An ensemble of mean-field coupled oscillators characterized by different frequencies can exhibit a highly complex collective dynamics. I discuss an example where the phase-response curve is derived by smoothing out the response of delayed leaky integrate-and-fire neurons. It turns out that even though the microscopic dynamics is linearly stable, the global (macroscopic) evolution is irregular (high-dimensional). This poses the question of how the two levels of description are actually connected to one another.

In this talk, I will describe the dynamics of a system of sinusoidally coupled phase oscillators on cubic graphs. The synchronous solution is always an attractor. However, as the graphs get larger (more nodes), it is possible to get other stable attractors. We study the basins, energy, and degree of stability of these non-synchronous attractors for all cubic graphs up to a certain number of nodes. We also use some techniques from computational algebraic geometry to show that for some graphs, the only attractor is synchrony.

Abstract not submitted.

A network epidemic model for cholera and other diseases that can be transmitted via the environment is developed by adapting the Miller model to include the environment. The person-to-person contacts are modeled by a random contact network, and the contagious environ- ment is modeled by an external node that connects to every individual. The dynamics of our model show excellent agreement with stochas- tic simulations. The basic reproduction number R0 is computed, and on a Poisson network shown to be the sum of the basic reproduc- tion numbers of the person-to-person and person-to-water-to-person transmission pathways, as in the homogeneous mixing limit. How- ever, on other networks, R0 depends nonlinearly on the transmission along the two pathways. Type reproduction numbers are computed and quantify measures to control cholera.

The master stability function is a powerful tool for determining synchrony in high dimensional networks of coupled limit cycle oscillators. In part this approach relies on the analysis of a low dimensional variational equation around a periodic orbit. For smooth dynamical systems this orbit is not generically available in closed form. However, many models in physics, engineering, and biology admit to piece-wise linear (pwl) caricatures which are also often nonsmooth, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably the master stability function cannot be immediately applied to networks of such elements if they are non-smooth. Here we show how to extend the master stability function to nonsmooth planar pwl systems, and in the process demonstrate that considerable insight into network dynamics can be obtained when choosing the dynamics of the nodes to be pwl. In illustration we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. We contrast this with node dynamics poised near a non-smooth Andronov-Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.

When taken with mass-action kinetics, which reaction networks admit multiple steady states? What structure must such a network possess? Mathematically, this question is: among certain parametrized families of polynomial systems, which families admit multiple positive roots (for some parameter values)? No complete answer is known, although various criteria now exist---some to answer the question in the affirmative and some in the negative. In this talk, we answer these questions for the smallest networks—those with only a few chemical species or reactions. Our results highlight the role played by the Newton polytope of a network (the convex hull of the reactant vectors). It has become apparent in recent years that analyzing this Newton polytope elucidates some aspects of the long-term dynamics and can be used to determine whether the network always admits {em at least one} steady state. What is new here is our use of the geometric objects to determine whether a network admits steady state. Finally, our work is motivated by recent results that connect the capacity for multistationarity of a given network to that of certain related networks which are typically smaller: we are therefore interested in classifying small multistationary networks, and our results form the first step in this direction.

The goal is to design an “intelligent chemical soup” that can do statistical inference. This may have niche technological applications in medicine and biological research, as well as provide fundamental insight into the workings of biochemical reaction pathways. As a first step towards our goal, we describe a scheme that exploits the remarkable mathematical similarity between log-linear models in statistics and chemical reaction networks. We present a simple scheme that encodes the information in a log-linear model as a chemical reaction network. Observed data is encoded as initial concentrations, and the equilibria of the corresponding mass-action system yield the maximum likelihood estimators. The simplicity of our scheme suggests that molecular environments, especially within cells, may be particularly well suited to performing statistical computations.

By the success of modern biology we have many examples of large networks which describe interactions between a large number of species of bio-molecules. On the other hand, we have a limited understanding for quantitative details of biological systems, like the regulatory functions, parameter values of reaction rates. To overcome this problem, we have developed structural theories for dynamics of network systems. By our theories, important aspects of the dynamical properties of the system can be derived from information on the network structure, only, without assuming other quantitative details. In this talk, I will introduce a new theory for chemical reaction networks.

