Sensorimotor Underpinnings of Mathematical Imagination: Qualitative Analysis.

Many mathematicians have a rich internal world of mental imagery. Using elementary mathematical skills, this study probes the mathematical imagination's sensorimotor foundations. Mental imagery is perturbed using body position: having the head and vestibular system in different positions with respect to gravity. No two mathematicians described the same imagery. Eight out of 11 habitually visualize, one uses sensorimotor imagery, and two do not habitually used mental imagery. Imagery was both intentional and partly autonomous. For example, coordinate planes rotated, drifted, wobbled, or slid down from vertical to horizontal. Parabolae slid into place or, on one side, a parabola arm reached upward in gravity. The sensorimotor foundation of imagery was evidenced in several ways. The imagery was placed with respect to the body. Further, the imagery had a variety of relationships to the body, such as the body being the coordinate system or the coordinate system being placed in front of the eyes for easy viewing by the mind's eye. The mind's eye, mind's arm, and awareness almost always obeyed the geometry of the real eye and arm. The imagery and body behaved as a dyad, so that the imagery moved or placed itself for the convenience of the mind's eye or arm, which in turn moved to follow the imagery. With eyes closed, participants created a peripersonal imagery space, along with the peripersonal space of the unseen environment. Although mathematics is fundamentally abstract, imagery was sometimes concrete or used a concrete substrate or was placed to avoid being inside concrete objects, such as furniture. Mathematicians varied in the numbers of components of mental imagery and the ways they interacted. The autonomy of the imagery was sometimes of mathematical interest, suggesting that the interaction of imagery habits and autonomy can be a source of mathematical creativity.

Symmetries of a generic utricular projection: neural connectivity and the distribution of utricular information.

Sensory contribution to perception and action depends on both sensory receptors and the organization of pathways (or projections) reaching the central nervous system. Unlike the semicircular canals that are divided into three discrete sensitivity directions, the utricle has a relatively complicated anatomical structure, including sensitivity directions over essentially 360° of a curved, two-dimensional disk. The utricle is not flat, and we do not assume it to be. Directional sensitivity of individual utricular afferents decreases in a cosine-like fashion from peak excitation for movement in one direction to a null or near null response for a movement in an orthogonal direction. Directional sensitivity varies slowly between neighboring cells except within the striolar region that separates the medial from the lateral zone, where the directional selectivity abruptly reverses along the reversal line. Utricular primary afferent pathways reach the vestibular nuclei and cerebellum and, in many cases, converge on target cells with semicircular canal primary afferents and afference from other sources. Mathematically, some canal pathways are known to be characterized by symmetry groups related to physical space. These groups structure rotational information and movement. They divide the target neural center into distinct populations according to the innervation patterns they receive. Like canal pathways, utricular pathways combine symmetries from the utricle with those from target neural centers. This study presents a generic set of transformations drawn from the known structure of the utricle and therefore likely to be found in utricular pathways, but not exhaustive of utricular pathway symmetries. This generic set of transformations forms a 32-element group that is a semi-direct product of two simple abelian groups. Subgroups of the group include order-four elements corresponding to discrete rotations. Evaluation of subgroups allows us to functionally identify the spatial implications of otolith and canal symmetries regarding action and perception. Our results are discussed in relation to observed utricular pathways, including those convergent with canal pathways. Oculomotor and other sensorimotor systems are organized according to canal planes. However, the utricle is evolutionarily prior to the canals and may provide a more fundamental spatial framework for canal pathways as well as for movement. The fullest purely otolithic pathway is likely that which reaches the lumbar spine via Deiters' cells in the lateral vestibular nucleus. It will be of great interest to see whether symmetries predicted from the utricle are identified within this pathway.

Face-infringement space: the frame of reference of the ventral intraparietal area.

