Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Location: Unitobler, F-105
53 seats, 70m^2
Date: Tuesday, 09/Jul/2019
10:00am - 12:00pmMS148, part 1: Algebraic neural coding
Unitobler, F-105 
 
10:00am - 12:00pm

Algebraic Neural Coding

Chair(s): Nora Youngs (Colby College), Zvi Rosen (Florida Atlantic University, United States of America)

Neuroscience aims to decipher how the brain represents information via the firing of neurons. Place cells of the hippocampus have been demonstrated to fire in response to specific regions of Euclidean space. Since this discovery, a wealth of mathematical exploration has described connections between the algebraic and combinatorial features of the firing patterns and the shape of the space of stimuli triggering the response. These methods generalize to other types of neurons with similar response behavior. At the SIAM AG meeting, we hope to bring together a group of mathematicians doing innovative work in this exciting field. This will allow experts in commutative algebra, combinatorics, geometry and topology to connect and collaborate on problems related to neural codes, neural rings, and neural networks.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Flexible Motifs in Threshold-Linear Networks

Carina Curto
The Pennsylvania State University

Threshold-linear networks (TLNs) are popular models of recurrent networks used to model neural activity in the brain. The state space in these networks is naturally partitioned into regions defined by an associated hyperplane arrangement. The combinatorial properties of this arrangement, as captured by an oriented matroid, provide strong constraints on the network's dynamics. In recent work, we have studied how the graph of a TLN constrains the possible fixed points of the network by providing constraints on the combinatorics of the hyperplane arrangement. Here we study the case of flexible motifs, where the graph allows multiple possibilities for the set of fixed points FP(W), depending on the choice of connectivity matrix W. In particular, we find that mutations of oriented matroids correspond naturally to bifurcations in the dynamics. Flexible motifs are interesting from a neuroscience perspective because they allow us to study the effects of sensory and state-dependent modulation on the dynamics of neural ensembles.

 

Robust Motifs in Threshold-Linear Networks

Katherine Morrison
University of Northern Colorado

Networks of neurons in the brain often exhibit complex patterns of activity that are shaped by the intrinsic structure of the network. How does the precise connectivity structure of the network influence these patterns of activity? We address this question in the context of threshold-linear networks, a commonly used model of recurrent neural networks. We identify constraints on the dynamics that arise from network architecture and are independent of the specific values of connection strengths. By appealing to an associated hyperplane arrangement, we find families of robust motifs, which are graphs where the collection of fixed points of the corresponding networks is fully determined by the graph structure, irrespective of the particular connection strengths. These motifs provide a direct link between network structure and function, and provide new insights into how connectivity may shape dynamics in real neural circuits.

 

An Algebraic Perceptron and the Neural Ideals

Vladimir Itskov
The Pennsylvania State University

Feedforward neural networks have been widely used in machine-learning and theoretical neuroscience. The paradigm of "deep learning", that makes use of many consecutive layers of feedforward networks, has achieved impressive engineering success in the past two decades. However, a theoretical understanding of many-layer feedforward networks is still mostly lacking. While each layer of a feedforward network can be understood via the geometry of an hyperplane arrangement, satisfactory understanding the mathematical properties of many-layered networks remains elusive.
We propose a generalization of the perceptron, i.e. a single layer of a feedforward network. This perceptron is best described via a neural ideal, i.e. an ideal in the ring of functions on the Boolean lattice. It turns out that many machine-learning problems can be converted into purely algebraic problems about neural ideals. This opens up a new avenue of developing a commutative algebra-based toolbox for machine-learning. In my talk I will explain the connection between these two subjects and also give a concrete example of translating a machine-learning problem into commutative algebra.

 

Properties of Hyperplane Neural Codes

Alexander Kunin
The Pennsylvania State University

The firing patterns of neurons in sensory systems give rise to combinatorial codes, i.e. subsets of the boolean lattice. These firing patterns represent the abstract intersection patterns of subsets of a Euclidean space, and an open problem is identifying the combinatorial properties of neural codes which distinguish the geometric properties of the corresponding subsets. We introduce the polar complex, a simplicial complex associated to any combinatorial code, and relate its associated Stanley-Reisner ring to the ring of $mathbb{F}_2$-valued functions on the code to identify some distinguishing characteristics of codes arising from feed-forward neural networks. In particular, we show the associated ring is Cohen-Macaulay, and make connections to other questions in the study of boolean functions.

