An Algebraic Analysis of Conchoids to Algebraic Curves

We study the conchoid to an algebraic affine plane curve C from the perspective of algebraic geometry, analyzing their main algebraic properties. Beside C, the notion of conchoid involves a point A in the affine plane (the focus) and a nonzero field element d (the distance).We introduce the formal definition of conchoid by means of incidence diagrams.We prove that the conchoid is a 1-dimensional algebraic set having atmost two irreducible components. Moreover, with the exception of circles centered at the focus A and taking d as its radius, all components of the corresponding conchoid have dimension 1. In addition, we introduce the notions of special and simple components of a conchoid. Furthermore we state that, with the exception of lines passing through A, the conchoid always has at least one simple component and that, for almost every distance, all the components of the conchoid are simple. We state that, in the reducible case, simple conchoid components are birationally equivalent to C, and we show how special components can be used to decide whether a given algebraic curve is the conchoid of another curve

Encoding and retrieval in a CA1 microcircuit model of the hippocampus

Recent years have witnessed a dramatic accumulation of knowledge about the morphological, physiological and molecular characteristics, as well as connectivity and synaptic properties of neurons in the mammalian hippocampus. Despite these advances, very little insight has been gained into the computational function of the different neuronal classes; in particular, the role of the various inhibitory interneurons in encoding and retrieval of information remains elusive. Mathematical and computational models of microcircuits play an instrumental role in exploring microcircuit functions and facilitate the dissection of operations performed by diverse inhibitory interneurons. A model of the CA1 microcircuitry is presented using biophysical representations of its major cell types: pyramidal, basket, axo-axonic, bistratified and oriens lacunosummoleculare cells. Computer simulations explore the biophysical mechanisms by which encoding and retrieval of spatio-temporal input patterns are achieved by the CA1 microcircuitry. The model proposes functional roles for the different classes of inhibitory interneurons in the encoding and retrieval cycles

Modeling of the condyle elements within a biomechanical knee model

The development of a computational multibody knee model able to capture some of the fundamental properties of the human knee articulation is presented. This desideratum is reached by including the kinetics of the real knee articulation. The research question is whether an accurate modeling of the condyle contact in the knee will lead to reproduction of the complex combination of flexion/extension, abduction/adduction and tibial rotation ob-served in the real knee? The model is composed by two anatomic segments, the tibia and the femur, whose characteristics are functions of the geometric and anatomic properties of the real bones. The biomechanical model characterization is developed under the framework of multibody systems methodologies using Cartesian coordinates. The type of approach used in the proposed knee model is the joint surface contact conditions between ellipsoids, represent-ing the two femoral condyles, and points, representing the tibial plateau and the menisci. These elements are closely fitted to the actual knee geometry. This task is undertaken by con-sidering a parameter optimization process to replicate experimental data published in the lit-erature, namely that by Lafortune and his co-workers in 1992. Then, kinematic data in the form of flexion/extension patterns are imposed on the model corresponding to the stance phase of the human gait. From the results obtained, by performing several computational simulations, it can be observed that the knee model approximates the average secondary mo-tion patterns observed in the literature. Because the literature reports considerable inter-individual differences in the secondary motion patterns, the knee model presented here is also used to check whether it is possible to reproduce the observed differences with reasonable variations of bone shape parameters. This task is accomplished by a parameter study, in which the main variables that define the geometry of condyles are taken into account. It was observed that the data reveal a difference in secondary kinematics of the knee in flexion ver-sus extension. The likely explanation for this fact is the elastic component of the secondary motions created by the combination of joint forces and soft tissue deformations. The proposed knee model is, therefore, used to investigate whether this observed behavior can be explained by reasonable elastic deformations of the points representing the menisci in the model.Fundação para a Ciência e a Tecnologia (FCT) - PROPAFE – Design and Development of a Patello-Femoral Prosthesis (PTDC/EME-PME/67687/2006), DACHOR - Multibody Dynamics and Control of Hybrid Active Orthoses MIT-Pt/BSHHMS/0042/2008, BIOJOINTS - Development of advanced biological joint models for human locomotion biomechanics (PTDC/EME-PME/099764/2008)

Imbalanced pattern completion vs. separation in cognitive disease: network simulations of synaptic pathologies predict a personalized therapeutics strategy

<p>Abstract</p> <p>Background</p> <p>Diverse Mouse genetic models of neurodevelopmental, neuropsychiatric, and neurodegenerative causes of impaired cognition exhibit at least four convergent points of synaptic malfunction: 1) Strength of long-term potentiation (LTP), 2) Strength of long-term depression (LTD), 3) Relative inhibition levels (Inhibition), and 4) Excitatory connectivity levels (Connectivity).</p> <p>Results</p> <p>To test the hypothesis that pathological increases or decreases in these synaptic properties could underlie imbalances at the level of basic neural network function, we explored each type of malfunction in a simulation of autoassociative memory. These network simulations revealed that one impact of impairments or excesses in each of these synaptic properties is to shift the trade-off between pattern separation and pattern completion performance during memory storage and recall. Each type of synaptic pathology either pushed the network balance towards intolerable error in pattern separation or intolerable error in pattern completion. Imbalances caused by pathological impairments or excesses in LTP, LTD, inhibition, or connectivity, could all be exacerbated, or rescued, by the simultaneous modulation of any of the other three synaptic properties.</p> <p>Conclusions</p> <p>Because appropriate modulation of any of the synaptic properties could help re-balance network function, regardless of the origins of the imbalance, we propose a new strategy of personalized cognitive therapeutics guided by assay of pattern completion vs. pattern separation function. Simulated examples and testable predictions of this theorized approach to cognitive therapeutics are presented.</p