Workshop 1: Dynamics in Networks with Special Properties

(January 25,2016 - January 29,2016 )

Organizers


Pete Ashwin
Department of Mathematics, University of Exeter
Gheorghe Craciun
Mathematics and Biomolecular Chemistry, University of Wisconsin-Madison
Tomas Gedeon
Department of Mathematical Sciences, Montana State University

Workshop 1 Discussion session

Network structures underlie models for the dynamical descriptions and bifurcations in a wide range of biological phenomena. These include models for subcellular genomic and signaling processes, neural models at single-cell or multiple-cell level, high-level cognitive models, and many forms of chemically reacting systems. Naturally arising networks often have properties that make them especially pliant mathematically:

  1. Networks with special overall architectures (having, for example, special symmetries or special node-coupling rules) enable one to pose and answer questions about network dynamics by techniques that exploit those architectures.
  2. Networks often have natural and precise restrictions on the behavior of nodes or edges (for example, mass action kinetics, piecewise linear dynamics, or Boolean dynamics) that can lead to fairly deep and general theorems about network behavior.
  3. Small networks, quotient networks, factor networks, or network motifs can be used as building blocks to understand the dynamics of larger networks, allowing one to think of network structures in a constructive and synthetic way.

This workshop will explore the current state of affairs and ways of unifying emerging mathematical techniques by focusing on a variety of special biologically motivated structures. A few examples of applications with such structures include:

a. Chemical reaction network theory: In recent years separate theories have been developed for the dynamics of chemical reaction networks and for general networks with symmetry. Both theories have yielded interesting and nontrivial results. What has not been explored, however, are networks of identical and interconnected chemical cells. Although the dynamics within the cells themselves might exhibit a high degree of stability, there remain questions about the dynamics of the full multi-cell assembly that can exhibit important behavior through symmetry-breaking pattern formation.

b. Neuroscience: Classic central pattern generators and recent Wilson networks describing generalized rivalry have symmetries that dictate preferred kinds of patterned oscillation.

c. Gene expression networks: New technologies are providing massive data concerning the connectivity and functional control of these networks. Theoretical models based on these data typically incorporate combinatorial logical control of gene expression in dynamical models.

d. Epidemiology: Heterosexual contact networks are best explored by (quasi-) bipartite graphs, usually with fairly strong restrictions on degree distributions within each partition. Exploring sexually transmitted infection dynamics relies strongly on both this architecture (especially as the global description is preserved over time while local descriptions shift), but also on the time ordering of edge presence.

Development and analysis of models like these provide a rich source of new mathematical problems involving classification of dynamic states, bifurcations, and reverse engineering of complex networks based on observed dynamics.

Accepted Speakers

Fernando Antoneli
Mathematics, Universidade Federal De São Paulo
Murad Banaji
mathematics, Middlesex University
Stephen Coombes
School of Mathematical Sciences, University of Nottingham
Patrick De Leenheer
Department of Mathematics, Oregon State University
Alicia Dickenstein
Mathematics, University of Buenos Aires/MSRI
Roderick Edwards
Mathematics and Statistics, University of Victoria
Bard Ermentrout
Department of Mathematics, University of Pittsburgh
Gilles Gnacadja
Applied Mathematics, Amgen
Marty Golubitsky
Mathematical Biosciences Institute, The Ohio State University
Manoj Gopalkrishnan
School of Technology and Computer Science, Tata Institute of Fundamental Research
Toni Guillamon
Mathematics, Polytechnical University of Catalu~na (Barcelona)
Matthew Johnston
Mathematics, San Jose State University
Heinz Koeppl
Bio-Inspired Communication Systems, Technische Universitat Darmstadt
Georgi Medvedev
Mathematics, Drexel University
Maya Mincheva
Mathematics, Northern Illinois University
Konstantin Mischaikow
Mathematics, Rutgers
Atsushi Mochizuki
Theoretical Biology Laboratory, RIKEN
Casian Pantea
Mathematics, West Virginia University
Srinivasan Parthasarathy
Computer Science and Engineering, The Ohio State University
Antonio Politi
Physics, University of Aberdeen
Stephen Proulx
EEMB, University of California, Santa Barbara
Jonathan Rubin
Mathematics, University of Pittsburgh
Santiago Schnell
Department of Molecular & Integrative Biology, University of Michigan Medical School
Andrey Shilnikov
Neuroscience Institute, Georgia State University
Anne Shiu
Mathematics, Texas A&M University
Stanislav Shvartsman
Chemical Engineering, Princeton University
Christophe Soule
C.N.R.S, Institut des Hautes Études Scientifiques
Krasimira Tsaneva-Atanasova
Mathematics, University of Exeter
Pauline van den Driessche
Mathematics and Statistics, University of Victoria
Yunjiao Wang
Department of Mathematics, Texas Southern University
Alexey Zaikin
Mathematics, University College
Monday, January 25, 2016
Time Session
07:45 AM

Shuttle to MBI

08:00 AM
08:30 AM

Breakfast

08:30 AM
08:45 AM

Welcome, overview, introductions: Marty Golubitsky

08:45 AM
09:00 AM

Introduction by Workshop Organizers

09:00 AM
09:30 AM
Antonio Politi - Nontrivial collective dynamics in a network of pulse-coupled oscillators
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.
09:30 AM
10:00 AM
Bard Ermentrout - Data mining the Kuramoto Equations on regular graphs
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.
10:00 AM
10:40 AM

Break

10:40 AM
11:10 AM
Peter Ashwin - Weak chimera states for small networks of oscillators
This talk will look at emergent "chimera" dynamics in coupled oscillator systems composed of identical and indistinguishable oscillators. We propose a checkable definition of a weak chimera state and give some basic results on systems that can/cannot have chimera states in their dynamics using this definition. These include chimera states for systems of at least four oscillators with two coupling strengths.
11:10 AM
11:40 AM
Alexey Zaikin - Noise and Intelligence in intracellular gene-regulatory networks
I will discuss results of theoretical modelling in very multi-disciplinary area between Systems Medicine, Synthetic Biology, Artificial Intelligence and Applied Mathematics. Multicellular systems, e.g. neural networks of a living brain, can learn and be intelligent. Some of the principles of this intelligence have been mathematically formulated in the study of Artificial Intelligence (AI), starting from the basic Rosenblatt€™s and associative Hebbian perceptrons and resulting in modern artificial neural networks with multilayer structure and recurrence. In some sense AI has mimicked the function of natural neural networks. However, relatively simple systems as cells are also able to perform tasks such as decision making and learning by utilizing their genetic regulatory frameworks. Intracellular genetic networks can be more intelligent than was first assumed due to their ability to learn. Such learning includes classification of several inputs or, the manifestations of this intelligence is the ability to learn associations of two stimuli within gene regulating circuitry: Hebbian type learning within the cellular life. However, gene expression is an intrinsically noisy process, hence, we investigate the effect of intrinsic and extrinsic noise on this kind of intracellular intelligence. During the talk I will also include brief introductions/tutorials about Synthetic Biology, modelling of genetic networks and noise-induced ordering.
11:40 AM
02:00 PM

