Homotopy Theory of Strong and Weak Topological Insulators

We use homotopy theory to extend the notion of strong and weak topological insulators to the non-stable regime (low numbers of occupied/empty energy bands). We show that for strong topological insulators in d spatial dimensions to be "truly d-dimensional", i.e. not realizable by stacking lower-dimensional insulators, a more restrictive definition of "strong" is required. However, this does not exclude weak topological insulators from being "truly d-dimensional", which we demonstrate by an example. Additionally, we prove some useful technical results, including the homotopy theoretic derivation of the factorization of invariants over the torus into invariants over spheres in the stable regime, as well as the rigorous justification of replacing $T^d$ by $S^d$ and $T^{d_k}\times S^{d_x}$ by $S^{d_k+d_x}$ as is common in the current literature.Comment: 11 pages, 3 figure

Quantum symmetric pairs and representations of double affine Hecke algebras of type C∨CnC^\vee C_n

We build representations of the affine and double affine braid groups and Hecke algebras of type $C^\vee C_n$, based upon the theory of quantum symmetric pairs $(U,B)$. In the case $U=U_q(gl_N)$, our constructions provide a quantization of the representations constructed by Etingof, Freund and Ma in arXiv:0801.1530, and also a type $BC$ generalization of the results in arXiv:0805.2766.Comment: Final version, to appear in Selecta Mathematic

Dynamical modeling of collective behavior from pigeon flight data: flock cohesion and dispersion

Several models of flocking have been promoted based on simulations with qualitatively naturalistic behavior. In this paper we provide the first direct application of computational modeling methods to infer flocking behavior from experimental field data. We show that this approach is able to infer general rules for interaction, or lack of interaction, among members of a flock or, more generally, any community. Using experimental field measurements of homing pigeons in flight we demonstrate the existence of a basic distance dependent attraction/repulsion relationship and show that this rule is sufficient to explain collective behavior observed in nature. Positional data of individuals over time are used as input data to a computational algorithm capable of building complex nonlinear functions that can represent the system behavior. Topological nearest neighbor interactions are considered to characterize the components within this model. The efficacy of this method is demonstrated with simulated noisy data generated from the classical (two dimensional) Vicsek model. When applied to experimental data from homing pigeon flights we show that the more complex three dimensional models are capable of predicting and simulating trajectories, as well as exhibiting realistic collective dynamics. The simulations of the reconstructed models are used to extract properties of the collective behavior in pigeons, and how it is affected by changing the initial conditions of the system. Our results demonstrate that this approach may be applied to construct models capable of simulating trajectories and collective dynamics using experimental field measurements of herd movement. From these models, the behavior of the individual agents (animals) may be inferred

Subcellular sequencing of single neurons reveals the dendritic transcriptome of GABAergic interneurons

Although mRNAs are localized in the processes of excitatory neurons, it is still unclear whether interneurons also localize a large population of mRNAs. In addition, the variability in the localized mRNA population within and between cell-types is unknown. Here we describe the unbiased transcriptomic characterization of the subcellular compartments of hundreds of single neurons. We separately profiled the dendritic and somatic transcriptomes of individual rat hippocampal neurons and investigated mRNA abundances in the soma and dendrites of single glutamatergic and GABAergic neurons. We found that, like their excitatory counterparts, interneurons contain a rich repertoire of ~4000 mRNAs. We observed more cell type-specific features among somatic transcriptomes than their associated dendritic transcriptomes. Finally, using cell-type specific metabolic labelling of isolated neurites, we demonstrated that the processes of Glutamatergic and, notably, GABAergic neurons were capable of local translation, suggesting mRNA localization and local translation is a general property of neurons

The translatome of neuronal cell bodies, dendrites,and axons

To form synaptic connections and store information, neurons continuously remodel their proteomes. The impressive length of dendrites and axons imposes logistical challenges to maintain synaptic proteins at locations remote from the transcription source (the nucleus). The discovery of thousands of messenger RNAs (mRNAs) near synapses suggested that neurons overcome distance and gain autonomy by producing proteins locally. It is not generally known, however, if, how, and when localized mRNAs are translated into protein. To investigate the translational landscape in neuronal subregions, we performed simultaneous RNA sequencing (RNA-seq) and ribosome sequencing (Ribo-seq) from microdissected rodent brain slices to identify and quantify the transcriptome and translatome in cell bodies (somata) as well as dendrites and axons (neuropil). Thousands of transcripts were differentially translated between somatic and synaptic regions, with many scaffold and signaling molecules displaying increased translation levels in the neuropil. Most translational changes between compartments could be accounted for by differences in RNA abundance. Pervasive translational regulation was observed in both somata and neuropil influenced by specific mRNA features (e.g., untranslated region [UTR] length, RNA-binding protein [RBP] motifs, and upstream open reading frames [uORFs]). For over 800 mRNAs, the dominant source of translation was the neuropil. We constructed a searchable and interactive database for exploring mRNA transcripts and their translation levels in the somata and neuropil [MPI Brain Research, The mRNA translation landscape in the synaptic neuropil. https://public.brain.mpg.de/dashapps/localseq/ Accessed 5 October 2021]. Overall, our findings emphasize the substantial contribution of local translation to maintaining synaptic protein levels and indicate that on-site translational control is an important mechanism to control synaptic strength