 
3:00pm - 5:00pmMS152: Stochastic chemical reaction networks
Unitobler, F-105 
 
3:00pm - 5:00pm

Stochastic chemical reaction networks

Chair(s): Michael Felix Adamer (University of Oxford, United Kingdom)

The focus of this minisymposium is on new algebraic and analytic methods for stochastic chemical reaction networks. In contrast to deterministic models, stochastic systems cannot be described by systems of ordinary differential equations and, hence, direct application of algebraic methods is often not possible. We are interested in when the deterministic and the stochastic behaviour of chemical reaction networks diverge and how to analyse this behaviour with a combination of algebra, stochastic analysis and chemical reaction network theory.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Piecewise linear Lyapunov functions for stochastic reaction networks

Daniele Cappelletti
ETHZ

Stochastic reaction networks are mathematical models heavily utilized to describe the time evolution of biological systems, when few active molecules are present. In this case, the system dynamics is stochastic, and the changes of the molecules counts are described by means of a continuous time Markov chain. Despite the large use of these models, simple questions concerning the existence of a stationary distribution are hard to answer to, except for few exceptions, and constitute an active area of research. Often, in order to prove the convergence of a model to a stationary distribution, a suitable Foster-Lyapunov function is sought. I will present a novel and fast convex programming technique to check whether conditions implying the existence of a piecewise linear Lyapunov function hold. Such technique utilizes the geometry of the network to divide the state space in different regions where the calculated Foster-Lyapunov function is linear.

 

Robust stochastic control of reaction networks

Tomislav Plesa
Imperial College

Synthetic biology is a rapidly growing interdisciplinary field of science and engineering that aims to design biochemical systems which behave in a desired manner. With the recent breakthroughs in nucleic-acid-based biochemistry, arbitrary reaction networks can be experimentally implemented using solely DNA molecules, with applications to areas such as medicine, industry and nanotechnology.

In this talk, I will focus on developing mathematical methods for designing reaction networks with predefined stochastic behaviors. In particular, the following fundamental problem in biochemical control theory will be considered: given any well-behaved mass-action kinetics input reaction network, whose structure and dynamics are at-most partially known, the goal is to introduce suitable additional biochemical species and reactions in a systematic manner, such that the resulting enlarged output network has a predefined stationary probability mass function (PMF). For experimental implementability in nucleic-acid-based synthetic biology, it is also required that the output stationary PMF is robust with respect to the initial conditions (ergodicity), and with respect to variations in the rate coefficients of the input network (robust perfect adaptation). I will present a solution to this problem, by embedding a faster controller network, called the stochastic morpher, into any given (slower) input network. Due to the introduced time-scale separation, it will be rigorously shown, using singular perturbation theory, that the controller overrides the firing of the input network, and morphs the input PMF into an output one with a desired form. The morphing will be performed at a lower-resolution level, by mapping the input PMF to the output one taking the form of a linear combination of Poisson PMFs, suitable for designing networks with predefined multi-modality (multi-stability). Higher-resolution morphing will also be presented, with arbitrary output PMFs. The results will be exemplified on relatively simple input networks, whose dynamics will be morphed to display noise-induced multi-modality and predefined PMFs.

 

One-dimensional stochastic reaction networks: Classification and dynamics

Chuang Xu
U Copenhagen

A crucial dynamical property to guarantee the existence of a stationary distribution is positive recurrence. However, it is not easy to provide checkable criteria for stochastic reaction networks, particularly with complex topological or graphical structures.

Motivated by this need, this talk contributes to stochastic dynamics of chemical reaction networks (CRNs) with one-dimensional stoichiometric subspace. I will first present a classification of the state space of the underlying continuous time Markov chain (CTMC) associated with the CRN by identifying all types of states: absorbing (neutral and trapping) as well as escaping states, and open repelling as well as closed attracting non-singleton communicating classes, on each stochastic stoichiometric compatibility class with all large initial states. I will also mention how to use this result to discuss the diversity of long-term dynamics of stochastic CRNs, and point out that limit distributions of CRNs with absolute concentration robustness (ACR) are not necessarily Poisson, which answers a question by Anderson in 2014.

Moreover, I will present checkable necessary and sufficient network conditions for various dynamical properties: Recurrence (positive and null), transience, (non)explosivity, (non)implosivity, as well as existence of moments of passage times of the associated CTMC of one-dimensional stochastic CRNs. As a byproduct, any one-dimensional weakly reversible CRN is positive recurrent, confirming the Positive Recurrence Conjecture proposed by Anderson and Kim in 2018 (in 1-d case). In addition, I will mention how to use these conditions to address a question by Anderson et al. in 2018 on non-explosivity.

Finally, I will emphasize results on one-species CRNs, regarding asymptotics of tails of stationary distributions as well as approximation of an arbitrary discrete distribution by either ergodic stationary distributions or quasi-stationary distributions of one-species mass-action CRNs, and present parameter regions for consistency and inconsistency of stochastic and deterministic one-species CRNs regarding various dynamical properties aforementioned.

