Brain-derived neurotrophic factor (BDNF) and its receptor TrkB are crucial for many forms of neuronal plasticity1, 2, 3, 4, 5, 6, including structural long-term potentiation (sLTP)7, 8, which is a correlate of an animal’s learning7, 9, 10, 11, 12. However, it is unknown whether BDNF release and TrkB activation occur during sLTP, and if so, when and where. Here, using a fluorescence resonance energy transfer-based sensor for TrkB and two-photon fluorescence lifetime imaging microscopy13, 14, 15, 16, we monitor TrkB activity in single dendritic spines of CA1 pyramidal neurons in cultured murine hippocampal slices. In response to sLTP induction9, 14, 15, 16, we find fast (onset < 1 min) and sustained (>20 min) activation of TrkB in the stimulated spine that depends on NMDAR (N-methyl-d-aspartate receptor) and CaMKII signalling and on postsynaptically synthesized BDNF. We confirm the presence of postsynaptic BDNF using electron microscopy to localize endogenous BDNF to dendrites and spines of hippocampal CA1 pyramidal neurons. Consistent with these findings, we also show rapid, glutamate-uncaging-evoked, time-locked BDNF release from single dendritic spines using BDNF fused to superecliptic pHluorin17, 18, 19. We demonstrate that this postsynaptic BDNF–TrkB signalling pathway is necessary for both structural and functional LTP20. Together, these findings reveal a spine-autonomous, autocrine signalling mechanism involving NMDAR–CaMKII-dependent BDNF release from stimulated dendritic spines and subsequent TrkB activation on these same spines that is crucial for structural and functional plasticity.

SUMMARYComputer-oriented mesh generators, which serve as pre-processors to finite element programs, have recently been developed by several investigators to alleviate the frustration and to reduce the amount of time involved in the tedious manual subdividing of a complex structure into finite elements. Our purpose here is to describe how the techniques of bivariate 'blending-function' interpolation, which were originally developed for, and applied to, geometric problems of computer-aided design and numerically controlled machining of free-form surfaces such as automobile exterior panels, can be adapted and applied to the problems of mesh generation for finite element analyses. We concentrate attention on the problem of curvilinearly co-ordinating simply connected planar domains W by constructing invertible maps of the unit square 9 = [0, 11 x [0, 11 onto 9. Extensions of the methods described herein to shells in 3-space is straightforward and is illustrated by a practical example taken from the automobile industry. Analogous mesh generators for three-dimensional solids can be developed on the basis of the trivariate 'blending-function' formulae found at the end of the second section. INTRODUCTION Inherent in any finite difference or finite element method for solving a boundary value problem over a non-rectangular domain 9 is the strategy for generating a partition or mesh on ~32.Whether rectangular, triangular, isoparametric,l transfinite2 or whatever other elements are to be used, the basic problem of where to place the nodes or mesh points must first be solved. If done manually, this involves a considerable expenditure of time and almost invariably results in a mesh for which a natural ordering of the nodes may not be apparent. The purpose of this paper is to describe an approach to automatic mesh generation based upon a systematic and unified class of schemes for treating the geometric aspects of the finite element method. Fundamental to our approach is the almost trivial observation that domains such as squares, ellipsoidal discs, spheres, circular cylinders and tori, for example, have intrinsic curvilinear co-ordinate systems (Reference 3, pp. 655-666) whose constant co-ordinate lines induce a natural rectilinear parametrization of the region. This much is classical. Very recently, Zienkiewicz and Phillips4 have shown how to construct curvilinear co-ordinate systems on regions whose boundaries are defined as low-order parametric polynomials. The essence of their scheme is the use of 'isoparametric' mappings to generate generalized co-ordinate systems on quadrilateral zones. In Reference 2 and below, the authors extend the work of Zienkiewicz and co-workers-and remove the 'element of chance' referred to in Reference 1, p. 109, from the derivation of the The problem of mesh generation is that of locating a set of nodal points rg €9 appropriate to the problem at hand. It is in this sense that the mapping techniques studied herein provide the basis for mesh generation in 9. To be more specific, let r9 be a colle...

The Rho GTPase proteins Rac1, RhoA and Cdc42 have a central role in regulating the actin cytoskeleton in dendritic spines1, thereby exerting control over the structural and functional plasticity of spines2, 3, 4, 5 and, ultimately, learning and memory6, 7, 8. Although previous work has shown that precise spatiotemporal coordination of these GTPases is crucial for some forms of cell morphogenesis9, the nature of such coordination during structural spine plasticity is unclear. Here we describe a three-molecule model of structural long-term potentiation (sLTP) of murine dendritic spines, implicating the localized, coincident activation of Rac1, RhoA and Cdc42 as a causal signal of sLTP. This model posits that complete tripartite signal overlap in spines confers sLTP, but that partial overlap primes spines for structural plasticity. By monitoring the spatiotemporal activation patterns of these GTPases during sLTP, we find that such spatiotemporal signal complementation simultaneously explains three integral features of plasticity: the facilitation of plasticity by brain-derived neurotrophic factor (BDNF), the postsynaptic source of which activates Cdc42 and Rac1, but not RhoA; heterosynaptic facilitation of sLTP, which is conveyed by diffusive Rac1 and RhoA activity; and input specificity, which is afforded by spine-restricted Cdc42 activity. Thus, we present a form of biochemical computation in dendrites involving the controlled complementation of three molecules that simultaneously ensures signal specificity and primes the system for plasticity.