A theory of geometric constraints on neural activity for natural three-dimensional movement

Synapse fits neuron: joint reduction by model inversion

In this paper, we introduce a novel simplification method for dealing with physical systems that can be thought to consist of two subsystems connected in series, such as a neuron and a synapse. The aim of our method is to help find a simple, yet convincing model of the full cascade-connected system, assuming that a satisfactory model of one of the subsystems, e.g., the neuron, is already given. Our method allows us to validate a candidate model of the full cascade against data at a finer scale. In our main example, we apply our method to part of the squid's giant fiber system. We first postulate a simple, hypothetical model of cell-to-cell signaling based on the squid's escape response. Then, given a FitzHugh-type neuron model, we derive the verifiable model of the squid giant synapse that this hypothesis implies. We show that the derived synapse model accurately reproduces synaptic recordings, hence lending support to the postulated, simple model of cell-to-cell signaling, which thus, in turn, can be used as a basic building block for network models.

The influence of proprioceptive state on learning control of reach dynamics

The motor system shows a remarkable capacity to generalize learned behavior to new contexts while simultaneously permitting learning of multiple and sometimes conflicting skills. To examine the influence of proprioceptive state on this capacity, we compared the effectiveness of changes in workspace location and limb orientation (horizontal vs. parasagittal plane posture) in facilitating learning of opposing dynamic force-field perturbations. When opposing fields were encountered in similar workspace positions and limb orientations, subjects failed to learn the two tasks. In contrast, differences in initial limb proprioceptive state were sufficient for significant learning to take place. The extent of learning was similar when the two fields were encountered in different arm orientations in a similar workspace location as compared to when learning took place in spatially separated workspace locations, consistent with the generalization of learning mainly in intrinsic joint coordinates. In keeping with these observations, examination of how trial-to-trial adaptation generalized showed that generalization tended to be greater across similar limb postures. However, when the two fields were encountered in distinct spatial locations, the extent of generalization of adaptation to one field depended on the limb orientation in which the other field was encountered. These results suggest that three-dimensional proprioceptive limb state plays an important role in modulating generalization patterns so as to permit the best compromise between broad generalization and the simultaneous learning of conflicting skills.

Modeling the shape hierarchy for visually guided grasping

The monkey anterior intraparietal area (AIP) encodes visual information about three-dimensional object shape that is used to shape the hand for grasping. We modeled shape tuning in visual AIP neurons and its relationship with curvature and gradient information from the caudal intraparietal area (CIP). The main goal was to gain insight into the kinds of shape parameterizations that can account for AIP tuning and that are consistent with both the inputs to AIP and the role of AIP in grasping. We first experimented with superquadric shape parameters. We considered superquadrics because they occupy a role in robotics that is similar to AIP, in that superquadric fits are derived from visual input and used for grasp planning. We also experimented with an alternative shape parameterization that was based on an Isomap dimension reduction of spatial derivatives of depth (i.e., distance from the observer to the object surface). We considered an Isomap-based model because its parameters lacked discontinuities between similar shapes. When we matched the dimension of the Isomap to the number of superquadric parameters, the superquadric model fit the AIP data somewhat more closely. However, higher-dimensional Isomaps provided excellent fits. Also, we found that the Isomap parameters could be approximated much more accurately than superquadric parameters by feedforward neural networks with CIP-like inputs. We conclude that Isomaps, or perhaps alternative dimension reductions of visual inputs to AIP, provide a promising model of AIP electrophysiology data. Further work is needed to test whether such shape parameterizations actually provide an effective basis for grasp control.

STIEFEL-MANIFOLD LEARNING BY IMPROVED RIGID-BODY THEORY APPLIED TO ICA

In previous contributions we presented a new class of algorithms for orthonormal learning of a linear neural network with p inputs and m outputs, based on the equations describing the dynamics of a massive rigid frame in a submanifold of ℛp. While exhibiting interesting features, such as intrinsic numerical stability, strongly binding to the orthonormal submanifolds, and good controllability of the learning dynamics, tested on principal/independent component analysis, the proposed algorithms were not completely satisfactory from a computational-complexity point of view. The main drawback was the need to repeatedly evaluate a matrix exponential map. With the aim to lessen the computational efforts pertaining to these algorithms, we propose here an improvement based on the closed-form Rodriguez formula for the exponential map. Such formula is available in the p=3 and m=3 case, which is discussed with details here. In particular, experimental results on independent component analysis (ICA), carried out with both synthetic and real-world data, help confirming the computational gain due to the proposed improvement.

