Douglas G. Kelly

Stability in contractive nonlinear neural networks

Models of the form mu x=-x+p+WF(x), where x=x(t) is a vector whose entries represent the electrical activities in the units of a neural network are considered. W is a matrix of synaptic weights, F is a nonlinear function, and p is a vector (constant or slowly varying over time) of inputs to the units. If the map WF(x) is a contraction, then the system has a unique equilibrium which is globally asymptotically stable; consequently, the network acts as a stable encoder in that its steady-state response to an input is independent of the initial state of the network. Considered are some relatively mild restrictions on W and F(x), involving the eigenvalues of W and the derivative of F, that are sufficient to ensure that WF(x) is a contraction. It is shown that, in the linear case with spatially homogeneous synaptic weights, the eigenvalues of W are simply related to the Fourier transform of the connection pattern. This relation makes it possible, given cortical activity patterns as measured by autoradiographic labeling, to construct a pattern of synaptic weights which produces steady-state patterns showing similar frequency characteristics.<<ETX>>

Utility of square-wave gratings to assess perioral spatial acuity

PURPOSE The usefulness of square-wave gratings to assess perioral spatial resolution acuity was evaluated. MATERIALS AND METHODS A psychophysical tracking procedure was used to estimate the threshold groove width for discriminating orientation (horizontal or vertical) of square-wave gratings pressed into the skin. Ten positionally matched sites on the two sides of the face of 36 right-handed, healthy young adults were studied. Commercially available gratings provided alternating ridge- and-groove stimuli with element widths from 0.35 to 3.0 mm. RESULTS Thirty-three of the 36 subjects could discriminate orientation at all six sites on the vermilion (threshold width averaged 1.06 mm for grooves and for ridges). Thresholds were lower on the mid-portion of the lower vermilion than on the mid-portion of the upper vermilion (P < .05). Moreover, thresholds were lower on the right side of the vermilion than on the left side (P < .02). In contrast to the vermilion, only 25 and 30 subjects could discriminate orientation on the left and right hairy upper lip, respectively; and two or fewer subjects, at each of 12 sites on the chin and cheeks. CONCLUSIONS Clinical use of small square-wave gratings with ridge and groove widths of 3 mm or smaller is limited to the vermilion. Moreover, baseline values are needed for individual patients to minimize false-positive diagnoses of sensory impairment. The size required of coarser gratings to test other perioral sites may preclude their use for evaluation of discrete, suspect skin areas.

Local receptive field diversity within cortical neuronal populations

The aim of this review paper is to draw attention to a little-appreciated but ubiquitous feature of cerebral cortical organization that we propose is fundamental to an understanding of cortical functioning. Specifically, we (i) describe evidence supporting the idea that neighboring cells in the cortex have richly diverse receptive fields, and (ii) suggest some of the possible mechanisms that generate this receptive field diversity. The possible roles of local receptive field diversity in cortical information processing are discussed.

Asymptotic properties of random subsets of projective spaces

Abstract : A random graph on n vertices is a random subgraph of the complete graph on n vertices. By analogy with this, the present paper studies the asymptotic properties of a random submatroid omega(r) of the projective geometry PG(r-1,q). The main result concerns Kr, the rank of the largest projective geometry occurring as a submatroid of omega(r). We show that with probability one, for sufficiently large r, Kr takes one of at most two values depending on r. This theorem is analogous to a result of Bollobas and Erdos on the clique number of a random graph. However, whereas from the matroid theorem one can essentially determine the critical exponent of omega(r), the graph theorem gives only a lower bound on the chromatic number of a random graph. (Author)

Evocation and characterization of percepts of apparent motion on the face

The percepts evoked by sequential stimulation of sites in close spatial proximity (<2.5 cm) on the face were studied. Both method-of-limits and magnitude-estimation procedures were used to identify and characterize alterations in the percepts produced by systematic changes in the temporal and spatial parameters of the sequence. Each site was stimulated by a vertically oriented row of miniature vibrating probes. Apparent motion was consistently perceived when the delay between the onsets of sequentially activated rows (interstimulus onset interval, or ISOI) fell within a relatively narrow range of values, the lower limit of which approximated 5 msec. Both the upper limit and the perceived smoothness and continuity of the motion percepts (goodness of motion) increased with the duration for which each row stimulated the skin over the range evaluated, 15–185 msec. For the successive activation of only two rows, goodness of motion was not influenced by changes in their separation from 0.4 to 2.5 cm. The ISOI values at which magnitude estimates of goodness of motion were highest increased with the duration for which each row stimulated the skin. As such, maximum goodness of motion decreased with increases in the apparent velocity of motion. When the number of sequentially activated rows was increased from two to four or more, the quality of the motion percepts improved. For the successive activation of multiple closely spaced rows, values of ISOI at which numerical estimates of goodness of motion were highest approximated integral fractions of the duration for which each row stimulated the skin. In this situation, the probes rose and fell in a regular, step-locked rhythm to simulate an edge-like or rectangular object moving across the skin. The goodness of motion so attained was relatively independent of the apparent velocity of motion.

