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Juliana Londono Alvarez

Publications and source records attributed to Juliana Londono Alvarez.

4 recordsLinked to original sources
arXiv2026

Fixed point compositionality via low-rank gluing rules in inhibition-dominated threshold-linear networks

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. In this work, we formally investigate how structural modularity supports 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. We prove that the global fixed points of these networks 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. We apply these results to graph-based networks, extending fixed point decomposition rules from combinatorial threshold-linear networks (CTLNs) to the more flexible family of generalized CTLNs (gCTLNs), thereby proving that these structural rules are more robust than initially posited. Finally, we demonstrate that these gluing rules provide a mathematically tractable recipe for engineering compositional dynamics, enabling the construction of networks with a combinatorially large repertoire of predictable attractors that can be understood from simpler component motifs, ranging from compositions of fixed points to compositional limit cycles.

q-bio.NC↗
arXiv2024

Attractor-based models for sequences and pattern generation in neural circuits

Neural circuits in the brain perform a variety of essential functions, including input classification, pattern completion, and the generation of rhythms and oscillations that support processes such as breathing and locomotion. There is also substantial evidence that the brain encodes memories and processes information via sequences of neural activity. In this dissertation, we are focused on the general problem of how neural circuits encode rhythmic activity, as in central pattern generators (CPGs), as well as the encoding of sequences. Traditionally, rhythmic activity and CPGs have been modeled using coupled oscillators. Here we take a different approach, and present models for several different neural functions using threshold-linear networks. Our approach aims to unify attractor-based models (e.g., Hopfield networks) which encode static and dynamic patterns as attractors of the network. In the first half of this dissertation, we present several attractor-based models. These include: a network that can count the number of external inputs it receives; two models for locomotion, one encoding five different quadruped gaits and another encoding the orientation system of a swimming mollusk; and, finally, a model that connects the fixed point sequences with locomotion attractors to obtain a network that steps through a sequence of dynamic attractors. In the second half of the thesis, we present new theoretical results, some of which have already been published. There, we established conditions on network architectures to produce sequential attractors. Here we also include several new theorems relating the fixed points of composite networks to those of their component subnetworks, as well as a new architecture for layering networks which produces "fusion" attractors by minimizing interference between the attractors of individual layers.

q-bio.NC↗
arXiv2022

Sequential attractors in combinatorial threshold-linear networks

Sequences of neural activity arise in many brain areas, including cortex, hippocampus, and central pattern generator circuits that underlie rhythmic behaviors like locomotion. While network architectures supporting sequence generation vary considerably, a common feature is an abundance of inhibition. In this work, we focus on architectures that support sequential activity in recurrently connected networks with inhibition-dominated dynamics. Specifically, we study emergent sequences in a special family of threshold-linear networks, called combinatorial threshold-linear networks (CTLNs), whose connectivity matrices are defined from directed graphs. Such networks naturally give rise to an abundance of sequences whose dynamics are tightly connected to the underlying graph. We find that architectures based on generalizations of cycle graphs produce limit cycle attractors that can be activated to generate transient or persistent (repeating) sequences. Each architecture type gives rise to an infinite family of graphs that can be built from arbitrary component subgraphs. Moreover, we prove a number of graph rules for the corresponding CTLNs in each family. The graph rules allow us to strongly constrain, and in some cases fully determine, the fixed points of the network in terms of the fixed points of the component subnetworks. Finally, we also show how the structure of certain architectures gives insight into the sequential dynamics of the corresponding attractor.

q-bio.NC↗