 

The geometry and dynamics of spatial networks subject external noise

Michael Adamer
University of Oxford

In this talk I will introduce a way of modelling external noise in chemical reaction networks.
Unlike internal noise, which arises from the finite number of particles present in a chemical reaction system, external noise can have a multitude of origins and also affects systems modelled by ODEs. Due to this fact many stochastic systems subject to external noise retain the geometric features, i.e. the steady state geometry, of a mass-action system. Hence, we propose to treat the system as a mass-action system at steady state with the noise acting as forcing term creating model dynamics. In certain circumstances this approach is mathematically equivalent to the modelling of internal noise and in this talk I am going to explain the assumptions and simplifications behind this.

Further, I am going to show how steady state geometry influences the modelling of external noise systems with particular regard to the multistationarity structure of the system.
To illustrate the dynamical features of external noise system I choose some particular types of noise such as white Gaussian noise and Ornstein-Uhlenbeck noise and show how noise correlations can help to form or destroy spatio-temporal patterns.

 

Date: Wednesday, 10/Jul/2019
10:00am - 12:00pmMS148, part 2: Algebraic neural coding
Unitobler, F-105 
 
10:00am - 12:00pm

Algebraic Neural Coding

Chair(s): Nora Youngs (Colby College), Zvi Rosen (Florida Atlantic University, United States of America)

Neuroscience aims to decipher how the brain represents information via the firing of neurons. Place cells of the hippocampus have been demonstrated to fire in response to specific regions of Euclidean space. Since this discovery, a wealth of mathematical exploration has described connections between the algebraic and combinatorial features of the firing patterns and the shape of the space of stimuli triggering the response. These methods generalize to other types of neurons with similar response behavior. At the SIAM AG meeting, we hope to bring together a group of mathematicians doing innovative work in this exciting field. This will allow experts in commutative algebra, combinatorics, geometry and topology to connect and collaborate on problems related to neural codes, neural rings, and neural networks.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Sunflowers of Convex Sets and New Obstructions to Convexity

R. Amzi Jeffs
University of Washington

Any collection of convex open sets in R^d gives rise to an associated neural code. The question of which codes can be realized in this way has been an open problem for a number of years, and although recent literature has described rich combinatorial and geometric obstructions to convexity, a full classification (even conjectural) is far out of reach. I will describe some new obstructions based on sunflowers of convex open sets, and show how these obstructions differ fundamentally from those which have been investigated previously.

 

Convex Codes and Oriented Matroids

Caitlin Lienkaemper
The Pennsylvania State University

Convex neural codes describe the intersection patterns of collections of convex open sets. Representable oriented matroids describe the intersection patterns of collections of half spaces—that is, of convex sets with convex complements. It is thus natural to view convex codes as a generalization of oriented matroids. In this talk, we will make this relationship precise. First, using a new notion of neural code morphism, we show that a code has a realization with convex polytopes if and only if it is the image of a representable matroid under such a morphism. This allows us to translate the problem of whether a code has a convex polytope realization into a matroid completion problem. Next, we enumerate all neural codes which are images of small representable matroids, and use the relationship between convex codes and oriented matroids to define new signatures of convexity and non-convexity. This is joint work with Alex Kunin and Zvi Rosen.

 

Sufficient Conditions for 1- and 2- Inductively Pierced Codes

Nida Obatake
Texas A&M University

Neural codes are binary codes in {0, 1}^n ; here we focus on the ones which represent the firing patterns of a type of neurons called place cells. There is much interest in determining which neural codes can be realized by a collection of convex sets. However, drawing representations of these convex sets, particularly as the number of neurons in a code increases, can be very difficult. Nevertheless, for a class of codes that are said to be k-inductively pierced for k = 0, 1, 2 there is an algorithm for drawing Euler diagrams. Here we use the toric ideal of a code to show sufficient conditions for a code to be 1- or 2-inductively pierced, so that we may use the existing algorithm to draw realizations of such codes.

 

Progress Toward a Classification of Inductively Pierced Codes via Polyhedra

Robert Davis
Harvey Mudd College

A difficult problem in the field of combinatorial neural codes is to determine when a given code can be represented in the plane as intersections of convex sets and their complements. If the code is 2-inductively pierced, then there exists a polynomial-time algorithm which constructs such a representation in the plane and which uses closed discs as the convex sets. Recently, Gross, Obatake, and Youngs provided a way to classify 2-inductively pierced codes for up to three neurons by considering a special weight order on ideals of polynomials associated to the codes. In this talk, we present progress toward extending their result for an arbitrary number of neurons. We focus on the use of state polytopes of homogeneous toric ideals, which encode their distinct reduced Gröbner bases. It is the properties of these bases that we aim to connect to being 2-inductively pierced.

