Conference Agenda

Research at the interface of topology and neuroscience is growing rapidly and has produced many remarkable results in the past five years. In this minisymposium, speakers will present a wide and exciting array of current applications of topology in neuroscience, including classification and synthesis of neuron morphologies, analysis of synaptic plasticity, and diagnosis of traumatic brain injuries.

The morphological diversity of neurons supports the complex information-processing capabilities of biological neuronal networks. A major challenge in neuroscience has been to reliably describe neuronal shapes with universal morphometrics that generalize across cell types and species. Inspired by algebraic topology, we have conceived a topological descriptor of trees that couples the topology of their complex arborization with the geometric features of its structure, retaining more information than traditional morphometrics. The topological morphology descriptor (TMD) has proved to be very powerful in categorizing neurons into concrete groups based on morphological grounds. The TMD algorithm has lead to the discovery of two distinct classes of pyramidal cells in the human cortex, and the identification of robust groups for rodent cortical neurons.

I will present some techniques from elementary algebraic topology that can in some cases be used to completely determine the homotopy types of directed flag complexes (or more generally, ordered simplicial complexes). These involve simplicial collapses, heuristic homology computations and coning operations. Then I will explain how to use these techniques to classify the homotopy types for certain large families of tournaments. Finally, I will show how to compute the homotopy type for the C. Elegans connectome.

Stroke has recently been called “the epidemic of the 21st century." Today, 1.5 million new strokes occur each year in Europe alone, and this number is expected to increase by a factor of 1.5 to 2 by the year 2050. Despite advances in treatment, only a small proportion of patients recover enough to re-enter normal life.

The brain is a highly networked organ; accordingly, many brain diseases, including stroke, are increasingly understood as network disorders. Stroke lesions cause impairment by disabling nearby nodes and edges in the structural network, which in turn affects the functional network and the corresponding behavior. Likewise, it is hypothesized that recovery from stroke can be expressed in terms of reorganization of both functional and structural networks.

The use of graph theory-based metrics to study brain networks is well established. Some metrics, such as degree or betweenness centrality, capture local characteristics; others, such as connection density or the small-world index, capture global characteristics of a network. In recent years, algebraic topology has become increasingly prominent for its ability to integrate local network characteristics to a global notion of shape.

We will present evidence that in vivo structural brain networks possess significantly more cavities than random networks with the same degree distribution, building on in silico evidence from Hess et al that information flow is organized by topological invariants. We will also attempt to use persistent homology to distinguish between two groups of patients: those who, within 3 months poststroke, recover roughly 70% of lost motor function (called “fitters") and those who do not (called “non-fitters").

Neural decoding is the process of determining which stimuli are driving the activity of neurons. For instance, head direction neurons fire depending on which direction the animal is looking. Determining this relationship, however, can be a tedious proces, where the researcher would have to track and process all kinds of stimuli that might be relevant for the neural activity.

I will demonstrate how we can use topological methods to decodethe relevant features from neural recordings alone, without having to observe the external stimuli or the behavior of the animal at all. In particular, we have decoded the head direction of mice with high accuracy, without even knowing that the neurons were coding for head direction. This demonstrates that topologi-cal methods provide a useful tool in neural decoding that could possibly be used to discover new types of neurons.