Bohm and Einstein-Sasaki Metrics, Black Holes and Cosmological Event Horizons

We study physical applications of the Bohm metrics, which are infinite sequences of inhomogeneous Einstein metrics on spheres and products of spheres of dimension 5 <= d <= 9. We prove that all the Bohm metrics on S^3 x S^2 and S^3 x S^3 have negative eigenvalue modes of the Lichnerowicz operator and by numerical methods we establish that Bohm metrics on S^5 have negative eigenvalues too. We argue that all the Bohm metrics will have negative modes. These results imply that higher-dimensional black-hole spacetimes where the Bohm metric replaces the usual round sphere metric are classically unstable. We also show that the stability criterion for Freund-Rubin solutions is the same as for black-hole stability, and hence such solutions using Bohm metrics will also be unstable. We consider possible endpoints of the instabilities, and show that all Einstein-Sasaki manifolds give stable solutions. We show how Wick rotation of Bohm metrics gives spacetimes that provide counterexamples to a strict form of the Cosmic Baldness conjecture, but they are still consistent with the intuition behind the cosmic No-Hair conjectures. We show how the Lorentzian metrics may be created ``from nothing'' in a no-boundary setting. We argue that Lorentzian Bohm metrics are unstable to decay to de Sitter spacetime. We also argue that noncompact versions of the Bohm metrics have infinitely many negative Lichernowicz modes, and we conjecture a general relation between Lichnerowicz eigenvalues and non-uniqueness of the Dirichlet problem for Einstein's equations.Comment: 53 pages, 11 figure

On the Importance of Countergradients for the Development of Retinotopy: Insights from a Generalised Gierer Model

During the development of the topographic map from vertebrate retina to superior colliculus (SC), EphA receptors are expressed in a gradient along the nasotemporal retinal axis. Their ligands, ephrin-As, are expressed in a gradient along the rostrocaudal axis of the SC. Countergradients of ephrin-As in the retina and EphAs in the SC are also expressed. Disruption of any of these gradients leads to mapping errors. Gierer's (1981) model, which uses well-matched pairs of gradients and countergradients to establish the mapping, can account for the formation of wild type maps, but not the double maps found in EphA knock-in experiments. I show that these maps can be explained by models, such as Gierer's (1983), which have gradients and no countergradients, together with a powerful compensatory mechanism that helps to distribute connections evenly over the target region. However, this type of model cannot explain mapping errors found when the countergradients are knocked out partially. I examine the relative importance of countergradients as against compensatory mechanisms by generalising Gierer's (1983) model so that the strength of compensation is adjustable. Either matching gradients and countergradients alone or poorly matching gradients and countergradients together with a strong compensatory mechanism are sufficient to establish an ordered mapping. With a weaker compensatory mechanism, gradients without countergradients lead to a poorer map, but the addition of countergradients improves the mapping. This model produces the double maps in simulated EphA knock-in experiments and a map consistent with the Math5 knock-out phenotype. Simulations of a set of phenotypes from the literature substantiate the finding that countergradients and compensation can be traded off against each other to give similar maps. I conclude that a successful model of retinotopy should contain countergradients and some form of compensation mechanism, but not in the strong form put forward by Gierer

A Novel Function of DELTA-NOTCH Signalling Mediates the Transition from Proliferation to Neurogenesis in Neural Progenitor Cells

A complete account of the whole developmental process of neurogenesis involves understanding a number of complex underlying molecular processes. Among them, those that govern the crucial transition from proliferative (self-replicating) to neurogenic neural progenitor (NP) cells remain largely unknown. Due to its sequential rostro-caudal gradients of proliferation and neurogenesis, the prospective spinal cord of the chick embryo is a good experimental system to study this issue. We report that the NOTCH ligand DELTA-1 is expressed in scattered cycling NP cells in the prospective chick spinal cord preceding the onset of neurogenesis. These Delta-1-expressing progenitors are placed in between the proliferating caudal neural plate (stem zone) and the rostral neurogenic zone (NZ) where neurons are born. Thus, these Delta-1-expressing progenitors define a proliferation to neurogenesis transition zone (PNTZ). Gain and loss of function experiments carried by electroporation demonstrate that the expression of Delta-1 in individual progenitors of the PNTZ is necessary and sufficient to induce neuronal generation. The activation of NOTCH signalling by DELTA-1 in the adjacent progenitors inhibits neurogenesis and is required to maintain proliferation. However, rather than inducing cell cycle exit and neuronal differentiation by a typical lateral inhibition mechanism as in the NZ, DELTA-1/NOTCH signalling functions in a distinct manner in the PNTZ. Thus, the inhibition of NOTCH signalling arrests proliferation but it is not sufficient to elicit neuronal differentiation. Moreover, after the expression of Delta-1 PNTZ NP continue cycling and induce the expression of Tis21, a gene that is upregulated in neurogenic progenitors, before generating neurons. Together, these experiments unravel a novel function of DELTA–NOTCH signalling that regulates the transition from proliferation to neurogenesis in NP cells. We hypothesize that this novel function is evolutionary conserved