# Fast state-space methods for inferring dendritic synaptic connectivity

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## Abstract

We present fast methods for filtering voltage measurements and performing optimal
inference of the location and strength of synaptic connections in large dendritic
trees. Given noisy, subsampled voltage observations we develop fast *l* _{1}-penalized regression methods for Kalman state-space models
of the neuron voltage dynamics. The value of the *l* _{1}-penalty parameter is chosen using cross-validation or, for
low signal-to-noise ratio, a Mallows’ *C* _{ p }-like criterion. Using low-rank approximations, we reduce the inference
runtime from cubic to linear in the number of dendritic compartments. We also
present an alternative, fully Bayesian approach to the inference problem using a
spike-and-slab prior. We illustrate our results with simulations on toy and real
neuronal geometries. We consider observation schemes that either scan the dendritic
geometry uniformly or measure linear combinations of voltages across several
locations with random coefficients. For the latter, we show how to choose the
coefficients to offset the correlation between successive measurements imposed by
the neuron dynamics. This results in a “compressed sensing” observation scheme, with
an important reduction in the number of measurements required to infer the synaptic
weights.

## Keywords

Lasso Dendrites Low-rank State-space Synapses Spike-and-slab Compressed sensing## Notes

### Acknowledgments

This work was supported by an NSF CAREER grant, a McKnight Scholar award, and by NSF grant IIS-0904353. This material is based upon work supported by, or in part by, the U. S. Army Research Laboratory and the U. S. Army Research Office under contract number W911NF-12-1-0594. JHH was partially supported by the Columbia College Rabi Scholars Program. AP was partially supported by the Swartz Foundation. The computer simulations were done in the Hotfoot HPC Cluster of Columbia University. We thank E. Pnevmatikakis for helpful discussions and comments.

## Conflict of interests

The authors declare that they have no conflict of interest.

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