Affine modifications and affine hypersurfaces with a very transitive automorphism group

We study a kind of modification of an affine domain which produces another affine domain. First appeared in passing in the basic paper of O. Zariski (1942), it was further considered by E.D. Davis (1967). The first named author applied its geometric counterpart to construct contractible smooth affine varieties non-isomorphic to Euclidean spaces. Here we provide certain conditions which guarantee preservation of the topology under a modification. As an application, we show that the group of biregular automorphisms of the affine hypersurface $X \subset C^{k+2}$ given by the equation $uv=p(x_1,...,x_k)$ where $p \in C[x_1,...,x_k],$ acts $m-$transitively on the smooth part reg$X$ of $X$ for any $m \in N.$ We present examples of such hypersurfaces diffeomorphic to Euclidean spaces.Comment: 39 Pages, LaTeX; a revised version with minor changes and correction

Active zone proteins are dynamically associated with synaptic ribbons in rat pinealocytes

Synaptic ribbons (SRs) are prominent organelles that are abundant in the ribbon synapses of sensory neurons where they represent a specialization of the cytomatrix at the active zone (CAZ). SRs occur not only in neurons, but also in neuroendocrine pinealocytes where their function is still obscure. In this study, we report that pinealocyte SRs are associated with CAZ proteins such as Bassoon, Piccolo, CtBP1, Munc13–1, and the motorprotein KIF3A and, therefore, consist of a protein complex that resembles the ribbon complex of retinal and other sensory ribbon synapses. The pinealocyte ribbon complex is biochemically dynamic. Its protein composition changes in favor of Bassoon, Piccolo, and Munc13–1 at night and in favor of KIF3A during the day, whereas CtBP1 is equally present during the night and day. The diurnal dynamics of the ribbon complex persist under constant darkness and decrease after stimulus deprivation of the pineal gland by constant light. Our findings indicate that neuroendocrine pinealocytes possess a protein complex that resembles the CAZ of ribbon synapses in sensory organs and whose dynamics are under circadian regulation

Differential Spatial Expression and Subcellular Localization of CtBP Family Members in Rodent Brain

C-terminal binding proteins (CtBPs) are well-characterized nuclear transcriptional co-regulators. In addition, cytoplasmic functions were discovered for these ubiquitously expressed proteins. These include the involvement of the isoform CtBP1-S/BARS50 in cellular membrane-trafficking processes and a role of the isoform RIBEYE as molecular scaffolds in ribbons, the presynaptic specializations of sensory synapses. CtBPs were suggested to regulate neuronal differentiation and they were implied in the control of gene expression during epileptogenesis. However, the expression patterns of CtBP family members in specific brain areas and their subcellular localizations in neurons in situ are largely unknown. Here, we performed comprehensive assessment of the expression of CtBP1 and CtBP2 in mouse brain at the microscopic and the ultra-structural levels using specific antibodies. We quantified and compared expression levels of both CtBPs in biochemically isolated brain fractions containing cellular nuclei or synaptic compartment. Our study demonstrates differential regional and subcellular expression patterns for the two CtBP family members in brain and reveals a previously unknown synaptic localization for CtBP2 in particular brain regions. Finally, we propose a mechanism of differential synapto-nuclear targeting of its splice variants CtBP2-S and CtBP2-L in neurons

Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory

We study D-branes and Ramond-Ramond fields on global orbifolds of Type II string theory with vanishing H-flux using methods of equivariant K-theory and K-homology. We illustrate how Bredon equivariant cohomology naturally realizes stringy orbifold cohomology. We emphasize its role as the correct cohomological tool which captures known features of the low-energy effective field theory, and which provides new consistency conditions for fractional D-branes and Ramond-Ramond fields on orbifolds. We use an equivariant Chern character from equivariant K-theory to Bredon cohomology to define new Ramond-Ramond couplings of D-branes which generalize previous examples. We propose a definition for groups of differential characters associated to equivariant K-theory. We derive a Dirac quantization rule for Ramond-Ramond fluxes, and study flat Ramond-Ramond potentials on orbifolds.Comment: 46 pages; v2: typos correcte