Seminarium Zakładu Biomatematyki i Teorii Gier

Sight is the most important sense that we posses. It is crucial to

understand the mechanics of vision in order to treat various diseases

that may occur and disturb regular seeing. The eye's main part

responsible for about two-thirds of refractive power is the cornea.

Cornea is transparent, shell-like structure situated in the frontal part

of the eye. It is one of the most sensitive parts of human body and its

various irregularities can cause many seeing disorders. Precise

knowledge of corneal shape is very important and accurate

mathematical models are necessary to fully understand biomechanics

of cornea. We present a new model of corneal geometry based on a

nonlinear membrane equation. We establish existence of solution and

provide some estimates. When fitting with data we use its simplied

form and find that mean error is of order of a few percent. Also, we

are concerned with determining some unknown parameters when the

solution is known (usually with a noise). This is one example of so

called Inverse Problems. They are usually more difficult to solve and

analyze than direct ones. Moreover, they often are ill-posed, that is,

not necessarily have unique solution which is continuous with respect

to the initial data. We propose some regularization methods and apply

them to the real corneal data. We struggle with two types of

different inverse problems. The first one concerns constant parameter

case. It turns out that determination of these constants is a nonlinear

problem in two unknowns to which solution we develop an iteration

scheme and prove rates of convergence. The second problem is linear

one and concerns the case when one of unknown parameters is not

necessarily constant.
We propose some regularization method based on classical ones. In

the end we obtain a stable method of determining unknown

parameters of our differential equation from the knowledge of

corneal shape. These parameters may have direct relation to some

biomechanical properties of the eye and can be used to provide some

insights of corneal structure.

The purpose of the talk is to discuss intricacies of quantifying

information in some problems arising at the interface between statistic

and biology.  Behaviour generated by interacting molecules, cells or

tissues is not random but focuses on sustaining life processes.

Sustaining life, however, is to the large extend based on information

storage, transmission and processing with examples ranging from the

iconic DNA double-helix through signal transduction pathways to

proteomic and metabolic networks. Understanding how information is

processed in living cells and organism is therefore one of the crucial

elements to understand how life is sustained.

The concept of information however is also inherent to statistics with

Fisher and Shannon as most well known concepts. It has been

originally associated with a precision  parameters can be estimated in

an experiment. Alternative approach by C. Shannon arose on the basic

of communication theory and was later  assimilated by statistical

sciences.

In the talk I will discuss relationships between such objects as

posterior distribution, Shannon Information, Fisher Information and

Jeffrey's prior. Using examples of  experimental design techniques and

signal transduction pathways I will demonstrate how statistical theory

can be used to better understand biological systems at the cellular

level.

I'm going to present my results on large games -- mainly games with a continuum of players and their economic applications. Games with a continuum of players may be regarded as an equivalent of continuum -- microscale models of physics. They were developed to model situation in which the number of interacting agents, called players (human beings, animals or even particles),is large enough to make the influence of a single player on some global variables negligible.

The main theoretical results from the point of view of mathematics, are various decomposition theorems allowing to simplify the procedure of finding an equilibrium in dynamic games: the original problem, joining dynamic optimization and fixed point procedure in a space of functions is reduced to a sequence of static optimization and fixed point procedures applied to static, one stage problems, coupled by some reccurential or differential equation.

I will also present economic applications of such games to exploitation of a common ecosystem and to modelling financial markets, mainly stock exchange.

This work is devoted to the study of a problem resulting from plasma physics: heat transfer of electrons in a plasma close to Maxwellian equilibrium. A formal derivation from the Vlasov equations is proposed. A hierarchy of intermediate models between the kinetic equations and the hydrodynamic limit is described. In particular, a new system
hydrodynamics, integro-differential in nature, is proposed. The system Schurtz and Nicolai appears as a simplification of the system resulting from the diversion. The local existence and uniqueness of the solution of the nonstationary system are established in a simplified framework. The last part is devoted to the implementation of a specific numerical scheme for solving these models. We propose a finite volume approach can be effective on unstructured grids. The accuracy of this scheme to capture specific effects such as kinetic, which may not be reproduced by the asymptotic Spitzer-Harm model.The consistency of this pattern with that of the Spitzer-Harm equation is highlighted, paving the way for a strategy of
coupling between the two models.

