Geometric Invariant Theory via Cox Rings

We consider actions of reductive groups on a varieties with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox ring all maximal open subsets such that the quotient is quasiprojective or embeddable into a toric variety. As applications, we obtain an explicit description of the chamber structure of the linearized ample cone and several Gelfand-MacPherson type correspondences relating quotients of reductive groups to quotients of torus actions. Moreover, our approach provides information on the geometry of many of the resulting quotient spaces.Comment: 27 pages, minor changes, Example 8.8 replaced, to appear in Journal of Pure and Applied Algebr

On embeddings of homogeneous spaces with small boundary

We study equivariant embeddings with small boundary of a given homogeneous space $G/H$, where $G$ is a connected, linear algebraic group with trivial Picard group and only trivial characters, and $H \subset G$ is an extension of a connected Grosshans subgroup by a torus. Under certain maximality conditions, like completeness, we obtain finiteness of the number of isomorphism classes of such embeddings, and we provide a combinatorial description the embbeddings and their morphisms. The latter allows a systematic treatment of examples and basic statements on the geometry of the equivariant embeddings of a given homogeneous space $G/H$.Comment: minor changes, 30 pages, to appear in J. Algebr

On the classification of Kahler-Ricci solitons on Gorenstein del Pezzo surfaces

We give a classification of all pairs (X,v) of Gorenstein del Pezzo surfaces X and vector fields v which are K-stable in the sense of Berman-Nystrom and therefore are expected to admit a Kahler-Ricci solition. Moreover, we provide some new examples of Fano threefolds admitting a Kahler-Ricci soliton.Comment: 21 pages, ancillary files containing calculations in SageMath; minor correction

Localized direction selective responses in the dendrites of visual interneurons of the fly

<p>Abstract</p> <p>Background</p> <p>The various tasks of visual systems, including course control, collision avoidance and the detection of small objects, require at the neuronal level the dendritic integration and subsequent processing of many spatially distributed visual motion inputs. While much is known about the pooled output in these systems, as in the medial superior temporal cortex of monkeys or in the lobula plate of the insect visual system, the motion tuning of the elements that provide the input has yet received little attention. In order to visualize the motion tuning of these inputs we examined the dendritic activation patterns of neurons that are selective for the characteristic patterns of wide-field motion, the lobula-plate tangential cells (LPTCs) of the blowfly. These neurons are known to sample direction-selective motion information from large parts of the visual field and combine these signals into axonal and dendro-dendritic outputs.</p> <p>Results</p> <p>Fluorescence imaging of intracellular calcium concentration allowed us to take a direct look at the local dendritic activity and the resulting local preferred directions in LPTC dendrites during activation by wide-field motion in different directions. These 'calcium response fields' resembled a retinotopic dendritic map of local preferred directions in the receptive field, the layout of which is a distinguishing feature of different LPTCs.</p> <p>Conclusions</p> <p>Our study reveals how neurons acquire selectivity for distinct visual motion patterns by dendritic integration of the local inputs with different preferred directions. With their spatial layout of directional responses, the dendrites of the LPTCs we investigated thus served as matched filters for wide-field motion patterns.</p

Universal Statistical Behavior of Neural Spike Trains

We construct a model that predicts the statistical properties of spike trains generated by a sensory neuron. The model describes the combined effects of the neuron's intrinsic properties, the noise in the surrounding, and the external driving stimulus. We show that the spike trains exhibit universal statistical behavior over short times, modulated by a strongly stimulus-dependent behavior over long times. These predictions are confirmed in experiments on H1, a motion-sensitive neuron in the fly visual system.Comment: 7 pages, 4 figure

Function of a Fly Motion-Sensitive Neuron Matches Eye Movements during Free Flight

Sensing is often implicitly assumed to be the passive acquisition of information. However, part of the sensory information is generated actively when animals move. For instance, humans shift their gaze actively in a sequence of saccades towards interesting locations in a scene. Likewise, many insects shift their gaze by saccadic turns of body and head, keeping their gaze fixed between saccades. Here we employ a novel panoramic virtual reality stimulator and show that motion computation in a blowfly visual interneuron is tuned to make efficient use of the characteristic dynamics of retinal image flow. The neuron is able to extract information about the spatial layout of the environment by utilizing intervals of stable vision resulting from the saccadic viewing strategy. The extraction is possible because the retinal image flow evoked by translation, containing information about object distances, is confined to low frequencies. This flow component can be derived from the total optic flow between saccades because the residual intersaccadic head rotations are small and encoded at higher frequencies. Information about the spatial layout of the environment can thus be extracted by the neuron in a computationally parsimonious way. These results on neuronal function based on naturalistic, behaviourally generated optic flow are in stark contrast to conclusions based on conventional visual stimuli that the neuron primarily represents a detector for yaw rotations of the animal

Distributed Dendritic Processing Facilitates Object Detection: A Computational Analysis on the Visual System of the Fly

Hennig P, Möller R, Egelhaaf M. Distributed Dendritic Processing Facilitates Object Detection: A Computational Analysis on the Visual System of the Fly. PLoS ONE. 2008;3(8): e3092.Background: Detecting objects is an important task when moving through a natural environment. Flies, for example, may land on salient objects or may avoid collisions with them. The neuronal ensemble of Figure Detection cells (FD-cells) in the visual system of the fly is likely to be involved in controlling these behaviours, as these cells are more sensitive to objects than to extended background structures. Until now the computations in the presynaptic neuronal network of FD-cells and, in particular, the functional significance of the experimentally established distributed dendritic processing of excitatory and inhibitory inputs is not understood. Methodology/Principal Findings: We use model simulations to analyse the neuronal computations responsible for the preference of FD-cells for small objects. We employed a new modelling approach which allowed us to account for the spatial spread of electrical signals in the dendrites while avoiding detailed compartmental modelling. The models are based on available physiological and anatomical data. Three models were tested each implementing an inhibitory neural circuit, but differing by the spatial arrangement of the inhibitory interaction. Parameter optimisation with an evolutionary algorithm revealed that only distributed dendritic processing satisfies the constraints arising from electrophysiological experiments. In contrast to a direct dendro-dendritic inhibition of the FD-cell (Direct Distributed Inhibition model), an inhibition of its presynaptic retinotopic elements (Indirect Distributed Inhibition model) requires smaller changes in input resistance in the inhibited neurons during visual stimulation. Conclusions/Significance: Distributed dendritic inhibition of retinotopic elements as implemented in our Indirect Distributed Inhibition model is the most plausible wiring scheme for the neuronal circuit of FD-cells. This microcircuit is computationally similar to lateral inhibition between the retinotopic elements. Hence, distributed inhibition might be an alternative explanation of perceptual phenomena currently explained by lateral inhibition networks