Lunch Break

02:00 PM
02:30 PM
Gheorghe Craciun - A proof of the Global Attractor Conjecture
In a groundbreaking 1972 paper Fritz Horn and Roy Jackson showed that a complex balanced mass-action system must have a unique locally stable equilibrium within any compatibility class. In 1974 Horn conjectured that this equilibrium is a global attractor, i.e., all solutions in the same compatibility class must converge to this equilibrium. Later, this claim was called the Global Attractor Conjecture, and it was shown that it has remarkable implications for the dynamics of large classes of polynomial and power-law dynamical systems, even if they are not derived from mass-action kinetics. Several special cases of this conjecture have been proved during the last decade. We describe a proof of the conjecture in full generality. In particular, it will follow that all detailed balanced mass action systems and all deficiency zero weakly reversible networks have the global attractor property.


We also mention some mathematical implications for robust stability of general polynomial dynamical systems, as well as some implications for biochemical mechanisms that implement noise filtering and cellular homeostasis.

02:30 PM
03:00 PM

Break

03:00 PM
04:00 PM
COLLOQUIUM: Marty Golubitsky - Properties of Solutions of Coupled Systems

Networks of differential equations can be defined by directed graphs. The graphs (or network architecture) indicate who is talking to whom and when they are saying the same thing. We ask: Which properties of solutions of coupled equations follow from network architecture. Answers include "patterns of synchrony" for equilibria and "patterns of phase-shift synchrony" for time-periodic solutions. We show how these properties can be used to explain surprising results in binocular rivalry experiments and we discuss how homeostasis can be thought of as a network phenomenon.

04:00 PM
05:30 PM

Reception and poster session in MBI lounge

05:30 PM

Shuttle pick-up from MBI

Tuesday, January 26, 2016
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:30 AM
Stephen Proulx - Evolution of gene networks in fluctuating environments
Gene networks in living organisms are part of a dynamical system whose output make up the traits of organisms and determine reproductive fitness. One of the roles that such networks play is to respond to variability in the environment. Because organisms are the product of past evolution, we expect that evolution will generally increase organismal fitness, but this is subject to some constraints and historical effects. In this talk, I will discuss models of fluctuating environments where the output of the gene network determines fitness. I compare the outcome of optimal control models with evolved gene networks and discuss how the networks parameters evolve.
09:30 AM
10:00 AM
Matthew Johnston - Applications of Generalized Networks to Biochemical Reaction Systems
Spurred by the rise of systems biology in the last decade and a half, network-based approaches have gained prominence as an efficient and insightful way to analyze complex biochemical reaction systems, such as MAPK signaling cascades and gene regulatory networks. Surprisingly, network-based methods are often able to make dynamical and steady state predictions independent of the initial conditions, rate parameters, and even rate form.

In this talk, I will outline some recent applications of generalized network theory to biochemical reaction systems. In a generalized network, there are two networks with the same topological structure: one for the stoichiometry, and one for the kinetics. Examples of biochemical reaction systems with dynamically equivalent but better structured generalized networks will be presented.
10:00 AM
10:40 AM

Break

10:40 AM
11:10 AM
Pauline van den Driessche - Model for Cholera Dynamics on a Random Network
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 environment is modeled by an external node that connects to every individual. The dynamics of our model show excellent agreement with stochastic simulations. The basic reproduction number R0 is computed, and on a Poisson network shown to be the sum of the basic reproduction 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.
11:10 AM
11:40 AM
Stephen Coombes - Synchrony in networks of coupled non smooth dynamical systems: Extending the master stability function
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.
11:40 AM
02:00 PM

Lunch Break

02:00 PM
02:30 PM
Anne Shiu - Which reaction networks are multistationary?
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 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.
02:30 PM
03:00 PM
Manoj Gopalkrishnan - Doing statistical inference in a chemical soup
The goal is to design an €œintelligent 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.
03:00 PM
03:30 PM

Break

03:30 PM
04:00 PM
Patrick De Leenheer - A patched Ross-Macdonald malaria model with human and mosquito movement: implications for control
Abstract not submitted
04:00 PM
04:30 PM
Stanislav Shvartsman - Dynamics of Multisite Phosphorylation
Multisite phosphorylation cycles are ubiquitous in cell regulation and are studied at multiple levels of complexity, with the ultimate goal to establish a quantitative view of phosphorylation networks in vivo. Achieving this goal is essentially impossible without mathematical models. Several models of multisite phosphorylation have been already proposed in the literature and received considerable attention from both experimentalists and theorists. Most of these models do not discriminate between distinct partially phosphorylated states of the regulated proteins and focus on two limiting regimes, distributive and processive, which differ in the number of enzyme substrate encounters needed for complete phosphorylation or dephosphorylation. Here we use the minimal model of ERK regulation to explore the dynamics of multisite phosphorylation in a reaction network that includes all essential phosphorylation states and varying levels of reaction processivity. In addition to bistability, which has been extensively studied in models with distributive mechanisms, this network can also generate oscillations, in which the relative abundances of the four phosphorylation states change in an ordered way. Both bistability and oscillations are suppressed at high levels of reaction processivity. Our work provides a general approach for large scale analysis of dynamics in multisite phosphorylation systems.
04:30 PM
05:00 PM
Santiago Schnell - Network motifs provide signatures that characterize metabolism of cellular organelles
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.
05:00 PM