 
3:00pm - 5:00pmMS183, part 1: Polyhedral geometry methods for biochemical reaction networks
Unitobler, F-105 
 
3:00pm - 5:00pm

Polyhedral geometry methods for biochemical reaction networks

Chair(s): Elisenda Feliu (University of Copenhagen, Denmark), Stefan Müller (University of Vienna)

This minisymposium focuses on geometric objects arising in the study of parametrized polynomial ODEs given by biochemical reaction networks. In particular, we consider recent work that employs techniques from convex, polyhedral, and tropical geometry in order to extract properties of interest from the ODE system and to relate them to the choice of parameter values.

Specific problems covered in the minisimposium include the analysis of forward-invariant regions of the ODE system, the determination of parameter regions for multistationarity or oscillations, the performance of model reduction close to metastable regimes, and the characterization of unique existence of equilibria using oriented matroids.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Endotactic Networks and Toric Differential Inclusions

Gheorghe Craciun, Abhishek Deshpande
University of Wisconsin (Madison)

An important dynamical property of biological interaction network models is persistence, which intuitively means that “no species goes extinct”. It has been conjectured that weakly reversible networks are persistent. The property of persistence is related to yet another conjecture called the Global Attractor Conjecture. Recently, Craciun has proposed a proof of the Global Attractor Conjecture. An important step in this proof is the embedding of weakly reversible dynamical systems into toric differential inclusions. We show that dynamical systems generated by the larger class of endotactic networks can be embedded into toric differential inclusions.

 

Approximating Convex Hulls of Curves by Polytopes

Nidhi Kaihnsa
MPI Leipzig

We study the convex hulls of trajectories of polynomial dynamical systems. Such trajec- tories also include real algebraic curves. The boundaries of the resulting convex bodies are stratified into families of faces. We approximate these convex hulls by a family of polytopes. We present numerical algorithms to identify the patches of the convex hull by classifying the facets of the polytope. An implementation based on the software Bensolve Tools is given. This is based an a joint work with Daniel Ciripoi, Andreas Lóhne and Bernd Sturmfels.

 

Multistationarity conditions in a network motif describing ERK activation

Carsten Conradi
HTW Berlin

ERK is an important signaling molecule that is activated by phosphorylation at two binding sites. In theory phosphorylation is either distributive or processive. It has been shown however that ERK phosphorylation is neither purely distributive nor purely processive but rather a mixture of both. While purely distributive processes are known to be multistationary, processive are not.

We study a network incorporating both mechanisms. By varying certain rate constants the contribution of the distributive mechanism can be controlled. As rate constants are hard to determine experimentally this network gives rise to a parametrized family of polynomials. In this context multistationarity refers to the existence of rate constants such that the polynomials have at least two positive solutions. Multistationarity is considered an important feature of this network and we want to understand the contribution of the distributive mechanism to the occurrence of multistationarity.

The corresponding variety admits a monomial parameterization and the family belongs to the class of systems described with Feliu, Mincheva and Wiuf. Thus multistationarity can be decided by studying the sign of the determinant of the Jacobian evaluated at this parameterization. We establish multistationarity and study whether multistationarity persists as the contribution from the distributive mechanism goes to zero.

 

Oscillations in a mixed phosphorylation mechanism

Maya Mincheva
Northern Illinois University

We will discuss the existence of oscillations in a phosphorylation mechanism where the phosphorylation is processive and the dephosphorylation is distributive. We show that in the three-dimensional space of total amounts, the border between systems with a stable versus unstable steady state is a surface that consists of points of Hopf bifurcations. The emergence of oscillations via a Hopf bifurcation is enabled by the catalytic and association constants of the distributive part of the mechanism: if these rate constants satisfy two inequalities, then the system admits a Hopf bifurcation.

This is a joint work with C. Conradi and A. Shiu.

 

Date: Thursday, 11/Jul/2019
10:00am - 12:00pmMS174, part 1: Algebraic aspects of biochemical reaction networks
Unitobler, F-105 
 
10:00am - 12:00pm

Algebraic aspects of biochemical reaction networks

Chair(s): Alicia Dickenstein (Universidad de Buenos Aires), Georg Regensburger (Johannes Kepler University Linz)

ODE models for biochemical reaction networks usually give rise to dynamical systems defined by polynomial or rational functions. These systems are often high-dimensional, very sparse, and involve many parameters. This minisymposium deals with recent progress on applying and adapting techniques from (real) algebraic geometry and computational algebra for analyzing such systems. The minisymposium consists of three parts focusing on positive steady states, multistationarity and the corresponding parameter regions, and dynamical aspects.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Network models and polynomial positivity

Murad Banaji
Middlesex University, London

A number of problems in chemical reaction network theory - and in the study of other networks with model structure - can be restated as questions about the positivity of polynomials, or more generally the emptiness or otherwise of semialgebraic sets. In particular, positivity claims can be used to deduce the absence of particular bifurcations in networks with certain structural features. Sometimes positivity can be determined trivially, but it is not uncommon for polynomials to arise which are positive but in nontrivial ways, namely they do not belong to the smallest "natural" cone of polynomials positive on the set in question. Some problems and examples will be surveyed, and a few results using techniques from real algebraic geometry and exterior algebra will be outlined.

