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For the past four years, the University of Alberta has hosted a summer school on mathematical biology, aimed at undergraduate students who have completed 2-3 years of study in mathematics or a similar quantitative science. The aim of this summer school is to introduce the students to mathematical modelling and analysis applied to real biological systems. In the span of 10 days, students attend lectures and exercise sessions, learn how to set up mathematical models, and use analytical and computational tools to relate them to biological data. They experience the modelling process by working on a research project. In this chapter, we explain our teaching philosophy, share some unique features of our summer school, and exemplify key course components [ABSTRACT FROM AUTHOR]

We study the Keller-Segel model for chemotaxis, consisting of a drift-diffusion equation describing the evolution of the cell density coupled to an equation for the chemoattractant. It is known that in the classical Keller-Segel model solutions can become unbounded in finite time. We present recent analytical results for this model, and compare its behavior in two space dimensions numerically to the behavior of a model accounting for the finite volume of cells. This modified Keller-Segel model relies on the assumption that cells stop aggregating when their density is too high, and thus allows for the global existence of solutions. We characterize the slow movement of a certain class of plateau-shaped solutions and perform numerical experiments for both models, showing that solutions of the classical (before blow-up) and of the density control model share common features: regions of high cell density are attracted by each other and, under suitable boundary conditions, by the domain boundaries. [ABSTRACT FROM AUTHOR]

Functional-structural plant models (FSPMs), combining the physiological function of a plant with its architecture, require precise and transparent specifications. This can be seen as a new challenge to the design of programming languages. Here we introduce, exemplarily for a model of young poplar trees, our new formalism of relational growth grammars (RGGs), which extend the well-known Lindenmayer (L-)systems to a specific sort of node- and edge-labelled graph grammars. The model has been written in the programming language XL, which extends standard Java by rule-based programming with RGGs and overcomes many of the disadvantages of L-systems. RGGs can bridge different scales: In our model, morphogenetic rules in L-system style are combined with rules describing a regulatory network of hormone biosynthesis and rules updating photosynthate concentrations of shoot modules, all in one and the same formalism. [ABSTRACT FROM AUTHOR]

Stochastic interacting particle systems (IPSs) are individual-based models, which include stochastic local interactions on a spatial lattice. In this respect an IPS works similarly to a cellular automaton. However, IPSs are continuous-time Markov processes, hence there is a large background of analytical methods. Further, one has the possibility to simulate the system on a finite lattice, which is what we focus on in this work. We explain the modelling steps broadly and by means of examples. Finally, we state the core of a simulation algorithm. The ideas is to convince the reader that IPSs can be used to set up and simulate sophisticated and applicable models but allow an analytical approach as well. [ABSTRACT FROM AUTHOR]

This chapter deals with the behavior of a branching population undergoing saturation effects when it becomes too large. We study in particular the limits of the prediction given in the setting of the deterministic dynamical system related to the stochastic branching process modeling the evolution of the population. We also generalize the usual Markovian branching processes of order one to size-dependent branching processes that may have a longer memory and give conditions leading to an almost sure extinction of the process while the dynamical system is persistent. The notion of reproductive rate is explained and generalized. We give some examples, in particular the amplification process in the polymerase chain reaction (PCR). [ABSTRACT FROM AUTHOR]

The quantitative polymerase chain reaction aims at determining the initial amount X0 of a specific portion of DNA molecules from the observation of the amplification process of the DNA molecules' quantity. This amplification process is achieved through successive replication cycles. It depends on the efficiency {pnn} of the replication of the molecules, pn being the probability that a molecule will duplicate at replication cycle n. Modelling the amplification process by a branching process and assuming pn = p for all n, we estimate the unknown parameter Θ = (p, X0) using Markov chain Monte Carlo methods under a Bayesian framework. [ABSTRACT FROM AUTHOR]

We report a three-variable simplified model of excitation fronts in human atrial tissue. The model is derived by novel asymptotic techniques from the biophysically realistic model of Courtemanche et al. [11] in an extension of our previous similar models. An iterative analytical solution of the model is presented which is in excellent quantitative agreement with the realistic model. It opens new possibilities for analytical studies as well as for efficient numerical simulation of this and other cardiac models of similar structure. [ABSTRACT FROM AUTHOR]

