from a discrete to a continuous model of biological cell movement the process by which one may take a discrete model of a biophysical process and construct a continuous model based upon it is of mathematical interest as well as being of practical use in this work we take the extended potts model applied to biological cell movement to its continuous limit beginning with a single cell moving in one dimension on a lattice and obeying potts model rules of movement we develop an expression for the diffusion coefficient of a collection of noninteracting cells which depends explicitly on the potts model parameters we show how this coefficient varies when the potts parameters for cell membrane elasticity and cell medium adhesion are varied and perform computer simulations which support our theoretical result we explain the relationship between the probability of occupancy of lattice points and the density profile in the continuous limit and extend our analysis by including interactions between the cells in so doing we are able to develop a set of coupled ordinary differential equations showing the evolution of a density profile in the presence of significant cell cell adhesion and show how increases in the strength of this adhesion modulates diffusion in so doing we develop some insights into how continuous models of physical systems can be based upon discrete models which describe the same system

the spectral geometry of foliations a classical topic in riemannian geometry seeks to relate the fundamental group of a compact manifold m with the possible metrics that m can carry an important result in this direction is the theorem of milnorsvarc 141 which relates growth properties of the fundamental group of m with growth properties of the riemannian universal cover of m it has been observed by a number of people 15 16 19 20 that results of this type may be generalized to results about leaves in foliations of compact manifolds it is not hard to see why this should be the caseleaves of foliations share with universal covers of compact manifolds the property that they are composed of constituent pieces on which only bounded geometric behavior can occur one can then hope that certain asymptotic behavior of the space in question is reflected only in the combinatorial structure of how these pieces are glued together in the case of the universal covering this combinatorial structure is given precisely by the fundamental group in the case of foliations it is given by a more complicated object which we call the fundamental pseudogroup and which is a subpseudogroup of the holonomy pseudogroup of 15 we explain this concept in section 1 below in contrast to the holonomy pseudogroup the fundamental pseudogroup is not intrinsic and depends on a suitable choice of flow boxes for the foliation however it is our main theme that certain algebraic notions concerning the fundamental pseudogroup translate into geometric properties of the leaves and hence are independent of the choices involved in constructing this pseudogroup the goal of the present paper is to generalize to foliations a result of the author 3 interpreting the amenability of the fundamental group as a spectral condition on the universal cover more precisely our main result theorem 2 1 of section 2 below gives a necessary and sufficient condition for the l2 spectrum of the laplacian of a leaf l to be bounded away from 0 in terms of the algebraic structure of the pseudogroup mentioned above

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

curvature of leaves in amenable foliations 1 statement of results if m is a smooth manifold a basic problem is to examine the relationship between the fundamental group ir1 m and the possible riemannian geometries on m a fundamental result in this direction due to s t yau 22 and d gromoll and j wolf 11 asserts that if m is compact with non positive sectional curvature and i1x m is solvable then m is flat and in particular r1r m is a finite extension of an abelian group this result has been generalized from the compact case to the finite volume case and in fact to certain infinite volume manifolds as well by s s chen and p eberlein 3 the point of this paper is to prove an analogue of these results for foliations

myopic random walkers and exclusion processes single and multispecies abstract a motility mechanism based on a simple exclusion process where the probability of movement of an agent depends on the number of unoccupied nearest neighbor sites is considered such interacting agents are termed myopic this problem is an extension of the famous blind or myopic ant in a labyrinth problem for the interacting agent models considered here each agent plays the role of an ant in a labyrinth where the paths of allowed sites though the labyrinth consist of sites not occupied by other agents we derive a nonlinear diffusion equation for the average occupancy of the discrete agent based model for myopic agents in contrast interacting blind agents have a constant probability of movement to each of their nearest neighbor sites giving rise to a linear diffusion equation insight into the various terms in the nonlinear diffusion coefficient is obtained from a study of multiple subpopulations of interacting myopic agents where an advection diffusion equation for each subpopulation is derived and from tracking an individual agent within the crowd where a motility coefficient is extracted averaged discrete simulation data compares very well with the solution to the continuum models we also compare the behavior of myopic and blind agents the myopic motility mechanism is biologically motivated to emulate information an individual cell gathers from environment cues the multispecies model developed and investigated here assists with the interpretation of experimental data involving the tracking subpopulations of cells within a total cell population