 

Some approaches to understand the parameter region of multistationarity

Elisenda Feliu
University of Copenhagen

In the context of chemical reaction networks, the dynamics of the concentrations in time are modelled by a system of parameter-dependent ordinary differential equations, which typically admit invariant linear subspaces, called stoichiometric compatibility classes. Multistationarity refers to the existence of two positive equilibrium points in some stoichiometric compatibility class. Numerous approaches exist to address the qualitative question of whether a network exhibits multistationarity for at least one choice of parameter values. However, tools and strategies to determine 'when' this is the case, that is, to determine for which parameter values the network is multistationary, have only recently emerged.

In this talk I will focus on conditions on the reaction rate constants that guarantee, or preclude, multistastionarity. I will discuss a result, joint with de Wolff, Kaihnsa, Sturmfels and Yürük, based on the use of Sums of Nonnegative Circuits (SONC). Our benchmark example is the n-site phosphorylation system.

 

On the bijectivity of families of exponential maps

Stefan Müller
University of Vienna

In the setting of generalized mass-action systems, uniqueness and existence of complex-balanced equilibria (in every compatibility class and for all rate constants) are equivalent to injectivity and surjectivity of a certain family of exponential maps. In previous work, we have shown that injectivity can be characterized in terms of sign vectors of the stoichiometric and kinetic-order subspaces, that is, of the coefficient and exponent subspaces given by the family of maps. The negation of the sign-vector condition is equivalent to the existence of multiple complex-balanced equilibria (in some compatibility class and for some rate constant). In this work, we characterize the existence of a unique complex-balanced equilibrium, that is, the bijectivity of the family of exponential maps. As it turns out, the conditions for bijectivity do not only involve sign vectors, but also the exponent subspace itself. Further, we provide sufficient conditions involving only sign vectors or the Newton polytope. In terms of generalized mass-action systems, we provide an extension of the classical deficiency zero theorem.

(Joint work with Josef Hofbauer and Georg Regensburger)

 

An algebraic approach to detecting bistability in chemical reaction networks

Angélica Torres
University of Copenhagen

In recent years, algebraic parameterizations of varieties have been used to study parameter regions where a Chemical Reaction Network has multistationarity. In this work we combine these algebraic parameterizations, the Hurwitz criterion for stability and structural reduction techniques for chemical reaction networks to additionally explore the existence of bistability. The procedure can be used to find parameter regions where bistability arises. In this talk I will present our approach, how to detect bistability in a special case and some examples from cell signaling where our procedure was successfully applied.

 
3:00pm - 5:00pmMS183, part 2: Polyhedral geometry methods for biochemical reaction networks
Unitobler, F-105 
 
3:00pm - 5:00pm

Polyhedral geometry methods for biochemical reaction networks

Chair(s): Elisenda Feliu (University of Copenhagen, Denmark), Stefan Müller (University of Vienna)

This minisymposium focuses on geometric objects arising in the study of parametrized polynomial ODEs given by biochemical reaction networks. In particular, we consider recent work that employs techniques from convex, polyhedral, and tropical geometry in order to extract properties of interest from the ODE system and to relate them to the choice of parameter values.

Specific problems covered in the minisimposium include the analysis of forward-invariant regions of the ODE system, the determination of parameter regions for multistationarity or oscillations, the performance of model reduction close to metastable regimes, and the characterization of unique existence of equilibria using oriented matroids.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Algorithmic Aspects of Computing Tropical Prevarieties Parametrically

Andreas Weber
University of Bonn

Tropical prevarieties of polynomial vector fields arising from chemical reaction networks have been found useful in the analysis of such systems. For fixed parameter values it has been shown that even for systems having 30 dimensions or more tropical prevarieties can be computed. As for biochemical reaction networks there is typically high parameter uncertainty (and the qualitative dependency of the system on parameters is also of high interest) parametric computations of tropical prevarieties are very desireable. Based on experiments with the PtCut software we compare the outcome (also in terms of needed computational ressources) of simple grid sampling strategies, for which the curse of dimensionality fully applies, against computations in a polyhedral setting including parameters. The presented results are joint work with Christoph Lüders

 

Empiric investigations on the number and structure of solution polytopes for tropical equilibration problems arising from biological networks

Christoph Lüders
University of Bonn

A quite recent approach to assist in solving ODE systems (with polynomial vector fields) lies in methods of tropical geometry. Tropical geometry transforms polynomials into piece-wise linear functions and still preserves some structure of the original polynomial (like the number of roots). The polynomial is transformed into a set of polyhedra and multiple of such sets can be intersected to find common roots. Thus tropical geometry problems are combinatorial problems.

