Transience and recurrence of random walks on percolation clusters in an ultrametric space

We study existence of percolation in the hierarchical group of order $N$, which is an ultrametric space, and transience and recurrence of random walks on the percolation clusters. The connection probability on the hierarchical group for two points separated by distance $k$ is of the form $c_k/N^{k(1+\delta)}, \delta>-1$, with $c_k=C_0+C_1\log k+C_2k^\alpha$, non-negative constants $C_0, C_1, C_2$, and $\alpha>0$. Percolation was proved in Dawson and Gorostiza (2013) for $\delta0$, with $\alpha>2$. In this paper we improve the result for the critical case by showing percolation for $\alpha>0$. We use a renormalization method of the type in the previous paper in a new way which is more intrinsic to the model. The proof involves ultrametric random graphs (described in the Introduction). The results for simple (nearest neighbour) random walks on the percolation clusters are: in the case $\delta<1$ the walk is transient, and in the critical case $\delta=1, C_2>0,\alpha>0$, there exists a critical $\alpha_c\in(0,\infty)$ such that the walk is recurrent for $\alpha<\alpha_c$ and transient for $\alpha>\alpha_c$. The proofs involve graph diameters, path lengths, and electric circuit theory. Some comparisons are made with behaviours of random walks on long-range percolation clusters in the one-dimensional Euclidean lattice.Comment: 27 page

Hierarchical equilibria of branching populations

The objective of this paper is the study of the equilibrium behavior of a population on the hierarchical group $\Omega_N$ consisting of families of individuals undergoing critical branching random walk and in addition these families also develop according to a critical branching process. Strong transience of the random walk guarantees existence of an equilibrium for this two-level branching system. In the limit $N\to\infty$ (called the hierarchical mean field limit), the equilibrium aggregated populations in a nested sequence of balls $B^{(N)}_\ell$ of hierarchical radius $\ell$ converge to a backward Markov chain on $\mathbb{R_+}$. This limiting Markov chain can be explicitly represented in terms of a cascade of subordinators which in turn makes possible a description of the genealogy of the population.Comment: 62 page

Percolation in a hierarchical random graph

We study asymptotic percolation as $N\to \infty$ in an infinite random graph ${\cal G}_N$ embedded in the hierarchical group of order $N$, with connection probabilities depending on an ultrametric distance between vertices. ${\cal G}_N$ is structured as a cascade of finite random subgraphs of (approximate) Erd\"os-Renyi type. We give a criterion for percolation, and show that percolation takes place along giant components of giant components at the previous level in the cascade of subgraphs for all consecutive hierarchical distances. The proof involves a hierarchy of random graphs with vertices having an internal structure and random connection probabilities.Comment: 19 pages and 1 figur

Self-similar stable processes arising from high-density limits of occupation times of particle systems

We extend results on time-rescaled occupation time fluctuation limits of the $(d,\alpha, \beta)$-branching particle system $(0<\alpha \leq 2, 0<\beta \leq 1)$ with Poisson initial condition. The earlier results in the homogeneous case (i.e., with Lebesgue initial intensity measure) were obtained for dimensions $d>\alpha / \beta$ only, since the particle system becomes locally extinct if $d\le \alpha / \beta$. In this paper we show that by introducing high density of the initial Poisson configuration, limits are obtained for all dimensions, and they coincide with the previous ones if $d>\alpha/\beta$. We also give high-density limits for the systems with finite intensity measures (without high density no limits exist in this case due to extinction); the results are different and harder to obtain due to the non-invariance of the measure for the particle motion. In both cases, i.e., Lebesgue and finite intensity measures, for low dimensions ($d<\alpha(1+\beta)/\beta$ and $d<\alpha(2+\beta)/(1+\beta)$, respectively) the limits are determined by non-L\'evy self-similar stable processes. For the corresponding high dimensions the limits are qualitatively different: ${\cal S}'(R^d)$-valued L\'evy processes in the Lebesgue case, stable processes constant in time on $(0,\infty)$ in the finite measure case. For high dimensions, the laws of all limit processes are expressed in terms of Riesz potentials. If $\beta=1$, the limits are Gaussian. Limits are also given for particle systems without branching, which yields in particular weighted fractional Brownian motions in low dimensions. The results are obtained in the setup of weak convergence of S'(R^d)$-valued processes.Comment: 28 page

Oscillatory Fractional Brownian Motion and Hierarchical Random Walks

We introduce oscillatory analogues of fractional Brownian motion, sub-fractional Brownian motion and other related long range dependent Gaussian processes, we discuss their properties, and we show how they arise from particle systems with or without branching and with different types of initial conditions, where the individual particle motion is the so-called c-random walk on a hierarchical group. The oscillations are caused by the discrete and ultrametric structure of the hierarchical group, and they become slower as time tends to infinity and faster as time approaches zero. We also give other results to provide an overall picture of the behavior of this kind of systems, emphasizing the new phenomena that are caused by the ultrametric structure as compared with results for analogous models on Euclidean space