The human brain is characterized by complex convolution patterns. Analyzing the variability of these patterns among human subjects can reveal information for the detection of diseases that affect the human brain. This chapter presents a novel method to visualize the brain surface and its folding pattern at different scales. The analysis steps involved are the transformation of the cortical surface from high resolution magnetic resonance tomography images (MRI) to an initial representation as a triangulated mesh and finally to a representation as a series of spherical harmonic basis functions. The spherical harmonic parameterization of the surface is translation, rotation and scaling invariant. The parametric representation gives a multidimensional coefficient vector for each cortical surface. The technique allows easier recognition of convolutional patterns. The method is a first step toward a statistical multi-scale analysis of the brain surface. [ABSTRACT FROM AUTHOR]

The auditory brainstem in mammals contains a neural circuit for sound localization. The exact functioning of this circuit is still under controversy. Two spike generation mechanisms studied previously, excitatory coincidence detection and inhibitory coincidence detection, are studied here regarding the input-output relationship of the spike time densities. We propose that synchronous binary multiplication operation on spikes is the underlying process of these two variants of coincidence detection. A derivation of time to the spike is shown, which enables us to estimate the contribution of the neural circuit in the auditory brainstem to the overall reaction time of sound localization. The brainstem contribution is minute compared to the conduction delays in the mammalian neocortex. Finally, the skewness of the resulting output spike time densities is discussed in both the excitatory and inhibitory cases and the inhibitory case is shown to be close to the normal density with a standard goodness-of-fit test for the normal probability density function. [ABSTRACT FROM AUTHOR]

Many complex biological, social, and economical networks show topologies drastically differing from random graphs. But what is a complex network, i.e., how can one quantify the complexity of a graph? Here the Offdiagonal Complexity (OdC), a new, and computationally cheap, measure of complexity is defined, based on the node-node link cross-distribution, whose nondiagonal elements characterize the graph structure beyond link distribution, cluster coefficient, and average path length. The OdC approach is applied to the Helicobacter pylori protein interaction network and randomly rewired surrogates thereof. In addition, OdC is used to characterize the spatial complexity of cell aggregates. We investigate the earliest embryo development states of Caenorhabditis elegans. The development states of the premorphogenetic phase are represented by symmetric binary-valued cell connection matrices with dimension growing from 4 to 385. These matrices can be interpreted as adjacency matrices of an undirected graph, or network. The OdC approach allows us to describe quantitatively the complexity of the cell aggregate geometry. [ABSTRACT FROM AUTHOR]

Over the last couple of years, it can be said that the focus of the computational aspects of neurons has moved from synaptic weight and firing rate encoding to temporal firing encoding. On the other hand, several elements of these models have been based on some conceptual assumptions that imply relative simple dynamic behaviour of neuronal membrane activity in an active-passive process. In line with recent advances that have produced a better understanding of the biochemical processes that occur within cells, it is proposed that the processes that are involved in a membrane depolarisation cascade are less static than have been assumed so far. In particular, the possibilities of low level computation at the membrane level need to be explored more extensively. In this chapter some computational properties of the spike generation processes are explored using phenomenological models. [ABSTRACT FROM AUTHOR]

In this chapter we describe the behavior of the muscle spindle by using a logistic regression model. The system receives input from a motoneuron and fires through the Ia sensory axon that transfers the information to the spinal cord and from there to the brain. Three functions which are of special interest are included in the model: the threshold, the recovery and the summation functions. The most favorable method of estimating the parameters of the muscle spindle is the maximum likelihood approach. However, there are cases when this approach fails to converge because some of the model's parameters are considered to be perfect predictors. In this case, the exact likelihood can be used, which succeeds in finding the estimates and the exact confidence intervals for the unknown parameters. This method has a main drawback: it is computationally very demanding, especially with large data sets. A good alternative in this case is a specific application of the Monte Carlo technique. [ABSTRACT FROM AUTHOR]

We use the differential entropy concept and methods related to differential entropy estimation in this chapter. In the beginning, we define the basic terms: entropy, differential entropy, the Kullback-Leibler distance and the refractory periods. We show relations between differential entropy and the Kullback-Leibler distance as well. Hereafter a detailed description of the methods used is given. These methods can be divided into three groups: parametric methods of entropy estimation, "plug-in" entropy estimators based on nonparametric density estimation and direct entropy estimators. The formulas for direct entropy estimation based on the first four sample moments are introduced. The results are illustrated by comparing the methods of the entropy estimation, combined with two refractory period estimates. We compare the estimates based on the histogram, the kernel density estimator, the sample spacing method, Vasicek's method, the nearest neighbor distance method and the methods based on sample moments. [ABSTRACT FROM AUTHOR]