We have developed the "PtCut" program to compute the tropical prevariety resp. tropical equilibrium of a polynomial system. Details of use and implementation of PtCut are presented. Large models can cause a lot of polyhedra to be created and calculating their intersection can be very slow. Several methods of remedy are shown.

Since ODE systems often arise in biology, we wrote a free SBML-parser and used PtCut to compute tropical solutions of the curated models listed in the BioModels database. Statistics about the number of solutions, their dimension, their number of connected components and run-time are presented.

 

Perturbations of exponents of exponential maps: robustness of bijectivity

Georg Regensburger
Johannes Kepler University Linz

For generalized mass-action systems, uniqueness and existence of complex-balanced equilibria (in every compatibility class and for all rate constants) are equivalent to injectivity and surjectivity of a certain family of exponential maps. In this talk, we discuss when the existence of a unique solution is robust with respect to small perturbations of the exponents. In particular, we give a characterization in terms of sign vectors of the stoichiometric and kinetic-order subspaces or, alternatively, in terms of maximal minors of the coefficient and exponent matrices. This characterization allows to formulate a robust deficiency zero theorem for generalized mass-action systems. As a corollary, we show that the classical deficiency theorem for mass-action kinetics by Horn, Jackson, and Feinberg is robust with respect to small perturbations of the kinetic orders (from the stoichiometric coefficients).


(Joint work with Josef Hofbauer and Stefan Müller)

 

Weakly reversible mass-action systems with infinitely many positive steady states

Balázs Boros
University of Vienna

In 2011, Deng, Feinberg, Jones, and Nachman investigated the number of positive steady states of weakly reversible mass-action systems. They proposed a proof of the existence of a positive steady state in each positive stoichiometric class. Furthermore, they claimed that there can only be finitely many positive steady states in each positive stoichiometric class. Recently, I provided a complete and clearer proof of the existence part. Moreover, together with Craciun and Yu, I constructed examples with infinitely many positive steady states within a positive stoichiometric class, thereby disproving the finiteness part. In this talk, I will sketch our method that produces such mass-action systems.

 

Date: Friday, 12/Jul/2019
10:00am - 12:00pmMS174, part 2: Algebraic aspects of biochemical reaction networks
Unitobler, F-105 
 
10:00am - 12:00pm

Algebraic aspects of biochemical reaction networks

Chair(s): Alicia Dickenstein (Universidad de Buenos Aires), Georg Regensburger (Johannes Kepler University Linz)

ODE models for biochemical reaction networks usually give rise to dynamical systems defined by polynomial or rational functions. These systems are often high-dimensional, very sparse, and involve many parameters. This minisymposium deals with recent progress on applying and adapting techniques from (real) algebraic geometry and computational algebra for analyzing such systems. The minisymposium consists of three parts focusing on positive steady states, multistationarity and the corresponding parameter regions, and dynamical aspects.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Expected number of positive real solutions to systems of polynomial equations arising from reaction networks

AmirHosein Sadeghimanesh
University of Copenhagen

Given a chemical reaction network that exhibits multistationarity, a natural question is how to determine the region in the parameter space where the network is multistationary or to divide the parameter region according to the number of positive steady states. We introduce a new approach based on the Kac-Rice formula and Monte-Carlo integration to approximate the multistationarity region in the parameter space. Furthermore, we apply our method to solve two related questions. First, we provide a measure to compare two points in the multistationarity region and decide what choice is more robust to small variations of the parameters. Second, we address the problem of finding a point in a given box in the parameter space for which the network is multistationary. We apply our approach to two relevant reaction networks, namely a simple model of a hybrid histidine-kinase and the 2-site sequential distributive phosphorylation network. Finally we compare our approach with the former existing methods of studying the multistationarity region. This is a joint work with Elisenda Feliu.

 

Absolute concentration robustness: an algebraic perspective

Anne Shiu
Texas A&M University

How do cells maintain homeostasis in fluctuating environments? Investigations into this question led Shinar and Feinberg to introduce in 2010 the concept of absolute concentration robustness (ACR). A biochemical system exhibits ACR in some species if the steady-state value of that species does not depend on initial conditions. Thus, a system with ACR can maintain a constant level of one species even as the environment changes. Despite a great deal of interest in ACR in recent years, the following basic question remains open: How can we determine quickly whether a given biochemical system has ACR? Although various approaches to this problem have been proposed, we show in this talk that they are incomplete. Accordingly, we present a new method for deciding ACR, which uses computational algebra. We illustrate our results on several biochemical signaling networks.