GHEP-ISFG collaborative exercise on mixture profiles of autosomal STRs (GHEP-MIX01, GHEP-MIX02 and GHEP-MIX03): results and evaluation

One of the main objectives of the Spanish and Portuguese-Speaking Group of the International Society for Forensic Genetics (GHEP-ISFG) is to promote and contribute to the development and dissemination of scientific knowledge in the area of forensic genetics. Due to this fact, GHEP-ISFG holds different working commissions that are set up to develop activities in scientific aspects of general interest. One of them, the Mixture Commission of GHEP-ISFG, has organized annually, since 2009, a collaborative exercise on analysis and interpretation of autosomal short tandem repeat (STR) mixture profiles. Until now, three exercises have been organized (GHEP-MIX01, GHEP-MIX02 and GHEP-MIX03), with 32, 24 and 17 participant laboratories respectively. The exercise aims to give a general vision by addressing, through the proposal of mock cases, aspects related to the edition of mixture profiles and the statistical treatment. The main conclusions obtained from these exercises may be summarized as follows. Firstly, the data show an increased tendency of the laboratories toward validation of DNA mixture profiles analysis following international recommendations (ISO/IEC 17025:2005). Secondly, the majority of discrepancies are mainly encountered in stutters positions (53.4%, 96.0% and 74.9%, respectively for the three editions). On the other hand, the results submitted reveal the importance of performing duplicate analysis by using different kits in order to reduce errors as much as possible. Regarding the statistical aspect (GHEP-MIX02 and 03), all participants employed the likelihood ratio (LR) parameter to evaluate the statistical compatibility and the formulas employed were quite similar. When the hypotheses to evaluate the LR value were locked by the coordinators (GHEP-MIX02) the results revealed a minor number of discrepancies that were mainly due to clerical reasons. However, the GHEP-MIX03 exercise allowed the participants to freely come up with their own hypotheses to calculate the LR value. In this situation the laboratories reported several options to explain the mock cases proposed and therefore significant differences between the final LR values were obtained. Complete information concerning the background of the criminal case is a critical aspect in order to select the adequate hypotheses to calculate the LR value. Although this should be a task for the judicial court to decide, it is important for the expert to account for the different possibilities and scenarios, and also offer this expertise to the judge. In addition, continuing education in the analysis and interpretation of mixture DNA profiles may also be a priority for the vast majority of forensic laboratories.Fil: Sala, Adriana Andrea. Universidad de Buenos Aires. Facultad de Farmacia y Bioquímica. Servicio de Huellas Digitales Genéticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Crespillo, M.. Instituto Nacional de Toxicología y Ciencias Forenses; EspañaFil: Barrio, P. A.. Instituto Nacional de Toxicología y Ciencias Forenses; EspañaFil: Luque, J. A.. Instituto Nacional de Toxicología y Ciencias Forenses; EspañaFil: Alves, Cíntia. Universidad de Porto; PortugalFil: Aler, M.. Servicio de Laboratorio. Sección de Genética Forense y Criminalística; EspañaFil: Alessandrini, F.. Università Politecnica delle Marche. Department of Biomedical Sciences and Public Health; ItaliaFil: Andrade, L.. Instituto Nacional de Medicina Legal e Ciências Forenses, Delegação do Centro. Serviço de Genética e Biologia Forenses; PortugalFil: Barretto, R. M.. Universidade Estadual Paulista Julio de Mesquita Filho; BrasilFil: Bofarull, A.. Instituto Nacional de Toxicología y Ciencias Forenses; EspañaFil: Costa, S.. Instituto Nacional de Medicina Legal y Ciencias Forenses; PortugalFil: García, M. A.. Servicio de Criminalística de la Guardia Civil. Laboratorio Central de Criminalística. Departamento de Biología; EspañaFil: García, O.. Basque Country Police. Forensic Genetics Section. Forensic Science Unit; EspañaFil: Gaviria, A.. Cruz Roja Ecuatoriana. Laboratorio de Genética Molecular; EcuadorFil: Gladys, A.. Corte Suprema de Justicia de la Nación; ArgentinaFil: Gorostiza, A.. Grupo Zeltia. Genomica S. A. U.. Laboratorio de Identificación Genética; EspañaFil: Hernández, A.. Instituto Nacional de Toxicología y Ciencias Forenses; EspañaFil: Herrera, M.. Laboratorio Genda S. A.; ArgentinaFil: Hombreiro, L.. Jefatura Superior de Policía de Galicia. Brigada de Policía Científica. Laboratorio Territorial de Biología – ADN; EspañaFil: Ibarra, A. A.. Universidad de Antioquia; ColombiaFil: Jiménez, M. J.. Policia de la Generalitat – Mossos d’Esquadra. Divisió de Policia Científica. Àrea Central de Criminalística. Unitat Central de Laboratori Biològic; EspañaFil: Luque, G. M.. Instituto Nacional de Toxicología y Ciencias Forenses; EspañaFil: Madero, P.. Centro de Análisis Genéticos; EspañaFil: Martínez Jarreta, B.. Universidad de Zaragoza; EspañaFil: Masciovecchio, M. Verónica. IACA Laboratorios; ArgentinaFil: Modesti, Nidia Maria. Provincia de Córdoba. Poder Judicial; ArgentinaFil: Moreno, F.. Servicio Médico Legal. Unidad de Genética Forense; ChileFil: Pagano, S.. Dirección Nacional de Policía Técnica. Laboratorio de Análisis de ADN para el CODIS; UruguayFil: Pedrosa, S.. Navarra de Servicios y Tecnologías S. A. U.; EspañaFil: Plaza, G.. Neodiagnostica S. L.; EspañaFil: Prat, E.. Comisaría General de Policía Científica. Laboratorio de ADN; EspañaFil: Puente, J.. Laboratorio de Genética Clínica S. L.; EspañaFil: Rendo, F.. Universidad del País Vasco; EspañaFil: Ribeiro, T.. Instituto Nacional de Medicina Legal e Ciências Forenses, Delegação Sul. Serviço de Genética e Biologia Forenses; PortugalFil: Santamaría, E.. Instituto Nacional de Toxicología y Ciencias Forenses; EspañaFil: Saragoni, V. G.. Servicio Médico Legal. Departamento de Laboratorios. Unidad de Genética Forense; ChileFil: Whittle, M. R.. Genomic Engenharia Molecular; Brasi

