use of an approximate inviscid flow intrinsic coordinate system to estimate only the streamline curvatures. This inviscid problem is formulated in terms of linearized perturbation potentials for ease of solution. The results of application of this approach to axisymmetric and high-aspect-ratio slot jet flows will be presented.

A new analytical method for the heat conduction across a bent plate

Due to the complicated geometry, the heat conduction across a bent plate has never been treated analytically before. In this paper we utilize an intrinsic coordinate system in order to obtain the temperature profile by perturbations. The curvature of the plate tends to decrease the local temperature in the vicinity of the bend.

Magnetic flow of viscous gas between rotating surfaces of revolution

In this work the steady laminar magnetic flow of viscous gas is considered in a narrow space (slot) between two surfaces of revolution rotating with constant angular velocities around a common axis of symmetry. The linearised equations of magnetic motion of the viscous gas flow for axial symmetry in the intrinsic curvilinear orthogonal coordinate system x, φ, y are used. The obtained solutions of the equations of motion have been illustrated by examples of gas flow through the slot of constant thickness between rotating and fixed conical surfaces, and between rotating and fixed spherical surfaces.

Secondary vorticity centres as centres of secondary (streamwise) vorticity

The theory of secondary vorticity is used to explain the essence of so-called secondary or upper vorticity centres, discovered on satellite photographs. The general equation of vorticity, which is then adjusted by means of the equations of state and motion, is transformed into a specially selected intrinsic co-ordinate system. Three-component equations of vorticity are discussed for cases important meteorologically. The conclusion could be drawn that the secondary vorticity centres may be identified with centres of so-called secondary vorticity and that the theoretical facts about their origin and occurrence may be used, together with satellite photographs, to improve weather forecasts.

Position operators in a (3+1) de Sitter space

Earlier studies of covariant position operators in special relativistic quantum theory are generalized here to the case of a de Sitter universe of positive curvature. The hyperplane formalism previously employed undergoes a natural generalization in this case to a spacelike hypersphere formalism. In the analysis of the underlying geometry, the minimal pseudo-Euclidean embedding space for the de Sitter universe plays a dominant role and suggests an intrinsic coordinate system of special interest for the representation of geodesics. The de Sitter analogs of the Minkowski space center of energy, center of spin, and center of inertia are constructed, and we find that de Sitter-center of spin to retain commuting components in our intrinsic coordinate system.

Exploratory Analysis of Nonuniform Plug Nozzle Flowfields

A method is presented by which the plug nozzle flowfield is calculated for the closed condition. The assumed flow model is divided into two basic components: the separated region or near wake and the inviscid flow adjacent to this near wake. The solution to the near wake is obtained by means of an integral analysis, and the rotational method of characteristics solution is employed in the adjacent inviscid region. Parametric studies for representative nozzle configurations are also included. These parameters include the effects of: plug truncation for an internal-external-expansion truncated plug nozzle, shroud truncation for an expansion-deflection nozzle, the over-all nozzle area ratio, nonisoenergetic mixing, the variation of the ambient pressure, base bleed, and boundary-layer thickness approaching separation. Comparison with a limited amount of experimental data is also presented. Nomenclature A = area H = nondimensional bleed number, Refs. 1, 5 or 6 L = plug length measured from geometric throat M = Mach number m = mass flow rate n = power law profile exponent P = pressure r = radius T — temperature u = velocity X,Y = coordinates of the reference (inviscid) coordinate system x,y = coordinates of the intrinsic (viscous) coordinate system y = ratio of specific heats 8 = boundary-layer thickness $** = boundary-layer momentum thickness e = 0 for planar flows; = 1 for axisymmetric flows 17 = dimensionless coordinate (=crylx) 0 = streamline angle p = density a = jet spread parameter (f> = velocity ratio, u/ui Subscripts