brief paper pole placement for linear time varying non lexicographically fixed mimo systems this document is the third in a series of recent eigenvalue placement studies by the authors in the first two publications valasek and olgac 1995a automatica 31 11 1605 1617 and 1995b iee control theory appl proc 142 5 451 458 the extension of ackermann s formula to time varying siso and later to time invariant and varying mimo systems were presented pole placement for linear time varying systems means that the closed loop system is equivalent via a lyapunov transformation to a linear time invariant system with poles at prescribed locations in addition to the controllability condition a critical property in this process is lexicographically fixedness which was taken as granted at all times in the prior work in this paper we deal with the pole placement of linear time varying mimo systems which are not lexicographically fixed this effort removes the last restriction in the pole placement of such systems aside from being controllable the original linear time varying mimo system which is not lexicographically fixed is augmented by appending additional state equations to ensure that the embedded system has the property of lexicographically fixedness the poles are then placed as desired for this augmented system using the techniques presented in our previous publications there is an added work of placing also the extra poles which appear due to the augmentation however they can be selected sufficiently away from the originally intended poles so that they do not influence the system behavior examples are provided to validate this claim

building macroscale models from microscale probabilistic models a general probabilistic approach for nonlinear diffusion and multispecies phenomena a discrete agent based model on a periodic lattice of arbitrary dimension is considered agents move to nearest neighbor sites by a motility mechanism accounting for general interactions which may include volume exclusion the partial differential equation describing the average occupancy of the agent population is derived systematically a diffusion equation arises for all types of interactions and is nonlinear except for the simplest interactions in addition multiple species of interacting subpopulations give rise to an advection diffusion equation for each subpopulation this work extends and generalizes previous specific results providing a construction method for determining the transport coefficients in terms of a single conditional transition probability which depends on the occupancy of sites in an influence region these coefficients characterize the diffusion of agents in a crowded environment in biological and physical processes

persistent exclusion processes inertia drift mixing and correlation in many biological systems motile agents exhibit random motion with short term directional persistence together with crowding effects arising from spatial exclusion we formulate and study a class of lattice based models for multiple walkers with motion persistence and spatial exclusion in one and two dimensions and use a mean field approximation to investigate relevant population level partial differential equations in the continuum limit we show that this model of a persistent exclusion process is in general well described by a nonlinear diffusion equation with reference to results presented in the current literature our results reveal that the nonlinearity arises from the combination of motion persistence and volume exclusion with linearity in terms of the canonical diffusion or heat equation being recovered in either the case of persistence without spatial exclusion or spatial exclusion without persistence we generalize our results to include systems of multiple species of interacting motion persistent walkers as well as to incorporate a global drift in addition to persistence these models are shown to be governed approximately by systems of nonlinear advection diffusion equations by comparing the prediction of the mean field approximation to stochastic simulation results we assess the performance of our results finally we also address the problem of inferring the presence of persistence from simulation results with a view to application to experimental cell imaging data

generalized cahn hilliard equation for biological applications recently we considered a stochastic discrete model which describes fronts of cells invading a wound e khain l m sander and c m schneider mizell j stat phys 128 209 2007 in the model cells can move proliferate and experience cell cell adhesion in this work we focus on a continuum description of this phenomenon by means of a generalized cahn hilliard equation gch with a proliferation term as in the discrete model there are two interesting regimes for subcritical adhesion there are propagating pulled fronts similar to those of the fisher kolmogorov equation the problem of front velocity selection is examined and our theoretical predictions are in the good agreement with a numerical solution of the gch equation for supercritical adhesion there is a nontrivial transient behavior where density profile exhibits a secondary peak to analyze this regime we investigated relaxation dynamics for the cahn hilliard equation without proliferation we found that the relaxation process exhibits self similar behavior the results of continuum and discrete models are in good agreement with each other for the different regimes we analyzed

