Optimal network topologies: Expanders, Cages, Ramanujan graphs, Entangled networks and all that

We report on some recent developments in the search for optimal network topologies. First we review some basic concepts on spectral graph theory, including adjacency and Laplacian matrices, and paying special attention to the topological implications of having large spectral gaps. We also introduce related concepts as ``expanders'', Ramanujan, and Cage graphs. Afterwards, we discuss two different dynamical feautures of networks: synchronizability and flow of random walkers and so that they are optimized if the corresponding Laplacian matrix have a large spectral gap. From this, we show, by developing a numerical optimization algorithm that maximum synchronizability and fast random walk spreading are obtained for a particular type of extremely homogeneous regular networks, with long loops and poor modular structure, that we call entangled networks. These turn out to be related to Ramanujan and Cage graphs. We argue also that these graphs are very good finite-size approximations to Bethe lattices, and provide almost or almost optimal solutions to many other problems as, for instance, searchability in the presence of congestion or performance of neural networks. Finally, we study how these results are modified when studying dynamical processes controlled by a normalized (weighted and directed) dynamics; much more heterogeneous graphs are optimal in this case. Finally, a critical discussion of the limitations and possible extensions of this work is presented.Comment: 17 pages. 11 figures. Small corrections and a new reference. Accepted for pub. in JSTA

A mouse model of autism implicates endosome pH in the regulation of presynaptic calcium entry.

Psychoactive compounds such as chloroquine and amphetamine act by dissipating the pH gradient across intracellular membranes, but the physiological mechanisms that normally regulate organelle pH remain poorly understood. Interestingly, recent human genetic studies have implicated the endosomal Na+/H+ exchanger NHE9 in both autism spectrum disorders (ASD) and attention deficit hyperactivity disorder (ADHD). Plasma membrane NHEs regulate cytosolic pH, but the role of intracellular isoforms has remained unclear. We now find that inactivation of NHE9 in mice reproduces behavioral features of ASD including impaired social interaction, repetitive behaviors, and altered sensory processing. Physiological characterization reveals hyperacidic endosomes, a cell-autonomous defect in glutamate receptor expression and impaired neurotransmitter release due to a defect in presynaptic Ca2+ entry. Acute inhibition of synaptic vesicle acidification rescues release but without affecting the primary defect due to loss of NHE9

Lamplighter model of a random copolymer adsorption on a line

We present a model of an AB-diblock random copolymer sequential self-packaging with local quenched interactions on a one-dimensional infinite sticky substrate. It is assumed that the A-A and B-B contacts are favorable, while A-B are not. The position of a newly added monomer is selected in view of the local contact energy minimization. The model demonstrates a self-organization behavior with the nontrivial dependence of the total energy, $E$ (the number of unfavorable contacts), on the number of chain monomers, $N$: $E\sim N^{3/4}$ for quenched random equally probable distribution of A- and B-monomers along the chain. The model is treated by mapping it onto the "lamplighter" random walk and the diffusion-controlled chemical reaction of $X+X\to 0$ type with the subdiffusive motion of reagents.Comment: 8 pages, 5 figure

Cages and anomalous diffusion in vibrated dense granular media

A vertically shaken granular medium hosts a blade rotating around a fixed vertical axis, which acts as a mesorheological probe. At high densities, independently from the shaking intensity, the blade's dynamics show strong caging effects, marked by transient sub-diffusion and a maximum in the velocity power density spectrum (vpds), at a resonant frequency $\sim 10$ Hz. Interpreting the data through a diffusing harmonic cage model allows us to retrieve the elastic constant of the granular medium and its collective diffusion coefficient. For high frequencies $f$, a tail $\sim 1/f$ in the vpds reveals non-trivial correlations in the intra-cage micro-dynamics. At very long times (larger than $10$ s), a super-diffusive behavior emerges, ballistic in the most extreme cases. Consistently, the distribution of slow velocity inversion times $\tau$ displays a power-law decay, likely due to persistent collective fluctuations of the host medium.Comment: 5 pages + 4 page of supplemental material, 6 figures, to be published on Phys. Rev. Let

Cholinergic Modulation of Locomotion and Striatal Dopamine Release Is Mediated by α6α4* Nicotinic Acetylcholine Receptors

Dopamine (DA) release in striatum is governed by firing rates of midbrain DA neurons, striatal cholinergic tone, and nicotinic ACh receptors (nAChRs) on DA presynaptic terminals. DA neurons selectively express α6* nAChRs, which show high ACh and nicotine sensitivity. To help identify nAChR subtypes that control DA transmission, we studied transgenic mice expressing hypersensitive α6^(L9’S*) receptors. α6^(L9’S) mice are hyperactive, travel greater distance, exhibit increased ambulatory behaviors such as walking, turning, and rearing, and show decreased pausing, hanging, drinking, and grooming. These effects were mediated by α6 α4* pentamers, as α6^(L9’S) mice lacking α4 subunits displayed essentially normal behavior. In α6^(L9’S) mice, receptor numbers are normal, but loss of α4 subunits leads to fewer and less sensitive α6* receptors. Gain-of-function nicotine-stimulated DA release from striatal synaptosomes requires α4 subunits, implicating α6α4β2* nAChRs in α6^(L9’S) mouse behaviors. In brain slices, we applied electrochemical measurements to study control of DA release by α6^(L9’S) nAChRs. Burst stimulation of DA fibers elicited increased DA release relative to single action potentials selectively in α6^(L9’S), but not WT or α4KO/ α6^(L9’S), mice. Thus, increased nAChR activity, like decreased activity, leads to enhanced extracellular DA release during phasic firing. Bursts may directly enhance DA release from α6^(L9’S) presynaptic terminals, as there was no difference in striatal DA receptor numbers or DA transporter levels or function in vitro. These results implicate α6α4β2* nAChRs in cholinergic control of DA transmission, and strongly suggest that these receptors are candidate drug targets for disorders involving the DA system

Infrared spectroscopy of small-molecule endofullerenes

Hydrogen is one of the few molecules which has been incarcerated in the molecular cage of C$_{60}$ and forms endohedral supramolecular complex H$_2$@C$_{60}$. In this confinement hydrogen acquires new properties. Its translational motion becomes quantized and is correlated with its rotations. We applied infrared spectroscopy to study the dynamics of hydrogen isotopologs H$_2$, D$_2$ and HD incarcerated in C$_{60}$. The translational and rotational modes appear as side bands to the hydrogen vibrational mode in the mid infrared part of the absorption spectrum. Because of the large mass difference of hydrogen and C$_{60}$ and the high symmetry of C$_{60}$ the problem is identical to a problem of a vibrating rotor moving in a three-dimensional spherical potential. The translational motion within the C$_{60}$ cavity breaks the inversion symmetry and induces optical activity of H$_2$. We derive potential, rotational, vibrational and dipole moment parameters from the analysis of the infrared absorption spectra. Our results were used to derive the parameters of a pairwise additive five-dimensional potential energy surface for H$_2$@C$_{60}$. The same parameters were used to predict H$_2$ energies inside C$_{70}$[Xu et al., J. Chem. Phys., {\bf 130}, 224306 (2009)]. We compare the predicted energies and the low temperature infrared absorption spectra of H$_2$@C$_{70}$.Comment: Updated author lis