Assessing the function of motor cortex: single-neuron models of how neural response is modulated by limb biomechanics

Do neurons in primary motor cortex encode the generative details of motor behavior, such as individual muscle activities, or do they encode high-level movement attributes? Resolving this question has proven difficult, in large part because of the sizeable uncertainty inherent in estimating or measuring the joint torques and muscle forces that underlie movements made by biological limbs. We circumvented this difficulty by considering single-neuron responses in an isometric task, where joint torques and muscle forces can be straightforwardly computed from limb geometry. The response for each neuron was modeled as a linear function of a "preferred" joint torque vector, and this model was fit to individual neural responses across variations in limb posture. The resulting goodness of fit suggests that neurons in motor cortex do encode the kinetics of motor behavior and that the neural response properties of "preferred direction" and "gain" are dual components of a unitary response vector.

Temporal complexity and heterogeneity of single-neuron activity in premotor and motor cortex

The relationship between neural activity in motor cortex and movement is highly debated. Although many studies have examined the spatial tuning (e.g., for direction) of cortical responses, less attention has been paid to the temporal properties of individual neuron responses. We developed a novel task, employing two instructed speeds, that allows meaningful averaging of neural responses across reaches with nearly identical velocity profiles. Doing so preserves fine temporal structure and reveals considerable complexity and heterogeneity of response patterns in primary motor and premotor cortex. Tuning for direction was prominent, but the preferred direction was frequently inconstant with respect to time, instructed-speed, and/or reach distance. Response patterns were often temporally complex and multiphasic, and varied with direction and instructed speed in idiosyncratic ways. A wide variety of patterns was observed, and it was not uncommon for a neuron to exhibit a pattern shared by no other neuron in our dataset. Response patterns of individual neurons rarely, if ever, matched those of individual muscles. Indeed, the set of recorded responses spanned a much higher dimensional space than would be expected for a model in which neural responses relate to a moderate number of factors-dynamic, kinematic, or otherwise. Complex responses may provide a basis-set representing many parameters. Alternately, it may be necessary to discard the notion that responses exist to "represent" movement parameters. It has been argued that complex and heterogeneous responses are expected of a recurrent network that produces temporally patterned outputs, and the present results would seem to support this view.

Statistical encoding model for a primary motor cortical brain-machine interface

A number of studies of the motor system suggest that the majority of primary motor cortical neurons represent simple movement-related kinematic and dynamic quantities in their time-varying activity patterns. An example of such an encoding relationship is the cosine tuning of firing rate with respect to the direction of hand motion. We present a systematic development of statistical encoding models for movement-related motor neurons using multielectrode array recordings during a two-dimensional (2-D) continuous pursuit-tracking task. Our approach avoids massive averaging of responses by utilizing 2-D normalized occupancy plots, cascaded linear-nonlinear (LN) system models and a method for describing variability in discrete random systems. We found that the expected firing rate of most movement-related motor neurons is related to the kinematic values by a linear transformation, with a significant nonlinear distortion in about 1/3 of the neurons. The measured variability of the neural responses is markedly non-Poisson in many neurons and is well captured by a "normalized-Gaussian" statistical model that is defined and introduced here. The statistical model is seamlessly integrated into a nearly-optimal recursive method for decoding movement from neural responses based on a Sequential Monte Carlo filter.

Relative uncertainty learning theory: an essay

The aim of this manuscript is to present a detailed analysis of the algebraic and geometric properties of relative uncertainty theory (RUT) applied to neural networks learning. Through the algebraic analysis of the original learning criterion, it is shown that RUT gives rise to principal-subspace-analysis-type learning equations. Through an algebraic-geometric analysis, the behavior of such matrix-type learning equations is illustrated, with particular emphasis to the existence of certain invariant manifolds.

Functional architecture of eye position gain fields in visual association cortex of behaving monkey

In the behaving monkey, inferior parietal lobe cortical neurons combine visual information with eye position signals. However, an organized topographic map of these neurons' properties has never been demonstrated. Intrinsic optical imaging revealed a functional architecture for the effect of eye position on the visual response to radial optic flow. The map was distributed across two subdivisions of the inferior parietal lobule, area 7a and the dorsal prelunate area, DP. Area 7a contains a representation of the lower eye position gain fields while area DP represents the upper eye position gain fields. Horizontal eye position is represented orthogonal to the vertical eye position across the medial lateral extents of the cortices. Similar topographies were found in three hemispheres of two monkeys; the horizontal and vertical gain field representations were not isotropic with a greater modulation found with the vertical. Monte Carlo methods demonstrated the significance of the maps, and they were verified in part using multiunit recordings. The novel topographic organization of this association cortex area provides a substrate for constructing representations of surrounding space for perception and the guidance of motor behaviors.

