Removable Singularities of Separately Harmonic Functions

Removable singularities of separately harmonic functions are considered. More precisely, we prove harmonic continuation property of a separately harmonic function u(x, y) in D \ S to the domain D, when D ⊂ Rn(x) × Rm(y), n,m > 1 and S is a closed subset of the domain D with nowhere dense projections S1 = {x ∈ Rn : (x, y) ∈ S} and S2 = {y ∈ Rm : (x, y) ∈ S}.

SynapsEM: Computer-Assisted Synapse Morphometry

The structural features of a synapse help determine its function. Synapses are extremely small and tightly packed with vesicles and other organelles. Visualizing synaptic structure requires imaging by electron microscopy, and the features in micrographs must be quantified, a process called morphometry. Three parameters are typically assessed from each specimen: (1) the sizes of individual vesicles and organelles; (2) the absolute number and densities of organelles; and (3) distances between organelles and key features at synapses, such as active zone membranes and dense projections. For data to be meaningful, the analysis must be repeated from hundreds to thousands of images from several biological replicates, a daunting task. Here we report a custom computer program to analyze key structural features of synapses: SynapsEM. In short, we developed ImageJ/Fiji macros to record x,y-coordinates of segmented structures. The coordinates are then exported as text files. Independent investigators can reload the images and text files to reexamine the segmentation using ImageJ. The Matlab program then calculates and reports key synaptic parameters from the coordinates. Since the values are calculated from coordinates, rather than measured from each micrograph, other parameters such as locations of docked vesicles relative to the center of an active zone can be extracted in Matlab by additional scripting. Thus, this program can accelerate the morphometry of synapses and promote a more comprehensive analysis of synaptic ultrastructure.

SynapsEM: computer-assisted synapse morphometry

The structural features of a synapse, in part, determine its output. Synapses are extremely small and tightly packed with vesicles and other organelles. Visualizing synaptic structure requires imaging by electron microscopy, and the features in micrographs must be quantified using morphometry. Three parameters are typically assessed from each specimen: 1) the sizes of individual vesicles and organelles, 2) the absolute number and densities of organelles, and 3) distances between organelles and key features at synapses such as active zone membranes and dense projections. For data to be valid, the analysis must be repeated from hundreds or thousands of images from several biological replicates, a daunting task. Here we report a custom computer program to analyze these features: SynapsEM. In short, we developed macros for ImageJ/Fiji to record x,y-coordinates of segmented structures; the coordinates are then exported as text files. Independent investigators can reload the images and text files to confirm or re-evaluate the segmentation using ImageJ. The Matlab program calculates and reports key synaptic parameters from the coordinates. Since the values are calculated, rather than measured from each micrograph, other parameters can be extracted in Matlab by additional scripting. Thus, this program can accelerate morphometry of synapses and promote a more comprehensive analysis of synaptic ultrastructure.

Bounding the covolume of lattices in products

We study lattices in a product $G=G_{1}\times \cdots \times G_{n}$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_{i}$ is non-compact and every closed normal subgroup of $G_{i}$ is discrete or cocompact (e.g. $G_{i}$ is topologically simple). We show that the set of discrete subgroups of $G$ containing a fixed cocompact lattice $\unicode[STIX]{x1D6E4}$ with dense projections is finite. The same result holds if $\unicode[STIX]{x1D6E4}$ is non-uniform, provided $G$ has Kazhdan’s property (T). We show that for any compact subset $K\subset G$, the collection of discrete subgroups $\unicode[STIX]{x1D6E4}\leqslant G$ with $G=\unicode[STIX]{x1D6E4}K$ and dense projections is uniformly discrete and hence of covolume bounded away from $0$. When the ambient group $G$ is compactly presented, we show in addition that the collection of those lattices falls into finitely many $\operatorname{Aut}(G)$-orbits. As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor. We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-non-compact tdlc group $G$ is a Chabauty limit of discrete subgroups, then some compact open subgroup of $G$ is an infinitely generated pro-$p$ group for some prime $p$. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.

Hypoxia activates a neuropeptidergic pathway from the paraventricular nucleus of the hypothalamus to the nucleus tractus solitarii

The paraventricular nucleus of the hypothalamus (PVN) contributes to both autonomic and neuroendocrine function. PVN lesion or inhibition blunts cardiorespiratory responses to peripheral chemoreflex activation, suggesting that the PVN is required for full expression of these effects. However, the role of efferent projections to cardiorespiratory nuclei and the neurotransmitters/neuromodulators that are involved is unclear. The PVN sends dense projections to the nucleus tractus solitarii (nTS), a region that displays neuronal activation following hypoxia. We hypothesized that acute hypoxia activates nTS-projecting PVN neurons. Using a combination of retrograde tracing and immunohistochemistry, we determined whether hypoxia activates PVN neurons that project to the nTS and examined the phenotype of these neurons. Conscious rats underwent 2 h normoxia (21% O2, n = 5) or hypoxia (10% O2, n = 6). Hypoxia significantly increased Fos immunoreactivity in nTS-projecting neurons, primarily in the caudal PVN. The majority of activated nTS-projecting neurons contained corticotropin-releasing hormone (CRH). In the nTS, fibers expressing the CRH receptor corticotropin-releasing factor receptor 2 (CRFR2) were colocalized with oxytocin (OT) fibers and were closely associated with hypoxia-activated nTS neurons. A separate group of animals that received a microinjection of adeno-associated virus type 2-hSyn-green fluorescent protein (GFP) into the PVN exhibited GFP-expressing fibers in the nTS; a proportion of these fibers displayed OT immunoreactivity. Thus, nTS CRFR2s appear to be located on the fibers of PVN OT neurons that project to the nTS. Taken together, our findings suggest that PVN CRH projections to the nTS may modulate nTS neuronal activation, possibly via OTergic mechanisms, and thus contribute to chemoreflex cardiorespiratory responses.