 

On the Stability of the Steady States in the n-site Futile Cycle

Carsten Wiuf
University of Copenhagen

The multiple or n-site futile cycle is a biological process that resides in the cell. Specifically, it is a phosphorylation system in which a molecular substrate might be phosphorylated sequentially n times by means of an enzymatic mechanism. The system has been studied mathematically using reaction network theory and ordinary differential equations. In its standard form it has 3n+3 variables (concentrations of species) and 6n parameters. It is known that the system might have at least as many as 2[n/2]+1 steady states (where [x] is the integer part of x) for particular choices of parameters. Furthermore, for the simple futile cycle (n=1) there is only one steady state which is globally stable. For the dual futile cycle (n=2) the stability of the steady states has been determined in the following sense: There exist parameter values for which the dual futile cycle admits two asymptotically stable and one unstable steady state. For general n, evidence that the possible number of asymptotically stable steady states increases with n has been given, which has led to the conjecture that parameter values can be chosen such that [n/2]+1 out of 2[n/2]+1 steady states are asymptotically stable and the remaining steady states are unstable.

We prove this conjecture here by first reducing the system to a smaller one, for which we find a choice of parameter values that give rise to a unique steady state with multiplicity 2[n/2]+1. Using arguments from geometric singular perturbation theory, and a detailed analysis of the centre manifold of this steady state, we achieve the desired result.

The work is joint with Alan Rendall (Mainz) and Elisenda Feliu (Copenhagen).

 

The DSR graph and dynamical properties of reaction networks

Casian Pantea
West Virginia University

A significant body of recent work in mathematical biology focuses on dynamical properties of biochemical reaction networks that are a function of the network topology alone (and are therefore independent of kinetics or parameter values). One way to represent network topology is via the DSR graph, introduced by Craciun and Banaji. The structure of the DSR graph may allow conclusions on dynamical properties of the network, including multistationarity, stability of steady states, and stability under delays. In this talk we survey some old and new results on the DSR graph, and discuss connections with other graphs stemming from reaction networks.

 
3:00pm - 5:00pmRoom free
Unitobler, F-105 

Date: Saturday, 13/Jul/2019
10:00am - 12:00pmMS174, part 3: Algebraic aspects of biochemical reaction networks
Unitobler, F-105 
 
10:00am - 12:00pm

Algebraic aspects of biochemical reaction networks

Chair(s): Alicia Dickenstein (Universidad de Buenos Aires), Georg Regensburger (Johannes Kepler University Linz)

ODE models for biochemical reaction networks usually give rise to dynamical systems defined by polynomial or rational functions. These systems are often high-dimensional, very sparse, and involve many parameters. This minisymposium deals with recent progress on applying and adapting techniques from (real) algebraic geometry and computational algebra for analyzing such systems. The minisymposium consists of three parts focusing on positive steady states, multistationarity and the corresponding parameter regions, and dynamical aspects.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Reduction of the number of parameters

János Tóth
Budapest University of Technology and Economics

We start from a kinetic differential equation and transform it using an extended positive diagonal transformation in such a way that we prescribe the values of some of the reaction rate coefficients in the transformed system. The problems to be studied are as follows.

1. Is it possible to prescribe the value of a given set of coefficients? If yes, does the transformation obeying the prescriptions uniquely determine the values of the other coefficients? In both cases: what are the new coefficients of the transformed equation?

2. Which are all the possible subsets of new coefficients that can be independently prescribed? What is the largest number of coefficient set(s) that can be prescribed?

3. Suppose we prescribe the values of some coefficients: they should be one. In this case we usually say we only have the remaining coefficient combinations as ''independent'' coefficients. Why?

We present some statements and examples as answers to the above questions and also we show a program to carry out the corresponding transformations. Furthermore, we mention some connections to the general problem of transforming kinetic differential equations.

 

"Good children" and "bad children"

Nicola Vassena
Free University Berlin

Equilibrium bifurcations arise from sign changes of Jacobian determinants, as parameters are varied. Therefore we address here the Jacobian determinant for metabolic networks with general reaction kinetics. Our approach is based on the concept of child selections: each (mother) metabolite is mapped, injectively, to one of those (child) reactions which it drives as an input.

Our analysis distinguishes reaction network Jacobians with constant sign from the bifurcation case, where that sign depends on specific reaction rates. In particular, we distinguish "good child" selections, which do not affect the sign, from more interesting / demanding / troublesome / mischievous "bad children", which gang up towards sign changes, instability, and bifurcations.