In the past attention has mainly been focused on neurons and the role they play, both individually and as parts of networks, in the functioning of the brain and nervous system. However, glial cells outnumber neurons in the brain, and it is now becoming apparent that, far from just performing supportive and housekeeping tasks, they are also actively engaged in information processing and possibly even learning. Communication in glial cells is manifested by waves of calcium ions (Ca2+) that are released from internal stores, and these waves are observed experimentally using fluorescent markers attached to the ions. The waves can be initiated by stimulation of a single cell, and initially it was assumed that the transmission mechanism involved the passage of an intercellular signalling agent passing through gap junctions connecting the cells. However, a surprising feature is that in many cases the calcium waves can cross cell-free zones, thus indicating the presence of an extracellular messenger. We have constructed a mathematical model of calcium wave propagation in networks of model astrocytes, these being a subclass of glial cells. The extracellular signalling agent is ATP (adenosine triphosphate) and it acts on metabotropic purinergic receptors on the astrocytes, initiating a G-protein cascade leading to the production of inositol trisphosphate (IP3) and the subsequent release of Ca2+ from intracellular stores via IP3-sensitive channels. Stimulation of one cell (by a pulse of ATP or by raising the IP3 level) leads to the regenerative release of ATP both from this cell and from neighbouring cells, and hence a Ca2+ wave. Results are given for the propagation of Ca2+ waves in two-dimensional arrays of model astrocytes and also in lanes with cell-free zones in between. These theoretical considerations support the concept of extracellular purinergic transmission in astrocyte networks. [ABSTRACT FROM AUTHOR]

A model for the generation of action potentials by a neuron is presented. This model is based on standard and commonly accepted properties of excitable cells (neurons). The novelty is that under quite natural assumptions the generation of action potentials is described as a special case of a general model for systems generating recurrent biological events. A formula for a density function of the membrane potential distribution in the firing times of the neuron is derived. An analysis of time intervals between spikes is of special interest. Three different interspike interval distributions are found, where one of them is close to the stable distribution. It is consistent with the well-known hypothesis that stable interspike intervals form part of the neural chain in which information is being preserved. [ABSTRACT FROM AUTHOR]

We consider the continuous field model of neural populations with the addition of a distribution of transmission speeds. The speed distribution arises as a result of the natural variability of the properties of axons, such as their degree of myelination. We analyze the stability and bifurcations of equilibrium solutions for the resulting field dynamics. Using a perturbation approach, we show that the speed distribution affects the frequency of bifurcating periodic solutions and the phase speed of traveling waves. The theoretical findings are illustrated by numerical calculations. [ABSTRACT FROM AUTHOR]

3Dmorphological data have been used to quantitatively characterize the morphological phenotype of pyramidal neurons in transgenic mice. We calculated the multiscale fractal dimension (MFD) of reconstructed neuronal cells. Changes in the complexity of neuronal morphology due to permanent activation of p21Ras in the primary somatosensory cortex of transgenic mice correlate with changes in the MFDof dendrites of pyramidal neurons. Transgenic neurons seem slightly less complex (i.e., have lower peak fractal dimension) if compared with the wild type. On the other hand, the enhanced p21Ras activity in transgenic mice may lead to greater variety in the cell morphological phenotype. [ABSTRACT FROM AUTHOR]

We present a simple model based on microscopic automata to describe the clonal expansion process. The model is based on a repertoire of antigens and T lymphocytes interacting via antigen-presenting cells which present the antigens peptides. Each cell is represented by an automaton moving randomly on a two-dimensional lattice.We use this simplified model in order to introduce local and spatial considerations in the mathematical models of clonal expansion based on differential equations, and at the same time to attempt an analytical interpretation of the results of computer simulations. For this reason we also derive a mean field theory, whose results are in good agreement with the solutions of the microscopic model, at least for situations that are not too far from equilibrium. This model may be used as the basis of a more realistic one that could follow the clonal expansion process on a simplified version of the lymphatic network. [ABSTRACT FROM AUTHOR]

The goal of this chapter is to describe the mechanism underlying the age-specific increase in death risk related to immunosenescence, to determine the cause-specific hazard rate as a function of immune system characteristics. A mathematical model that allows for the estimation of the age-specific risk of death caused by infectious diseases has been developed. The model consists of three compartments: (1) a model of immunosenescence, (2) a model of infectious disease, and (3) a model giving the relationship between disease severity and the risk of death. The proposed model makes it possible to analyze age-specific mortality from infectious diseases and to predict future changes in mortality due to public health activity. At the same time it can be used for individualized risk assessment. [ABSTRACT FROM AUTHOR]

