# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

All Issues

## Mathematical Biosciences & Engineering

2005 , Volume 2 , Issue 1

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2005, 2(1): 1-23 doi: 10.3934/mbe.2005.2.1 +[Abstract](765) +[PDF](486.6KB)
Abstract:
A basic task in understanding the neural mechanism of learning and adaptation is to detect and characterize neural interactions and their changes in response to new experiences. Recent experimental work has indicated that neural interactions in the primary motor cortex of the monkey brain tend to change their preferred directions during adaptation to an external force field. To quantify such changes, it is necessary to develop computational methodology for data analysis. Given that typical experimental data consist of spike trains recorded from individual neurons, probing the strength of neural interactions and their changes is extremely challenging. We recently reported in a brief communication [Zhu et al., Neural Computations 15 , 2359 (2003)] a general procedure to detect and quantify the causal interactions among neurons, which is based on the method of directed transfer function derived from a class of multivariate, linear stochastic models. The procedure was applied to spike trains from neurons in the primary motor cortex of the monkey brain during adaptation, where monkeys were trained to learn a new skill by moving their arms to reach a target under external perturbations. Our computation and analysis indicated that the adaptation tends to alter the connection topology of the underlying neural network, yet the average interaction strength in the network is approximately conserved before and after the adaptation. The present paper gives a detailed account of this procedure and its applicability to spike-train data in terms of the hypotheses, theory, computational methods, control test, and extensive analysis of experimental data.
2005, 2(1): 25-42 doi: 10.3934/mbe.2005.2.25 +[Abstract](727) +[PDF](383.9KB)
Abstract:
This paper deals with an almost global convergence result for Lotka-Volterra systems with predator-prey interactions. These systems can be written as (negative) feedback systems. The subsystems of the feedback loop are monotone control systems, possessing particular input-output properties. We use a small-gain theorem, adapted to a context of systems with multiple equilibrium points to obtain the desired almost global convergence result, which provides sufficient conditions to rule out oscillatory or more complicated behavior that is often observed in predator-prey systems.
2005, 2(1): 43-51 doi: 10.3934/mbe.2005.2.43 +[Abstract](645) +[PDF](171.6KB)
Abstract:
Living systems represent a local exception, albeit transient, to the second law of thermodynamics, which requires entropy or disorder to increase with time. Cells maintain a stable ordered state by generating a steep transmembrane entropy gradient in an open thermodynamic system far from equilibrium through a variety of entropy exchange mechanisms. Information storage in DNA and translation of that information into proteins is central to maintenance thermodynamic stability, through increased order that results from synthesis of specific macromolecules from monomeric precursors while heat and other reaction products are exported into the environment. While the genome is the most obvious and well-defined source of cellular information, it is not necessarily clear that it is the only cellular information system. In fact, information theory demonstrates that any cellular structure described by a nonrandom density distribution function may store and transmit information. Thus, lipids and polysaccharides, which are both highly structured and non-randomly distributed increase cellular order and potentially contain abundant information as well as polynucleotides and polypeptides. Interestingly, there is no known mechanism that allows information stored in the genome to determine the highly regulated structure and distribution of lipids and polysacchariedes in the cellular membrane suggesting these macromolecules may store and transmit information not contained in the genome. Furthermore, transmembrane gradients of H$^+$, Na$^+$, K$^+$, Ca$^+$, and Cl$^-$ concentrations and the consequent transmembrane electrical potential represent significant displacements from randomness and, therefore, rich potential sources of information.Thus, information theory suggests the genome-protein system may be only one component of a larger ensemble of cellular structures encoding and transmitting the necessary information to maintain living structures in an isoentropic steady state.
2005, 2(1): 53-77 doi: 10.3934/mbe.2005.2.53 +[Abstract](875) +[PDF](1049.4KB)
Abstract:
In this paper, we investigate the role of topology on synchronization, a fundamental feature of many technological and biological fields. We study it in Hindmarsh-Rose neural networks, with electrical and chemical synapses, where neurons are placed on a bi-dimensional lattice, folded on a torus, and the synapses are set according to several topologies. In addition to the standard topologies used in other studies, we introduce a new model that generalizes the Barabási-Albert scale-free model, taking into account the physical distance between nodes. Such a model, because of its plausibility both in the static characteristics and in the dynamical evolution, is a good representation for those real networks (such as a network of neurons) whose edges are not costless. We investigate synchronization in several topologies; the results strongly depend on the adopted synapse model.
2005, 2(1): 79-96 doi: 10.3934/mbe.2005.2.79 +[Abstract](814) +[PDF](2008.9KB)
Abstract:
