EXACT SPIKE TRAIN INFERENCE VIA ℓ0 OPTIMIZATION

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 ℓ₁ penalty was proposed for this task. In this paper we slightly modify that recent proposal by replacing the ℓ₁ penalty with an ℓ₀ penalty. In stark contrast to the conventional wisdom that ℓ₀ 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 ℓ₁ proposal, in simulations as well as on two calcium imaging datasets. 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.

Keywords: Neuroscience; calcium imaging; changepoint detection; dynamic programming.

Fig. 6.

Spike detection for cell 2002 of the Chen et al. (2013) data. In each panel, the observed fluorescence () and true spikes () are displayed. Estimated spikes from problem (13) are shown in (), and the estimated spikes from the ℓ₀ problem (5) with λ = 0.6 are shown in (). Times 0 s–35 s are shown in the top row; the second row zooms in on times 5 s–10 s to illustrate behavior around a large increase in calcium concentration. Columns correspond to different values of s_min.

Fig. 1.

A toy simulated data example. In each panel the x-axis represents time. Observed fluorescence values are displayed in (). (a): Unobserved calcium concentrations () and true spike times (). Data were generated according to the model (1). (b): Estimated calcium concentrations () and spike times () that result from solving the ℓ₁ optimization problem (3) with the value of λ that yields the true number of spikes. This value of λ leads to very poor estimation of both the underlying calcium dynamics and the spikes. (c): Estimated calcium concentrations () and spike times () that result from solving the ℓ₁ optimization problem (3) with the largest value of λ that results in at least one estimated spike within the vicinity of each true spike. This value of λ results in 19 estimated spikes, which is far more than the true number of spikes. The poor performance of the ℓ₁ optimization problem in panels (b) and (c) is a consequence of the fact that the ℓ₁ penalty performs shrinkage as well as spike estimation; this is discussed further in Section 1.2. (d): Estimated calcium concentrations () and spike times () that result from solving the ℓ₀ optimization problem (5). (e): The four spikes in panel (c) associated with the largest estimated increase in calcium (); we refer to this in the text as the post-thresholding ℓ₁ estimator. Since the estimated calcium is not well defined after post-thresholding, we do not plot the estimated calcium concentration.

Fig. 2.

Timing results for solving (5) for the global optimum, using Algorithms 1 () and 2 (). The x-axis displays the length of the time series (T), and the y-axis displays the average running time in seconds. Each panel, from left to right, corresponds to data simulated according to (1) with s_t ~_i.i.d. Poisson(θ), with θ ∈ {0.001, 0.01, 0.1}. Standard errors are on average < 0.1% of the mean compute time. Additional details are provided in Section 2.4.

Fig. 3.

Simulation study to assess the error in spike detection and calcium estimation, for the ℓ₁ (3), post-thresholded ℓ₁ (9) and ℓ₀ (4) problems. (a): Error in spike detection measured using van Rossum distance. (b): Error in spike detection, measured using Victor-Purpura distance. (c): Error in calcium estimation (10). Simulation details are provided in Section 3.

Fig. 4.

Spike detection for cell 2002 of the Chen et al. (2013) data. The observed fluorescence () and true spikes () are displayed. Estimated spike times from the ℓ₀ problem (4) are shown in (), estimated spike times from the ℓ₁ problem (3) are shown in (), and estimated spike times from the post-thresholding estimator (9) are shown in (). Times 0s–35s are shown in the top row; the second row zooms into time 30s–40s in order to illustrate the behavior around a large increase in calcium concentration.

Fig. 5.

The first 10,000 timesteps from the second ROI in NWB 510221121 from the Allen Brain Observatory. Each panel displays the DF/F-transformed fluorescence (), the estimated spikes from the ℓ₀ problem () (5), the estimated spikes from the ℓ₁ problem () (3), and the estimated spikes from post-thresholding the ℓ₁ problem () (9). The panels display results from applying the ℓ₁ and ℓ₀ methods with tuning parameter λ chosen to yield (a): 27 spikes for each method; (b): 49 spikes for each method; and (c): 128 spikes for each method. The post-thresholding estimator was obtained by applying the ℓ₁ method with λ = 1, and thresholding the result to obtain 27, 49 or 128 spikes. (d)–(f): As in (a)–(c), but zoomed in on 200–250 seconds.