On April 11, the World Parkinson's Disease Day is celebrated; and this year the celebration is more special, since it has been 200 years since the first description of the disease, by James Parkinson. "We have progressed in the treatment of the disease, since then and especially in the last ten years, but we are still far away from finding a cure," says Maria Elisa Pimentel Piemonte, from USP and NeuroMat. *Carolina Marins Santos, Jornal da USP, 4/12/2017. (In Portuguese)*

*João Guilherme Ferreira Campos, Ariadne de Andrade Costa, Mauro Copelli and Osame Kinouchi*

In a recent work, mean-field analysis and computer simulations were employed to analyze critical self-organization in networks of excitable cellular automata where randomly chosen synapses in the network were depressed after each spike (the so-called annealed dynamics). Calculations agree with simulations of the annealed version, showing that the nominal branching ratio σ converges to unity in the thermodynamic limit, as expected of a self-organized critical system. However, the question remains whether the same results apply to the biological case where only the synapses of firing neurons are depressed (the so-called quenched dynamics). We show that simulations of the quenched model yield significant deviations from σ = 1 due to spatial correlations. However, the model is shown to be critical, as the largest eigenvalue of the synaptic matrix approaches unity in the thermodynamic limit, that is, λc = 1. We also study the finite size effects near the critical state as a function of the parameters of the synaptic dynamics.

*Andressa Cerqueira; Daniel Fraiman; Claudia D. Vargas and Florencia Leonardi*

The theory of random graphs has been applied in recent years to model neural interactions in the brain. While the probabilistic properties of random graphs has been extensively studied, the development of statistical inference methods for this class of objects has received less attention. In this work we propose a non-parametric test of hypotheses to test if a sample of random graphs was generated by a given probability distribution (one-sample test) or if two samples of random graphs were originated from the same probability distribution (two-sample test). We prove a Central Limit Theorem providing the asymptotic distribution of the test statistics and we propose a method to compute the quantiles of the finite sample distributions by simulation. The test makes no assumption on the specific form of the distributions and it is consistent against any alternative hypothesis that differs from the sample distribution on at least one edge-marginal. Moreover, we show that the test is a Kolmogorov-Smirnov type test, for a given distance between graphs, and we study its performance on simulated data. We apply it to compare graphs of brain functional network interactions built from electroencephalographic (EEG) data collected during the visualization of point light displays depicting human locomotion.

*Roberto Imbuzeiro Oliveira*

This paper introduces the concept of random context representations for the transition probabilities of a finite-alphabet stochastic process. Processes with these representations generalize context tree processes (also known as variable length Markov chains), and are proved to coincide with processes whose transition probabilities are almost surely continuous functions of the (infinite) past. This is similar to a classical result by Kalikow about continuous transition probabilities. Existence and uniqueness of a minimal random context representation are shown, in the sense that there exists a unique representation that looks into the past as little as possible in order to determine the next symbol. Both this representation and the transition probabilities can be consistently estimated from data, and some finite sample adaptivity properties are also obtained (including an oracle inequality). In particular, the estimator achieves minimax performance, up to logarithmic factors, for the class of binary renewal processes whose arrival distributions have bounded moments of order 2 + γ.

*Fernanda S. Matias, Leonardo L. Gollo, Pedro V. Carelli, Claudio R. Mirasso and Mauro Copelli*

We investigate the synchronization properties between two excitatory coupled neurons in the presence of an inhibitory loop mediated by an interneuron. Dynamic inhibition together with noise independently applied to each neuron provide phase diversity in the dynamics of the neuronal motif. We show that the interplay between the coupling strengths and the external noise controls the phase relations between the neurons in a counterintuitive way. For a master-slave configuration (unidirectional coupling) we find that the slave can anticipate the master, on average, if the slave is subject to the inhibitory feedback. In this nonusual regime, called anticipated synchronization (AS), the phase of the postsynaptic neuron is advanced with respect to that of the presynaptic neuron. We also show that the AS regime survives even in the presence of unbalanced bidirectional excitatory coupling. Moreover, for the symmetric mutually coupled situation, the neuron that is subject to the inhibitory loop leads in phase.