Colloids as Mobile Substrates for the Implantation and Integration of Differentiated Neurons into the Mammalian Brain

Neuronal degeneration and the deterioration of neuronal communication lie at the origin of many neuronal disorders, and there have been major efforts to develop cell replacement therapies for treating such diseases. One challenge, however, is that differentiated cells are challenging to transplant due to their sensitivity both to being uprooted from their cell culture growth support and to shear forces inherent in the implantation process. Here, we describe an approach to address these problems. We demonstrate that rat hippocampal neurons can be grown on colloidal particles or beads, matured and even transfected in vitro, and subsequently transplanted while adhered to the beads into the young adult rat hippocampus. The transplanted cells have a 76% cell survival rate one week post-surgery. At this time, most transplanted neurons have left their beads and elaborated long processes, similar to the host neurons. Additionally, the transplanted cells distribute uniformly across the host hippocampus. Expression of a fluorescent protein and the light-gated glutamate receptor in the transplanted neurons enabled them to be driven to fire by remote optical control. At 1-2 weeks after transplantation, calcium imaging of host brain slice shows that optical excitation of the transplanted neurons elicits activity in nearby host neurons, indicating the formation of functional transplant-host synaptic connections. After 6 months, the transplanted cell survival and overall cell distribution remained unchanged, suggesting that cells are functionally integrated. This approach, which could be extended to other cell classes such as neural stem cells and other regions of the brain, offers promising prospects for neuronal circuit repair via transplantation of in vitro differentiated, genetically engineered neurons

Optical Control of Metabotropic Glutamate Receptors

G-protein coupled receptors (GPCRs), the largest family of membrane signaling proteins, respond to neurotransmitters, hormones and small environmental molecules. The neuronal function of many GPCRs has been difficult to resolve because of an inability to gate them with subtype-specificity, spatial precision, speed and reversibility. To address this, we developed an approach for opto-chemical engineering native GPCRs. We applied this to the metabotropic glutamate receptors (mGluRs) to generate light-agonized and light-antagonized “LimGluRs”. The light-agonized “LimGluR2”, on which we focused, is fast, bistable, and supports multiple rounds of on/off switching. Light gates two of the primary neuronal functions of mGluR2: suppression of excitability and inhibition of neurotransmitter release. The light-antagonized “LimGluR2block” can be used to manipulate negative feedback of synaptically released glutamate on transmitter release. We generalize the optical control to two additional family members: mGluR3 and 6. The system works in rodent brain slice and in zebrafish in vivo, where we find that mGluR2 modulates the threshold for escape behavior. These light-gated mGluRs pave the way for determining the roles of mGluRs in synaptic plasticity, memory and disease