# Exact Spike Train Inference Via $\ell_0$ Optimization

## TL;DR

This paper introduces an efficient dynamic programming method for exact spike train inference from fluorescence data by replacing an $\\ell_1$ penalty with an $\\ell_0$ penalty, significantly improving accuracy and computational efficiency.

## Contribution

The paper demonstrates that an $\\ell_0$ optimization approach for spike inference is computationally feasible and more accurate than previous $\\ell_1$ methods, with practical software implementation.

## Key findings

- The $\\ell_0$ optimization problem can be solved efficiently with dynamic programming.
- The proposed method outperforms previous $\\ell_1$ based approaches in simulations and real data.
- The algorithm runs in minutes on large datasets with 100,000 timesteps.

## Abstract

In recent years, new technologies in neuroscience have made it possible to measure the activities of large numbers of neurons simultaneously in behaving animals. For each neuron, a fluorescence trace is measured; this can be seen as a first-order approximation of the neuron's activity over time. Determining the exact time at which a neuron spikes on the basis of its fluorescence trace is an important open problem in the field of computational neuroscience. Recently, a convex optimization problem involving an $\ell_1$ penalty was proposed for this task. In this paper, we slightly modify that recent proposal by replacing the $\ell_1$ penalty with an $\ell_0$ penalty. In stark contrast to the conventional wisdom that $\ell_0$ optimization problems are computationally intractable, we show that the resulting optimization problem can be efficiently solved for the global optimum using an extremely simple and efficient dynamic programming algorithm. Our R-language implementation of the proposed algorithm runs in a few minutes on fluorescence traces of $100,000$ timesteps. Furthermore, our proposal leads to substantial improvements over the previous $\ell_1$ proposal, in simulations as well as on two calcium imaging data sets. R-language software for our proposal is available on CRAN in the package LZeroSpikeInference. Instructions for running this software in python can be found at https://github.com/jewellsean/LZeroSpikeInference.