CHANI: Correlation-based Hawkes Aggregation of Neurons with bio-Inspiration

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 dataset. 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.

On the convergence of the MLE as an estimator of the learning rate in the Exp3 algorithm

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.

Sliding window strategy for convolutional spike sorting with Lasso : Algorithm, theoretical guarantees and complexity

We present a fast algorithm for the resolution of the Lasso for convolutional models in high dimension, with a particular focus on the problem of spike sorting in neuroscience. Making use of biological properties related to neurons, we explain how the particular structure of the problem allows several optimizations, leading to an algorithm with a temporal complexity which grows linearly with respect to the size of the recorded signal and can be performed online. Moreover the spatial separability of the initial problem allows to break it into subproblems, further reducing the complexity and making possible its application on the latest recording devices which comprise a large number of sensors. We provide several mathematical results: the size and numerical complexity of the subproblems can be estimated mathematically by using percolation theory. We also show under reasonable assumptions that the Lasso estimator retrieves the true support with large probability. Finally the theoretical time complexity of the algorithm is given. Numerical simulations are also provided in order to illustrate the efficiency of our approach.

Event-scheduling algorithms with Kalikow decomposition for simulating potentially infinite neuronal networks

Event-scheduling algorithms can compute in continuous time the next occurrence of points (as events) of a counting process based on their current conditional intensity. In particular event-scheduling algorithms can be adapted to perform the simulation of finite neuronal networks activity. These algorithms are based on Ogata's thinning strategy \cite{Oga81}, which always needs to simulate the whole network to access the behaviour of one particular neuron of the network. On the other hand, for discrete time models, theoretical algorithms based on Kalikow decomposition can pick at random influencing neurons and perform a perfect simulation (meaning without approximations) of the behaviour of one given neuron embedded in an infinite network, at every time step. These algorithms are currently not computationally tractable in continuous time. To solve this problem, an event-scheduling algorithm with Kalikow decomposition is proposed here for the sequential simulation of point processes neuronal models satisfying this decomposition. This new algorithm is applied to infinite neuronal networks whose finite time simulation is a prerequisite to realistic brain modeling.

Large scale Lasso with windowed active set for convolutional spike sorting

Spike sorting is a fundamental preprocessing step in neuroscience that is central to access simultaneous but distinct neuronal activities and therefore to better understand the animal or even human brain. But numerical complexity limits studies that require processing large scale datasets in terms of number of electrodes, neurons, spikes and length of the recorded signals. We propose in this work a novel active set algorithm aimed at solving the Lasso for a classical convolutional model. Our algorithm can be implemented efficiently on parallel architecture and has a linear complexity w.r.t. the temporal dimensionality which ensures scaling and will open the door to online spike sorting. We provide theoretical results about the complexity of the algorithm and illustrate it in numerical experiments along with results about the accuracy of the spike recovery and robustness to the regularization parameter.

Microscopic approach of a time elapsed neural model

The spike trains are the main components of the information processing in the brain. To model spike trains several point processes have been investigated in the literature. And more macroscopic approaches have also been studied, using partial differential equation models. The main aim of the present article is to build a bridge between several point processes models (Poisson, Wold, Hawkes) that have been proved to statistically fit real spike trains data and age-structured partial differential equations as introduced by Pakdaman, Perthame and Salort.