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

Overview

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., 2019).

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,

$$\begin{align*} y_t &= c_t + \epsilon_t, \quad \epsilon_t \sim_\text{ind.} (0, \sigma^2), \quad t = 1, \ldots, T; \nonumber\\ c_t &= \gamma c_{t-1} + z_t, \quad t = 2,\ldots, T, \end{align*}$$

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

$$\underset{c_1,\ldots,c_T, z_2,\ldots,z_T}{\mathrm{minimize}} \frac{1}{2} \sum_{t=1}^T \left( y_t - c_t \right)^2 + \lambda \sum_{t=2}^T 1_{\left( z_t \neq 0 \right) },$$

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

$$\underset{c_1,\ldots,c_T, z_2,\ldots,z_T}{\mathrm{minimize}} \frac{1}{2} \sum_{t=1}^T \left( y_t - c_t \right)^2 + \lambda \sum_{t=2}^T 1_{\left( z_t \neq 0 \right) },$$

as well as the related problem

$$\underset{c_1,\ldots,c_T, z_2,\ldots,z_T}{\mathrm{minimize}} \frac{1}{2} \sum_{t=1}^T \left( y_t - c_t \right)^2 + \lambda \sum_{t=2}^T 1_{\left( z_t \neq 0 \right) } \mbox{ subject to } z_t = c_t - \gamma c_{t-1} \geq 0,$$

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., 2019) 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 https://github.com/AllenInstitute/AllenSDK/wiki/Release-Notes-(0.16.0) for additional information.

Funding

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.

References

1. Jewell, S., & Witten, D. (2018). Exact spike train inference via L0 optimization. Ann. Appl. Statist., 12(4), 2457–2482. https://doi.org/10.1214/18-AOAS1162
   Bibtex

   @article{jewell2018exact,
     author = {Jewell, Sean and Witten, Daniela},
     title = {Exact spike train inference via L0 optimization},
     journal = {Ann. Appl. Statist.},
     fjournal = {Annals of Applied Statistics},
     year = {2018},
     volume = {12},
     number = {4},
     pages = {2457-2482},
     issn = {1932-6157},
     doi = {10.1214/18-AOAS1162},
     sici = {1932-6157(2018)12:4<2457:ESTIVE>2.0.CO;2-H},
     preprint = {https://arxiv.org/pdf/1703.08644.pdf},
     code = {https://github.com/jewellsean/LZeroSpikeInference}
   }
   

2. Jewell, S. W., Hocking, T. D., Fearnhead, P., & Witten, D. M. (2019). Fast nonconvex deconvolution of calcium imaging data. Biostatistics. https://doi.org/10.1093/biostatistics/kxy083
   Bibtex

   @article{jewell2018fast,
     author = {Jewell, Sean W and Hocking, Toby Dylan and Fearnhead, Paul and Witten, Daniela M},
     title = {{Fast nonconvex deconvolution of calcium imaging data}},
     year = {2019},
     month = feb,
     journal = {Biostatistics},
     doi = {10.1093/biostatistics/kxy083},
     url = {https://doi.org/10.1093/biostatistics/kxy083},
     preprint = {https://arxiv.org/pdf/1802.07380.pdf},
     code = {https://github.com/jewellsean/FastLZeroSpikeInference}
   }
   

3. de Vries, S. E. J., Lecoq, J., Buice, M. A., Groblewski, P. A., Ocker, G. K., Oliver, M., … Koch, C. (2019). A large-scale, standardized physiological survey reveals functional organization of the mouse visual cortex. To Appear in Nature Neuroscience.
   Bibtex

   @article{deVries2018large,
     author = {de Vries, S E J and Lecoq, J and Buice, M A and Groblewski, P A and Ocker, G K and Oliver, M and Feng, D and Cain, N and Ledochowitsch, P and Millman, D and Roll, K and Garrett, M and Keenan, T and Kuan, L and Mihalas, S and Olsen, S and Thompson, C and Wakeman, W and Waters, J and Williams, D and Barber, C and Berbesque, N and Blanchard, B and Bowles, N and Caldejon, S and Casal, L and Cho, A and Cross, S and Dang, C and Dolbeare, T and Edwards, M and Galbraith, J and Gaudreault, N and Griffin, F and Hargrave, P and Howard, R and Huang, L and Jewell, S and Keller, N and Knoblich, U and Larkin, J and Larsen, R and Lau, C and Lee, E and Lee, F and Leon, A and Li, L and Long, F and Luviano, J and Mace, K and Nguyen, T and Perkins, J and Robertson, M and Seid, S and Shea-Brown, E and Shi, J and Sjoquist, N and Slaughterbeck, C and Sullivan, D and Valenza, R and White, C and Williford, A and Witten, D and Zhuang, J and Zeng, H and Farrell, C and Ng, L and Bernard, A and Phillips, J W and Reid, R C and Koch, C},
     title = {A large-scale, standardized physiological survey reveals functional organization of the mouse visual cortex},
     journal = {To appear in Nature Neuroscience},
     year = {2019},
     preprint = {https://www.biorxiv.org/content/early/2018/06/29/359513.full.pdf}
   }