Shuttle pick-up from MBI

Wednesday, January 27, 2016
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:30 AM
Konstantin Mischaikow - Database for Dynamic Signatures of Gene Regulatory Networks: Theory
Experimental data on gene regulation is mostly qualitative, where the only information available about pairwise interactions is the presence of either up-or down- regulation. Quantitative data is often subject to large uncertainty and is mostly in terms of fold differences. Given these realities, it is very difficult to make reliable predictions using mathematical models. The current approach of choosing reasonable parameter values, a few initial conditions and then making predictions based on resulting solutions is severely subsampling both the parameter and phase space. This approach does not produce provable and reliable predictions.
We present a new approach that uses continuous time Boolean networks as a platform for qualitative studies of gene regulation. In this talk we focus on the theoretical justification for the approach that we are taking.
09:30 AM
10:00 AM
Thomas Gedeon - Database for Dynamic Signatures of Gene Regulatory Networks: Applications
Experimental data on gene regulation is mostly qualitative, where the only information available about pairwise interactions is the presence of either up-or down- regulation. Quantitative data is often subject to large uncertainty and is mostly in terms of fold differences. Given these realities, it is very difficult to make reliable predictions using mathematical models. The current approach of choosing reasonable parameter values, a few initial conditions and then making predictions based on resulting solutions is severely subsampling both the parameter and phase space. This approach does not produce provable and reliable predictions.
We present a new approach that uses continuous time Boolean networks as a platform for qualitative studies of gene regulation. In this talk we show how we plan to use this approach in applications ranging from cell cycle dynamics to malaria.
10:00 AM
10:40 AM

Break

10:40 AM
11:10 AM
Atsushi Mochizuki - Structural approach for sensitivity of chemical reaction networks
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.
11:10 AM
11:40 AM
Christophe Soule - Admissible circuits
The Thomas rule about gene networks asserts that, for such a network to have several steady states, it is necessary that its reaction graph contains a positive circuit.
Applied to (bio)chemical networks, the Thomas rule says very little because its condition is easily satisfied. Soliman found a stronger condition : the influence graph must contain an admissible positive circuit.
In a joint work with M.Kaufman, we found a new condition, stronger than the one of Soliman's.
11:40 AM
12:10 PM
Srinivasan Parthasarathy - Towards Trustworthy Network Analysis
Trust as noted by the classic dictionary definition represents €œthe belief that someone or something is reliable, effective or honest€?. In this talk I will examine some recent ideas that deal with the issue of €œtrust€? in network analysis. Specifically we will look at ideas drawn from ensemble learning, and the propagation and importance of trust in biological network analysis.
12:10 PM
02:00 PM

Lunch Break

02:00 PM
02:30 PM
Maya Mincheva - Multistationarity in a MAPK network model
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.
02:30 PM
03:00 PM
Krasimira Tsaneva-Atanasova - Effects of time-delay in a model of intra- and inter-personal motor coordination
Motor coordination is an important feature of intra- and inter-personal interactions, and several scenarios - from finger tapping to human-computer interfaces - have been investigated experimentally. In the 1980, Haken, Kelso and Bunz (HKB) formulated a two coupled nonlinear oscillator model, which has been shown to describe many observed ascpects of coordination tasks. However all previous studies have followed the line of analysis based on the slow-varying-amplitude and rotating-wave approximations. These approximations lead to a reduced system comprised of a single differential equation representing the evolution of the relative phase of the two coupled oscillators. Here we take a different approach and systematically investigate the behaviour in the full system. We perform detailed numerical bifurcation analyses and reveal that the HKB model supports previously unreported dynamical regimes as well as bi-stability between a variety of coordination patterns. Furthermore we also perform a bifurcation study of this model, where we consider a delay in the coupling. The delay is shown to have a significant effect on the observed dynamics. In particular, we find a much larger degree of bi-stablility between in-phase and anti-phase oscillations in the presence of a frequency detuning.
03:00 PM
03:30 PM

Break

03:30 PM
04:00 PM
Matthias Wolfrum - Emergence of collective behavior in coupled oscillator systems
Systems of coupled oscillators show a variety of collective dynamical regimes. We present a collection of such nonlinear phenomena, including non-universal transitions to synchrony in globally coupled oscillators, self-organized patterns of coherence and incoherence, called "chimera states" in spatially extended systems, and the emergence of macroscopic spatio-temporal chaos in such systems.
04:00 PM
04:30 PM
Jonathan Rubin - In search of network-level respiratory burst synchronization mechanisms
Quite a bit of work over many decades has gone into exploring respiratory rhythm-generation mechanisms. These studies have established an important role for the pre-Botzinger (pBC) complex in the mammalian brainstem and has investigated properties of single pBC neurons and their synaptic interactions. I will present work arising from efforts to understand how synchronous bursts of activity emerge across the network of pBC respiratory neurons. This work includes some computational approaches to systematically study how burst synchrony depends on network properties, some analytical approaches to estimate impacts of the prevalence of architectural motifs on the spread of activity in a network, and some rigorous analysis of graphicality and graph enumeration that are relevant to testing the motif-based ideas computationally.
04:30 PM
05:00 PM
Badal Joshi - Atoms of multistationarity in reaction networks
It is an open problem to identify reaction networks that admit multiple positive steady states. Criteria such as deficiency theory and Jacobian criterion help rule out the possibility of multiple steady states. But these tests are not sufficient to establish multistationarity. For fully open networks, we can establish multistationarity by relating the steady states of a reaction network with those of its component €œembedded networks€?. We refer to the multistationary fully open networks that are minimal with respect to the embedding relation as atoms of multistationarity. We identify some families of atoms of multistationarity and show that there exist arbitrarily large (in species, reactions) such atoms.
05:00 PM

Shuttle pick-up from MBI

Thursday, January 28, 2016
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:30 AM
Yunjiao Wang - Extending Levelt's Propositions to perceptual multistability involving interocular grouping
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.
09:30 AM
10:00 AM
Georgi Medvedev - The Kuramoto model on Cayley and random graphs
In this talk, I will discuss the continuum limit for coupled dynamical systems on large graphs and applications to stability of spatial patterns in the Kuramoto model of coupled phase oscillators.
10:00 AM
10:40 AM

