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

TL;DR

This paper introduces a fast, scalable convolutional Lasso-based algorithm for spike sorting in neuroscience, leveraging biological insights and mathematical guarantees to enable online processing of large, multi-sensor recordings.

Contribution

The authors develop a novel, efficient algorithm for convolutional spike sorting that exploits biological properties and problem structure, with theoretical guarantees and reduced complexity.

Findings

Algorithm has linear temporal complexity and can operate online.

Mathematical estimates of subproblem size using percolation theory.

High probability of accurately retrieving spike times with the proposed method.

Abstract

Spike sorting is a class of algorithms used in neuroscience to attribute the time occurences of particular electric signals, called action potential or spike, to neurons. We rephrase this problem as a particular optimization problem : Lasso for convolutional models in high dimension. Lasso (i.e. least absolute shrinkage and selection operator) is a very generic tool in machine learning that help us to look for sparse solutions (here the time occurrences). However, for the size of the problem at hand in this neuroscience context, the classical Lasso solvers are failing. We present here a new and much faster algorithm. 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 time occurrences of the spikes {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.