Hierarchies & Lower Bounds in Theoretical Connectomics

Connectomics is a sub-field of Neuroscience aimed at determining connectomes - exact structures of neurons and their synaptic connections in nervous systems. A number of ongoing initiatives at the present time are working towards the goal of ascertaining the connectomes or parts thereof of various organisms. Determining the detailed physiological response properties of all the neurons in these connectomes is out of reach of current experimental technology. It is therefore unclear, to what extent knowledge of the connectome alone will advance a mechanistic understanding of computation occurring in these neuronal circuits, especially when the high-level function(s) of the said circuit is unknown. We are pursuing a research program to build theory in order to investigate these issues. In previously published work [1], towards this end, we have developed a theory of connectomic constraints for feedforward networks of neurons. Specifically, for feedforward networks equipped with neurons that obey a deterministic spiking neuron model, we asked if just by knowing the structure of a network, we could rule out spike-timed computations that it could be doing, no matter what response properties each of its neurons may have. Our neurons were abstract mathematical objects that satisfied a small number of axioms that correspond to certain broadly-obeyed properties of neurons. Here, we develop additional theoretical tools and notions to address these questions. The idea is to study the space of all possible spike-train to spike-train transformations. We are interested in asking how the subset of transformations spanned by networks of specific architectures can be related to hierarchical subsets of the space that are characterized by particular mathematical properties of transformations. In particular, given such a hierarchy of subsets, what is the "smallest" subset that contains the set of transformations spanned by networks of a specific class of architectures? Even if one cannot establish such a subset exactly, proving bounds on it (according to the hierarchy) might offer insight. After setting up the mathematical framework to make these notions precise, we construct explicit classes of hierarchies and prove a number of such lower bounds.