Break

10:40 AM
11:10 AM
Casian Pantea - Some aspects of injectivity and multistationarity in networks of interacting elements
The capacity of reaction network system to exhibit two or more steady states has been the focus of considerable recent work. The question of multistationarity is closely related to that of injectivity of the corresponding vector field. In this talk we give an overview of some old and new results on injectivity and multistationarity in vector fields associated with interaction networks, under more or less general assumptions on the nature of the network and the kinetic laws. This is joint work with Murad Banaji.
11:10 AM
11:40 AM
Murad Banaji - Some combinatorial and algebraic problems arising from the study of chemical reaction networks
The combinatorial structure of a chemical reaction network (CRN) may determine various behaviours of the associated dynamical systems. From network structure we may gain information about multistationarity, oscillation, bifurcations, persistence, distances between trajectories, orderings of variables, etc. A large number of questions naturally take the form of decision problems (formal languages): can a given network have some particular behaviour? Pursuing results in this area we are led to problems in graph theory, linear and exterior algebra, analysis, and geometry, sometimes of interest beyond their immediate application and intersecting work in other domains. Even where there are elegant characterisations of networks with some property, questions about the complexity of the associated decision problems often remain. I will outline some results and open problems in this area.
11:40 AM
02:00 PM

Lunch Break

02:00 PM
02:30 PM
Roderick Edwards - A model framework for transcription-translation dynamics
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.
02:30 PM
03:00 PM
Alicia Dickenstein - Steady States of MESSI biological systems
We introduce a general framework for biological systems that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, which we call MESSI systems. Examples of MESSI systems are the sequential distributive or processive multisite phosphorylation networks, phosphorylation cascades, and the bacterial EnvZ/OmpR network. Assuming mass-action kinetics, we simplify the study of steady states and conservation laws of these systems by explicit elimination of intermediate complexes (inspired by [Feliu and Wiuf 2013, Thomson and Gunawardena 2009]). We also describe an important subclass of MESSI systems with toric steady states, for which we give combinatorial conditions to determine multistationarity and the occurrence of relevant boundary steady states. Joint work with Mercedes Pérez Millán.
03:00 PM
03:15 PM

Break

03:15 PM
04:15 PM

Discussion

04:30 PM

Shuttle pick-up from MBI

06:30 PM
07:00 PM

Cash Bar

07:00 PM
09:00 PM

Banquet @ Crowne Plaza Hotel

Friday, January 29, 2016
Time Session
08:00 AM

Shuttle to MBI

08:15 AM
09:00 AM

Breakfast

09:00 AM
09:30 AM
Andrey Shilnikov - Basic Bifurcation Theory for [small] Neural Networks - CPGs
09:30 AM
10:00 AM
Gilles Gnacadja - Species Composition and Reversibility in Chemical Reaction Network Theory
Chemical Reaction Network Theory (CRNT) is a field of research that uses mathematics to investigate the many questions surrounding chemical reactions. At the foundation of this research lies a specific definition of a reaction network, which both expresses the reality of chemistry and facilitates mathematical reasoning. The definition however is quite generous; it allows systems that have no chemical interpretation. This can be for good reasons. For instance, population models can be studied through CRNT. There are also drawbacks. We observe that some basic specificities of chemistry are often ignored, and consequently there could be missed opportunities for findings of immediate application in chemistry. With this in mind, we proposed a notion of species composition to augment the definition of a reaction network when suitable. We also proposed a notion of reversibility which is weaker than strict reversibility. We posit that it reflect non-strict reversibility better than weak reversibility as defined in CRNT. Our talk will focus on explaining and making relevant these notions.
10:00 AM
10:40 AM

Break

10:40 AM
11:10 AM
Heinz Koeppl - Stochastic Decoupling of Biomolecular Networks
11:10 AM
11:40 AM
Fernando Antoneli - From motifs of regulatory networks to coupled stochastic systems
12:00 PM

Shuttle pick-up from MBI (One to airport and one back to hotel)