During the talk we will recall/summarise methods allowing to construct Lyapunov functionals for ODEs. We will start from linear ODEs and then switch to some non-linear examples.

The seminar is devoted to the theory of delay differential equations with distributed delays or in other words with memory. The Linear Chain Trick is discussed and I show under which condition an equation with distributed delays is equivalent to the system of ordinary differential equations. The stability of steady states will be discussed. I show that the equation with distributed delays is more stable than with a discrete delay in the sense that steady states are stable for a wider range of parameters.

How neural activity in the brain generates a particular behavior is not well understood. The focus of the talk will be on modeling neural circuit responsible for forward and backward locomotion in the nematode C. elegans. In particular, I will discuss an optimal pattern of connections
between neurons that gives the best match to the locomotory data for this animal.

During the talk I want to describe key aspects of cellular
calcium dynamics e.g. calcium channels, pumps and exchangers
responsible for calcium flows and the role of mitochondria and endoplasmic reticulum in calcium homeostasis.
I want say a few words about modeling those mechanisms from
microscopic and macroscopic point of view.

In this talk I shall present an efficient numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. We develop a spectral approach for the pantograph type-delay differential equations. A Legendre spectral-collocation method is employed to obtain highly accurate numerical approximations to the exact solution. The exponential convergence was obtained theoretically which was confirmed by some numerical examples, provided the data in the given pantograph delay differential equation are smooth. We also discuss briefly the case of nonlinear and system of pantograph-type delay
differential equations.

We consider a spatially-extended model describing mutual
phosphorylation of cytosolic kinase molecules and membrane receptors. The analyzed regulatory subsystem is a part of signal transduction mechanisms characteristic for immune receptors and Src family kinases. It couples differential equations defined in a domain and on its boundary via
nonlinear Robin boundary conditions. Assuming a spherically symmetric framework, our approach is to consider an auxiliary problem in which a uniform Dirichlet boundary condition is imposed on the external boundary of the spherical shell  instead of the Robin boundary condition. This method allows us to find the stationary spherically-symmetric solutions, both stable and unstable.

Experimental observations show the relevance of Th1-Th2 cell balance in hypersensitive reactions. We propose a theoretical model of T cell mediated immune response focusing on Th1-Th2 cell balance, in the mathematical framework of the theory of reacting fluid mixtures with proliferative events. In our model the proliferative events, i.e. events which are not mass preserving, are the clonal expansions of Th1 and Th2 cells. Smooth approximate solutions of the resulting PDE's system are analyzed by using a double-scale approach enlightening some features regarding the multi-scale complexity of the phenomenon under observation.

Modified van der Pol equation can be used to describe the activity of elements of electrical conduction system of human heart: sinoatrial node (SA), atrioventicular node (AV), and His Purkinje system. This model has a number of interesting properties allowing do reconstruct phenomena observed in physiological experiments as well as in Holter
electrocardiographic recordings. The aim of the presented work was totake into consideration the influence of feedback and delay which occur in normal heart action mode as well as in pathological modes. Usage of certain values for feedback and delay parameters in this model allows to simulate the heart action in situation when an accessory conducting
pathway is present (Wolff - Parkinson - White syndrome).

So far fragmentation-coagulation equations with strong coagulation have been analysed using weak compactness techniques which only yielded weak solutions. Recently we have proved that the fragmentation operator is sectorial which, combined with some interpolation techniques, allowed for proving classical solvability of such equations for large classes of fragmentation and coagulation rates.

The evolution of an individual-based spatial ecological model with dispersion and competition is studied. In the model, an infinite number of individuals, point particles in R^d, reproduce themselves, compete, and die at random. These events are described by a Markov generator, which determines the evolution of states understood as probability measures on the space of particle configurations. The main result is a statement that the corresponding correlation functions evolve in a scale of Banach spaces and remain sub-Poissonian, and hence no clustering occurs, if the
dispersion is subordinate to the competition.

We consider a model of SIRC type, taking into account four groups: susceptible, infected, removed and carriers. We study the dynamics in two interesting cases: when a positive steady state does not exist and when it exists. Efficient vaccination strategies are proposed which are based on
impulsive modelling.