Experimental studies have shown that responses of ventral intraparietal area (VIP) neurons specialize in head movements and the environment near the head. VIP neurons respond to visual, auditory, and tactile stimuli, smooth pursuit eye movements, and passive and active movements of the head. This study demonstrates mathematical structure on a higher organizational level created within VIP by the integration of a complete set of variables covering face-infringement. Rather than positing dynamics in an a priori defined coordinate system such as those of physical space, we assemble neuronal receptive fields to find out what space of variables VIP neurons together cover. Section 1 presents a view of neurons as multidimensional mathematical objects. Each VIP neuron occupies or is responsive to a region in a sensorimotor phase space, thus unifying variables relevant to the disparate sensory modalities and movements. Convergence on one neuron joins variables functionally, as space and time are joined in relativistic physics to form a unified spacetime. The space of position and motion together forms a neuronal phase space, bridging neurophysiology and the physics of face-infringement. After a brief review of the experimental literature, the neuronal phase space natural to VIP is sequentially characterized, based on experimental data. Responses of neurons indicate variables that may serve as axes of neural reference frames, and neuronal responses have been so used in this study. The space of sensory and movement variables covered by VIP receptive fields joins visual and auditory space to body-bound sensory modalities: somatosensation and the inertial senses. This joining of allocentric and egocentric modalities is in keeping with the known relationship of the parietal lobe to the sense of self in space and to hemineglect, in both humans and monkeys. Following this inductive step, variables are formalized in terms of the mathematics of graph theory to deduce which combinations are complete as a multidimensional neural structure that provides the organism with a complete set of options regarding objects impacting the face, such as acceptance, pursuit, and avoidance. We consider four basic variable types: position and motion of the face and of an external object. Formalizing the four types of variables allows us to generalize to any sensory system and to determine the necessary and sufficient conditions for a neural center (for example, a cortical region) to provide a face-infringement space. We demonstrate that VIP includes at least one such face-infringement space.

Phase-linking and the perceived motion during off-vertical axis rotation.

Human off-vertical axis rotation (OVAR) in the dark typically produces perceived motion about a cone, the amplitude of which changes as a function of frequency. This perception is commonly attributed to the fact that both the OVAR and the conical motion have a gravity vector that rotates about the subject. Little-known, however, is that this rotating-gravity explanation for perceived conical motion is inconsistent with basic observations about self-motion perception: (a) that the perceived vertical moves toward alignment with the gravito-inertial acceleration (GIA) and (b) that perceived translation arises from perceived linear acceleration, as derived from the portion of the GIA not associated with gravity. Mathematically proved in this article is the fact that during OVAR these properties imply mismatched phase of perceived tilt and translation, in contrast to the common perception of matched phases which correspond to conical motion with pivot at the bottom. This result demonstrates that an additional perceptual rule is required to explain perception in OVAR. This study investigates, both analytically and computationally, the phase relationship between tilt and translation at different stimulus rates-slow (45 degrees /s) and fast (180 degrees /s), and the three-dimensional shape of predicted perceived motion, under different sets of hypotheses about self-motion perception. We propose that for human motion perception, there is a phase-linking of tilt and translation movements to construct a perception of one's overall motion path. Alternative hypotheses to achieve the phase match were tested with three-dimensional computational models, comparing the output with published experimental reports. The best fit with experimental data was the hypothesis that the phase of perceived translation was linked to perceived tilt, while the perceived tilt was determined by the GIA. This hypothesis successfully predicted the bottom-pivot cone commonly reported and a reduced sense of tilt during fast OVAR. Similar considerations apply to the hilltop illusion often reported during horizontal linear oscillation. Known response properties of central neurons are consistent with this ability to phase-link translation with tilt. In addition, the competing "standard" model was mathematically proved to be unable to predict the bottom-pivot cone regardless of the values used for parameters in the model.

Motion parallax contribution to perception of self-motion and depth.