Name Email Affiliation
Antoneli, Fernando fernando.antoneli@unifesp.br Mathematics, Universidade Federal De São Paulo
Ashwin, Pete p.ashwin@exeter.ac.uk Department of Mathematics, University of Exeter
Banaji, Murad murad.banaji@port.ac.uk mathematics, Middlesex University
Brunner, James jdbrunner@math.wisc.edu Mathematics, University of Wisconsin-Madison
Buckalew, Richard buckalew.2@mbi.osu.edu Mathematical Biosciences Institute, The Ohio State University
Cifuentes, Patricia cifuentesgarcia.1@osu.edu College of Public Health - department of Biomedical Informatics, The Ohio State University
Coombes, Stephen stephen.coombes@nottingham.ac.uk School of Mathematical Sciences, University of Nottingham
Craciun, Gheorghe craciun@math.wisc.edu Mathematics and Biomolecular Chemistry, University of Wisconsin-Madison
Cummins, Breschine breecummins@gmail.com Department of Mathematical Sciences, Montana State University
De Leenheer, Patrick deleenhp@math.oregonstate.edu Department of Mathematics, Oregon State University
Dickenstein, Alicia alidick@dm.uba.ar Mathematics, University of Buenos Aires/MSRI
Edwards, Roderick edwards@uvic.ca Mathematics and Statistics, University of Victoria
Ermentrout, Bard bard@pitt.edu Department of Mathematics, University of Pittsburgh
Gedeon, Tomas gedeon@math.montana.edu Department of Mathematical Sciences, Montana State University
Gnacadja, Gilles gnacadja@amgen.com Applied Mathematics, Amgen
Goldwyn, Joshua jhgoldwyn@gmail.com Mathematics, The Ohio State University
Golubitsky, Marty mg@mbi.osu.edu Mathematical Biosciences Institute, The Ohio State University
Gopalkrishnan, Manoj manojg@tifr.res.in School of Technology and Computer Science, Tata Institute of Fundamental Research
Guillamon, Antoni antoni.guillamon@upc.edu Mathematics, Polytechnical University of Catalu~na (Barcelona)
Harris, Pamela p83v@yahoo.com Mathematical Sciences, United States Military Academy
Johnston, Matthew matthew.johnston@sjsu.edu Mathematics, San Jose State University
Joshi, Badal bjoshi@csusm.edu Mathematics, California State University, San Marcos
Kim, Jae Kyoung jaekkim@kaist.ac.kr Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST)
Koeppl, Heinz heinz.koeppl@bcs.tu-darmstadt.de Bio-Inspired Communication Systems, Technische Universitat Darmstadt
Leite, Maria mcleite@mail.usf.edu Mathematics and Statistics, University of South Florida
Leung, KaYin k.y.leung@uu.nl Mathematical Institute, University of Stockholm
Mazza, Christian christian.mazza@unifr.ch Department of mathematics 23 chemin du Musée CH-1700 Fribourg, Department of Mathematics University of Fribourg
Medvedev, Georgi medvedev@drexel.edu Mathematics, Drexel University
Mincheva, Maya mincheva@math.niu.edu Mathematics, Northern Illinois University
Mischaikow, Konstantin mischaik@math.rutgers.edu Mathematics, Rutgers
Mochizuki, Atsushi mochi@riken.jp Theoretical Biology Laboratory, RIKEN
Oliveira, Juliane juliane.oliveira@fc.up.pt Mathematics, University of Porto
Pantea, Casian cpantea@math.wvu.edu Mathematics, West Virginia University
Parthasarathy, Srinivasan srini@cse.ohio-state.edu Computer Science and Engineering, The Ohio State University
Politi, Antonio a.politi@abdn.ac.uk Physics, University of Aberdeen
Poll, Daniel dbpoll@math.uh.edu Mathematics, University of Houston
Prieto Langarica, Alicia aprietolangarica@ysu.edu Mathematics and Statistics, Youngstown State University
Proulx, Stephen proulx@lifesci.ucsb.edu EEMB, University of California, Santa Barbara
Ramchandran, Sundaram sundarramchandran@hotmail.com
Rubin, Jonathan rubin@math.pitt.edu Mathematics, University of Pittsburgh
Schnell, Santiago schnells@umich.edu Department of Molecular & Integrative Biology, University of Michigan Medical School
Schult, Dan dschult@colgate.edu Mathematics, Colgate University
Shah, Rushina rushina@mit.edu Mechanical Engineering, Massachusetts Institute of Technology
Shilnikov, Andrey ashilnikov@gsu.edu Neuroscience Institute, Georgia State University
Shiu, Anne annejls@math.tamu.edu Mathematics, Texas A&M University
Shvartsman, Stanislav stas@princeton.edu Chemical Engineering, Princeton University
Sivakoff, David sivakoff.2@osu.edu Statistics and Mathematics, The Ohio State University
Soule, Christophe soule@ihes.fr C.N.R.S, Institut des Hautes Études Scientifiques
Tang, Evelyn evelynt@seas.upenn.edu Bioengineering, University of Pennsylvania
Taslim, Cenny taslim.2@osu.edu Statistics, The Ohio State University
Thompson, Daniel thompson@math.osu.edu Mathematics, The Ohio State University
Tsaneva-Atanasova , Krasimira K.Tsaneva-Atanasova@exeter.ac.uk Mathematics, University of Exeter
van den Driessche, Pauline pvdd@math.uvic.ca Mathematics and Statistics, University of Victoria
Wang, Yunjiao wangyx@tsu.edu Department of Mathematics, Texas Southern University
Wolfrum, Matthias wolfrum@wias-berlin.de Research Group Laser Dynamics, Weierstrass-Institut f""ur Angewandte Analysis und Stochastik (WIAS)
Young, Todd youngt@ohio.edu Mathematics, Ohio University
Zaikin, Alexey alexey.zaikin@ucl.ac.uk Mathematics, University College
Zavala, Eder e.zavala@exeter.ac.uk College of Engineering, Mathematics and Physical Sciences, University of Exeter
Zhu, Yunzhang zhu.219@osu.edu Statistics, The Ohio State University
From motifs of regulatory networks to coupled stochastic systems
Weak chimera states for small networks of oscillators
This talk will look at emergent "chimera" dynamics in coupled oscillator systems composed of identical and indistinguishable oscillators. We propose a checkable definition of a weak chimera state and give some basic results on systems that can/cannot have chimera states in their dynamics using this definition. These include chimera states for systems of at least four oscillators with two coupling strengths.
Some combinatorial and algebraic problems arising from the study of chemical reaction networks
The combinatorial structure of a chemical reaction network (CRN) may determine various behaviours of the associated dynamical systems. From network structure we may gain information about multistationarity, oscillation, bifurcations, persistence, distances between trajectories, orderings of variables, etc. A large number of questions naturally take the form of decision problems (formal languages): can a given network have some particular behaviour? Pursuing results in this area we are led to problems in graph theory, linear and exterior algebra, analysis, and geometry, sometimes of interest beyond their immediate application and intersecting work in other domains. Even where there are elegant characterisations of networks with some property, questions about the complexity of the associated decision problems often remain. I will outline some results and open problems in this area.
Synchrony in networks of coupled non smooth dynamical systems: Extending the master stability function
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.
A proof of the Global Attractor Conjecture
In a groundbreaking 1972 paper Fritz Horn and Roy Jackson showed that a complex balanced mass-action system must have a unique locally stable equilibrium within any compatibility class. In 1974 Horn conjectured that this equilibrium is a global attractor, i.e., all solutions in the same compatibility class must converge to this equilibrium. Later, this claim was called the Global Attractor Conjecture, and it was shown that it has remarkable implications for the dynamics of large classes of polynomial and power-law dynamical systems, even if they are not derived from mass-action kinetics. Several special cases of this conjecture have been proved during the last decade. We describe a proof of the conjecture in full generality. In particular, it will follow that all detailed balanced mass action systems and all deficiency zero weakly reversible networks have the global attractor property.


We also mention some mathematical implications for robust stability of general polynomial dynamical systems, as well as some implications for biochemical mechanisms that implement noise filtering and cellular homeostasis.