The mechanism that promotes coexistence of species has not been completely clarified yet. We propose that the amount of nutrient can be one of the factors that promote coexistence of species. Plant species have to reproduce seeds to produce descendants. Even if plant species do reproduce seeds, it is not ensured that every seed will bud. The amount of seeds that can bud successfully depends on the amount of nutrient: if the nutrient is scarce, then every seed cannot bud, but if the nutrient is rich, then every seed can bud. We also assume that the amount of seeds reproduced by one plant individual depends on the amount of nutrient.We show that, in this situation, the population dynamics of plants exhibits a complex behavior, which promotes coexistence of species. [ABSTRACT FROM AUTHOR]

B lymphocytes express on their surface receptors (antibodies) of a given specificity (idiotype). Crosslinking these receptors by complementary structures, antigens or antibodies, stimulates the lymphocyte. Thus a large functional network of interacting lymphocytes, the idiotypic network, emerges. Idiotypic networks, conceived by Niels Jerne 30 years ago, experience a renewed interest, e.g., in the context of autoimmune diseases. In a previously proposed minimalistic model idiotypes are represented by bit strings. The population dynamics of the idiotype clones is reduced to a zero-one scheme. An idiotype survives only if it meets enough but not too many complementary structures. We investigate the random evolution of the network towards a highly organized functional architecture which is driven by the influx of new idiotypes, randomly generated in the bone marrow. The vertices can be classified into different groups, which are clearly distinguished, e.g., by the mean life time of the occupied vertices. They include densely connected core groups and peripheral groups of isolated vertices, resembling the central and peripheral parts of the biological network. We determine the building principles of the observed patterns and propose a description of their architecture, which is easily transferable to other patterns and applicable to different system sizes. [ABSTRACT FROM AUTHOR]

A previously proposed mathematical model based on a simplified scheme of Th1- Th2 regulation mediated by the cytokine network which describes the population dynamics of allergen-specific naive T cells, Th1 and Th2 cells, autocrine and cross-suppressive cytokines, and allergen is more closely investigated. The model provides a theoretical explanation of the switch from a Th2-dominated response to a Th1-dominated response to allergen in allergic individuals as a result of a hyposensitization therapy. We focus here on the bifurcation analysis of the nonautonomous dynamical system driven by periodic allergen injections. The stability of the fixed points of a stroboscopic map is investigated. The set of unstable fixed points forms the dynamical separatrix between the regions of Th2-dominated response and Th1-dominated response which is crossed during a successful therapy. The maintenance phase of the therapy holds the system near the stable fixed point of the stroboscobic map. [ABSTRACT FROM AUTHOR]

Bacterial biofilms are communities of microorganisms that develop on interfaces in aqueous environments. We formulate a density-dependent diffusion-reaction model for the growth of a dual-species biofilm. Both bacteria respond differently to their environment and develop different types of biofilms: One is a classical aerobic biofilm former that produces the characteristic cluster-and-channel biofilm morphology, the other one also develops under anaerobic conditions and tends to form flat, creeping biofilms. A previously developed nonstandard finite-difference scheme is adapted for the computer simulation. In a numerical experiment it is shown how variations of a single parameter (growth rate) can trigger different spatial structure and organisation of the biofilm community. [ABSTRACT FROM AUTHOR]

Rapid species extinctions and the loss of other biodiversity features worldwide have prompted the development of a systematic planning framework for the conservation of biodiversity. Limited resources (~ 40 million USDannually) are available for conservation, particularly in the developing countries that contain many of the world's hotspots of species diversity. Thus, conservation planning problems are often represented as mathematical programs in which the objective is to select sites to serve as conservation areas so that the cost of the plan is as small as possible and adequate habitat is protected for each species. Here, we generalize this approach to allow for uncertainty in the planning process. In particular, we assume that the species to be protected disperse after the conservation areas are established and that planners cannot anticipate with certainty the species' future locations when selecting the conservation areas. This uncertainty is modeled by including random variables in the mathematical program. We illustrate the approach by designing a network of conservation areas for birds in southern Quebec. [ABSTRACT FROM AUTHOR]

According to the tenet of Darwinian selection, a phenotype will spread only if its fitness is greater than the mean fitness of the entire population. It is, therefore, natural to introduce the notion of relative advantage of a replicator, which is defined as the expected fitness of this replicator minus the average fitness of the entire replicator population. For the general replicator dynamics, it is shown that the relative advantage of an offspring population over its parent population is proportional to the variance in fitness. The relationship between the proposed and earlier versions of the fundamental theorem of natural selection is also discussed. [ABSTRACT FROM AUTHOR]

