The present work aims at proving mathematically that a neural network inspired by biology can learn a classification task thanks to local transformations only. In this purpose, we propose a spiking neural network named CHANI (Correlation-based Hawkes Aggregation of Neurons with bio-Inspiration), whose neurons activity is modeled by Hawkes processes. Synaptic weights are updated thanks to an expert aggregation algorithm, providing a local and simple learning rule. We were able to prove that our network can learn on average and asymptotically. Moreover, we demonstrated that it automatically produces neuronal assemblies in the sense that the network can encode several classes and that a same neuron in the intermediate layers might be activated by more than one class, and we provided numerical simulations on synthetic datasets. This theoretical approach contrasts with the traditional empirical validation of biologically inspired networks and paves the way for understanding how local learning rules enable neurons to form assemblies able to represent complex concepts.

Learning for animals or humans is the process that leads to behaviors better adapted to the environment. This process highly depends on the individual that learns and is usually observed only through the individual’s actions. This article presents ways to use this individual behavioral data to find the model that best explains how the individual learns. We propose two model selection methods: a general hold-out procedure and an AIC-type criterion, both adapted to non-stationary dependent data. We provide theoretical error bounds for these methods that are close to those of the standard i.i.d. case. To compare these approaches, we apply them to contextual bandit models and illustrate their use on both synthetic and experimental learning data in a human categorization task.

We propose a simple network of Hawkes processes as a cognitive model capable of learning to classify objects. Our learning algorithm, named HAN for Hawkes Aggregation of Neurons, is based on a local synaptic learning rule based on spiking probabilities at each output node. We were able to use local regret bounds to prove mathematically that the network is able to learn on average and even asymptotically under more restrictive assumptions.

When fitting the learning data of an individual to algorithm-like learning models, the observations are so dependent and non-stationary that one may wonder what the classical Maximum Likelihood Estimator (MLE) could do, even if it is the usual tool applied to experimental cognition. Our objective in this work is to show that the estimation of the learning rate cannot be efficient if the learning rate is constant in the classical Exp3 (Exponential weights for Exploration and Exploitation) algorithm. Secondly, we show that if the learning rate decreases polynomially with the sample size, then the prediction error and in some cases the estimation error of the MLE satisfy bounds in probability that decrease at a polynomial rate.

We present a new algorithm to efficiently simulate random models of large neural networks satisfying the property of time asynchrony. The model parameters (average firing rate, number of neurons, synaptic connection probability, and postsynaptic duration) are of the order of magnitude of a small mammalian brain or of human brain areas. Through the use of activity tracking and procedural connectivity (dynamical regeneration of synapses), computational and memory complexities of this algorithm are proved to be theoretically linear with the number of neurons. These results are experimentally validated by sequential simulations of millions of neurons and billions of synapses running in a few minutes using a single thread of an equivalent desktop computer.

Sparse linear inverse problems appear in a variety of settings, but often the noise contaminating observations cannot accurately be described as bounded by or arising from a Gaussian distribution. Poisson observations in particular are a characteristic feature of several real-world applications. Previous work on sparse Poisson inverse problems encountered several limiting technical hurdles. This paper describes a novel alternative analysis approach for sparse Poisson inverse problems that 1) sidesteps the technical challenges present in previous work, 2) admits estimators that can readily be computed using off-the-shelf LASSO algorithms, and 3) hints at a general framework for broad classes of noise in sparse linear inverse problems. At the heart of this new approach lies a weighted LASSO estimator for which data-dependent weights are based on Poisson concentration inequalities. Unlike previous analyses of the weighted LASSO, the proposed analysis depends on conditions which can be checked or shown to hold in general settings with high probability. 2000 Math Subject Classification: 60E15, 62G05, 62G08, and 94A12.

We investigate several distribution-free dependence detection procedures, all based on a shuffling of the trials, from a statistical point of view. The mathematical justification of such procedures lies in the bootstrap principle and its approximation properties. In particular, we show that such a shuffling has mainly to be done on centered quantities-that is, quantities with zero mean under independence-to construct correct p-values, meaning that the corresponding tests control their false positive (FP) rate. Thanks to this study, we introduce a method, named permutation UE, which consists of a multiple testing procedure based on permutation of experimental trials and delayed coincidence count. Each involved single test of this procedure achieves the prescribed level, so that the corresponding multiple testing procedure controls the false discovery rate (FDR), and this with as few assumptions as possible on the underneath distribution, except independence and identical distribution across trials. The mathematical meaning of this assumption is discussed, and it is in particular argued that it does not mean what is commonly referred in neuroscience to as cross-trials stationarity. Some simulations show, moreover, that permutation UE outperforms the trial-shuffling of Pipa and Grün ( 2003 ) and the MTGAUE method of Tuleau-Malot et al. ( 2014 ) in terms of single levels and FDR, for a comparable amount of false negatives. Application to real data is also provided.

The unitary events (UE) method is one of the most popular and efficient methods used over the past decade to detect patterns of coincident joint spike activity among simultaneously recorded neurons. The detection of coincidences is usually based on binned coincidence count (Grün, 1996 ), which is known to be subject to loss in synchrony detection (Grün, Diesmann, Grammont, Riehle, & Aertsen, 1999 ). This defect has been corrected by the multiple shift coincidence count (Grün et al., 1999 ). The statistical properties of this count have not been further investigated until this work, the formula being more difficult to deal with than the original binned count. First, we propose a new notion of coincidence count, the delayed coincidence count, which is equal to the multiple shift coincidence count when discretized point processes are involved as models for the spike trains. Moreover, it generalizes this notion to nondiscretized point processes, allowing us to propose a new gaussian approximation of the count. Since unknown parameters are involved in the approximation, we perform a plug-in step, where unknown parameters are replaced by estimated ones, leading to a modification of the approximating distribution. Finally the method takes the multiplicity of the tests into account via a Benjamini and Hochberg approach (Benjamini & Hochberg, 1995 ), to guarantee a prescribed control of the false discovery rate. We compare our new method, MTGAUE (multiple tests based on a gaussian approximation of the unitary events) and the UE method proposed in Grün et al. ( 1999 ) over various simulations, showing that MTGAUE extends the validity of the previous method. In particular, MTGAUE is able to detect both profusion and lack of coincidences with respect to the independence case and is robust to changes in the underlying model. Furthermore MTGAUE is applied on real data.