Topological conditions for propagation of spatially-distributed neural activity

An important task of the nervous system is to transmit information faithfully and reliably across brain regions, a process that involves the coordinated activity of a relatively large population of neurons. In topographically organized networks, where the entering and exiting axons of neurons terminate in confined areas, successful propagation depends on the spatial patterns of activity: the firing neurons in a presynaptic or source layer must be located sufficiently close to each other to ensure that cells in the postsynaptic or target layer receive the requisite number of convergent inputs to fire. Here, we use principles of topology to define the conditions for transmitting information across layers. We show that simplicial complexes formed by source neurons can be used to: 1) determine whether target neurons receive suprathreshold inputs; 2) identify neurons within the active population that contribute to firing; and 3) discriminate between single and multiple active clusters of neurons.

Higher-Order Networks

Higher-order networks describe the many-body interactions of a large variety of complex systems, ranging from the the brain to collaboration networks. Simplicial complexes are generalized network structures which allow us to capture the combinatorial properties, the topology and the geometry of higher-order networks. Having been used extensively in quantum gravity to describe discrete or discretized space-time, simplicial complexes have only recently started becoming the representation of choice for capturing the underlying network topology and geometry of complex systems. This Element provides an in-depth introduction to the very hot topic of network theory, covering a wide range of subjects ranging from emergent hyperbolic geometry and topological data analysis to higher-order dynamics. This Elements aims to demonstrate that simplicial complexes provide a very general mathematical framework to reveal how higher-order dynamics depends on simplicial network topology and geometry.

Graded Betti Numbers of Balanced Simplicial Complexes

We prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings S/I, where S is a polynomial ring and $I\subseteq S$ I ⊆ S is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.