arbitrary eigenvalue assignments for linear time varying multivariable control systems the problem of eigenvalue assignments for a class of linear time varying multi variable systems is considered using matrix operators and canonical transformations it is shown that a time varying system that is lexicography fixedly controllable can be made via state feedback to be equivalent to a time invariant system whose eigenvalues are arbitrarily assignable a simple algorithm for the design of the state feedback is provided

wavefront propagation in a competition equation with a new motility term modelling contact inhibition between cell populations linear diffusion is an established model for spatial spread in biological systems including movement of cell populations however for interacting closely packed cell populations simple diffusion is inappropriate because different cell populations will not move through one another rather a cell will stop moving when it encounters another cell in this paper i introduce a nonlinear diffusion term that reflects this phenomenon known as contact inhibition of migration i study this term in the context of two competing cell populations one of which has a proliferative advantage over the other this is motivated by the very early stages of solid tumour growth i focus in particular on travelling wave solutions corresponding to moving interfaces between the two cell populations numerical simulations indicate that there are wavefront solutions for wave speeds above a critical minimum value and i present linear analysis that explains the selection of wave speeds by initial conditions i obtain an approximation to the shape of these waves for high speeds and show that the minimum speed arises via quite new behaviour in the travelling wave equations with the proportion of cells of each type approaching a step function as the wave speed decreases towards the minimum exploiting this structure i use singular perturbation theory to investigate the wave shape for speeds close to the minimum

stabilisation of time varying linear systems via lyapunov differential equations this article studies stabilisation problem for time varying linear systems via state feedback two types of controllers are designed by utilising solutions to lyapunov differential equations the first type of feedback controllers involves the unique positive definite solution to a parametric lyapunov differential equation which can be solved when either the state transition matrix of the open loop system is exactly known or the future information of the system matrices are accessible in advance different from the first class of controllers which may be difficult to implement in practice the second type of controllers can be easily implemented by solving a state dependent lyapunov differential equation with a given positive definite initial condition in both cases explicit conditions are obtained to guarantee the exponentially asymptotic stability of the associated closed loop systems numerical examples show the effectiveness of the proposed approaches

from a discrete model of chemotaxis with volume filling to a generalized patlak keller segel model we present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased rand

field theoretic approach to metastability in the contact process a quantum field theoretic formulation of the dynamics of the contact process on a regular graph of degree z is introduced a perturbative calculation in powers of 1 z of the effective potential for the density of particles phi t and an instantonic field psi t emerging from the quantum formalism is performed corrections to the mean field distribution of densities of particles in the out of equilibrium stationary state are derived in powers of 1 z results for typical e g average density and rare fluctuation e g lifetime of the metastable state properties are in very good agreement with numerical simulations carried out on d dimensional hypercubic z 2d and cayley lattices

multi species simple exclusion processes abstract a motility mechanism based on a simple exclusion process where the movement of discrete agents on a lattice is either unbiased symmetric or biased asymmetric is considered estimates of diffusivities from tracking data do not describe the population level response of the system this mismatch between the individual level and population level behaviour can be resolved by averaging the individual level mechanism in terms of an expected site occupancy new insight into simple exclusion processes is obtained by representing the system as a series of interacting subpopulations this formalism leads to a system of nonlinear advection diffusion equations which can be interpreted in terms of the agent fluxes these interactions have consequences for both agent based modelling and continuum modelling in cell biology such as tracking subpopulations of cells within a total cell population