A patched Ross-Macdonald malaria model with human and mosquito movement: implications for control
Abstract not submitted
Steady States of MESSI biological systems
We introduce a general framework for biological systems that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, which we call MESSI systems. Examples of MESSI systems are the sequential distributive or processive multisite phosphorylation networks, phosphorylation cascades, and the bacterial EnvZ/OmpR network. Assuming mass-action kinetics, we simplify the study of steady states and conservation laws of these systems by explicit elimination of intermediate complexes (inspired by [Feliu and Wiuf 2013, Thomson and Gunawardena 2009]). We also describe an important subclass of MESSI systems with toric steady states, for which we give combinatorial conditions to determine multistationarity and the occurrence of relevant boundary steady states. Joint work with Mercedes Pérez Millán.
A model framework for transcription-translation dynamics
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.
Data mining the Kuramoto Equations on regular graphs
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.
Database for Dynamic Signatures of Gene Regulatory Networks: Applications
Experimental data on gene regulation is mostly qualitative, where the only information available about pairwise interactions is the presence of either up-or down- regulation. Quantitative data is often subject to large uncertainty and is mostly in terms of fold differences. Given these realities, it is very difficult to make reliable predictions using mathematical models. The current approach of choosing reasonable parameter values, a few initial conditions and then making predictions based on resulting solutions is severely subsampling both the parameter and phase space. This approach does not produce provable and reliable predictions.
We present a new approach that uses continuous time Boolean networks as a platform for qualitative studies of gene regulation. In this talk we show how we plan to use this approach in applications ranging from cell cycle dynamics to malaria.
Species Composition and Reversibility in Chemical Reaction Network Theory
Chemical Reaction Network Theory (CRNT) is a field of research that uses mathematics to investigate the many questions surrounding chemical reactions. At the foundation of this research lies a specific definition of a reaction network, which both expresses the reality of chemistry and facilitates mathematical reasoning. The definition however is quite generous; it allows systems that have no chemical interpretation. This can be for good reasons. For instance, population models can be studied through CRNT. There are also drawbacks. We observe that some basic specificities of chemistry are often ignored, and consequently there could be missed opportunities for findings of immediate application in chemistry. With this in mind, we proposed a notion of species composition to augment the definition of a reaction network when suitable. We also proposed a notion of reversibility which is weaker than strict reversibility. We posit that it reflect non-strict reversibility better than weak reversibility as defined in CRNT. Our talk will focus on explaining and making relevant these notions.
Properties of Solutions of Coupled Systems

Networks of differential equations can be defined by directed graphs. The graphs (or network architecture) indicate who is talking to whom and when they are saying the same thing. We ask: Which properties of solutions of coupled equations follow from network architecture. Answers include "patterns of synchrony" for equilibria and "patterns of phase-shift synchrony" for time-periodic solutions. We show how these properties can be used to explain surprising results in binocular rivalry experiments and we discuss how homeostasis can be thought of as a network phenomenon.

Doing statistical inference in a chemical soup
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.
Bifurcations analysis of oscillating hypercycles

TBD

Applications of Generalized Networks to Biochemical Reaction Systems
Spurred by the rise of systems biology in the last decade and a half, network-based approaches have gained prominence as an efficient and insightful way to analyze complex biochemical reaction systems, such as MAPK signaling cascades and gene regulatory networks. Surprisingly, network-based methods are often able to make dynamical and steady state predictions independent of the initial conditions, rate parameters, and even rate form.

In this talk, I will outline some recent applications of generalized network theory to biochemical reaction systems. In a generalized network, there are two networks with the same topological structure: one for the stoichiometry, and one for the kinetics. Examples of biochemical reaction systems with dynamically equivalent but better structured generalized networks will be presented.
Atoms of multistationarity in reaction networks
It is an open problem to identify reaction networks that admit multiple positive steady states. Criteria such as deficiency theory and Jacobian criterion help rule out the possibility of multiple steady states. But these tests are not sufficient to establish multistationarity. For fully open networks, we can establish multistationarity by relating the steady states of a reaction network with those of its component “embedded networks�. We refer to the multistationary fully open networks that are minimal with respect to the embedding relation as atoms of multistationarity. We identify some families of atoms of multistationarity and show that there exist arbitrarily large (in species, reactions) such atoms.
Network Designs for Robust Biological Oscillators

While a single negative feedback loop is enough to generate rhythms, additional feedback loops exist in many biological oscillators. Many theoretical and experimental studies have shown that including fast additional positive feedback loop, but not negative feedback loop can lead to robust rhythms over a wide range of conditions. However, the periods of rhythms generated with this design are sensitive to parameter change. Here, we present a novel network design for robust oscillators with robust periods by investigating circadian (~24hr) clocks whose periods are robust over a wide range of conditions. In the new design, protein sequestration is used to close the negative feedback loop in contrast to the Hill-type repression, which is widely used in previous studies. With the protein sequestration based repression, surprisingly, including slow additional negative feedback loop, not positive feedback loop leads to robust rhythms with robust periods. This work indicates that the property of network design can be considerably change depending on repression mechanisms.

Stochastic Decoupling of Biomolecular Networks
Dangerous connections: on binding site models for infectious disease dynamics

We formulate models for the spread of infection on networks that are amenable to analysis in the large population limit. We distinguish three different levels: (1) binding sites, (2) individuals, and (3) the population. In the tradition of Physiologically Structured Population Models, the formulation starts on the individual level. Influences from the ‘outside world’ on an individual are captured by environmental variables. These environmental variables are population level quantities. A key characteristic of the network models is that individuals can be decomposed into a number of conditionally independent components: each individual has a fixed number of ‘binding sites’ for partners. The Markov chain dynamics of binding sites are described by only a few equations. In particular, individual-level probabilities are obtained from binding-site-level probabilities by combinatorics while population-level quantities are obtained by averaging over individuals in the population. Thus we are able to characterize population-level epidemiological quantities, such as R0, r, the final size, and the endemic equilibrium, in terms of the corresponding variables.

The Kuramoto model on Cayley and random graphs
In this talk, I will discuss the continuum limit for coupled dynamical systems on large graphs and applications to stability of spatial patterns in the Kuramoto model of coupled phase oscillators.
Multistationarity in a MAPK network model
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.
Database for Dynamic Signatures of Gene Regulatory Networks: Theory
Experimental data on gene regulation is mostly qualitative, where the only information available about pairwise interactions is the presence of either up-or down- regulation. Quantitative data is often subject to large uncertainty and is mostly in terms of fold differences. Given these realities, it is very difficult to make reliable predictions using mathematical models. The current approach of choosing reasonable parameter values, a few initial conditions and then making predictions based on resulting solutions is severely subsampling both the parameter and phase space. This approach does not produce provable and reliable predictions.
We present a new approach that uses continuous time Boolean networks as a platform for qualitative studies of gene regulation. In this talk we focus on the theoretical justification for the approach that we are taking.
Structural approach for sensitivity of chemical reaction networks
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.
Dimensionality of Pattern Formation in Reaction Diffusion Systems

In the study of pattern formation in symmetric physical systems a 3-dimensional structure in thin domains is often modelled as 2-dimensional one. As a contrast, in this work we use the full 3-dimensionality of the problem to give a theoretical interpretation and possibly decide whether the pattern seen in Reaction Diffusion systems naturally occur in either 2- or 3- dimension. For this purpose, we are concerned with functions in 3-dimention that are invariant under the action of a crystallographic group and the symmetries of their projections into a function defined on a plane. In particular, we give a formalism to explain the formation of unexpected patterns on CIMA (chlorite-iodide-malonic acid) reaction and how results related to the study of forced symmetry breaking on the hexagonal lattice is applied to the theory of projection.