Mirosław Lachowicz (MIMUW)
Introduction to Tatiana's talk


In the talk, the problem of existence and uniqueness of equilibrium solutions to a microscopic model of population dynamics will be discussed. The model is a modified Liouville equation corresponding to a Markov jump process. The condidtions guarateeing uniqueness or non-uniqueness of equilibrium solutions will be proposed.

Existing models of fish predation are either descriptive or lack realistic assumptions on predator's motion. Presented model fills this gap. The pure optimization approach gives answers not only about prey mortality, but also about predator's trajectory and speed.

Minority game (MG) was designed as a microscopic model of adaptive behavior observed in multi-agent systems where being in minority is profitable. It was shown that the MG exhibits different modes of behavior, depending on the game parameters: the random, cooperation, and herd. The latter case is characterized by small strategy space compared to the overall number of agents. In the first part of our seminar we will incorporate the utility function to study phenomenology of MGs in their herd regime. We will prove that the utility is bounded and the number of states is finite. We can represent the game as a Markov process and we can substantially reduce the number of states and calculate their probabilities. Then such interesting features of an important extensive random variable called an aggregated demand, like its strong inhomogeneity and presence of patterns in time, can be easily interpreted. In the second part, the generalization of the minority game to more than one market will be considered. We will find that if the payoff function allows for strong fluctuation of the utility then market occupancies can differ significantly, with preference given to this market where the fluctuation occurred first. There exists a critical size of agent population above which agents on bigger market behave collectively. In this regime, there always exists a history of decisions for which all agents on a bigger market react identically. We developed a model that explains these phenomena. In the last part of the seminar we will focus on applications. We will present an improved version of the MG predictor. We will apply  it to both the synthetic and real time series using data from financial markets. The statistical analysis of these data will be performed and the theory how to optimize predictor's parameters will be presented.

We study a system of particles, in general d-dimensional space, that interact by means of pair potential and adjust their positions according to the gradient flow dynamics induced by the total energy of the system. We consider the case when the range of the interaction is of the same order as the mean interparticle distance. It is also assumed that particles, locally, are located close to some crystallographic lattice. An appropriate system of equations that describes the evolution of macroscopic deformation of the crystallographic lattice, as well as the system that describe the evolution of the main crystallographic directions is derived. Well posedness of the derived system is studied as well as the stability of the particle system. Same examples of potentials that yield stable and unstable systems are given.

Mathematical description of fluid and solute transport between blood and
dialysis fluid in the peritoneal cavity has not been formulated fully yet,
in spite of the well known basic physical laws for such transport. Recent
mathematical, theoretical and numerical studies introduced new concepts on
peritoneal transport and yielded better results for the transport of fluid
and osmotic agent [1, 2, 3]. However, the problem of a combined
description of osmotic ultra filtration to the peritoneal cavity,
absorption of osmotic agent from the peritoneal cavity and leak of
macromolecules (proteins, e.g., albumin) from blood to the peritoneal
cavity has not been addressed yet. Therefore, we present here a new
extended model for these phenomena
and investigate its mathematical structure [4]. The model is based on a
three-component nonlinear system of two-dimensional partial differential
equations with the relevant boundary and initial conditions.
The non-constant steady-state solutions of the model are studied.
The restrictions on the parameters arising in the model were established
with the aim to obtain exact formulae for non-constant steady-state
solutions. As the result, exact formulae for the density of fluid
flux from blood to tissue and the volumetric flux across the tissue were
constructed and two linear autonomous ordinary differential equations for
glucose and albumin concentrations were derived. The analytical results
were checked for their applicability for the description of fluid-
glucose-albumin transport in peritoneal dialysis.

References

[1] Cherniha, R., Waniewski, J.: Exact solutions of a mathematical model
for fluid transport in peritoneal dialysis. Ukrainian Math. J., 57,
1112{1119 (2005)
[2] R. Cherniha, V.Dutka, J.Stachowska-Pietka and J.Waniewski.
Fluid transport in peritoneal dialysis: a mathematical model and numerical
solutions. //Mathematical Modeling of Biological Systems, Vol.I. Ed. by
A.Deutsch et al., Birkhaeuser, P.291-298, 2007
[3] Waniewski J, Dutka V, Stachowska-Pietka J, Cherniha R:
Distributed modeling of glucose-induced osmotic flow.
Adv Perit Dial 2007;23:2-6.
[4] Cherniha, R., Waniewski, J.:New mathematical model for
fluid-glucose-albumin transportin peritoneal dialysis. ArXiv:1110.1283v1 5
oct.2011

This talk deals with analytical methods in statistical mechanics. It's
goal is to give a deductive presentation of the statistical mechanics of
nonequilibrium  systems based on BBGKY hierarchy studying - description of
the evolution of states - as well as to treat the basic principles of
interplay of statistical mechanic systems with the classical mechanics. A
compact pesentation was prefered wherever possible. The matherial is made
as understandable as possible by inclusion of the mathematical steps and a
detailed presentation of some intermediate culculations.