when push comes to shove exclusion processes with nonlocal consequences abstract stochastic agent based models are useful for modelling collective movement of biological cells lattice based random walk models of interacting agents where each site can be occupied by at most one agent are called simple exclusion processes an alternative motility mechanism to simple exclusion is formulated in which agents are granted more freedom to move under the compromise that interactions are no longer necessarily local this mechanism is termed shoving a nonlinear diffusion equation is derived for a single population of shoving agents using mean field continuum approximations a continuum model is also derived for a multispecies problem with interacting subpopulations which either obey the shoving rules or the simple exclusion rules numerical solutions of the derived partial differential equations compare well with averaged simulation results for both the single species and multispecies processes in two dimensions while some issues arise in one dimension for the multispecies case

the diffusion limit of transport equations ii chemotaxis equations in this paper we use the diffusion limit expansion of transport equations developed earlier t hillen and h g othmer siam j appl math 61 2000 pp 751 775 to study the limiting equation under a variety of external biases imposed on the motion when applied to chemotaxis or chemokinesis these biases produce modification of the turning rate the movement speed or the preferred direction of movement depending on the strength of the bias it leads to anisotropic diffusion to a drift term in the flux or to both in the parabolic limit we show that the classical chemotaxis equation which we call the patlak keller segel alt pksa equation arises only when the bias is sufficiently small using this general framework we derive phenomenological models for chemotaxis of flagellated bacteria of slime molds and of myxobacteria we also show that certain results derived earlier for one dimensional motion can easily be generalized to two or three dimensional motion as well

brief paper output feedback pole placement for linear time varying systems with application to the control of nonlinear systems the output feedback pole placement problem is solved in an input output algebraic formalism for linear time varying ltv systems the recent extensions of the notions of transfer matrices and poles of the system to the case of ltv systems are exploited here to provide constructive solutions based as in the linear time invariant lti case on the solutions of diophantine equations also differences with the results known in the lti case are pointed out especially concerning the possibilities to assign specific dynamics to the closed loop system and the conditions for tracking and disturbance rejection this approach is applied to the control of nonlinear systems by linearization around a given trajectory several examples are treated in detail to show the computation and implementation issues

efficient eigenvalue assignments for general linear mimo systems abstract this paper deals with the transformation of linear multi input multi output mimo systems into frobenius canonical form with the ultimate objective of developing a new computationally efficient methodology for a poleplacement procedure both time invariant and time varying systems are considered the conventional pole placement steps for time invariant siso single input single output systems are generalized for both classes this is a unique study of the expansion of the pole placement capability in particular for time varying mimo systems this depth of generalization has been neglected in the past due to its complex formulation the practical advantage of the proposed technique is that it does not require the coefficients of the characteristic polynomial the eigenvalues of the original system or the coefficients of the characteristic polynomial of the transformed system the repercussions of such a development are expected in nonlinear systems theory as well considering the fact that some recently developed techniques yield exact i o linearization with time varying coefficients

scaling limits for interacting diffusions we consider a large number of particles diffusing on a circle interacting through a drift resulting from the gradient of a pair potential whose support is of the order of the interparticle distance we derive a nonlinear bulk diffusion equation for the density of the particle distribution on the circle the diffusion coefficient is determined as a function of density in terms of standard thermodynamical objects

lattice free models of directed cell motility directed cell migration often occurs when individual cells move in response to an external chemical stimulus cells can respond by moving in either the direction of increasing chemoattraction or decreasing chemorepulsion concentration many previous models of directed cell migration use a lattice based framework where agents undergo a lattice based random walk and the direction of nearest neighbour motility events is biased in a preferred direction such lattice based models can lead to unrealistic configurations of agents since the agents always move on an artificial lattice structure which is never observed experimentally we present a lattice free model of directed cell migration that incorporates two key features first agents move on a continuous domain with the possibility that there is some preferred direction of motion second to be consistent with experimental observations we enforce a crowding mechanism so that motility events that would lead to agent overlap are not permitted we compare simulation data from the new lattice free model with a more traditional lattice based model to provide additional insight into the lattice free model we construct an approximate conservation statement which corresponds to a nonlinear advection diffusion equation in the continuum limit the solution of this mean field model compares well with averaged data from the individual based model