Thalamostriatal projections from the medial posterior and parafascicular nuclei have distinct topographic and physiologic properties

The dorsolateral striatum (DLS) is critical for executing sensorimotor behaviors that depend on stimulus-response (S-R) associations. In rats, the DLS receives it densest inputs from primary somatosensory (SI) cortex, but it also receives substantial input from the thalamus. Much of rat DLS is devoted to processing whisker-related information, and thalamic projections to these whisker-responsive DLS regions originate from the parafascicular (Pf) and medial posterior (POm) nuclei. To determine which thalamic nucleus is better suited for mediating S-R associations in the DLS, we compared their input-output connections and neuronal responses to repetitive whisker stimulation. Tracing experiments demonstrate that POm projects specifically to the DLS, but the Pf innervates both dorsolateral and dorsomedial parts of the striatum. The Pf nucleus is innervated by whisker-sensitive sites in the superior colliculus, and these sites also send dense projections to the zona incerta, a thalamic region that sends inhibitory projections to the POm. These data suggest that projections from POm to the DLS are suppressed by incertal inputs when the superior colliculus is activated by unexpected sensory stimuli. Simultaneous recordings with two electrodes indicate that POm neurons are more responsive and habituate significantly less than Pf neurons during repetitive whisker stimulation. Response latencies are also shorter in POm than in Pf, which is consistent with the fact that Pf receives its whisker information via synaptic relays in the superior colliculus. These findings indicate that, compared with the Pf nucleus, POm transmits somatosensory information to the DLS with a higher degree of sensory fidelity.

Liprin-α/SYD-2 determines the size of dense projections in presynaptic active zones in C. elegans

Synaptic vesicle (SV) release is spatially and temporally regulated by a network of proteins that form the presynaptic active zone (AZ). The hallmark of most AZs is an electron-dense projection (DP) surrounded by SVs. Despite their importance for our understanding of triggered SV release, high-resolution analyses of DP structures are limited. Using electron microscopy, we show that DPs at Caenorhabditis elegans neuromuscular junctions (NMJs) were highly structured, composed of building units forming bays in which SVs are docked to the AZ membrane. Furthermore, larger ribbonlike DPs that were multimers of the NMJ building unit are found at synapses between inter- and motoneurons. We also demonstrate that DP size is determined by the activity of the AZ protein SYD-2/Liprin-α. Whereas loss of syd-2 function led to smaller DPs, syd-2 gain-of-function mutants displayed larger ribbonlike DPs through increased recruitment of ELKS-1/ELKS. Therefore, our data suggest that a main role of SYD-2/Liprin-α in synaptogenesis is to regulate the polymerization of DPs.

Maturation of active zone assembly by Drosophila Bruchpilot

Synaptic vesicles fuse at active zone (AZ) membranes where Ca2+ channels are clustered and that are typically decorated by electron-dense projections. Recently, mutants of the Drosophila melanogaster ERC/CAST family protein Bruchpilot (BRP) were shown to lack dense projections (T-bars) and to suffer from Ca2+ channel–clustering defects. In this study, we used high resolution light microscopy, electron microscopy, and intravital imaging to analyze the function of BRP in AZ assembly. Consistent with truncated BRP variants forming shortened T-bars, we identify BRP as a direct T-bar component at the AZ center with its N terminus closer to the AZ membrane than its C terminus. In contrast, Drosophila Liprin-α, another AZ-organizing protein, precedes BRP during the assembly of newly forming AZs by several hours and surrounds the AZ center in few discrete punctae. BRP seems responsible for effectively clustering Ca2+ channels beneath the T-bar density late in a protracted AZ formation process, potentially through a direct molecular interaction with intracellular Ca2+ channel domains.

MI Neuronal Responses to Peripheral Whisker Stimulation: Relationship to Neuronal Activity in SI Barrels and Septa

The whisker region in the rodent primary motor (MI) cortex receives dense projections from neurons aligned with the layer IV septa in the whisker region of the primary somatosensory (SI) cortex. To compare whisker-induced responses in MI with respect to the SI responses in the septa and adjoining barrel regions, we used several experimental approaches in anesthetized rats. Reversible inactivation of SI and the surrounding cortex suppressed the magnitude of whisker-induced responses in the MI whisker region by 80%. Subsequent laminar analysis of MI responses to electrical or mechanical stimulation of the whisker pad revealed that the most responsive MI neurons were located ≥1.0 mm below the pia. When layer IV neurons in SI were recorded simultaneously with deep MI neurons during low-frequency (2-Hz) deflections of the whiskers, the neurons in the SI barrels responded 2–6 ms earlier than those in MI. Barrel neurons displayed similar response latencies at all stimulus frequencies, but the response latencies in MI and the SI septa increased significantly when the whiskers were deflected at frequencies of 8 Hz. Finally, cross-correlation analysis of neuronal activity in SI and MI revealed greater amounts of time-locked coordination among septa–MI neuron pairs than among barrel–MI neuron pairs. These results suggest that the somatosensory corticocortical inputs to MI cortex convey information processed by the SI septal circuits.