Little is known about the functional connectivity between astrocytes in the CNS. To explore this issue we photo-released glutamate onto a single astrocyte in murine hippocampal slices and imaged calcium responses. Photo-release of glutamate causes a metabotropic glutamate receptor (mGluR)-dependent increase in internal calcium in the stimulated astrocyte and delayed calcium elevations in neighboring cells. The delayed elevation in calcium was not caused by either neuronal activity following synaptic transmission or by glutamate released from astrocytes. However, it was reduced by flufenamic acid (FFA), which is consistent with a role for adenosine triphosphate (ATP) release from astrocytes as an intercellular messenger. Exogenous ligands such as ATP (1 µM) increased the number of astrocytes that were recruited into coupled astrocytic networks, indicating that extracellular accumulation of neurotransmitters modulates neuronal excitability, synaptic transmission and functional coupling between astrocytes.

A systematic study is made of the isomorphic properties of the Banach space ${\cal C}_0(\Upsilon)$ of continuous functions, vanishing at infinity, on a tree $\Upsilon$, equipped with its natural locally compact topology.Necessary and sufficient conditions, expressed in terms of the combinatorial structure of $\Upsilon$, are obtained for ${\cal C}_0(\Upsilon)$ to possess equivalent norms with various good properties of smoothness and strict convexity.These characterizations, together with the construction of appropriate trees, lead to counter-examples refuting a number of conjectures about renormings.It is shown that the existence of a Fr\'echet-smooth renorming is not inherited by quotients, that strict convexifiability is not a three-space property and that neither the Kadec property nor the MLUR property implies the existence of an equivalent norm which is locally uniformly rotund.An example is also given of a space with a smooth norm but no equivalent strictly convex norm.Finally, it is shown that ${\cal C}_0(\Upsilon)$ always admits a ${\cal C}^\infty$ `bump-function', even in cases where no good norms exist.

Many positive results are known to hold for the class of Banach spaces known as Asplund spaces and it was for a time conjectured that Asplund spaces should admit equivalent norms with good smoothness and strict convexity properties. A counterexample to these conjectures, in the form of a space of continuous real-valued functions on a suitably chosen tree, was presented in [5]. In this paper we show that the bad behaviour of that example is shared by a wider class of Banach spaces, associated with a wider class of trees. The immediate aim of this extension of the original result is to answer a question posed by Deville and Godefroy [3]. They introduced and studied a subclass of Asplund spaces, those with Corson compact bidual balls, and asked whether this additional assumption is enough to guarantee the existence of nice renormings. We show that it is not.

A compact space K is said to have the Namioka Property, or to belong to the class *, if, for every Baire space B and every separately continuous function Ψ:B × K → ℝ, there is dense δ subset H of B such that Ψ is (jointly) continuous at all points of H × K. Although the terminology is more recent, the idea of looking at properties of this kind goes back to Namioka's paper [6] on separate and joint continuity. Talagrand [8] gave the first example of a compact space that is not in * and it is now known [4] that there are even examples of scattered compact spaces that are not in *. On the other hand, many good classes of compact spaces have been shown to be contained in *, probably the most general being the class of continuous images of Valdivia compacts [2]. The aim of this note is to prove the following stability result: a compact space which is a countable union of closed subsets with the Namioka Property does itself possess that property.

We construct a Banach space X which has no equivalent (wLUR) norm but which has no subspace isomorphic to l ∞.

In a series of recent papers ((10), (9) and (11)) Rosenthal and Odell have given a number of characterizations of Banach spaces that contain subspaces isomorphic (that is, linearly homeomorphic) to the space l1 of absolutely summable series. The methods of (9) and (11) are applicable only in the case of separable Banach spaces and some of the results there were established only in this case. We demonstrate here, without the separability assumption, one of these characterizations:

a Banach space B contains no subspace isomorphic to l1 if and only if every weak* compact convex subset of B* is the norm closed convex hull of its extreme points.