Convex polytopes and quantization of symplectic manifolds

Quantum mechanics associate to some symplectic manifolds M a quantum model Q(M), which is a Hilbert space. The space Q(M) is the quantum mechanical analogue of the classical phase space M. We discuss here relations between the volume of M and the dimension of the vector space Q(M). Analogues for convex polyhedra are considered.

Intrastriatal injection of an adenoviral vector expressing glial-cell-line-derived neurotrophic factor prevents dopaminergic neuron degeneration and behavioral impairment in a rat model of Parkinson disease

Glial-cell-line-derived neurotrophic factor (GDNF) is a potent neurotrophic factor for adult nigral dopamine neurons in vivo. GDNF has both protective and restorative effects on the nigro-striatal dopaminergic (DA) system in animal models of Parkinson disease. Appropriate administration of this factor is essential for the success of its clinical application. Since it cannot cross the blood–brain barrier, a gene transfer method may be appropriate for delivery of the trophic factor to DA cells. We have constructed a recombinant adenovirus (Ad) encoding GDNF and injected it into rat striatum to make use of its ability to infect neurons and to be retrogradely transported by DA neurons. Ad-GDNF was found to drive production of large amounts of GDNF, as quantified by ELISA. The GDNF produced after gene transfer was biologically active: it increased the survival and differentiation of DA neurons in vitro. To test the efficacy of the Ad-mediated GDNF gene transfer in vivo, we used a progressive lesion model of Parkinson disease. Rats received injections unilaterally into their striatum first of Ad and then 6 days later of 6-hydroxydopamine. We found that mesencephalic nigral dopamine neurons of animals treated with the Ad-GDNF were protected, whereas those of animals treated with the Ad-β-galactosidase were not. This protection was associated with a difference in motor function: amphetamine-induced turning was much lower in animals that received the Ad-GDNF than in the animals that received Ad-β-galactosidase. This finding may have implications for the development of a treatment for Parkinson disease based on the use of neurotrophic factors.