## 68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section -04 in that area)

### Refine

#### Year of publication

- 2016 (2) (remove)

#### Keywords

- ADOL-C (1)
- Automatic differentiation (1)
- Computational graph (1)
- Generalized Hermite interpolation (1)
- Lipschitz continuity (1)
- NBLAST (1)
- cell type (1)
- clustering (1)
- neuroinformatics (1)
- neuron similarity (1)

#### Institute

It is shown how piecewise differentiable functions \(F: R^n → R^m\) that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points x̌ and x̂. We show that the discrepancy between function and model at any point x is of the bilinear order O(||x − x̌|| ||x − x̂||). This is a little surprising since x ∈ R^n may vary over the whole Euclidean space, and we utilize only two function samples F̌ = F(x̌) and F̂ = F(x̂), as well as the intermediates computed during their evaluation. As an application of the piecewise linearization procedure we devise a generalized Newton’s method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equaling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.

Neural circuit mapping is generating datasets of 10,000s of labeled neurons. New computational tools are needed to search and organize these data. We present NBLAST, a sensitive and rapid algorithm, for measuring pairwise neuronal similarity. NBLAST considers both position and local geometry, decomposing neurons into short segments; matched segments are scored using a probabilistic scoring matrix defined by statistics of matches and non-matches.
We validated NBLAST on a published dataset of 16,129 single Drosophila neurons. NBLAST can distinguish neuronal types down to the finest level (single identified neurons) without a priori information. Cluster analysis of extensively studied neuronal classes identified new types and unreported topographical features. Fully automated clustering organized the validation dataset into 1052 clusters, many of which map onto previously described neuronal types. NBLAST supports additional query types including searching neurons against transgene expression patterns. Finally we show that NBLAST is effective with data from other invertebrates and zebrafish.