Analysis of growth cone extension in standardized coordinates highlights self-organization rules during wiring of the Drosophila visual system

A fascinating question in neuroscience is how ensembles of neurons, originating from different locations, extend to the proper place and by the right time to create precise circuits. Here, we investigate this question in the Drosophila visual system, where photoreceptors re-sort in the lamina to form the crystalline-like neural superposition circuit. The repeated nature of this circuit allowed us to establish a data-driven, standardized coordinate system for quantitative comparison of sparsely perturbed growth cones within and across specimens. Using this common frame of reference, we investigated the extension of the R3 and R4 photoreceptors, which is the only pair of symmetrically arranged photoreceptors with asymmetric target choices. Specifically, we found that extension speeds of the R3 and R4 growth cones are inherent to their cell identities. The ability to parameterize local regularity in tissue organization facilitated the characterization of ensemble cellular behaviors and dissection of mechanisms governing neural circuit formation.

Properties of the free boundary near the fixed boundary of the double obstacle problems

In this paper, we study the tangential touch and [Formula: see text] regularity of the free boundary near the fixed boundary of the double obstacle problem for Laplacian and fully nonlinear operator. The main idea to have the properties is regarding the upper obstacle as a solution of the single obstacle problem. Then, in the classification of global solutions of the double problem, it is enough to consider only two cases for the upper obstacle, [Formula: see text] The second one is a new type of upper obstacle, which does not exist in the study of local regularity of the free boundary of the double problem. Thus, in this paper, a new type of difficulties that come from the second type upper obstacle is mainly studied.

On the regularity of distributions via the convergence of the continuous shearlet transform in two dimensions

The local convergence of the continuous shearlet transform (CST) in two dimensions is used to prove the local regularity of functions [Formula: see text]. Moreover, by means of the regularity theorem of distributions [Formula: see text] and the results for functions in [Formula: see text], the local regularity of distributions [Formula: see text] with compact support is also proved via the local convergence of any derivative of the CST.

Forecasting Value-at-Risk in turbulent stock markets via the local regularity of the price process

A new computational approach based on the pointwise regularity exponent of the price time series is proposed to estimate Value at Risk. The forecasts obtained are compared with those of two largely used methodologies: the variance-covariance method and the exponentially weighted moving average method. Our findings show that in two very turbulent periods of financial markets the forecasts obtained using our algorithm decidedly outperform the two benchmarks, providing more accurate estimates in terms of both unconditional coverage and independence and magnitude of losses.

Non-uniformly parabolic equations and applications to the random conductance model

We study local regularity properties of linear, non-uniformly parabolic finite-difference operators in divergence form related to the random conductance model on $$\mathbb Z^d$$ Z d . In particular, we provide an oscillation decay assuming only certain summability properties of the conductances and their inverse, thus improving recent results in that direction. As an application, we provide a local limit theorem for the random walk in a random degenerate and unbounded environment.

Local regularity for concave homogeneous complex degenerate elliptic equations dominating the Monge–Ampère equation

In this paper, we establish a local regularity result for $$W^{2,p}_{{\mathrm {loc}}}$$ W loc 2 , p solutions to complex degenerate nonlinear elliptic equations $$F(D^2_{\mathbb {C}}u)=f$$ F ( D C 2 u ) = f when they dominate the Monge–Ampère equation. Notably, we apply our result to the so-called k-Monge–Ampère equation.

Hierarchical Learning of Statistical Regularities over Multiple Timescales of Sound Sequence Processing: A Dynamic Causal Modeling Study

Our understanding of the sensory environment is contextualized on the basis of prior experience. Measurement of auditory ERPs provides insight into automatic processes that contextualize the relevance of sound as a function of how sequences change over time. However, task-independent exposure to sound has revealed that strong first impressions exert a lasting impact on how the relevance of sound is contextualized. Dynamic causal modeling was applied to auditory ERPs collected during presentation of alternating pattern sequences. A local regularity (a rare p = .125 vs. common p = .875 sound) alternated to create a longer timescale regularity (sound probabilities alternated regularly creating a predictable block length), and the longer timescale regularity changed halfway through the sequence (the regular block length became shorter or longer). Predictions should be revised for local patterns when blocks alternated and for longer patterning when the block length changed. Dynamic causal modeling revealed an overall higher precision for the error signal to the rare sound in the first block type, consistent with the first impression. The connectivity changes in response to errors within the underlying neural network were also different for the two blocks with significantly more revision of predictions in the arrangement that violated the first impression. Furthermore, the effects of block length change suggested errors within the first block type exerted more influence on the updating of longer timescale predictions. These observations support the hypothesis that automatic sequential learning creates a high-precision context (first impression) that impacts learning rates and updates to those learning rates when predictions arising from that context are violated. The results further evidence automatic pattern learning over multiple timescales simultaneously, even during task-independent passive exposure to sound.