We investigate competitive exclusion between two reproductively isolated yearclasses in the Leslie matrix model for a semelparous biennial population. Our results show that competitive exclusion occurs if competition is more severe between than within year-classes. Our criterion is applicable even if the model exhibits complex behavior. [ABSTRACT FROM AUTHOR]

This chapter is devoted to the analysis of single-species population dynamics models with overlapping and non-overlapping generations. Within the framework of all models it is assumed that there are no activities of individuals during the wintertime (as, for example, is the case for forest insect populations in the boreal zone), and changes in population size at these moments are described with a broken trajectory ("jump down"). Also, it is assumed that the fecundity of individuals is constant and that the quota of individuals surviving winter depends on the within-year population dynamics. The dynamics of the models, which are determined by the influence of winter conditions on the survival of individuals and by the influence of intrapopulation self-regulative mechanisms, are analyzed. For some particular cases the conditions for population extinction and for stabilization at a non-zero level are determined; it is shown numerically that chaotic regimes can also be realized in some models. The conditions for the reduction of the models under consideration to some well-known discrete models are obtained. [ABSTRACT FROM AUTHOR]

We present a review of the Tangled Nature model. The model is developed to focus on the effect of evolution and multiple interactions on ecological and evolutionary observables. The model is individual based and ecological structures, such as species, are emergent quantities. The dynamics consists of a simplistic mutation-prone multiplication in which the probability of producing an offspring is determined by the occupancy in genotype space. The macroscopic long time dynamics is intermittent and exhibits a slow decrease in the macroscopic extinction rate. Ecological quantities such as the species abundance distribution and the species-area relationship compare qualitatively well with observations, as does the relation between interaction and diversity. The effect of correlations between parents and mutants has been studied, as has the effect of a conserved resource. [ABSTRACT FROM AUTHOR]

Parameter estimation in nosocomial infections poses specific problems for estimation techniques. The mathematical description of the spread of nosocomial infections incorporates transmission as a dynamic part; the outcome is discrete and the amount of available information is usually small. We transfer an estimation technique developed previously for plant epidemics to nosocomial infections and demonstrate its application to a data set related to methicillin-resistant Staphyloccocus aureus. [ABSTRACT FROM AUTHOR]

Recent breakthroughs in malaria vaccines have given new hope that a safe, effective malaria vaccine may be found. The following epidemiological questions are addressed: 1. What level of vaccination coverage is required to offset the limitations of an imperfect diseasemodifying vaccine? 2. Could the introduction of a low-efficacy malaria vaccine lead to an increase in the number of secondary infections? 3.What characteristics of such a vaccine will have the greatest effect on the outcome? A mathematical model is developed for a disease-modifying malaria vaccine that is given once prior to infection, and the minimum coverage level for disease eradication is established. There is a threshold depending on the relative rate of infection, the efficacy of the vaccine and the duration of infection. Vaccines which reduce the rate and duration of infection will always result in a decrease in secondary infections. More surprisingly, there is a duration "shoulder," such that vaccines that increase the duration of infection slightly will still lead to a decrease in secondary infections, even if the rate of infection is unchanged. Beyond this, the number of secondary infections will increase unless the rate of infection is sufficiently lowered. This is critical for low-efficacy vaccines. [ABSTRACT FROM AUTHOR]

Many species possess defences (such as toxins) against predator attack which cannot be observed by the predator prior to attack, but which it might be beneficial for the predator to avoid. Often, such animals are brightly coloured or have some other way of signalling that they are defended (aposematism). In one of our papers we examined the evolution and maintenance of defence and conspicuousness, the brightness of the defence signal, in such prey species using a game theoretic model. In this chapter we develop the model further and in particular expand on the more theoretical results with examples demonstrating the type of solutions which can occur. We categorise eight possible configurations of solution states for simple solutions. Finally there is another class of solutions possible where there is strong between-individual variation in appearance between conspicuous, poorly defended prey, and we demonstrate one example of this complex solution. [ABSTRACT FROM AUTHOR]

The spatiotemporal spreading of a fungal disease over a vineyard is investigated using a susceptible-exposed-infected-removed (SEIR)-type model coupled with a set of partial differential equations describing the dispersal of the spores. The model takes into account both short and long range dispersal of spores and growth of the foliar surface. Results of numerical simulations are presented. A mathematical result for the asymptotic behavior of the solutions is given as well. [ABSTRACT FROM AUTHOR]