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

modeling tumor cell migration from microscopic to macroscopic models it has been shown experimentally that contact interactions may influence the migration of cancer cells previous works have modelized this thanks to stochastic discrete models cellular automata at the cell level however for the study of the growth of real size tumors with several million cells it is best to use a macroscopic model having the form of a partial differential equation pde for the density of cells the difficulty is to predict the effect at the macroscopic scale of contact interactions that take place at the microscopic scale to address this we use a multiscale approach starting from a very simple yet experimentally validated microscopic model of migration with contact interactions we derive a macroscopic model we show that a diffusion equation arises as is often postulated in the field of glioma modeling but it is nonlinear because of the interactions we give the explicit dependence of diffusivity on the cell density and on a parameter governing cell cell interactions we discuss in detail the conditions of validity of the approximations used in the derivation and we compare analytic results from our pde to numerical simulations and to some in vitro experiments we notice that the family of microscopic models we started from includes as special cases some kinetically constrained models that were introduced for the study of the physics of glasses supercooled liquids and jamming systems

nonlinear diffusion and exclusion processes with contact interactions exclusion processes on a regular lattice are used to model many biological and physical systems at a discrete level the average properties of an exclusion process may be described by a continuum model given by a partial differential equation we combine a general class of contact interactions with an exclusion process we determine that many different types of contact interactions at the agent level always give rise to a nonlinear diffusion equation with a vast variety of diffusion functions d c we find that these functions may be dependent on the chosen lattice and the defined neighborhood of the contact interactions mild to moderate contact interaction strength generally results in good agreement between discrete and continuum models while strong interactions often show discrepancies between the two particularly when d c takes on negative values we present a measure to predict the goodness of fit between the discrete and continuous model and thus the validity of the continuum description of a motile contact interacting population of agents this work has implications for modeling cell motility and interpreting cell motility assays giving the ability to incorporate biologically realistic cell cell interactions and develop global measures of discrete microscopic data

distinguishing between mean field moment dynamics and stochastic descriptions of birth death movement processes abstract mathematical descriptions of birth death movement processes are often calibrated to measurements from cell biology experiments to quantify tissue growth rates here we describe and analyze a discrete model of a birth death movement process applied to a typical two dimensional cell biology experiment we present three different descriptions of the system i a standard mean field description which neglects correlation effects and clustering ii a moment dynamics description which approximately incorporates correlation and clustering effects and iii averaged data from repeated discrete simulations which directly incorporates correlation and clustering effects comparing these three descriptions indicates that the mean field and moment dynamics approaches are valid only for certain parameter regimes and that both these descriptions fail to make accurate predictions of the system for sufficiently fast birth and death rates where the effects of spatial correlations and clustering are sufficiently strong without any method to distinguish between the parameter regimes where these three descriptions are valid it is possible that either the mean field or moment dynamics model could be calibrated to experimental data under inappropriate conditions leading to errors in parameter estimation in this work we demonstrate that a simple measurement of agent clustering and correlation based on coordination number data provides an indirect measure of agent correlation and clustering effects and can therefore be used to make a distinction between the validity of the different descriptions of the birth death movement process

optimal control for a new class of singularly perturbed linear systems singularly perturbed linear control systems have been traditionally studied with one vector input that controls both the slow and fast state variables motivated by examples of real physical systems in this paper we extend the study to corresponding two vector input systems we have identified several classes of real physical singularly perturbed systems that can be more efficiently controlled using two vector inputs in such systems one vector input is used to control the slow state space variables and the other vector input is used to control the fast state variables this control structure offers more flexibility controlling given systems and appears to be more effective we study the linearquadratic regulator lqr problem for this new class of singularly perturbed systems and develop a numerical algorithm based on fixed point iterations to obtain its solution the latter is obtained in terms of reduced order sub problems which avoid possible numerical stiffness issues and are computationally advantageous for large scale systems