We present a new algorithm for segmenting organs in CT scans for radiotherapy treatment planning. Given a contour of an organ that is segmented in one image, our algorithm proceeds to segment contours that identify the same organ in the consecutive images. Our technique combines partial differential equations-based morphing active contours with algorithms for joint segmentation and registration. The coupling between these different techniques is done in order to deal with the complexity of segmenting ''real'' images, where boundaries are not always well defined, and the initial contour is not an isophote of the image.
2005, 2(1): 97-109 doi: 10.3934/mbe.2005.2.97 +[Abstract](662) +[PDF](295.1KB)
Abstract:
We investigate how the dynamics of a mathematical model of a nephron depend on the precise form of the delay in the tubuloglomerular feedback loop. Although qualitative behavioral similarities emerge for different orders of delay, we find that significant quantitative differences occur. Without more knowledge of the form of the delay, this places restrictions on how reasonable it is to expect close quantitative agreement between the mathematical model and experimental data.
2005, 2(1): 111-132 doi: 10.3934/mbe.2005.2.111 +[Abstract](718) +[PDF](1021.1KB)
Abstract:
In this paper, we consider the population growth of a single species living in a two-dimensional spatial domain. New reaction-diffusion equation models with delayed nonlocal reaction are developed in two-dimensional bounded domains combining different boundary conditions. The important feature of the models is the reflection of the joint effect of the diffusion dynamics and the nonlocal maturation delayed effect. We consider and analyze numerical solutions of the mature population dynamics with some well-known birth functions. In particular, we observe and study the occurrences of asymptotically stable steady state solutions and periodic waves for the two-dimensional problems with nonlocal delayed reaction. We also investigate numerically the effects of various parameters on the period, the peak and the shape of the periodic wave as well as the shape of the asymptotically stable steady state solution.
2005, 2(1): 133-152 doi: 10.3934/mbe.2005.2.133 +[Abstract](1116) +[PDF](341.5KB)
Abstract:
A simple model incorporating demographic and epidemiological processes is explored. Four re-parameterized quantities the basic demographic reproductive number ($\R_d$), the basic epidemiological reproductive number ($\R_0$), the ratio ($\nu$) between the average life spans of susceptible and infective class, and the relative fecundity of infectives ($\theta$), are utilized in qualitative analysis. Mathematically, non-analytic vector fields are handled by blow-up transformations to carry out a complete and global dynamical analysis. A family of homoclinics is found, suggesting that a disease outbreak would be ignited by a tiny number of infectious individuals.
2005, 2(1): 153-168 doi: 10.3934/mbe.2005.2.153 +[Abstract](665) +[PDF](914.7KB)
Abstract:
The immune system maintains a highly diverse T-cell repertoire, which is shaped by active interactions between developing thymocytes and endogenous peptide/MHC molecules through the principle of positive and negative selections. Detours et al. developed a quantitative model addressing key immunologic notions such as selection, alloreactivity, and self-restriction. The model was based on the assumption that the clone size is uniformly distributed in the naive T-cell repertoire. However, recent biological findings have indicated that the naive T-cell repertoire is highly skewed, due to the uneven proliferation of premigrant single-positive thymocytes. In this paper, the model is revised to include these new findings. The effects of the uneven clonal expansion are investigated in detail and their biological significance is discussed. It is found that the uneven clonal expansion can significantly enhance the self-MHC restriction, while avoiding decreasing the alloreactivity. The clonal expansion therefore appears to be an additional selection event, resulting in fine tuning of the repertoire. In this way, T-cells reaching the periphery pool can fulfill maximum competence: both high self-restriction and high alloreactivity.
2005, 2(1): 169-206 doi: 10.3934/mbe.2005.2.169 +[Abstract](750) +[PDF](631.7KB)
Abstract:
The dynamics of microbial growth is a problem of fundamental interest in microbiology, microbial ecology, and biotechnology. The pioneering work of Jacob Monod, served as a starting point for developing a wealth of mathematical models that address different aspects of microbial growth in batch and continuous cultures. A number of phenomenological models have appeared in the literature over the last half century. These models can capture the steady-state behavior of pure and mixed cultures, but fall short of explaining most of the complex dynamic phenomena. This is because the onset of these complex dynamics is invariably driven by one or more intracellular variables not accounted for by phenomenological models.
In this paper, we provide an overview of the experimental data, and introduce a different class of mathematical models that can be used to understand microbial growth dynamics. In addition to the standard variables such as the cell and substrate concentrations, these models explicitly include the dynamics of the physiological variables responsible for adaptation of the cells to environmental variations. We present these physiological models in the order of increasing complexity. Thus, we begin with models of single-species growth in environments containing a single growth-limiting substrate, then advance to models of single-species growth in mixed-substrate media, and conclude with models of multiple-species growth in mixed-substrate environments. Throughout the paper, we discuss both the analytical and simulation techniques to illustrate how these models capture and explain various experimental phenomena. Finally, we also present open questions and possible directions for future research that would integrate these models into a global physiological theory of microbial growth.

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