We show that finite-state Markov chains may be approximated by fast
diffusions on finite graphs with semi-permeable membranes on vertices.
The approximation involves a singular perturbation with singularity in
both operator and boundary/transmission conditions. The result is
motivated by recent models of synaptic depression.

Przedstawiamy się cd

1. Tatiana Ryabukha Institute of Mathematics, National Academy of Sciences of Ukraine On solutions of classical and dual BBGKY hierarchies

2. Jacek Miękisz On games, genes, and delays O grach, genach i czasowych opóźnieniach

3. Urszula Foryś Delays in different natural phenomena Opóźnienia w różnych procesach naturalnych

4. Monika Piotrowska About the fascinating delays and cellular automata O fascynujących opóźnieniach i automatach komórkowych

5. Jan Poleszczuk Cancer stem cell hypothesis, tumor angiogenesis and influence of delay on biochemical networks Hipoteza komórek macierzystych nowotworu, angiogeneza oraz wpływ opóźnień na przebieg reakcji chemicznych   

Przedstawiamy się

1. Tatiana Ryabukha, Institute of Mathematics, National Academy of Sciences of Ukraine, On interplay between statistical mechanics and life sciences in science globalization perspective

2. Mirosław Lachowicz, Markov jumps everywhere, Markowskie skoki wszędzie

3. Tadeusz Płatkowski, Some models of cooperation, O kilku modelach kooperacji

4. Agnieszka Wiszniewska-Matyszkiel, Dynamic games, continuum of players and expectations, Gry dynamiczne, continuum graczy i oczekiwania

5. Jacek Miękisz, On games, genes, and delays, O grach, genach i czasowych opóźnieniach

6. Urszula Foryś, Delays in different natural phenomena, Opóźnienia w różnych procesach naturalnych

7. Monika Piotrowska, About the fascinating delays and cellular automata, O fascynujących opóźnieniach i automatach komórkowych

8. Jan Poleszczuk, Cancer stem cell hypothesis, tumor angiogenesis and influence of delay on biochemical networks, Hipoteza komórek macierzystych nowotworu, angiogeneza oraz wpływ opóźnień na przebieg reakcji chemicznych

Wszechobecność roślin zielonych zainspirowała Hairstona, Smitha i Slobodkina do opracowania ogólnej teorii kontroli populacji lądowych. W Hipotezie Zielonej Ziemi (HSS) zauważyli oni, że do znacznego uszczuplenia populacji roślin dochodzi niezmiernie rzadko, liczebność (biomasa) roślinożerców zatem nie może być limitowana przez zasoby. Z drugiej strony nie ogranicza ich również środowisko abiotyczne, stąd czynnikiem ograniczającym jest drapieżnictwo. Stworzony został komputerowy model ewolucji testujący hipotezę, że względne dostosowanie roślinożerców jest niższe niż roślin (i szczytowych drapieżników), co wyklucza osiągnięcie przez nie wysokiego zagęszczenia, bliskiego pojemności środowiska. Założono istnienie trzech populacji, należących do jednego łańcucha troficznego (populacje roślin, roślinożerców i mięsożerców). Osobniki w każdej z populacji określono za pomocą binarnych sekwencji ich genotypów. Dostosowanie osobników, wyrażające ich zdolność do korzystania z zasobów oraz obrony przed drapieżnikami, oszacowano licząc zgodne allele w odpowiednich genotypach. Rozrodczość oraz śmiertelność w każdym kroku czasowym były funkcją dostosowania i liczebności populacji. Analiza zmienności genetycznej pokazała, że najwyższa różnorodność genetyczna występowała w populacji roślinożerców. Zgodnie z przewidywaniami, dostosowanie roślinożerców oraz poziom wypełnienia ich pojemności środowiska były istotnie niższe niż wartości otrzymywane dla skrajnych poziomów troficznych (roślin i drapieżników). Model powstał we współpracy z prof. Piotrem Dawidowiczem oraz prof. Dariuszem Wrzoskiem.