macroscopic dynamics of biological cells interacting via chemotaxis and direct contact a connection is established between discrete stochastic model describing microscopic motion of fluctuating cells and macroscopic equations describing dynamics of cellular density cells move towards chemical gradient process called chemotaxis with their shapes randomly fluctuating nonlinear diffusion equation is derived from microscopic dynamics in dimensions one and two using excluded volume approach nonlinear diffusion coefficient depends on cellular volume fraction and it is demonstrated to prevent collapse of cellular density a very good agreement is shown between monte carlo simulations of the microscopic cellular potts model and numerical solutions of the macroscopic equations for relatively large cellular volume fractions combination of microscopic and macroscopic models were used to simulate growth of structures similar to early vascular networks

box and ball system with a carrier and ultradiscrete modified kdv equation a new soliton cellular automaton is proposed it is defined by an array of an infinite number of boxes a finite number of balls and a carrier of balls moreover it reduces to a discrete equation obtained from the discrete modified korteweg de vries equation through a limit an algebraic expression of soliton solutions is also proposed

diffusion in a crowded environment we analyze a pair of diffusion equations which are derived in the infinite system size limit from a microscopic individual based stochastic model deviations from the conventional fickian picture are found which ultimately relate to the depletion of resources on which the particles rely the macroscopic equations are studied both analytically and numerically and are shown to yield anomalous diffusion which does not follow a power law with time as is frequently assumed when fitting data for such phenomena these anomalies are here understood within a consistent dynamical picture which applies to a wide range of physical and biological systems underlining the need for clearly defined mechanisms which are systematically analyzed to give definite predictions

continuum models of cohesive stochastic swarms the effect of motility on aggregation patterns abstract mathematical models of swarms of moving agents with non local interactions have many applications and have been the subject of considerable recent interest for modest numbers of agents cellular automata or related algorithms can be used to study such systems but in the present work instead of considering discrete agents we discuss a class of one dimensional continuum models in which the agents possess a density x t at location x at time t the agents are subject to a stochastic motility mechanism and to a global cohesive inter agent force the motility mechanisms covered include classical diffusion nonlinear diffusion which may be used to model in a phenomenological way volume exclusion or other short range local interactions and a family of linear redistribution operators related to fractional diffusion equations a variety of exact analytic results are discussed including equilibrium solutions and criteria for unimodality of equilibrium distributions full time dependent solutions and transitions between asymptotic collapse and asymptotic escape we address the behaviour of the system for diffusive motility in the low diffusivity limit for both smooth and singular interaction potentials and show how this elucidates puzzling behaviour in fully deterministic non local particle interaction models we conclude with speculative remarks about extensions and applications of the models

modeling biological tissue growth discrete to continuum representations there is much interest in building deterministic continuum models from discrete agent based models governed by local stochastic rules where an agent represents a biological cell in developmental biology cells are able to move and undergo cell division on and within growing tissues a growing tissue is itself made up of cells which undergo cell division thereby providing a significant transport mechanism for other cells within it we develop a discrete agent based model where domain agents represent tissue cells each agent has the ability to undergo a proliferation event whereby an additional domain agent is incorporated into the lattice if a probability distribution describes the waiting times between proliferation events for an individual agent then the total length of the domain is a random variable the average behavior of these stochastically proliferating agents defining the growing lattice is determined in terms of a fokker planck equation with an advection and diffusion term the diffusion term differs from the one obtained landman and binder j theor biol 259 541 2009 when the rate of growth of the domain is specified but the choice of agents is random this discrepancy is reconciled by determining a discrete time master equation for this process and an associated asymmetric nonexclusion random walk together with consideration of synchronous and asynchronous updating schemes all theoretical results are confirmed with numerical simulations this study furthers our understanding of the relationship between agent based rules their implementation and their associated partial differential equations since tissue growth is a significant cellular transport mechanism during embryonic growth it is important to use the correct partial differential equation description when combining with other cellular functions