The object of this study is to mathematically specify important characteristics of visual flow during translation of the eye for the perception of depth and self-motion. We address various strategies by which the central nervous system may estimate self-motion and depth from motion parallax, using equations for the visual velocity field generated by translation of the eye through space. Our results focus on information provided by the movement and deformation of three-dimensional objects and on local flow behavior around a fixated point. All of these issues are addressed mathematically in terms of definite equations for the optic flow. This formal characterization of the visual information presented to the observer is then considered in parallel with other sensory cues to self-motion in order to see how these contribute to the effective use of visual motion parallax, and how parallactic flow can, conversely, contribute to the sense of self-motion. This article will focus on a central case, for understanding of motion parallax in spacious real-world environments, of monocular visual cues observable during pure horizontal translation of the eye through a stationary environment. We suggest that the global optokinetic stimulus associated with visual motion parallax must converge in significant fashion with vestibular and proprioceptive pathways that carry signals related to self-motion. Suggestions of experiments to test some of the predictions of this study are made.

Constructive perception of self-motion.

This review focusses attention on a ragged edge of our knowledge of self-motion perception, where understanding ends but there are experimental results to indicate that present approaches to analysis are inadequate. Although self-motion perception displays processes of "top-down" construction, it is typically analyzed as if it is nothing more than a deformation of the stimulus, using a "bottom-up" and input/output approach beginning with the transduction of the stimulus. Analysis often focusses on the extent to which passive transduction of the movement stimulus is accurate. Some perceptual processes that deform or transform the stimulus arise from the way known properties of sensory receptors contribute to perceptual accuracy or inaccuracy. However, further constructive processes in self-motion perception that involve discrete transformations are not well understood. We introduce constructive perception with a linguistic example which displays familiar discrete properties, then look closely at self-motion perception. Examples of self-motion perception begin with cases in which constructive processes transform particular properties of the stimulus. These transformations allow the nervous system to compose whole percepts of movement; that is, self-motion perception acts at a whole-movement level of analysis, rather than passively transducing individual cues. These whole-movement percepts may be paradoxical. In addition, a single stimulus may give rise to multiple perceptions. After reviewing self-motion perception studies, we discuss research methods for delineating principles of the constructed perception of self-motion. The habit of viewing self-motion illusions only as continuous deformations of the stimulus may be blinding the field to other perceptual phenomena, including those best characterized using the mathematics of discrete transformations or mathematical relationships relating sensory modalities in novel, sometimes discrete ways. Analysis of experiments such as these is required to mathematically formalize elements of self-motion perception, the transformations they may undergo, consistency principles, and logical structure underlying multiplicity of perceptions. Such analysis will lead to perceptual rules analogous to those recognized in visual perception.

Spatial symmetries in vestibular projections to the uvula-nodulus.

The discharge of secondary vestibular neurons relays the activity of the vestibular endorgans, occasioned by movements in three-dimensional physical space. At a slightly higher level of analysis, the discharge of each secondary vestibular neuron participates in a multifiber projection or pathway from primary afferents via the secondary neurons to another neuronal population. The logical organization of this projection determines whether characteristics of physical space are retained or lost. The logical structure of physical space is standardly expressed in terms of the mathematics of group theory. The logical organization of a projection can be compared to that of physical space by evaluating its symmetry group. The direct projection from the semicircular canal nerves via the vestibular nuclei to neck motor neurons has a full three-dimensional symmetry group, allowing it to maintain a three-dimensional coordinate frame. However, a projection may embed only a subgroup of the symmetry group of physical space, which incompletely mirrors the properties of physical space. The major visual and vestibular projections in the rabbit via the inferior olive to the uvula-nodulus carry three degrees of freedom-rotations about one vertical and two horizontal axes-but do not have full three dimensional symmetry. Instead, the vestibulo-olivo-nodular projection has symmetries corresponding to a product of two-dimensional vestibular and one-dimensional optokinetic spaces. This combination of projection symmetries provides the foundation for distinguishing horizontal from vertical rotations within a three dimensional space. In this study, we evaluate the symmetry group given by the physiological organization of the vestibulo-olivo-nodular projection. Although it acts on the same sets of elements and mirrors the rotations that occur in physical space, the physiological transformation group is distinct from the spatial group. We identify symmetries as products of physiological and spatial transformations. The symmetry group shapes the information the projection conveys to the uvula-nodulus; this shaping may depend on a physiological choice of generators, in the same way that function depends on the physiological choice of coordinates. We discuss the implications of the symmetry group for uvula-nodulus function, evolution, and functions of the vestibular system in general.

