On a toy model of interacting neurons

Nicolas Fournier, Eva Löcherbach

We continue the study of a stochastic system of interacting neurons introduced in De Masi-Galves-Löcherbach-Presutti (2014). The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to 0 and all other neurons receive an additional amount 1/N of potential. Moreover, electrical synapses induce a deterministic drift of the system towards its center of mass. We prove propagation of chaos of the system, as N tends to infinity, to a limit nonlinear jumping stochastic differential equation.

Predicting upcoming actions by observation: some facts, models and challenges

C. D. Vargas, M. L. Rangel, A. Galves

Predicting another person's upcoming action to build an appropriate response is a regular occurrence in the domain of motor control. In this review we discuss conceptual and experimental approaches aiming at the neural basis of predicting and learning to predict upcoming movements by their observation.

On the Number of Orientations of Random Graphs with No Directed Cycles of a Given Length

P. Allen, Y. Kohayakawa, G. O. Mota, R. F. Parente

Let H⃗ be an orientation of a graph H. Alon and Yuster proposed the problem of determining or estimating D(n,m,H⃗), the maximum number of H⃗-free orientations a graph with n vertices and m edges may have. We consider the maximum number of H⃗-free orientations of typical graphs G(n,m) with n vertices and m edges. Suppose H⃗ =C↻ℓ is the directed cycle of length ℓ≥3. We show that if m≫n^(1+1/(ℓ−1)), then this maximum is 2^o(m), while if m≪n^(1+1/(ℓ−1)), then it is 2^(1−o(1))m.

Tight Hamilton cycles in random hypergraphs, Random Structures Algorithms

Allen, P.; Böttcher, J.; Kohayakawa, Y. and Person Y.

We give an algorithmic proof for the existence of tight Hamilton cycles in a random r-uniform hypergraph with edge probability p=n^{-1+eps} for every eps>0. This partly answers a question of Dudek and Frieze [Random Structures Algorithms], who used a second moment method to show that tight Hamilton cycles exist even for p=omega(n)/n (r>2) where omega(n) tends to infinity arbitrary slowly, and for p=(e+o(1))/n (r>3). The method we develop for proving our result applies to related problems as well.

Stochastically Perturbed Chains of Variable Memory

Garcia, N. L. and Moreira, L. - In this paper, we study inference for chains of variable order under two distinct contamination regimes. Consider a chain of variable memory on a finite alphabet containing zero, at each instant of time an independent coin is flipped and if it turns head a contamination occurs.

Undersampled Critical Branching Processes on Small-World and Random Networks Fail to Reproduce the Statistics of Spike Avalanches

Ribeiro, T. L. ; Ribeiro, S. ; Belchior, H. ; Caixeta, F. and Copelli, M. - The power-law size distributions obtained experimentally for neuronal avalanches are an important evidence of criticality in the brain. This evidence is supported by the fact that a critical branching process exhibits the same exponent τ=3/2. Models at criticality have been employed to mimic avalanche propagation and explain the statistics observed experimentally.

Hydrodynamic limit for interacting neurons

De Masi, A. ; Galves, A. ; Löcherbach, E. and Presutti, E. - This paper studies the hydrodynamic limit of a stochastic process describing the time evolution of a system with N neurons with mean-field interactions produced both by chemical and by electrical synapses.

Independence tests for continuous random variables based on the longest increasing subsequence

García, J. E. and González-López, V. A. - We propose a new class of nonparametric tests for the supposition of independence between two continuous random variables X and Y. Given a size n sample, let π be the permutation which maps the ranks of the X observations on the rank of the Y observations.

Modeling of acoustic signal energies with a generalized Frank copula. A linguistic conjecture is reviewed

García, J. E. and González-López, V. A. - In this paper was selected a generalized Frank copula to model the dependence between the energy on two frequency bands of the speech signal, coming from eight languages. Was developed an algorithm that uses maximum likelihood to choose the best fitting copula's parameters.

Loss of Memory of Hidden Markov Models and Lyapunov Exponents

Collet, P. and Leonardi, F. - In this paper we prove that the asymptotic rate of exponential loss of memory of a finite state hidden Markov model is bounded above by the difference of the first two Lyapunov exponents of a certain product of matrices.




The Research, Innovation and Dissemination Center for Neuromathematics is hosted by the University of São Paulo and funded by FAPESP (São Paulo Research Foundation).


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