Tropical Geometry of Statistical Models

This paper presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are coordinates of a point in the variety, and the sum-product algorithm is an efficient tool for evaluating specific coordinates. The question addressed here is how the solutions to various inference problems depend on the model parameters. The proposed answer is expressed in terms of tropical algebraic geometry. A key role is played by the Newton polytope of a statistical model. Our results are applied to the hidden Markov model and to the general Markov model on a binary tree.Comment: 14 pages, 3 figures. Major revision. Applications now in companion paper, "Parametric Inference for Biological Sequence Analysis

Interrelations Between the Neutron's Magnetic Interactions and the Magnetic Aharonov-Bohm Effect

It is proved that the phase shift of a polarized neutron interacting with a spatially uniform time-dependent magnetic field, demonstrates the same physical principles as the magnetic Aharonov-Bohm effect. The crucial role of inert objects is explained, thereby proving the quantum mechanical nature of the effect. It is also proved that the nonsimply connectedness of the field-free region is not a profound property of the system and that it cannot be regarded as a sufficient condition for a nonzero phase shift.Comment: 18 pages, 1 postscript figure, Late

Prostate Cancer Screening Trends After United States Preventative Services Task Force Guidelines in an Underserved Population

Purpose: Prostate cancer screening is a controversial topic. We examined trends in Prostate Specific Antigen (PSA) testing in an underserved population before and after the United States Preventative Services Task Force (USPSTF) recommendation against screening. Methods: Data were collected on all PSA and cholesterol screening tests from 2008 to 2014. We examined the trend of these tests and prostate biopsies while comparing this data to lipid panel data to adjust for changes in patient population. Results: A decrease in PSA screening was observed from 2010 through 2014, with the greatest decline in 2012. The age group most affected was patients aged 55-69 years. The amount of prostate biopsies during this period decreased as well. Conclusions: Decreased rates of PSA screening were observed in our urban hospital population that preceded the publication of the USPSTF guidelines. The incidence of prostate biopsies decreased in this timeframe. It now remains to be demonstrated whether decreased PSA screening rates impact the diagnosis of and ultimately the survival from prostate cancer

Double deletion of Panx1 and Panx3 affects skin and bone but not hearing

Pannexins (Panxs), large-pore channel forming glycoproteins, are expressed in a wide variety of tissues including the skin, bone, and cochlea. To date, the use of single knock-out mouse models of both Panx1 and Panx3 have demonstrated their roles in skin development, bone formation, and auditory phenotypes. Due to sequence homology between Panx1 and Panx3, when one Panx is ablated from germline, the other may be upregulated in a compensatory mechanism to maintain tissue homeostasis and function. To evaluate the roles of Panx1 and Panx3 in the skin, bone, and cochlea, we created the first Panx1/Panx3 double knock-out mouse model (dKO). These mice had smaller litters and reduced body weight compared to wildtype controls. The dKO dorsal skin had decreased epidermal and dermal area as well as decreased hypodermal area in neonatal but not in older mice. In addition, mouse skull shape and size were altered, and long bone length was decreased in neonatal dKO mice. Finally, auditory tests revealed that dKO mice did not exhibit hearing loss and were even slightly protected against noise-induced hearing damage at mid-frequency regions. Taken together, our findings suggest that Panx1 and Panx3 are important at early stages of development in the skin and bone but may be redundant in the auditory system. Key messages Panx double KO mice had smaller litters and reduced body weight. dKO skin had decreased epidermal and dermal area in neonatal mice. Skull shape and size changed plus long bone length decreased in neonatal dKO mice. dKO had no hearing loss and were slightly protected against noise-induced damage

State-dependent activity dynamics of hypothalamic stress effector neurons

The stress response necessitates an immediate boost in vital physiological functions from their homeostatic operation to an elevated emergency response. However, the neural mechanisms underlying this state-dependent change remain largely unknown. Using a combination of in vivo and ex vivo electrophysiology with computational modeling, we report that corticotropin releasing hormone (CRH) neurons in the paraventricular nucleus of the hypothalamus (PVN), the effector neurons of hormonal stress response, rapidly transition between distinct activity states through recurrent inhibition. Specifically, in vivo optrode recording shows that under non-stress conditions, CRHPVN neurons often fire with rhythmic brief bursts (RB), which, somewhat counterintuitively, constrains firing rate due to long (~2 s) interburst intervals. Stressful stimuli rapidly switch RB to continuous single spiking (SS), permitting a large increase in firing rate. A spiking network model shows that recurrent inhibition can control this activity-state switch, and more broadly the gain of spiking responses to excitatory inputs. In biological CRHPVN neurons ex vivo, the injection of whole-cell currents derived from our computational model recreates the in vivo-like switch between RB and SS, providing direct evidence that physiologically relevant network inputs enable state-dependent computation in single neurons. Together, we present a novel mechanism for state-dependent activity dynamics in CRHPVN neurons

Four lectures on secant varieties

This paper is based on the first author's lectures at the 2012 University of Regina Workshop "Connections Between Algebra and Geometry". Its aim is to provide an introduction to the theory of higher secant varieties and their applications. Several references and solved exercises are also included.Comment: Lectures notes to appear in PROMS (Springer Proceedings in Mathematics & Statistics), Springer/Birkhause

Hyperbolic planforms in relation to visual edges and textures perception

We propose to use bifurcation theory and pattern formation as theoretical probes for various hypotheses about the neural organization of the brain. This allows us to make predictions about the kinds of patterns that should be observed in the activity of real brains through, e.g. optical imaging, and opens the door to the design of experiments to test these hypotheses. We study the specific problem of visual edges and textures perception and suggest that these features may be represented at the population level in the visual cortex as a specific second-order tensor, the structure tensor, perhaps within a hypercolumn. We then extend the classical ring model to this case and show that its natural framework is the non-Euclidean hyperbolic geometry. This brings in the beautiful structure of its group of isometries and certain of its subgroups which have a direct interpretation in terms of the organization of the neural populations that are assumed to encode the structure tensor. By studying the bifurcations of the solutions of the structure tensor equations, the analog of the classical Wilson and Cowan equations, under the assumption of invariance with respect to the action of these subgroups, we predict the appearance of characteristic patterns. These patterns can be described by what we call hyperbolic or H-planforms that are reminiscent of Euclidean planar waves and of the planforms that were used in [1, 2] to account for some visual hallucinations. If these patterns could be observed through brain imaging techniques they would reveal the built-in or acquired invariance of the neural organization to the action of the corresponding subgroups.Comment: 34 pages, 11 figures, 2 table