Spatial symmetry groups as sensorimotor guidelines.

While some aspects of neuroanatomical organization are related to packing and access rather than to function, other aspects of anatomical/physiological organization are directly related to function. The mathematics of symmetry groups can be used to determine logical structure in projections and to relate it to function. This paper reviews two studies of the symmetry groups of vestibular projections that are related to the spatial functions of the vestibular complex, including gaze, posture, and movement. These logical structures have been determined by finding symmetry groups of two vestibular projections directly from physiological and anatomical data. Logical structures in vestibular projections are distinct from mapping properties such as the ability to maintain two- and three-dimensional coordinate systems; rather, they provide anatomical/physiological foundations for these mapping properties. The symmetry group of the direct projection from the semicircular canal primary afferents to neck motor neurons is that of the cube (O, the octahedral group), which can serve as a discrete skeleton for coordinate systems in three-dimensional space. The symmetry group of the canal projection from the secondary vestibular afferents to the inferior olive and thence to the cerebellar uvula-nodulus is that of the square (D8), which can support coordinates for the horizontal plane. While the mathematical relationship between these symmetry groups and functions of the vestibular complex are clear, these studies open a larger question: what is the causal logic by which neural centers and their intrinsic organization affect each other and behavior? The relationship of vestibular projection symmetry groups to spatial function make them ideal projections for investigating this causal logic. The symmetry group results are discussed in relationship to possible ways they communicate spatial structure to other neural centers and format spatial functions such as body movements. These two projection symmetry groups suggest that all vestibular projections may have symmetry groups significantly related to function, perhaps all to spatial function.

Head tilt-translation combinations distinguished at the level of neurons.

Angular and linear accelerations of the head occur throughout everyday life, whether from external forces such as in a vehicle or from volitional head movements. The relative timing of the angular and linear components of motion differs depending on the movement. The inner ear detects the angular and linear components with its semicircular canals and otolith organs, respectively, and secondary neurons in the vestibular nuclei receive input from these vestibular organs. Many secondary neurons receive both angular and linear input. Linear information alone does not distinguish between translational linear acceleration and angular tilt, with its gravity-induced change in the linear acceleration vector. Instead, motions are thought to be distinguished by use of both angular and linear information. However, for combined motions, composed of angular tilt and linear translation, the infinite range of possible relative timing of the angular and linear components gives an infinite set of motions among which to distinguish the various types of movement. The present research focuses on motions consisting of angular tilt and horizontal translation, both sinusoidal, where the relative timing, i.e. phase, of the tilt and translation can take any value in the range -180 degrees to 180 degrees . The results show how hypothetical neurons receiving convergent input can distinguish tilt from translation, and that each of these neurons has a preferred combined motion, to which the neuron responds maximally. Also shown are the values of angular and linear response amplitudes and phases that can cause a neuron to be tilt-only or translation-only. Such neurons turn out to be sufficient for distinguishing between combined motions, with all of the possible relative angular-linear phases. Combinations of other neurons, as well, are shown to distinguish motions. Relative response phases and in-phase firing-rate modulation are the key to identifying specific motions from within this infinite set of combined motions.

Variables contributing to the coordination of rapid eye/head gaze shifts.

In this article results of several published studies are synthesized in order to address the neural system for the determination of eye and head movement amplitudes of horizontal eye/head gaze shifts with arbitrary initial head and eye positions. Target position, initial head position, and initial eye position span the space of physical parameters for a planned eye/head gaze saccade. The principal result is that a functional mechanism for determining the amplitudes of the component eye and head movements must use the entire space of variables. Moreover, it is shown that amplitudes cannot be determined additively by summing contributions from single variables. Many earlier models calculate amplitudes as a function of one or two variables and/or restrict consideration to best-fit linear formulae. Our analysis systematically eliminates such models as candidates for a system that can generate appropriate movements for all possible initial conditions. The results of this study are stated in terms of properties of the response system. Certain axiom sets for the intrinsic organization of the response system obey these properties. We briefly provide one example of such an axiomatic model. The results presented in this article help to characterize the actual neural system for the control of rapid eye/head gaze shifts by showing that, in order to account for behavioral data, certain physical quantities must be represented in and used by the neural system. Our theoretical analysis generates predictions and identifies gaps in the data. We suggest needed experiments.

