skip to main content


The NSF Public Access Repository (NSF-PAR) system and access will be unavailable from 11:00 PM ET on Thursday, June 13 until 2:00 AM ET on Friday, June 14 due to maintenance. We apologize for the inconvenience.

Title: Simplicial cascades are orchestrated by the multidimensional geometry of neuronal complexes
Abstract Cascades over networks (e.g., neuronal avalanches, social contagions, and system failures) often involve higher-order dependencies, yet theory development has largely focused on pairwise-interaction models. Here, we develop a ‘simplicial threshold model’ (STM) for cascades over simplicial complexes that encode dyadic, triadic and higher-order interactions. Focusing on small-world models containing both short- and long-range k -simplices, we explore spatio-temporal patterns that manifest as a frustration between local and nonlocal propagations. We show that higher-order interactions and nonlinear thresholding coordinate to robustly guide cascades along a k -dimensional generalization of paths that we call ‘geometrical channels’. We also find this coordination to enhance the diversity and efficiency of cascades over a simplicial-complex model for a neuronal network, or ‘neuronal complex’. We support these findings with bifurcation theory and data-driven approaches based on latent geometry. Our findings provide fruitful directions for uncovering the multiscale, multidimensional mechanisms that orchestrate the spatio-temporal patterns of nonlinear cascades.  more » « less
Award ID(s):
Author(s) / Creator(s):
Date Published:
Journal Name:
Communications Physics
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    To understand the governing mechanisms of bio-inspired swimming has always been challenging due to intense interactions between flexible bodies of natural aquatic species and water around them. Advanced modal decomposition techniques provide us with tools to develop more in-depth understating about these complex dynamical systems. In this paper, we employ proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) techniques to extract energetically strongest spatio-temporal orthonormal components of complex kinematics of a Crevalle jack (Caranx hippos) fish. Then, we present a computational framework for handling fluid–structure interaction related problems in order to investigate their contributions towards the overall dynamics of highly nonlinear systems. We find that the undulating motion of this fish can be described by only two standing-wave like spatially orthonormal modes. Constructing the data set from our numerical simulations for flows over the membranous caudal fin of the jack fish, our modal analyses reveal that only the first few modes receive energy from both the fluid and structure, but the contribution of the structure in the remaining modes is minimal. For the viscous and transitional flow conditions considered here, both spatially and temporally orthonormal modes show strikingly similar coherent flow structures. Our investigations are expected to assist in developing data-driven reduced-order mathematical models to examine the dynamics of bio-inspired swimming robots and develop new and effective control strategies to bring their performance closer to real fish species.

    more » « less
  2. Abstract Systems of activator–inhibitor reaction–diffusion equations posed on an infinite line are studied using a variety of analytical and numerical methods. A canonical form is considered, which contains all known models with simple cubic autocatalytic nonlinearity and arbitrary constant and linear kinetics. Restricting attention to models that have a unique homogeneous equilibrium, this class includes the classical Schnakenberg and Brusselator models, as well as other systems proposed in the literature to model morphogenesis. Such models are known to feature Turing instability, when activator diffuses more slowly than inhibitor, leading to stable spatially periodic patterns. Conversely in the limit of small feed rates, semi-strong interaction asymptotic analysis shows existence of isolated spike-like patterns. This paper describes the broad bifurcation structures that connect these two regimes. A certain universal two-parameter state diagram is revealed in which the Turing bifurcation becomes sub-critical, leading to the onset of homoclinic snaking. This regime then morphs into the spike regime, with the outer-fold being predicted by the semi-strong asymptotics. A rescaling of parameters and field concentrations shows how this state diagram can be studied independently of the diffusion rates. Temporal dynamics is found to strongly depend on the diffusion ratio though. A Hopf bifurcation occurs along the branch of stable spikes, which is subcritical for small diffusion ratio, leading to collapse to the homogeneous state. As the diffusion ratio increases, this bifurcation typically becomes supercritical and interacts with the homoclinic snaking and also with a supercritical homogeneous Hopf bifurcation, leading to complex spatio-temporal dynamics. The details are worked out for a number of different models that fit the theory using a mixture of weakly nonlinear analysis, semi-strong asymptotics and different numerical continuation algorithms. 
    more » « less
  3. Simplicial neural networks (SNNs) have recently emerged as a new direction in graph learning which expands the idea of convolutional architectures from node space to simplicial complexes on graphs. Instead of predominantly assessing pairwise relations among nodes as in the current practice, simplicial complexes allow us to describe higher-order interactions and multi-node graph structures. By building upon connection between the convolution operation and the new block Hodge-Laplacian, we propose the first SNN for link prediction. Our new Block Simplicial Complex Neural Networks (BScNets) model generalizes existing graph convolutional network (GCN) frameworks by systematically incorporating salient interactions among multiple higher-order graph structures of different dimensions. We discuss theoretical foundations behind BScNets and illustrate its utility for link prediction on eight real-world and synthetic datasets. Our experiments indicate that BScNets outperforms the state-of-the-art models by a significant margin while maintaining low computation costs. Finally, we show utility of BScNets as a new promising alternative for tracking spread of infectious diseases such as COVID-19 and measuring the effectiveness of the healthcare risk mitigation strategies. 
    more » « less
  4. The integration of environmental DNA (eDNA) within management strategies for lotic organisms requires translating eDNA detection and quantification data into inferences of the locations and abundances of target species. Understanding how eDNA is distributed in space and time within the complex environments of rivers and streams is a major factor in achieving this translation. Here we study bidimensional eDNA signals in streams to predict the position and abundance of Atlantic salmon ( Salmo salar ) juveniles. We use data from sentinel cages with a range of abundances (3–63 juveniles) that were deployed in three coastal streams in New Brunswick, Canada. We evaluate the spatial patterns of eDNA dispersal and determine the effect of discharge on the dilution rate of eDNA. Our results show that eDNA exhibits predictable plume dynamics downstream from sources, with eDNA being initially concentrated and transported in the midstream, but eventually accumulating in stream margins with time and distance. From these findings we developed a fish detection and distribution prediction model based on the eDNA ratio in midstream versus bankside sites for a variety of fish distribution scenarios. Finally, we advise that sampling midstream at every 400 m is sufficient to detect a single fish at low velocity, but sampling efforts need to be increased at higher water velocity (every 100 m in the systems surveyed in this study). Studying salmon eDNA spatio-temporal patterns in lotic environments is essential to developing strong quantitative population assessment models that successfully leverage eDNA as a tool to protect salmon populations. 
    more » « less
  5. Recurrent neural networks (RNNs) are nonlinear dynamical models commonly used in the machine learning and dynamical systems literature to represent complex dynamical or sequential relationships between variables. Recently, as deep learning models have become more common, RNNs have been used to forecast increasingly complicated systems. Dynamical spatio-temporal processes represent a class of complex systems that can potentially benefit from these types of models. Although the RNN literature is expansive and highly developed, uncertainty quantification is often ignored. Even when considered, the uncertainty is generally quantified without the use of a rigorous framework, such as a fully Bayesian setting. Here we attempt to quantify uncertainty in a more formal framework while maintaining the forecast accuracy that makes these models appealing, by presenting a Bayesian RNN model for nonlinear spatio-temporal forecasting. Additionally, we make simple modifications to the basic RNN to help accommodate the unique nature of nonlinear spatio-temporal data. The proposed model is applied to a Lorenz simulation and two real-world nonlinear spatio-temporal forecasting applications. 
    more » « less