Abstract
In recent years, the problem of reconstructing the connectivity in large neural circuits (“connectomics”) has re-emerged as one of the main objectives of neuroscience. Classically, reconstructions of neural connectivity have been approached anatomically, using electron or light microscopy and histological tracing methods. This paper describes a statistical approach for connectivity reconstruction that relies on relatively easy-to-obtain measurements using fluorescent probes such as synaptic markers, cytoplasmic dyes, transsynaptic tracers, or activity-dependent dyes. We describe the possible design of these experiments and develop a Bayesian framework for extracting synaptic neural connectivity from such data. We show that the statistical reconstruction problem can be formulated naturally as a tractable L 1-regularized quadratic optimization. As a concrete example, we consider a realistic hypothetical connectivity reconstruction experiment in C. elegans, a popular neuroscience model where a complete wiring diagram has been previously obtained based on long-term electron microscopy work. We show that the new statistical approach could lead to an orders of magnitude reduction in experimental effort in reconstructing the connectivity in this circuit. We further demonstrate that the spatial heterogeneity and biological variability in the connectivity matrix—not just the “average” connectivity—can also be estimated using the same method.
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Notes
Note that with i and j we can refer to both individual neurons as well as, more generally, entire classes of neurons, such as morphological or genetic classes, geometrically defined areas of the brain, etc., with respective changes in the interpretation of C ij as describing the count of synapses between such groups of neurons.
The nonnegativity constraint M ij ≥ 0 may be additionally enforced if the mean connectivity matrix M ij is expected to be nonnegative, e.g. if it describes the expected count of (unsigned) synapses between neurons, which clearly can not be negative. Such a constraint can significantly simplify the task of numerically solving Eq. (8) but, in general, nonnegativity cannot be imposed if the connectivity matrix is not expected to be sign definite, for example, such as if M ij describes the post-synaptic potential strength of neural connections in an excitatory and inhibitory network, as discussed in Section 4 below.
To be precise, the LASSO problem corresponds to the case that the variance parameter \(\sigma_t^2\) does not depend on the mean parameter M. As discussed above, for some connectivity distributions p({C ij }) \(\sigma_t^2\) might depend weakly on M. For example, in the Poisson case, \(\sigma_t^2\) corresponds to a one-dimensional projection of the very high-dimensional parameter M. These cases can be handled with straightforward extensions of the LASSO approach; however, in this paper we will focus on the model with \(\sigma_t^2\) independent of M.
In greater detail, a condition on the matrix \(\{e_{ij}^k\}\) sufficient to ensure exact reconstruction is the so-called Restricted Isometry Property (RIP): if a constant δ S exists such that \((1-\delta_S)\|{\bf x}_S\|^2_2 \leq \|{\bf E}{\bf x}_S\|^2_2 \leq (1+\delta_S)\|{\bf x}_S\|^2_2\) for every S-sparse vector x S , then the RIP holds for the matrix E. Although for a specific matrix E it is typically very difficult to verify RIP, RIP has been proven to hold with high probability in a number of classes of large random matrices (Romberg 2008; Candes and Wakin 2008).
More specifically, the compressed sensing problem for a set of measurements O and sensing matrix E can be defined as follows (Candes and Wakin 2008; Romberg 2008):
$$ \hat {\bf M}^{CS} = \arg \min \|{\bf M}\|_{1} ~ s.t. ~ {\bf O}={\bf E}\cdot{\bf M}. $$If the measurements O and the matrix \({\bf E}=\{e^k_{ij} \}\) are such that there exist an M * such that O = E·M * identically for that M *, then we can equivalently rewrite this optimization program as
$$ \min \|{\bf M}\|_{1} ~ s.t. ~ ({\bf O} - {\bf E} \cdot {\bf M})^2 = 0. $$At the same time, we can use the standard Lagrange multipliers trick to rewrite the penalized LASSO optimization for the same problem,
$$ \min \left[ ({\bf O}-{\bf E}\cdot{\bf M})^2 + \lambda \|{\bf M}\|_{1} \right], $$in the form of a constrained optimization,
$$ \min \|{\bf M}\|_{1}\ s.t.\ ({\bf O}-{\bf E}\cdot{\bf M})^2 \leq \epsilon^2(\lambda), $$thus observing that \(\|\hat {\bf M}^{CS}-\hat {\bf M}^{LASSO}\|^2_{2}=O(\epsilon^2(\lambda))\). ϵ 2(λ) can be bounded by using the equivalence of the above two optimization problems so that \(\|\hat {\bf M}^{CS}-\hat {\bf M}^{LASSO}\|^2_{2} \leq B \|{\bf M}^*\|_{1} \lambda\), for a suitable constant B, and \(\hat {\bf M}^{LASSO}(\lambda)\rightarrow \hat {\bf M}^{CS}\) continuously as λ→0.
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Acknowledgements
The authors are grateful to Prof. David Hall for his re-examination of the original print data from White et al. (1986), and for providing the statistics about the spatial distribution of synapses in C. elegans used in Section 3.4. This work was supported by an NSF CAREER grant, a McKnight Scholar award, and by NSF grant IIS-0904353.
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Mishchenko, Y., Paninski, L. A Bayesian compressed-sensing approach for reconstructing neural connectivity from subsampled anatomical data. J Comput Neurosci 33, 371–388 (2012). https://doi.org/10.1007/s10827-012-0390-z
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DOI: https://doi.org/10.1007/s10827-012-0390-z