Cognitive-vestibular interactions: a review of patient difficulties and possible mechanisms.

Cognitive deficits such as poor concentration and short-term memory loss are known by clinicians to occur frequently among patients with vestibular abnormalities. Although direct scientific study of such deficits has been limited, several types of investigations do lend weight to the existence of vestibular-cognitive effects. In this article we review a wide range of studies indicating a vestibular influence on the ability to perform certain cognitive functions. In addition to tests of vestibular patient abilities, these studies include dual-task studies of cognitive and balance functions, studies of vestibular contribution to spatial perception and memory, and works demonstrating a vestibular influence on oculomotor and motor coordination abilities that are involved in the performance of everyday cognitive tasks. A growing literature on the physiology of the vestibular system has demonstrated the existence of projections from the vestibular nuclei to the cerebral cortex. The goals of this review are to both raise awareness of the cognitive effects of vestibular disease and to focus scientific attention on aspects of cognitive-vestibular interactions indicated by a wide range of results in the literature.

Rotations in a vertebrate setting: evaluation of the symmetry group of the disynaptic canal-neck projection.

Organizational structures intrinsic to nervous systems can be more precisely analyzed and compared with other logical structures once they are expressed in mathematical languages. A standard mathematical language for expressing organizational structure is that of groups. Groups are especially well suited to organizational structures involving multiple symmetries such as spatial structures. The vestibular system is widely believed to mediate many neural functions involving spatial structure. The vestibular nuclei receive direct projections from the vestibular endorgans, the semicircular canals and the otolith organs. The near-orthogonal directions of the semicircular canals are embedded in the bone. However, those canal directions are external to the nervous system. This study addresses the way the three-dimensional space of rotations is also embedded in the group structure of neural connectivity. Although we know a great deal about physical rotation, it is not clear that nervous systems organize rotations in the same way as physicists do. It would make sense for nervous systems to organize rotations in such a way as to provide physiologically relevant information about performing or compensating for rotations. The vestibular nuclei, which might be expected to display an organization that binds rotations into a rotation space, do not give a clear organization. This may be because of the multiplicity of spatial functions performed by the vestibular nuclei; rather than one spatial organization, the vestibular nuclei are likely to accommodate multiple, related spatial organizations. This study evaluates one particular data set from the literature that specifies the organization of the disynaptic canal-neck projection; other projections and neuronal populations may have other intrinsic organizations. The data are evaluated directly for their symmetry group. In the symmetry group, the vertebrate requirement that physiology have a right and left is found to be satisfied in two ways: (i) by a hexagonal symmetry arising from the right-left doubling of front and back, (ii) along with separate organizations on the two sides that may be required to operate independently to some extent. The eight observed muscle innervation patterns from the data are the complete set of possible combinations of inhibitory/excitatory polarities from three canal pairs. These eight innervation patterns are organized as the vertices of a cube. The two types of side muscles provide the vertical direction. As the head rotates in physical space, the cube rotates in sensorimotor space. Like the canal-neck projection, otolith projections and proprioceptive afferents contact both the vestibular nuclei and neck motoneurons. They may have a similar organization, perhaps with extensions of the same pattern. Otherwise, like a checkerboard superimposed over a paisley, they will form an overlapping organization with the disynaptic canal-neck projection. Further research is required to determine whether the sensorimotor spatial structure of the canal-neck projection is widespread in nervous systems or whether there are several complete structures that are fragmented and reintegrated.

Dynamics of everyday life: rigorous modular modeling in neurobiology based on Bloch's dynamical theorem.

