Motivated by the desire to automate classification of neuron morphologies, we designed a topological signature, the Topological Morphology Descriptor (TMD), that assigns a "barcode" to any any finite binary tree embedded in ${\mathbb R}^3$. Using the TMD we performed an objective, stable classification of pyramidal cells in the rat neocortex, based only on the shape of their dendrites.

In this talk, I will introduce the TMD, then focus on a very recent application to comparing mouse and human cortical neurons and characterizing the differences between them. I'll also briefly discuss the role of machine learning in our work.

This talk is based on collaborations led by Lida Kanari of the Blue Brain Project.

Deep learning's successes are often attributed to its ability to automatically discover new representations of the data, rather than relying on handcrafted features like other learning methods. In this talk, however, I will show that deep networks learned by the standard gradient descent algorithm are in fact mathematically approximately equivalent to kernel machines, a learning method that simply memorizes the data and uses it directly for prediction via a similarity function (the kernel). This greatly enhances the interpretability of deep network weights, by elucidating that they are effectively a superposition of the training examples. The network architecture incorporates knowledge of the target function into the kernel. The talk will include a discussion of both the main ideas behind this result and some of its more startling consequences for deep learning, kernel machines, and machine learning at large.