Some aspects of injectivity and multistationarity in networks of interacting elements
The capacity of reaction network system to exhibit two or more steady states has been the focus of considerable recent work. The question of multistationarity is closely related to that of injectivity of the corresponding vector field. In this talk we give an overview of some old and new results on injectivity and multistationarity in vector fields associated with interaction networks, under more or less general assumptions on the nature of the network and the kinetic laws. This is joint work with Murad Banaji.
Towards Trustworthy Network Analysis
Trust as noted by the classic dictionary definition represents “the belief that someone or something is reliable, effective or honest�. In this talk I will examine some recent ideas that deal with the issue of “trust� in network analysis. Specifically we will look at ideas drawn from ensemble learning, and the propagation and importance of trust in biological network analysis.
Nontrivial collective dynamics in a network of pulse-coupled oscillators
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.
Sensory Feedback in a Bump Attractor Model of Path Integration

Several experiments have demonstrated that a mammal's representation of space is sharpened in the presence of sensory cues. This suggests mammalian spatial navigation systems utilize several difference sensory information channels. This information is converted into a neural code that represents the animal's current position in space by engaging place cell, grid cell, and head direction cell networks. In particular, sensory landmark (allothetic) cues can be utilized in concert with an animal's knowledge of its own velocity (idiothetic) cues to generate a more accurate representation of position than path integration provides on its own. Starting with a continuous bump attractor model, we explore the impact of synaptic spatial asymmetry and heterogeneity, which disrupt the position code of the path integration process. We use asymptotic analysis to reduce the bump attractor model to a single scalar equation whose potential represents the impact of asymmetry and heterogeneity. Such imperfections cause errors to build up when the network performs path integration, but these errors can be corrected by an external control signal representing the effects of sensory cues. We demonstrate that there is an optimal strength and decay rate of the control signal when cues appear either periodically or randomly. A similar analysis is performed when errors in path integration arise from dynamic noise fluctuations. Again, there is an optimal strength and decay of discrete control that minimizes the path integration error.

Evolution of gene networks in fluctuating environments
Gene networks in living organisms are part of a dynamical system whose output make up the traits of organisms and determine reproductive fitness. One of the roles that such networks play is to respond to variability in the environment. Because organisms are the product of past evolution, we expect that evolution will generally increase organismal fitness, but this is subject to some constraints and historical effects. In this talk, I will discuss models of fluctuating environments where the output of the gene network determines fitness. I compare the outcome of optimal control models with evolved gene networks and discuss how the networks parameters evolve.
In search of network-level respiratory burst synchronization mechanisms
Quite a bit of work over many decades has gone into exploring respiratory rhythm-generation mechanisms. These studies have established an important role for the pre-Botzinger (pBC) complex in the mammalian brainstem and has investigated properties of single pBC neurons and their synaptic interactions. I will present work arising from efforts to understand how synchronous bursts of activity emerge across the network of pBC respiratory neurons. This work includes some computational approaches to systematically study how burst synchrony depends on network properties, some analytical approaches to estimate impacts of the prevalence of architectural motifs on the spread of activity in a network, and some rigorous analysis of graphicality and graph enumeration that are relevant to testing the motif-based ideas computationally.
Network motifs provide signatures that characterize metabolism of cellular organelles
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.
Basic Bifurcation Theory for [small] Neural Networks - CPGs
Which reaction networks are multistationary?
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 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.
Dynamics of Multisite Phosphorylation
Multisite phosphorylation cycles are ubiquitous in cell regulation and are studied at multiple levels of complexity, with the ultimate goal to establish a quantitative view of phosphorylation networks in vivo. Achieving this goal is essentially impossible without mathematical models. Several models of multisite phosphorylation have been already proposed in the literature and received considerable attention from both experimentalists and theorists. Most of these models do not discriminate between distinct partially phosphorylated states of the regulated proteins and focus on two limiting regimes, distributive and processive, which differ in the number of enzyme substrate encounters needed for complete phosphorylation or dephosphorylation. Here we use the minimal model of ERK regulation to explore the dynamics of multisite phosphorylation in a reaction network that includes all essential phosphorylation states and varying levels of reaction processivity. In addition to bistability, which has been extensively studied in models with distributive mechanisms, this network can also generate oscillations, in which the relative abundances of the four phosphorylation states change in an ordered way. Both bistability and oscillations are suppressed at high levels of reaction processivity. Our work provides a general approach for large scale analysis of dynamics in multisite phosphorylation systems.
Admissible circuits
The Thomas rule about gene networks asserts that, for such a network to have several steady states, it is necessary that its reaction graph contains a positive circuit.
Applied to (bio)chemical networks, the Thomas rule says very little because its condition is easily satisfied. Soliman found a stronger condition : the influence graph must contain an admissible positive circuit.
In a joint work with M.Kaufman, we found a new condition, stronger than the one of Soliman's.
Effects of time-delay in a model of intra- and inter-personal motor coordination
Motor coordination is an important feature of intra- and inter-personal interactions, and several scenarios - from finger tapping to human-computer interfaces - have been investigated experimentally. In the 1980, Haken, Kelso and Bunz (HKB) formulated a two coupled nonlinear oscillator model, which has been shown to describe many observed ascpects of coordination tasks. However all previous studies have followed the line of analysis based on the slow-varying-amplitude and rotating-wave approximations. These approximations lead to a reduced system comprised of a single differential equation representing the evolution of the relative phase of the two coupled oscillators. Here we take a different approach and systematically investigate the behaviour in the full system. We perform detailed numerical bifurcation analyses and reveal that the HKB model supports previously unreported dynamical regimes as well as bi-stability between a variety of coordination patterns. Furthermore we also perform a bifurcation study of this model, where we consider a delay in the coupling. The delay is shown to have a significant effect on the observed dynamics. In particular, we find a much larger degree of bi-stablility between in-phase and anti-phase oscillations in the presence of a frequency detuning.
Model for Cholera Dynamics on a Random Network
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 environment is modeled by an external node that connects to every individual. The dynamics of our model show excellent agreement with stochastic simulations. The basic reproduction number R0 is computed, and on a Poisson network shown to be the sum of the basic reproduction 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.
Extending Levelt's Propositions to perceptual multistability involving interocular grouping
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.
Emergence of collective behavior in coupled oscillator systems
Systems of coupled oscillators show a variety of collective dynamical regimes. We present a collection of such nonlinear phenomena, including non-universal transitions to synchrony in globally coupled oscillators, self-organized patterns of coherence and incoherence, called "chimera states" in spatially extended systems, and the emergence of macroscopic spatio-temporal chaos in such systems.
Noise and Intelligence in intracellular gene-regulatory networks
I will discuss results of theoretical modelling in very multi-disciplinary area between Systems Medicine, Synthetic Biology, Artificial Intelligence and Applied Mathematics. Multicellular systems, e.g. neural networks of a living brain, can learn and be intelligent. Some of the principles of this intelligence have been mathematically formulated in the study of Artificial Intelligence (AI), starting from the basic Rosenblatt’s and associative Hebbian perceptrons and resulting in modern artificial neural networks with multilayer structure and recurrence. In some sense AI has mimicked the function of natural neural networks. However, relatively simple systems as cells are also able to perform tasks such as decision making and learning by utilizing their genetic regulatory frameworks. Intracellular genetic networks can be more intelligent than was first assumed due to their ability to learn. Such learning includes classification of several inputs or, the manifestations of this intelligence is the ability to learn associations of two stimuli within gene regulating circuitry: Hebbian type learning within the cellular life. However, gene expression is an intrinsically noisy process, hence, we investigate the effect of intrinsic and extrinsic noise on this kind of intracellular intelligence. During the talk I will also include brief introductions/tutorials about Synthetic Biology, modelling of genetic networks and noise-induced ordering.
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Doing statistical inference in a chemical soup
Manoj Gopalkrishnan The goal is to design an €œintelligent 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 th