In living cells a large number of reactions are connected by sharing substrates or product chemicals, forming complex network systems like metabolic network. One experimental approach to the dynamics of such systems examines their sensitivity: each enzyme mediating a reaction in the system is increased/decreased or knocked out separately, and the responses in the concentrations of chemicals or their fluxes are observed. However, due to the complexity of the systems, it has been unclear how the network structures influence/determine the responses of the systems. In this study, we present a mathematical method, named structural sensitivity analysis, to determine the sensitivity of reaction systems from information on the network alone. We investigate how the sensitivity responses of chemicals in a reaction network depend on the structure of the network, and on the position of the perturbed reaction in the network. We establish and prove a general law which connects the network topology and the sensitivity patterns of metabolite responses directly. Our theorem explains two prominent features of network in sensitivity: localization and hierarchy in response pattern. We apply our method to several hypothetical and real life chemical reaction networks, including the metabolic network of the E. coli TCA cycle. The theorem is useful, practically, when examining real biological networks based on sensitivity experiments.

The MAPK network is a principal component of many intracellular signaling modules. Multistability (the existence of multiple stable steady states) is considered an important property of such networks. Theoretical studies have established parameter values for multistability for many models of MAPK networks. Deciding if a given model has the capacity for multistationarity (the existence of multiple steady states) usually requires an extensive search of the parameter space. Two simple parameter inequalities will be presented. If the first inequality is satisfied, multistationarity, and hence the potential for multistability, is guaranteed. If the second inequality is satisfied, the uniqueness of a steady state, and hence the absence of multistability, is guaranteed. The method also allows for the direct computation of the total concentration values such that multistationarity occurs. Multistability in the ERK -- MEK -- MKP model will be presented. Some possible generalizations of this method will be discussed. This is a joint work with Carsten Conradi.

Multistable perception phenomena have been widely used for examining visual awareness and its underlying cortical mechanisms. Plausible models can explain binocular rivalry – the perceptual switching between two conflicting stimuli presented to each eye. Human subjects also report rivalry between percepts formed by grouping complementary patches from images presented to either eye. The dynamics of rivalry between such integrated percepts is not completely understood, and it is unclear whether models that explain binocular rivalry can be generalized. Classical models rely on mutual inhibition between distinct populations whose activity corresponds to each percept, with switches driven by adaptation or noise. Such models do not reflect the more complex patterns of neural activity necessary to describe interocular grouping. Moreover, the switching dynamics between more than two percepts is characterized by the sequence of perceptual states in addition to dominance times. Mechanistic models of multistable rivalry need to explain such dynamics.

We studied the effect of color saturation on the dynamics of four-state perceptual rivalry. We presented subjects with split-grating stimuli composed of a half green grating and half red orthogonal grating to each eye. Subjects reliably reported four percepts: the two stimuli presented to each eye, as well as two coherent images resulting form interocular grouping. We hypothesized that an increase in color saturation would provide a strong cue to group the coherent halves, and would increase the dominance of grouped percepts. Experiments confirmed that this was the case. Further analysis showed that the increase in the fraction of time grouped stimuli were perceived was partly due to a decrease in single-eye dominance durations and partly due to an increase in the number of visits to grouped percepts. We used a computational model to show that our experimental observations can be reproduced by combining three mechanisms: mutual inhibition, recurrent excitation, and adaptation.

A theory for qualitative models of gene regulatory networks has been developed over several decades, generally considering transcription factors to regulate directly the expression of other transcription factors, without any intermediate variables. In fact, gene expression always involves transcription, which produces mRNA molecules, followed by translation, which produces protein molecules, and which can then act as transcription factors for other genes. Here we explore a class of models that explicitly includes both transcription and translation, keeping track of both mRNA and protein concentrations. We mainly deal with transcription regulation functions that are steep sigmoids or step functions, as is often done in protein-only models, though translation is governed by a linear term. We extend many aspects of the protein-only theory to this new context, including properties of fixed points, mappings between switching points, qualitative analysis via a state-transition diagram, and a result on periodic orbits for negative feedback loops. We find that while singular behaviour in switching domains is largely avoided, non-uniqueness of solutions can still occur in the step-function limit.

Motifs are repeating patterns that determine the local properties of networks. In this work, we characterized all 3-node motifs using enzyme commission numbers of the International Union of Biochemistry and Molecular Biology to show that motif abundance is related to biochemical function. Further, we present a comparative analysis of motif distributions in the metabolic networks of 21 species across six kingdoms of life. We found the distribution of motif abundances to be similar between species, but unique across cellular organelles. We also show that motifs are able to capture inter-species differences in metabolic networks and that molecular differences between some biological species are reflected by the distribution of motif abundances in metabolic networks. Our metabolic network analysis can be used to gain insights into evolutionary origin of cellular organelles.