Evolution of cooperation, within a framework of Prisoner's Dilemma game, on scale-free graphs will be discussed. We will present coevolutionary stochastic dynamics which lead to scale-free graphs with various exponents. 

I will summarize results of my papers with Andrzej Kolinski. We have designed a minimal model of protein folding that reproduces in a qualitative way the most pronounced features of globular proteins. Using Monte Carlo simulations we have estimated a set of parameters for which the native state is the global minimum of conformational energy. This implies the minimality of our force field. Then we have found such interaction parameters for which the model satisfies the thermodynamic hypothesis and folding transition is maximally cooperative. Contrary to H-P models, long range interactions are consistent with statistical contact potentials within an average protein environment and not with the transfer energies of residues from water. Cooperativity depends on protein architecture: it is the highest for the beta motif and the lowest for the alpha motif.

Few remarks on the possible mathematical description of the process of denaturation of DNA

The seminar will present possibilities of applying Gaussian networks in realistic animation of virtual characters and optimal redeployment in multiagent environments. It will be achieved by solving the variational equations for Freidlin-Wentzell action functionals.

Połączone seminarium:  Równania Fizyki Matematycznej i Biomatematyka i Teoria Gier

We will discuss a family of models with delays describing the process of angiogenesis, that is a physiological process involving the growth of new blood vessels from pre-existing ones. This family includes the well known models of tumour angiogenesis proposed by Hahnfeldt et al. and Ergun et al. and is based on the Gompertz type of the tumour growth. The family of considered models depends on two time delays and a parameter which reflects how strongly the vessels dynamics depends on the ratio between tumour and vessels volume. 

Będziemy rozważać Zamieć Śnieżną na grafach regularnych i z losowym doborem graczy oraz Dylemat Więźniaa bezskalowych grafach losowych Barabasi-Alberty.

Główne pytanie: jak struktura grafu wpływa na poziom kooperacji to znaczy postać miary stacjonarnej odpowiedniego łańcucha Markowa? Odpowiedź na seminarium.
Będziemy rozważać Zamieć Śnieżną na sieciach regularnych i z losowym doborem graczy oraz Dylemat Więźnia na bezskalowych sieciach Barabasi-Alberty.

Od pewnego czasu wiadomo, że struktura przestrzenna sprzyja zachowaniom kooperacyjnym. Przedstawię wyniki dotyczące poziomu kooperacji w grach typu Dylemat Więźnia, Zamieć Śnieżna oraz Jeleń-Zając.
Pokażę, że poziom kooperacji w stanie stacjonarym odpowiedniego łańcucha Markowa zależy od sposobu doboru graczy. Przedyskutuję zależność długoterminowego zachowania graczy od poziomu szumu i wielkości populacji.

This work is devoted to the development and justification of averaging
circuit for systems of differential equations with slow and fast variables
which pass through resonances in the course of evolution, the resonance
relation for frequencies dependant on retardation of argument in fast
variables is introduced. We build the uniform estimates for oscillation
integrals that are appropriate to the multifrequency systems under the
condition of invariable and variable delay. The case of systems with
linear delay is considered in detail. Under imposed conditions the
asymptotics of estimates is unimprovable. On the basis of obtained
estimates, we prove new
theorems on justification of averaging method for systems with invariable
and variable delay when frequencies depend on time lag or slow variables.
We
propose and justify the averaging circuit for systems with delay when
multipoint or integral boundary conditions are given. The averaging
procedure is also applied to integral boundary conditions.

We propose a new mechanism of interactions between game - theoretical
agents in which the weights of the connections between interacting
individuals are dynamical, payoff - dependent variables.
Their evolution depends on the difference between the payoff of the agents
from a given type of encounter and their average payoff. The mechanism is
studied in the framework of two models: agents distributed on a random
graph
and a mean field model. Symmetric and asymmetric connections between the
agents are introduced. Long time behavior of both systems is discussed for
the Prisoner's Dilemma and the Snow Drift games.