 

Tikhonov-Fenichel parameter values for chemical reaction networks

Sebastian Walcher
RWTH Aachen

The chemical reaction networks under consideration here are described by polynomial ordinary differential equations depending on positive parameters, with the positive orthant as a positively invariant set. In order to determine parameter regions where singular perturbation reduction (in the sense of Tikhonov and Fenichel) is possible, the notion of Tikhonov-Fenichel parameter values (TFPV) was introduced some time ago in Alexandra Goeke's dissertation and subsequent publications. A TFPV is characterized by the property that small perturbations give rise to a singular perturbation reduction. It is known that the TFPV of a given system form a semi-algebraic subset, and the defining equations may be determined algorithmically by standard elimination theory.

After reviewing the above notions and results, we discuss three types of questions arising for TFPV:

(i) Singular perturbation reduction versus "classical" quasi-steady state reduction.

(ii) The unreasonable simplicity of TFPV for CRN, and the unreasonable feasibility of their computation.

(iii) Nested TFPV for multiscale systems.

The talk reports on recent joint work with Elisenda Feliu, Niclas Kruff, Christian Lax and Carsten Wiuf.

 

Parameter geography

Jeremy Gunawardena
Department of Systems Biology, Harvard Medical School

I will discuss numerical observations of the parameteric region in which a two-site, post-translational modification system exhibits bi-stationarity. Aside from general features of connectedness and near convexity, we find a substantial difference in region volume between Michaelis-Menten and realistic enzymatic assumptions, which suggests that bi-stationarity may be rare under biological conditions. We uncover a previously unsuspected parameteric relationship underlying this phenomenon. We also find that boundary parameter points move back and forth between mono-stationarity and bi-stationarity (“blinking”), as conserved quantities increase, despite apparent monotonic increase in region volume. These results rely on combining the linear framework, which allows algebraic reduction of the steady state, with numerical algebraic geometry using Bertini, which permits sampling of approximately 10^9 points in parameter space.

 
3:00pm - 5:00pmMS199, part 2: Applications of topology in neuroscience
Unitobler, F-105 
 
3:00pm - 5:00pm

Applications of topology in neuroscience

Chair(s): Kathryn Hess Bellwald (Laboratory for topology and neuroscience, EPFL, Switzerland), Ran Levi (University of Aberdeen, UK)

Research at the interface of topology and neuroscience is growing rapidly and has produced many remarkable results in the past five years. In this minisymposium, speakers will present a wide and exciting array of current applications of topology in neuroscience, including classification and synthesis of neuron morphologies, analysis of synaptic plasticity, and diagnosis of traumatic brain injuries.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Simplicial convolutional neural networks for in-painting of cochains

Gard Spreemann
Laboratory for topology and neuroscience, EPFL, Switzerland

We use the simplicial Laplacian to define convolutional neural networks over simplicial complexes in a way that naturally generalizes classical CNNs. This provides us with the tools to build networks that perform in-painting of simplicial cochains while respecting the underlying topological structure. This is joint work with Stefania Ebli and Michaël Defferrard.

 

Using topological data analysis to classify certain stimuli in the Blue Brain reconstruction

Jason Smith
University of Abedeen , UK

The Blue Brain Project's digital reconstruction of a rat's neocortical column allows us to study the effect of certain stimuli on the brain. The insertion of a stimulus into the model causes information to propagate through the column, creating activity patterns that are not well understood. Using techniques from applied topology and combinatorics we attempt to characterise the firing patterns of different stimuli. Using this characterisation we then apply the methods to an unknown sequence of stimuli of the same type and attempt a classification of those stimuli.

 

Topology and neuroscience

Daniela Egas Santander
Laboratory for topology and neuroscience, EPFL, Switzerland

I will present some of the applications of topology and topological data analysis to neuroscience through an exploration of the collaboration between the applied topology group at EPFL and the Blue Brain Project. In particular, I will describe how we are using topology to further understand learning or simulations of voltage sensitive dye experiments.

 

Application of topological data analysis to the detection of mild cognitive impairment

Alice Patania
Indiana University

Identifying subjects with cognitive deficits as early as possible is critical in pursuing treatments for Alzheimer’s Disease. However, in the mildly symptomatic stages, pathological brain atrophy can be subtle and overpowered in signal by aging. Applying persistent homology, we are able to build coarse descriptors of the overall cortical thickness of each subject and isolate atrophy features that are indicative of MCI. These 0- and 1-persistence features can be used to build integrated persistent homological kernels which retain the meaningful homological information of brain atrophy. Using a support vector machine approach, we show how building a coarse descriptor of the cortical topology improves discriminative power of whole brain atrophy biomarkers at the MCI stage and homological features prove useful in identifying individuals with early stages of cognitive impairment.