Positive feedback plays a key role in the ability of signaling molecules to form highly localized clusters in the membrane or cytosol of cells. Such clustering can occur in the absence of localizing mechanisms such as pre-existing spatial cues, diffusional barriers, or molecular cross-linking. What prevents positive feedback from amplifying inevitable biological noise when an un-clustered "off" state is desired? And, what limits the spread of clusters when an "on" state is desired? Here, we show that a minimal positive feedback circuit provides the general principle for both suppressing and amplifying noise: below a critical density of signaling molecules, clustering switches off; above this threshold, highly localized clusters are recurrently generated. Clustering occurs only in the stochastic regime, suggesting that finite sizes of molecular populations cannot be ignored in signal transduction networks. The emergence of a dominant cluster for finite numbers of molecules is partly a phenomenon of random sampling, analogous to the fixation or loss of neutral mutations in finite populations. We refer to our model as the "neutral drift polarity model." Regulating the density of signaling molecules provides a simple mechanism for a positive feedback circuit to robustly switch between clustered and un-clustered states. The intrinsic ability of positive feedback both to create and suppress clustering is a general mechanism that could operate within diverse biological networks to create dynamic spatial organization.

In this paper, we propose an image segmentation technique based on augmenting the conformal (or geodesic) active contour framework with directional information. In the isotropic case, the Euclidean metric is locally multiplied by a scalar conformal factor based on image information such that the weighted length of curves lying on points of interest (typically edges) is small. The conformal factor which is chosen depends only upon position and is in this sense isotropic. While directional information has been studied previously for other segmentation frameworks, here we show that if one desires to add directionality in the conformal active contour framework, then one gets a well-defined minimization problem in the case that the factor defines a Finsler metric. Optimal curves may be obtained using the calculus of variations or dynamic programming based schemes. Finally we demonstrate the technique by extracting roads from aerial imagery, blood vessels from medical angiograms, and neural tracts from diffusion-weighted magnetic resonance imagery.

In this paper, we consider a novel three-dimensional (3-D) visualization technique based on surface flattening for virtual colonoscopy. Such visualization methods could be important in virtual colonoscopy because they have the potential for noninvasively determining the presence of polyps and other pathologies. Further, we demonstrate a method that presents a surface scan of the entire colon as a cine, and affords the viewer the opportunity to examine each point on the surface without distortion. We use certain angle-preserving mappings from differential geometry to derive an explicit method for flattening surfaces obtained from 3-D colon computed tomography (CT) imagery. Indeed, we describe a general method based on a discretization of the Laplace-Beltrami operator for flattening a surface into the plane in a conformal manner. From a triangulated surface representation of the colon, we indicate how the procedure may be implemented using a finite element technique, which takes into account special boundary conditions. We also provide simple formulas that may be used in a real-time cine to correct for distortion.

We give an explicit method for mapping any simply connected surface onto the sphere in a manner which preserves angles. This technique relies on certain conformal mappings from differential geometry. Our method provides a new way to automatically assign texture coordinates to complex undulating surfaces. We demonstrate a finite element method that can be used to apply our mapping technique to a triangulated geometric description of a surface.

In this paper, using certain conformal mappings from uniformization theory, we give an explicit method for flattening the brain surface in a way which preserves angles. From a triangulated surface representation of the cortex, we indicate how the procedure may be implemented using finite elements. Further, we show how the geometry of the brain surface may be studied using this approach.