A novel approach for studying direction discrimination

The use of frictionless moving stimuli to study direction discrimination limits the peripheral neural mechanisms by which information can be encoded. The utility of such stimuli, provided by a novel dense-array tactile stimulator, to address basic psychophysical questions regarding cutaneous motion processing is demonstrated. Data collected to date suggest that direction of simulated motion is discriminated when the product of the length of skin traversed and the duration of stimulation attains or exceeds a criterion -- reflecting a perfect space-time tradeoff or reciprocity at threshold.

The cortical pyramidal cell as a set of interacting error backpropagating dendrites: mechanism for discovering nature's order

Central to our ability to have behaviors adapted to environmental circumstances is our skill at recognizing and making use of the orderly nature of the environment. This is a most remarkable skill, considering that behaviorally significant environmental regularities are not easy to discern: they are complex, operating among multi-level nonlinear combinations of environmental conditions, which are orders of complexity removed from raw sensory inputs. How the brain is able to recognize such high-order conditional regularities is, arguably, the most fundamental question facing neuroscience. We propose that the brains basic mechanism for discovering such complex regularities is implemented at the level of individual pyramidal cells in the cerebral cortex. The proposal has three essential components: 1. Pyramidal cells have 5-8 principal dendrites. Each such dendrite is a functional analog of an error backpropagating network, capable of learning complex, nonlinear input-to-output transfer functions. 2. Each dendrite is trained, in learning its transfer function, by all the other principal dendrites of the same cell. These dendrites teach each other to respond to their separate inputs with matching outputs. 3. Exposed to different but related information about the sensory environment, principal dendrites of the same cell tune to different nonlinear combinations of environmental conditions that are predictably related. As a result, the cell as a whole tunes to a set of related combinations of environmental conditions that define an orderly feature of the environment.Single pyramidal cells, of course, are not omnipotent as to the complexity of orderly relations they can discover in their sensory environments. However, when organized into feed-forward/feedback layers, they can build their discoveries on the discoveries of other cells, thus cooperatively unraveling natures more and more complex regularities. If correct, this new understanding of the pyramidal cells functional nature offers a fundamentally new insight into the brains function and identifies what might be one of the key neural computational operations underlying the brains tremendous cognitive and behavioral capabilities.

The Higgs factorization of a geometric strong map

Abstract The Higgs factorization of a strong map between matroids on a fixed set is that factorization into elementary maps in which each matroid is the Higgs lift of its successor. This factorization is characterized by properties of the modular filters which induce the elementary maps of the factorizations in two different ways. It is also shown to be minimal in a natural order on factorizations arising from the weak-map partial order on matroids. The notion of essential nullity of flats of a matroid is introduced; this quantity is nonzero precisely for the cyclic flats, and is shown to be related to the minimal flats of the modular filters inducing the maps of the Higgs factorization.

Markov processes with infinitely divisible limit distributions: some examples

A set of examples is described which suggests that members of a certain class of Markov processes have infinitely divisible limit distributions. Counter examples rule out such a possibility and raise the question of what further restrictions are required to guarantee infinitely divisible limits. Some related examples illustrate the same occurrence of infinitely divisible limit distributions. For both settings, an easily checked necessary and sufficient condition is obtained for the existence of a limit distribution.

University of North Carolina (Chapel Hill) Department of Statistics and Operations Research

The Department of Statistics and Operations Research of the University of North Carolina at Chapel Hill (UNC) had its beginning in 1946 as the Department of Mathematical Statistics, under Harold Hotelling. It was then part of the Institute of Statistics, begun by Gertrude Cox in 1942, which established the Triangle area of North Carolina as a focus of statistical research and applications. The department’s expansion and growth included merger with the Department of Operations Research in 2003. Its research and teaching activities now include most areas of theory and applications in probability, statistics, and operations research.