A computational model of muscle recruitment for wrist movements

To execute a movement, the CNS must appropriately select and activate the set of muscles that will produce the desired movement. This problem is particularly difficult because a variety of muscle subsets can usually be used to produce the same joint motion. The motor system is therefore faced with a motor redundancy problem that must be resolved to produce the movement. In this paper, we present a model of muscle recruitment in the wrist step-tracking task. Muscle activation levels for five muscles are selected so as to satisfy task constraints (moving to the designated target) while also minimizing a measure of the total effort in producing the movement. Imposing these constraints yields muscle activation patterns qualitatively similar to those observed experimentally. In particular, the model reproduces the observed cosine-like recruitment of muscles as a function of movement direction and also appropriately predicts that certain muscles will be recruited most strongly in movement directions that differ significantly from their direction of action. These results suggest that the observed recruitment behavior may not be an explicit strategy employed by the nervous system, but instead may result from a process of movement optimization.

Cosine tuning minimizes motor errors

Cosine tuning is ubiquitous in the motor system, yet a satisfying explanation of its origin is lacking. Here we argue that cosine tuning minimizes expected errors in force production, which makes it a natural choice for activating muscles and neurons in the final stages of motor processing. Our results are based on the empirically observed scaling of neuromotor noise, whose standard deviation is a linear function of the mean. Such scaling predicts a reduction of net force errors when redundant actuators pull in the same direction. We confirm this prediction by comparing forces produced with one versus two hands and generalize it across directions. Under the resulting neuromotor noise model, we prove that the optimal activation profile is a (possibly truncated) cosine--for arbitrary dimensionality of the workspace, distribution of force directions, correlated or uncorrelated noise, with or without a separate cocontraction command. The model predicts a negative force bias, truncated cosine tuning at low muscle cocontraction levels, and misalignment of preferred directions and lines of action for nonuniform muscle distributions. All predictions are supported by experimental data.

Unsupervised neural learning on lie group

The present paper aims at introducing the concepts and mathematical details of unsupervised neural learning with orthonormality constrains. The neural structures considered are single non-linear layers and the learnable parameters are organized in matrices, as usual, which gives the parameters spaces the geometrical structure of the Euclidean manifold. The constraint of orthonormality for the connection-matrices further restricts the parameters spaces to differential manifolds such as the orthogonal group, the compact Stiefel manifold and its extensions. For these reasons, the instruments for characterizing and studying the behavior of learning equations for these particular networks are provided by the differential geometry of Lie groups. In particular, two sub-classes of the general Lie-group learning theories are studied in detail, dealing with first-order (gradient-based) and second-order (non-gradient-based) learning. Although the considered class of learning theories is very general, in the present paper special attention is paid to unsupervised learning paradigms.

Computational approaches to motor control

New concepts and computational models that integrate behavioral and neurophysiological observations have addressed several of the most fundamental long-standing problems in motor control. These problems include the selection of particular trajectories among the large number of possibilities, the solution of inverse kinematics and dynamics problems, motor adaptation and the learning of sequential behaviors.

Kinematic coordinates in which motor cortical cells encode movement direction

During goal-directed reaching in primates, a sensorimotor transformation generates a dynamical pattern of muscle activation. Within the context of this sensorimotor transformation, a fundamental question concerns the coordinate systems in which individual cells in the primary motor cortex (MI) encode movement direction. This article develops a mathematical framework that computes, as a function of the coordinate system in which an individual cell is hypothesized to operate, the spatial preferred direction (pd) of that cell as the arm configuration and hand location vary. Three coordinate systems are explicitly modeled: Cartesian spatial, shoulder-centered, and joint angle. The computed patterns of spatial pds are distinct for each of these three coordinate systems, and experimental approaches are described that can capitalize on these differences to compare the empirical adequacy of each coordinate hypothesis. One particular experiment involving curved motion was analyzed from this perspective. Out of the three coordinate systems tested, the assumption of joint angle coordinates best explained the observed cellular response properties. The mathematical framework developed in this paper can also be used to design new experiments that are capable of disambiguating between a given set of specified coordinate hypotheses.