The Brownian transport map.

Contraction properties of transport maps between probability measures play an important role in the theory of functional inequalities. The actual construction of such maps, however, is a non-trivial task and, so far, relies mostly on the theory of optimal transport. In this work, we take advantage of the infinite-dimensional nature of the Gaussian measure and construct a new transport map, based on the Föllmer process, which pushes forward the Wiener measure onto probability measures on Euclidean spaces. Utilizing the tools of the Malliavin and stochastic calculus in Wiener space, we show that this Brownian transport map is a contraction in various settings where the analogous questions for optimal transport maps are open. The contraction properties of the Brownian transport map enable us to prove functional inequalities in Euclidean spaces, which are either completely new or improve on current results. Further and related applications of our contraction results are the existence of Stein kernels with desirable properties (which lead to new central limit theorems), as well as new insights into the Kannan-Lovász-Simonovits conjecture. We go beyond the Euclidean setting and address the problem of contractions on the Wiener space itself. We show that optimal transport maps and causal optimal transport maps (which are related to Brownian transport maps) between the Wiener measure and other target measures on Wiener space exhibit very different behaviors.

Size and Depth of Monotone Neural Networks: Interpolation and Approximation.

We study monotone neural networks with threshold gates where all the weights (other than the biases) are nonnegative. We focus on the expressive power and efficiency of the representation of such networks. Our first result establishes that every monotone function over [0,1]d can be approximated within arbitrarily small additive error by a depth-4 monotone network. When , we improve upon the previous best-known construction, which has a depth of d+1 . Our proof goes by solving the monotone interpolation problem for monotone datasets using a depth-4 monotone threshold network. In our second main result, we compare size bounds between monotone and arbitrary neural networks with threshold gates. We find that there are monotone real functions that can be computed efficiently by networks with no restriction on the gates, whereas monotone networks approximating these functions need exponential size in the dimension.

ATP binding to synaspsin IIa regulates usage and clustering of vesicles in terminals of hippocampal neurons.

Synaptic transmission is expensive in terms of its energy demands and was recently shown to decrease the ATP concentration within presynaptic terminals transiently, an observation that we confirm. We hypothesized that, in addition to being an energy source, ATP may modulate the synapsins directly. Synapsins are abundant neuronal proteins that associate with the surface of synaptic vesicles and possess a well defined ATP-binding site of undetermined function. To examine our hypothesis, we produced a mutation (K270Q) in synapsin IIa that prevents ATP binding and reintroduced the mutant into cultured mouse hippocampal neurons devoid of all synapsins. Remarkably, staining for synaptic vesicle markers was enhanced in these neurons compared with neurons expressing wild-type synapsin IIa, suggesting overly efficient clustering of vesicles. In contrast, the mutation completely disrupted the capability of synapsin IIa to slow synaptic depression during sustained 10 Hz stimulation, indicating that it interfered with synapsin-dependent vesicle recruitment. Finally, we found that the K270Q mutation attenuated the phosphorylation of synapsin IIa on a distant PKA/CaMKI consensus site known to be essential for vesicle recruitment. We conclude that ATP binding to synapsin IIa plays a key role in modulating its function and in defining its contribution to hippocampal short-term synaptic plasticity.

Monitoring and quantifying dynamic physiological processes in live neurons using fluorescence recovery after photobleaching.

The direct visualization of subcellular dynamic processes is often hampered by limitations in the resolving power achievable with conventional microscopy techniques. Fluorescence recovery after photobleaching has emerged as a highly informative approach to address this challenge, permitting the quantitative measurement of the movement of small organelles and proteins in living functioning cells, and offering detailed insights into fundamental cellular phenomena of physiological importance. In recent years, its implementation has benefited from the increasing availability of confocal microscopy systems and of powerful labeling techniques based on genetically encoded fluorescent proteins or other chemical markers. In this review, we present fluorescence recovery after photobleaching and related techniques in the context of contemporary neurobiological research and discuss quantitative and semi-quantitative approaches to their interpretation.