Morgan André and Léo Planche
We consider a continuous-time stochastic model of spiking neurons. In this model, we have a finite or countable number of neurons which are vertices in some graph G where the edges indicate the synaptic connection between them. We focus on metastability, understood as the property for the time of extinction of the network to be asymptotically memory-less, and we prove that this model exhibits two different behaviors depending on the nature of the specific underlying graph of interaction G that is chosen. This model depends on a leakage parameter γ, and it was previously proven that when the graph G is the infinite one-dimensional lattice, this model presents a phase transition with respect to γ. It was also proven that, when γ is small enough, the renormalized time of extinction (the first time at which all neurons have a null membrane potential) of a finite version of the system converges in law toward an exponential random variable when the number of neurons goes to infinity. The present article is divided into two parts. First we prove that, in the finite one-dimensional lattice, this last result doesn't hold if γ is not small anymore, in fact we prove that for γ>1 the renormalized time of extinction is asymptotically deterministic. Then we prove that conversely, if G is the complete graph, the result of metastability holds for any positive γ.
Cecilia Romaro, Fernando Araujo Najman, William W Lytton, Antonio C. Roque and Salvador Dura-Bernal
The Potjans-Diesmann cortical microcircuit model is a widely used model originally implemented in NEST. Here, we re-implemented the model using NetPyNE, a high-level Python interface to the NEURON simulator, and reproduced the findings of the original publication. We also implemented a method for rescaling the network size which preserves first and second-order statistics, building on existing work on network theory. The new implementation enables using more detailed neuron models with multi-compartment morphologies and multiple biophysically realistic channels. This opens the model to new research, including the study of dendritic processing, the influence of individual channel parameters, and generally multiscale interactions in the network. The rescaling method provides flexibility to increase or decrease the network size if required when running these more realistic simulations. Finally, NetPyNE facilitates modifying or extending the model using its declarative language; optimizing model parameters; running efficient large-scale parallelized simulations; and analyzing the model through built-in methods, including local field potential calculation and information flow measures.
A conjecture in neurobiology that dates back to Helmholtz in the XIX century states that the brain can unconsciously identify statistical regularities in sequences of stimuli. Motivated by this claim, a NeuroMat group led by Antonio Galves and Claudia Vargas, with the participation of Aline Duarte, Ricardo Fraiman and Guilherme Ost, have successfully applied mathematical techniques to retrieve from EEG measurements in people the structure of stochastic chains with memory of variable length (called context trees) that generate auditory input stimuli (Duarte et al., 2019). The brains of the experiment subjects had ongoing spiking activity patterns (arguably somehow related to the EEG signals) that were perturbed by the input stimuli in a way that allowed the mathematical retrieval tools to work satisfactorily.
Thus, from a theoretical point of view the following question can be posed: is the brain machinery, with all its intricate web of molecular and cellular processes, necessary for the efficient retrieval of context trees? Or simpler, brain-inspired networks of spiking elements can also encode in their spiking activity a signature of the context tree that can be identified by the same mathematical tools?
Antonio Galves, Charlotte Galves, Jesús E. García, Nancy L. Garcia, Florencia Leonardi
The starting point of this article is the question "How to retrieve fingerprints of rhythm in written texts?" We address this problem in the case of Brazilian and European Portuguese. These two dialects of Modern Portuguese share the same lexicon and most of the sentences they produce are superficially identical. Yet they are conjectured, on linguistic grounds, to implement different rhythms. We show that this linguistic question can be formulated as a problem of model selection in the class of variable length Markov chains. To carry on this approach, we compare texts from European and Brazilian Portuguese. These texts are previously encoded according to some basic rhythmic features of the sentences which can be automatically retrieved. This is an entirely new approach from the linguistic point of view. Our statistical contribution is the introduction of the smallest maximizer criterion which is a constant free procedure for model selection. As a by-product, this provides a solution for the problem of optimal choice of the penalty constant when using the BIC to select a variable length Markov chain. Besides proving the consistency of the smallest maximizer criterion when the sample size diverges, we also make a simulation study comparing our approach with both the standard BIC selection and the Peres-Shields order estimation. Applied to the linguistic sample constituted for our case study, the smallest maximizer criterion assigns different context-tree models to the two dialects of Portuguese. The features of the selected models are compatible with current conjectures discussed in the linguistic literature.
Artem Logachov, A. Mogulskii and Anatoly Yambartsev
Here we obtain the exact asymptotics for large and moderate deviations, strong law of large numbers and central limit theorem for chains with unbounded variable length memory.
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