Speaker
Description
Brains routinely generate highly flexible and complex behaviors on a relatively stable structure and limited resources. A key mechanism underlying this ability is compositionality, which allows the brain to efficiently decompose complex tasks into simpler, reusable primitives. While network modularity has often been linked to compositionality in biological and artificial networks, a rigorous mathematical characterization of this relationship in nonlinear networks is still lacking.
We investigate how structural modularity enables functional compositionality in inhibition-dominated threshold-linear networks (TLNs). We introduce a novel class of modular network assembly called low-rank gluings, where component subnetworks with arbitrary internal connectivity are connected via specific low-rank couplings. Low-rank intercomponent connectivity has recently garnered interest in neuroscience as a model for low-dimensional communication subspaces between brain regions. For these networks, we show that their global fixed points are constrained to be combinations of the local fixed points of their constituent modules. For a more structured subclass, called rank-1 gluings, we provide a complete characterization that determines which combinations of local fixed points yield global ones.
By applying this framework to graph-based TLNs, we show that these gluing rules provide a mathematically tractable recipe to build architectures with combinatorially many attractors ranging from compositions of discrete fixed points to compositional limit cycles. This framework expands the modeler's toolbox and suggests candidate architectures by which biological networks might generate a range of complex behaviors from a limited set of building blocks.
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