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Dynamics of Multisite Phosphorylation
Stanislav Shvartsman Multisite phosphorylation cycles are ubiquitous in cell regulation and are studied at multiple levels of complexity, with the ultimate goal to establish a quantitative view of phosphorylation networks in vivo. Achieving this goal is essentially impos

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Emergence of collective behavior in coupled oscillator systems
Matthias Wolfrum Systems of coupled oscillators show a variety of collective dynamical regimes. We present a collection of such nonlinear phenomena, including non-universal transitions to synchrony in globally coupled oscillators, self-organized patterns of coherence

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Database for Dynamic Signatures of Gene Regulatory Networks: Theory
Konstantin Mischaikow Experimental data on gene regulation is mostly qualitative, where the only information available about pairwise interactions is the presence of either up-or down- regulation. Quantitative data is often subject to large uncertainty and is mostly in te

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Database for Dynamic Signatures of Gene Regulatory Networks: Applications
Tomas Gedeon Experimental data on gene regulation is mostly qualitative, where the only information available about pairwise interactions is the presence of either up-or down- regulation. Quantitative data is often subject to large uncertainty and is mostly in te

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Nontrivial collective dynamics in a network of pulse-coupled oscillators
Antonio Politi 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 integ

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Structural approach for sensitivity of chemical reaction networks
Atsushi Mochizuki 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

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Data mining the Kuramoto Equations on regular graphs
Bard Ermentrout 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 attr

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Multistationarity in a MAPK network model
Maya Mincheva 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 va

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Weak chimera states for small networks of oscillators
Pete Ashwin This talk will look at emergent "chimera" dynamics in coupled oscillator systems composed of identical and indistinguishable oscillators. We propose a checkable definition of a weak chimera state and give some basic results on systems that

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In search of network-level respiratory burst synchronization mechanisms
Jonathan Rubin Quite a bit of work over many decades has gone into exploring respiratory rhythm-generation mechanisms. These studies have established an important role for the pre-Botzinger (pBC) complex in the mammalian brainstem and has investigated properties of

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Noise and Intelligence in intracellular gene-regulatory networks
Alexey Zaikin I will discuss results of theoretical modelling in very multi-disciplinary area between Systems Medicine, Synthetic Biology, Artificial Intelligence and Applied Mathematics. Multicellular systems, e.g. neural networks of a living brain, can learn and

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Atoms of multistationarity in reaction networks
Badal Joshi It is an open problem to identify reaction networks that admit multiple positive steady states. Criteria such as deficiency theory and Jacobian criterion help rule out the possibility of multiple steady states. But these tests are not sufficient to e

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Properties of Solutions of Coupled Systems
Marty Golubitsky

Networks of differential equations can be defined by directed graphs. The graphs (or network architecture) indicate who is talking to whom and when they are saying the same thing. We ask: Which properties of solutions of coupled equations follow f

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Extending Levelt's Propositions to perceptual multistability involving interocular grouping
Yunjiao Wang 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

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Evolution of gene networks in fluctuating environments
Stephen Proulx Gene networks in living organisms are part of a dynamical system whose output make up the traits of organisms and determine reproductive fitness. One of the roles that such networks play is to respond to variability in the environment. Because organi

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The Kuramoto model on Cayley and random graphs
Georgi Medvedev In this talk, I will discuss the continuum limit for coupled dynamical systems on large graphs and applications to stability of spatial patterns in the Kuramoto model of coupled phase oscillators.

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Applications of Generalized Networks to Biochemical Reaction Systems
Matthew Johnston Spurred by the rise of systems biology in the last decade and a half, network-based approaches have gained prominence as an efficient and insightful way to analyze complex biochemical reaction systems, such as MAPK signaling cascades and gene regulat

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Some aspects of injectivity and multistationarity in networks of interacting elements
Casian Pantea The capacity of reaction network system to exhibit two or more steady states has been the focus of considerable recent work. The question of multistationarity is closely related to that of injectivity of the corresponding vector field. In this talk w

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Model for Cholera Dynamics on a Random Network
Pauline van den Driessche 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 c

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Some combinatorial and algebraic problems arising from the study of chemical reaction networks
Murad Banaji The combinatorial structure of a chemical reaction network (CRN) may determine various behaviours of the associated dynamical systems. From network structure we may gain information about multistationarity, oscillation, bifurcations, persistence, dis

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Synchrony in networks of coupled non smooth dynamical systems: Extending the master stability function
Stephen Coombes 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

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A model framework for transcription-translation dynamics
Roderick Edwards 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. I

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Which reaction networks are multistationary?
Anne Shiu 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

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Steady States of MESSI biological systems
Alicia Dickenstein We introduce a general framework for biological systems that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, which we call MESSI systems. Examples of MESSI systems are the sequential distributive or processive multisite ph