Natural, everyday sensorimotor behaviors, such as rising from sitting, typically have an intrinsic organization of several levels of analysis. Taking this intrinsic organization as key to understanding neural dynamics is neither a top-down nor a bottom-up approach, but rather a meshing of multiple centers and levels of analysis. Motor control requires body dynamics that are consistent with physical dynamics, besides the more microscopic levels of neural dynamics. The dynamics of separate movements have been investigated as if the ends can be capped off, separated from the rest of the individual's life. Is this dynamically correct? Even chaotic behavior is deterministic. However, the mathematics of nonlinear oscillations is not all of dynamics. This paper relates Bloch's dynamical theorem to the modular, conditional approach to sensorimotor and other neural functioning. Bloch's dynamical theorem lays a foundation for the piecewise study of structurally accurate dynamics in theoretical neurobiology. Piecewise studies can be used as a modeling option complementary to the methods of nonlinear oscillator dynamics. By applying Bloch's theorem, dynamics of movements analyzed piecewise can be extended into a smooth flow on any manifold, either as a whole or conditionally. Conditional dynamics makes dynamical modeling options explicit, often depending on what variables the organism can control, and allows one to take different modeling options at different junctures in analyzing the same phenomenon. For example, this approach allows the study of complex motor control problems to be reduced to modular constructions using singularities and flow lines. Dynamical contingencies are expressed using the mathematics of ordered structures. This paper presents Bloch's dynamical theorem and its relevance to model construction in theoretical neurobiology. Specific examples, integrated into physiological and behavioral context, are cited from the literature.

Mathematics reflecting sensorimotor organization.

This review combines short presentations of several mathematical approaches that conceptualize issues in sensorimotor neuroscience from different perspectives and levels of analysis. The intricate organization of neural structures and sensorimotor performance calls for characterization using a variety of mathematical approaches. This review points out the prospects for mathematical neuroscience: in addition to computational approaches, there is a wide variety of mathematical approaches that provide insight into the organization of neural systems. By starting from the perspective that provides the greatest clarity, a mathematical approach avoids specificity that is inaccurate in characterizing the inherent biological organization. Approaches presented include the mathematics of ordered structures, motion-phase space, subject-coincident coordinates, equivalence classes, topological biodynamics, rhythm space metric, and conditional dynamics. Issues considered in this paper include unification of levels of analysis, response equivalence, convergence, relationship of physics to motor control, support of rhythms, state transitions, and focussing on low-dimensional subspaces of a high-dimensional sensorimotor space.

Sensorimotor coordination and the structure of space.

Embedded in neural and behavioral organization is a structure of sensorimotor space. Both this embedded spatial structure and the structure of physical space inform sensorimotor control. This paper reviews studies in which the gravitational vertical and horizontal are crucial. The mathematical expressions of spatial geometry in these studies indicate methods for investigating sensorimotor control in freefall. In freefall, the spatial structure introduced by gravitation - the distinction between vertical and horizontal - does not exist. However, an astronaut arriving in space carries the physiologically-embedded distinction between horizontal and vertical learned on earth. The physiological organization based on this distinction collapses when the strong otolith activity and other gravitational cues for sensorimotor behavior become unavailable. The mathematical methods in this review are applicable in understanding the changes in physiological organization as an astronaut adapts to sensorimotor control in freefall. Many mathematical languages are available for characterizing the logical structures in physiological organization. Here, group theory is used to characterize basic structure of physical and physiological spaces. Dynamics and topology allow the grouping of trajectory ranges according to the outcomes or attractors. The mathematics of ordered structures express complex orderings, such as in multiphase movements in which different parts of the body are moving in different phase sequences. Conditional dynamics, which combines dynamics with the mathematics of ordered structures, accommodates the parsing of movement sequences into trajectories and transitions. Studies reviewed include those of the sit-to-stand movement and early locomotion, because of the salience of gravitation in those behaviors. Sensorimotor transitions and the conditions leading to them are characterized in conditional dynamic control structures that do not require thinking of an organism as an input-output device. Conditions leading to sensorimotor transitions on earth assume the presence of a gravitational vertical which is lacking in space. Thus, conditions used on earth for sensorimotor transitions may become ambiguous in space. A platform study in which sensorimotor transition conditions are ambiguous and are related to motion sickness is reviewed.