Self-switching random walks on Erd\"os-R\'enyi random graphs feel the phase transition

We study random walks on Erd\"os-R\'enyi random graphs in which, every time the random walk returns to the starting point, first an edge probability is independently sampled according to a priori measure $\mu$, and then an Erd\"os-R\'enyi random graph is sampled according to that edge probability. When the edge probability $p$ does not depend on the size of the graph $n$ (dense case), we show that the proportion of time the random walk spends on different values of $p$ -- {\it occupation measure} -- converges to the a priori measure $\mu$ as $n$ goes to infinity. More interestingly, when $p=\lambda/n$ (sparse case), we show that the occupation measure converges to a limiting measure with a density that is a function of the survival probability of a Poisson branching process. This limiting measure is supported on the supercritial values for the Erd\"os-R\'enyi random graphs, showing that self-witching random walks can detect the phase transition

Retrieving the structure of probabilistic sequences of auditory stimuli from EEG data

Using a new probabilistic approach we model the relationship between sequences of auditory stimuli generated by stochastic chains and the electroencephalographic (EEG) data acquired while 19 participants were exposed to those stimuli. The structure of the chains generating the stimuli are characterized by rooted and labeled trees whose leaves, henceforth called contexts, represent the sequences of past stimuli governing the choice of the next stimulus. A classical conjecture claims that the brain assigns probabilistic models to samples of stimuli. If this is true, then the context tree generating the sequence of stimuli should be encoded in the brain activity. Using an innovative statistical procedure we show that this context tree can effectively be extracted from the EEG data, thus giving support to the classical conjecture.Comment: 16 pages, 7 figure

Caveolins/caveolae protect adipocytes from fatty acid-mediated lipotoxicity

Mice and humans lacking functional caveolae are dyslipidemic and have reduced fat stores and smaller fat cells. To test the role of caveolins/caveolae in maintaining lipid stores and adipocyte integrity, we compared lipolysis in caveolin-1 (Cav1)-null fat cells to that in cells reconstituted for caveolae by caveolin-1 re-expression. We find that the Cav1-null cells have a modestly enhanced rate of lipolysis and reduced cellular integrity compared with reconstituted cells as determined by the release of lipid metabolites and lactic dehydrogenase, respectively, into the media. There are no apparent differences in the levels of lipolytic enzymes or hormonally stimulated phosphorylation events in the two cell lines. In addition, acute fasting, which dramatically raises circulating fatty acid levels in vivo, causes a significant upregulation of caveolar protein constituents. These results are consistent with the hypothesis that caveolae protect fat cells from the lipotoxic effects of elevated levels fatty acids, which are weak detergents at physiological pH, by virtue of the property of caveolae to form detergentresistant membrane domains

Fluctuations for Spatially Extended Hawkes Processes

International audienceIn a previous paper, it has been shown that the mean-field limit of spatially extended Hawkes processes is characterized as the unique solution u(t,x) of a neural field equation (NFE). The value u(t,x) represents the membrane potential at time t of a typical neuron located in position x, embedded in an infinite network of neurons. In the present paper, we complement this result by studying the fluctuations of such a stochastic system around its mean field limit u(t,x). Our first main result is a central limit theorem stating that the spatial distribution associated to these fluctuations converges to the unique solution of some stochastic differential equation driven by a Gaussian noise. In our second main result we show that the solutions of this stochastic differential equation can be well approximated by a stochastic version of the neural field equation satisfied by u(t,x). To the best of our knowledge, this result appears to be new in the literature

Hydrodynamic limit for spatially structured interacting neurons

Nessa tese, estudamos o limite hidrodinâmico de um sistema estocástico de neurônios cujas interações são dadas por potenciais de Kac que imitam sinapses elétricas e químicas, e as correntes de vazamento. Esse sistema consiste de \\ep^ neurônios imersos em $[0,1)^2$, cada um disparando aleatoriamente de acordo com um processo pontual com taxa que depende tanto do seu potential de membrana como da posição. Quando o neurônio $i$ dispara, seu potential de membrana é resetado para $0$, enquanto que o potencial de membrana do neurônio $j$ é aumentado por um valor positivo $\\ep^2 a(i,j)$, se $i$ influencia $j$. Além disso, entre disparos consecutivos, o sistema segue uma movimento determinístico devido às sinapses elétricas e às correntes de vazamento. As sinapses elétricas estão envolvidas na sincronização do potencial de membrana dos neurônios, enquanto que as correntes de vazamento inibem a atividade de todos os neurônios, atraindo simultaneamente todos os potenciais de membrana para $0$. No principal resultado dessa tese, mostramos que a distribuição empírica dos potenciais de membrana converge, quando o parâmetro $\\ep$ tende à 0 , para uma densidade de probabilidade $ho_t(u,r)$ que satisfaz uma equação diferencial parcial nâo linear do tipo hiperbólica .We study the hydrodynamic limit of a stochastic system of neurons whose interactions are given by Kac Potentials that mimic chemical and electrical synapses and leak currents. The system consists of \\ep^ neurons embedded in $[0,1)^2$, each spiking randomly according to a point process with rate depending on both its membrane potential and position. When neuron $i$ spikes, its membrane potential is reset to $0$ while the membrane potential of $j$ is increased by a positive value $\\ep^2 a(i,j)$, if $i$ influences $j$. Furthermore, between consecutive spikes, the system follows a deterministic motion due both to electrical synapses and leak currents. The electrical synapses are involved in the synchronization of the membrane potentials of the neurons, while the leak currents inhibit the activity of all neurons, attracting simultaneously their membrane potentials to 0. We show that the empirical distribution of the membrane potentials converges, as $\\ep$ vanishes, to a probability density $ho_t(u,r)$ which is proved to obey a nonlinear PDE of Hyperbolic type