TY - CONF
AB - A finite set N ⊃ Rd is a weak ε-net for an n-point set X ⊃ Rd (with respect to convex sets) if N intersects every convex set K with |K ∩ X| ≥ εn. We give an alternative, and arguably simpler, proof of the fact, first shown by Chazelle et al. [7], that every point set X in Rd admits a weak ε-net of cardinality O(ε-d polylog(1/ε)). Moreover, for a number of special point sets (e.g., for points on the moment curve), our method gives substantially better bounds. The construction yields an algorithm to construct such weak ε-nets in time O(n ln(1/ε)). We also prove, by a different method, a near-linear upper bound for points uniformly distributed on the (d - 1)-dimensional sphere.
AU - Matoušek, Jiří
AU - Uli Wagner
ID - 2423
TI - New constructions of weak epsilon-nets
ER -
TY - CONF
AB - We introduce the adaptive neighborhood graph as a data structure for modeling a smooth manifold M embedded in some (potentially very high-dimensional) Euclidean space ℝd. We assume that M is known to us only through a finite sample P ⊂ M, as it is often the case in applications. The adaptive neighborhood graph is a geometric graph on P. Its complexity is at most min{2O(k)(n, n2}, where n = |P| and k = dim M, as opposed to the n⌈d/2⌉ complexity of the Delaunay triangulation, which is often used to model manifolds. We show that we can provably correctly infer the connectivity of M and the dimension of M from the adaptive neighborhood graph provided a certain standard sampling condition is fulfilled. The running time of the dimension detection algorithm is d2O(k7 log k) for each connected component of M. If the dimension is considered constant, this is a constant-time operation, and the adaptive neighborhood graph is of linear size. Moreover, the exponential dependence of the constants is only on the intrinsic dimension k, not on the ambient dimension d. This is of particular interest if the co-dimension is high, i.e., if k is much smaller than d, as is the case in many applications. The adaptive neighborhood graph also allows us to approximate the geodesic distances between the points in P.
AU - Giesen, Joachim
AU - Uli Wagner
ID - 2424
TI - Shape dimension and intrinsic metric from samples of manifolds with high co-dimension
ER -
TY - JOUR
AB - Patients with Hodgkin's disease can develop paraneoplastic cerebellar ataxia because of the generation of autoantibodies against mGluR1 (mGluR1-Abs). Yet, the pathophysiological mechanisms underlying their motor coordination deficits remain to be elucidated. Here, we show that application of IgG purified from the patients' serum to cerebellar slices of mice acutely reduces the basal activity of Purkinje cells, whereas application to the flocculus of mice in vivo evokes acute disturbances in the performance of their compensatory eye movements. In addition, the mGluR1-Abs block induction of long-term depression in cultured mouse Purkinje cells, whereas the cerebellar motor learning behavior of the patients is affected in that they show impaired adaptation of their saccadic eye movements. Finally, postmortem analysis of the cerebellum of a paraneoplastic cerebellar ataxia patient showed that the number of Purkinje cells was significantly reduced by approximately two thirds compared with three controls. We conclude that autoantibodies against mGluR1 can cause cerebellar motor coordination deficits caused by a combination of rapid effects on both acute and plastic responses of Purkinje cells and chronic degenerative effects.
AU - Coesmans, Michiel P
AU - Sillevis-Smitt, Peter A
AU - Linden, David J
AU - Ryuichi Shigemoto
AU - Hirano, Tomoo
AU - Yamakawa, Yoshinori
AU - Van Alphen, Adriaan M
AU - Luo, Chongde
AU - Van Der Geest, Jos N
AU - Kros, Johan M
AU - Gaillard, Carlo A
AU - Frens, Maarten A
AU - De Zeeuw, Chris I
ID - 2623
IS - 3
JF - Annals of Neurology
TI - Mechanisms underlying cerebellar motor deficits due to mGluR1-autoantibodies
VL - 53
ER -
TY - JOUR
AB - Metabotropic glutamate receptor 1 (mGluR1) plays a crucial role in synaptic plasticity and motor learning in the cerebellum. We have studied activity-dependent changes in mGluR1 function in mouse cultured Purkinje neurons. Depolarizing stimulation potentiated Ca2+ and current responses to an mGluR1 agonist for several hours in the cultured Purkinje neurons. It also blocked internalization of mGluR1 and increased the number of mGluR1s on the cell membrane. We found that depolarization simultaneously increased transcription of Homer1a in Purkinje neurons. Homer1a inhibited internalization and increased cell-surface expression of mGluR1 when coexpressed in human embryonic kidney (HEK)-293 cells. Depolarization-induced Homer1a expression in Purkinje neurons was blocked by a mitogen-activated protein kinase (MAPK) inhibitor. Changes in internalization and mGluR1-mediated Ca2+ response were also blocked by inhibition of MAPK activity, suggesting that localization and activity of mGluR1 were regulated in the same signalling pathway as Homer1a expression. It is thus suggested that depolarization of the Purkinje neuron leads to the increment in mGluR1 responsiveness through MAPK activity and induction of Homer1a expression, which increases active mGluR1 on the cell surface by blocking internalization of mGluR1.
AU - Minami, Itsunari
AU - Kengaku, Mineko
AU - Smitt, Sillevis P
AU - Ryuichi Shigemoto
AU - Hirano, Tomoo
ID - 2625
IS - 5
JF - European Journal of Neuroscience
TI - Long-term potentiation of mGluR1 activity by depolarization-induced Homer1a in mouse cerebellar Purkinje neurons
VL - 17
ER -
TY - JOUR
AB - The expression pattern of metabotropic glutamate receptor Iα (mGluR1α) was immunohistochemically investigated in substantia nigra dopaminergic neurons of the macaque monkey. In normal monkeys, mGluR1α immunoreactivity was weakly observed in the dorsal tier of the substantia nigra pars compacta (SNc-d) where calbindin-D28k-containing dopaminergic neurons invulnerable to parkinsonian degeneration are specifically located. On the other hand, mGluR1α was strongly expressed in the ventral tier of the substantia nigra pars cornpacta (SNc-v). In monkeys treated with the parkinsonism-inducing drug, I-methyl-4-phenyl-1,2,3,6-tetrahydropyridine (MPTP), mGluR1α expression was decreased in dopaminergic neurons in the SNc-v that were spared its toxic action. These results suggest that mGluR1α expression may be involved at least partly in the vulnerability of dopaminergic neurons to parkinsonian insults.
AU - Kaneda, Katsuyuki
AU - Imanishi, Michiko
AU - Nambu, Atsushi
AU - Ryuichi Shigemoto
AU - Takada, Masahiko
ID - 2626
IS - 7
JF - Neuroreport
TI - Differential expression patterns of mGluR1α in monkey nigral dopamine neurons
VL - 14
ER -