The use of heuristics to assess the convergence and compress the output of Markov chain Monte Carlo can be sub-optimal in terms of the empirical approximations that are produced. Here we consider the problem of retrospectively selecting a subset of states, of fixed cardinality, from the sample path such that the approximation provided by their empirical distribution is close to optimal. A novel method is proposed, based on greedy minimisation of a kernel Stein discrepancy, that is suitable for problems where heavy compression is required. Theoretical results guarantee consistency of the method and its effectiveness is demonstrated in the challenging context of parameter inference for ordinary differential equations. Software is available in the Stein Thinning package in Python, R and MATLAB.

Motivated by the desire to automate classification of neuron morphologies, we designed a topological signature, the Topological Morphology Descriptor (TMD),  that assigns a so-called “barcode" to any geometric tree (i.e, any finite binary tree embedded in R^3). We showed that the TMD effectively determines  reliable clusterings of random and neuronal trees. Moreover, using the TMD we performed an objective, stable classification of pyramidal cells in the rat neocortex, based only on the shape of their dendrites.

We have also reverse-engineered the TMD, in order to digitally synthesize dendrites, to compensate for the relatively small number of available biological reconstructions. The algorithm we developed, called Topological Neuron Synthesis (TNS), stochastically generates a geometric tree from a barcode, in a biologically grounded manner. The synthesized neurons are statistically indistinguishable from real neurons of the same type, in terms of morpho-electrical properties and  connectivity. We synthesized networks of structurally altered neurons, revealing principles linking branching properties to the structure of large-scale networks.  We have also successfully applied these classification and synthesis techniques to microglia and astrocytes, two other types of cells that populate the brain.

In this talk I will provide an overview of the TMD and the TNS and then describe the results of our theoretical and computational analysis of their behavior and properties.

This talk is based on work in collaborations led by Lida Kanari at the Blue Brain Project.