Fast deconvolution of calcium imaging data via an $\ell_0$ penalty



This website provides software and tutorials to perform fast $\ell_0$ deconvolution of calcium imaging data, as described in these two papers (Jewell & Witten, 2018) and (Jewell, Hocking, Fearnhead, & Witten, 2019), and as implemented in the Allen Brain Observatory platform paper (de Vries et al., 2018).

Technical details

We assume that the observed fluorescence trace $y_{t}$ is a noisy version of the unobserved calcium concentration $c_{t}$. Furthermore, we assume that the calcium concentration decays exponentially, unless there is a spike, in which case the calcium concentration increases instantaneously. More precisely,

where $z_t\geq 0$, and where $z_{t} >0$ indicates the presence of a spike at the $t$th timestep.

Based on this generative model, our earlier work (Jewell & Witten, 2018) proposed solving the $\ell_0$ optimization problem

where $\lambda>0$ is a tuning parameter that balances the tradeoff between the total number of spikes, $\sum_{t}z_t$, and the goodness-of-fit, $\frac{1}{2} \sum_{t=1}^T \left( y_t - c_t \right)^2$.

In (Jewell, Hocking, Fearnhead, & Witten, 2019), we develop a very efficient algorithm for solving

as well as the related problem

that ensures a spike results in an increase in the calcium concentration. This website introduces the efficient algorithm of our work (Jewell, Hocking, Fearnhead, & Witten, 2019).

Allen Institute for Brain Science

The Allen Brain Observatory platform paper (de Vries et al., 2018) performs all spike deconvolution using the algorithm described on this website.

If you’re working with the AIBS data, you may be interested to know that the AIBS also released an update to their software development kit that provides users with the output from Algorithm 2 for close to 60,000 neurons during different experimental conditions. See for additional information.


This work was partially supported by the NIH BRAIN Initiative, R01EB026908, NIH grants DP5OD009145 and R01DA047869, NSF CAREER DMS-1252624, and a Simons Investigator Award in Mathematical Modeling of Living Systems to D. Witten. The following sources also provided support: the Natural Sciences and Engineering Research Council of Canada to S. Jewell, the Natural Sciences and Engineering Research Council of Canada grant RGPGR 448167-2013 and Canadian Institutes of Health Research grants EP1-120608 and EP1-120609 to T. Hocking, and by the Engineering and Physical Sciences Research Council Grant EP/N031938/1 to P. Fearnhead.


  1. Jewell, S., & Witten, D. (2018). Exact spike train inference via L0 optimization. Ann. Appl. Statist., 12(4), 2457–2482.
  2. Jewell, S. W., Hocking, T. D., Fearnhead, P., & Witten, D. M. (2019). Fast nonconvex deconvolution of calcium imaging data. Biostatistics.
  3. de Vries, S. E. J., Lecoq, J., Buice, M. A., Groblewski, P. A., Ocker, G. K., Oliver, M., … Koch, C. (2018). A large-scale, standardized physiological survey reveals higher order coding throughout the mouse visual cortex. BioRxiv:359513.