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Abstract In this paper we present a rigorous analysis of a class of coupled dynamical systems in which two distinct types of components, one excitatory and the other inhibitory, interact with one another. These network models are finite in size but can be arbitrarily large. They are inspired by real biological networks, and possess features that are idealizations of those in biological systems. Individual components of the network are represented by simple, much studied dynamical systems. Complex dynamical patterns on the network level emerge as a result of the coupling among its constituent subsystems. Appealing to existing techniques in (nonuniform) hyperbolic theory, we study their Lyapunov exponents and entropy, and prove that large time network dynamics are governed by physical measures with the SRB property. 

Biologically detailed models of brain circuitry are challenging to build and simulate due to the large number of neurons, their complex interactions, and the many unknown physiological parameters. Simplified mathematical models are more tractable, but harder to evaluate when too far removed from neuroanatomy/physiology. We propose that a multiscale model, coarse-grained (CG) while preserving local biological details, offers the best balance between biological realism and computability. This paper presents such a model. Generally, CG models focus on the interaction between groups of neurons—here termed “pixels”—rather than individual cells. In our case, dynamics are alternately updated at intra- and interpixel scales, with one informing the other, until convergence to equilibrium is achieved on both scales. An innovation is how we exploit the underlying biology: Taking advantage of the similarity in local anatomical structures across large regions of the cortex, we model intrapixel dynamics as a single dynamical system driven by “external” inputs. These inputs vary with events external to the pixel, but their ranges can be estimateda priori. Precomputing and tabulating all potential local responses speed up the updating procedure significantly compared to direct multiscale simulation. We illustrate our methodology using a model of the primate visual cortex. Except for local neuron-to-neuron variability (necessarily lost in any CG approximation) our model reproduces various features of large-scale network models at a tiny fraction of the computational cost. These include neuronal responses as a consequence of their orientation selectivity, a primary function of visual neurons. 

This paper is about a class of stochastic reaction networks. Of interest are the dynamics of interconversion among a finite number of substances through reactions that consume some of the substances and produce others. The models we consider are continuous-time Markov jump processes, intended as idealizations of a broad class of biological networks. Reaction rates depend linearly on “enzymes,” which are among the substances produced, and a reaction can occur only in the presence of sufficient upstream material. We present rigorous results for this class of stochastic dynamical systems, the mean-field behaviors of which are described by ordinary differential equations (ODEs). Under the assumption of exponential network growth, we identify certain ODE solutions as being potentially traceable and give conditions on network trajectories which, when rescaled, can with high probability be approximated by these ODE solutions. This leads to a complete characterization of the ω -limit sets of such network solutions (as points or random tori). Dimension reduction is noted depending on the number of enzymes. The second half of this paper is focused on depletion dynamics, i.e., dynamics subsequent to the “phase transition” that occurs when one of the substances becomes unavailable. The picture can be complex, for the depleted substance can be produced intermittently through other network reactions. Treating the model as a slow–fast system, we offer a mean-field description, a first step to understanding what we believe is one of the most natural bifurcations for reaction networks. 

Key features of biological activity can often be captured by transitions between a finite number of semi-stable states that correspond to behaviors or decisions. We present here a broad class of dynamical systems that are ideal for modeling such activity. The models we propose are chaotic heteroclinic networks with nontrivial intersections of stable and unstable manifolds. Due to the sensitive dependence on initial conditions, transitions between states are seemingly random. Dwell times, exit distributions, and other transition statistics can be built into the model through geometric design and can be controlled by tunable parameters. To test our model’s ability to simulate realistic biological phenomena, we turned to one of the most studied organisms, C. elegans, well known for its limited behavioral states. We reconstructed experimental data from two laboratories, demonstrating the model’s ability to quantitatively reproduce dwell times and transition statistics under a variety of conditions. Stochastic switching between dominant states in complex dynamical systems has been extensively studied and is often modeled as Markov chains. As an alternative, we propose here a new paradigm, namely, chaotic heteroclinic networks generated by deterministic rules (without the necessity for noise). Chaotic heteroclinic networks can be used to model systems with arbitrary architecture and size without a commensurate increase in phase dimension. They are highly flexible and able to capture a wide range of transition characteristics that can be adjusted through control parameters. 

This paper contains a theoretical study of epidemic control. It is inspired by current events but not intended to be an accurate depiction of the SARS-CoV-2 pandemic. We consider the emergence of a highly transmissible pathogen, focusing on metropolitan areas. To ensure some degree of realism, we present a conceptual model of the outbreak and early attempts to stave off the onslaught, including the use of lockdowns. Model outputs show strong qualitative—in some respects even quantitative—resemblance to the events of Spring 2020 in many cities worldwide. We then use this model to project forward in time to examine different paths in epidemic control after the initial surge is tamed and before the arrival of vaccines. Three very different control strategies are analyzed, leading to vastly different outcomes in terms of economic recovery and total infected population (or progress toward herd immunity). Our model, which is a version of the SEIQR model, is a time-dependent dynamical system with feedback-control. One of the main conclusions of this analysis is that the course of the epidemic is not entirely dictated by the virus: how the population responds to it can play an equally important role in determining the eventual outcome. 

Exponentially growing systems are prevalent in nature, spanning all scales from biochemical reaction networks in single cells to food webs of ecosystems. How exponential growth emerges in nonlin- ear systems is mathematically unclear. Here, we describe a general theoretical framework that reveals underlying principles of long- term growth: scalability of flux functions and ergodicity of the rescaled systems. Our theory shows that nonlinear fluxes can gen- erate not only balanced growth but also oscillatory or chaotic growth modalities, explaining nonequilibrium dynamics observed in cell cycles and ecosystems. Our mathematical framework is broadly useful in predicting long-term growth rates from natural and synthetic networks, analyzing the effects of system noise and perturbations, validating empirical and phenomenological laws on growth rate, and studying autocatalysis and network evolution. 

Exponentially growing systems are prevalent in nature, spanning all scales from biochemical reaction networks in single cells to food webs of ecosystems. How exponential growth emerges in nonlin- ear systems is mathematically unclear. Here, we describe a general theoretical framework that reveals underlying principles of long- term growth: scalability of flux functions and ergodicity of the rescaled systems. Our theory shows that nonlinear fluxes can gen- erate not only balanced growth but also oscillatory or chaotic growth modalities, explaining nonequilibrium dynamics observed in cell cycles and ecosystems. Our mathematical framework is broadly useful in predicting long-term growth rates from natural and synthetic networks, analyzing the effects of system noise and perturbations, validating empirical and phenomenological laws on growth rate, and studying autocatalysis and network evolution.