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Beginning with realistic mathematical or verbal models of physical
or biological phenomena, the author derives tractable models for
further mathematical analysis or computer simulations. For the most
part, derivations are based on perturbation methods, and the
majority of the text is devoted to careful derivations of implicit
function theorems, the method of averaging, and quasi-static state
approximation methods. The duality between stability and
perturbation is developed and used, relying heavily on the concept
of stability under persistent disturbances. Relevant topics about
linear systems, nonlinear oscillations, and stability methods for
difference, differential-delay, integro-differential and ordinary
and partial differential equations are developed throughout the
book. For the second edition, the author has restructured the
chapters, placing special emphasis on introductory materials in
Chapters 1 and 2 as distinct from presentation materials in
Chapters 3 through 8. In addition, more material on bifurcations
from the point of view of canonical models, sections on randomly
perturbed systems, and several new computer simulations have been
added.
Devoted to local and global analysis of weakly connected systems
with applications to neurosciences, this book uses bifurcation
theory and canonical models as the major tools of analysis. It
presents a systematic and well motivated development of both weakly
connected system theory and mathematical neuroscience, addressing
bifurcations in neuron and brain dynamics, synaptic organisations
of the brain, and the nature of neural codes. The authors present
classical results together with the most recent developments in the
field, making this a useful reference for researchers and graduate
students in various branches of mathematical neuroscience.
This book develops methods for describing random dynamical systems,
and it illustrats how the methods can be used in a variety of
applications. Appeals to researchers and graduate students who
require tools to investigate stochastic systems.
The result of lectures given by the authors at New York University,
the University of Utah, and Michigan State University, the material
is written for students who have had only one term of calculus, but
it contains material that can be used in modeling courses in
applied mathematics at all levels through early graduate courses.
Numerous exercises are given as well as solutions to selected
exercises, so as to lead readers to discover interesting extensions
of that material. Throughout, illustrations depict physiological
processes, population biology phenomena, corresponding models, and
the results of computer simulations. Topics covered range from
population phenomena to demographics, genetics, epidemics and
dispersal; in physiological processes, including the circulation,
gas exchange in the lungs, control of cell volume, the renal
counter-current multiplier mechanism, and muscle mechanics; to
mechanisms of neural control. Each chapter is graded in difficulty,
so a reading of the first parts of each provides an elementary
introduction to the processes and their models.
Mathematics in Medicine and the Life Sciences grew from lectures given by the authors at New York University, the University of Utah, and Michigan State University. The material is written for students who have had but one term of calculus, but it contains material that can be used in modeling courses in applied mathematics at all levels through early graduate courses. Numerous exercises are given as well, and solutions to selected exercises are included. Numerous illustrations depict physiological processes, population biology phenomena, models of them, and the results of computer simulations. Mathematical models and methods are becoming increasingly important in medicine and the life sciences. This book provides an introduction to a wide diversity of problems ranging from population phenomena to demographics, genetics, epidemics and dispersal; in physiological processes, including the circulation, gas exchange in the lungs, control of cell volume, the renal counter-current multiplier mechanism, and muscle mechanics; to mechanisms of neural control. Each chapter is graded in difficulty, so a reading of the first parts of each provides an elementary introduction to the processes and their models. Materials that deal with the same topics but in greater depth are included later. Finally, exercises and some solutions are given to test the reader on important parts of the material in teh text, or to lead the reader to the discovery of interesting extensions of that material.
This book covers the impact of noise on models that are widely used in science and engineering applications. It applies perturbed methods, which assume noise changes on a faster time or space scale than the system being studied. The book is written in two parts. The first part presents a careful development of mathematical methods needed to study random perturbations of dynamical systems. The second part presents non-random problems in a variety of important applications. Such problems are reformulated to account for both external and system random noise.
Beginning with realistic mathematical or verbal models of physical
or biological phenomena, the author derives tractable mathematical
models that are amenable to further mathematical analysis or to
elucidating computer simulations. For the most part, derivations
are based on perturbation methods. Because of this, the majority of
the text is devoted to careful derivations of implicit function
theorems, the method of averaging, and quasi-static state
approximation methods. The duality between stability and
perturbation is developed and used, relying heavily on the concept
of stability under persistent disturbances. This explains why
stability results developed for quite simple problems are often
useful for more complicated, even chaotic, ones. Relevant topics
about linear systems, nonlinear oscillations, and stability methods
for difference, differential-delay, integro- differential and
ordinary and partial differential equations are developed
throughout the book. For the second edition, the author has
restructured the chapters, placing special emphasis on introductory
materials in Chapters 1 and 2 as distinct from presentation
materials in Chapters 3 through 8. In addition, more material on
bifurcations from the point of view of canonical models, sections
on randomly perturbed systems, and several new computer simulations
have been added.
Devoted to local and global analysis of weakly connected systems with applications to neurosciences, this book uses bifurcation theory and canonical models as the major tools of analysis. It presents a systematic and well motivated development of both weakly connected system theory and mathematical neuroscience, addressing bifurcations in neuron and brain dynamics, synaptic organisations of the brain, and the nature of neural codes. The authors present classical results together with the most recent developments in the field, making this a useful reference for researchers and graduate students in various branches of mathematical neuroscience.
Complex diseases involve most aspects of population biology,
including genetics, demographics, epidemiology, and ecology.
Mathematical methods, including differential, difference, and
integral equations, numerical analysis, and random processes, have
been used effectively in all of these areas. The aim of this book
is to provide sufficient background in such mathematical and
computational methods to enable the reader to better understand
complex systems in biology, medicine, and the life sciences. It
introduces concepts in mathematics to study population phenomena
with the goal of describing complicated aspects of a disease, such
as malaria, involving several species. The book is based on a
graduate course in computational biology and applied mathematics
taught at the Courant Institute of Mathematical Sciences in fall
2010. The mathematical level is kept to essentially advanced
undergraduate mathematics, and the results in the book are intended
to provide readers with tools for performing more in-depth analysis
of population phenomena.
This book is based on a course on advanced topics in differential
equations given in Spring 2010 at the Courant Institute of
Mathematical Sciences. It describes aspects of mathematical
modeling, analysis, computer simulation, and visualization in the
mathematical sciences and engineering that involve singular
perturbations. There is a large literature devoted to singular
perturbation methods for ordinary and partial differential
equations, but there are not many studies that deal with difference
equations, Volterra integral equations, and purely nonlinear
gradient systems where there is no dominant linear part. Designed
for a one-semester course for students in applied mathematics, it
is the purpose of this book to present sufficient rigorous methods
and examples to position the reader to investigate singular
perturbation problems in such equations. Titles in this series are
co-published with the Courant Institute of Mathematical Sciences at
New York University.|This book is based on a course on advanced
topics in differential equations given in Spring 2010 at the
Courant Institute of Mathematical Sciences. It describes aspects of
mathematical modeling, analysis, computer simulation, and
visualization in the mathematical sciences and engineering that
involve singular perturbations. There is a large literature devoted
to singular perturbation methods for ordinary and partial
differential equations, but there are not many studies that deal
with difference equations, Volterra integral equations, and purely
nonlinear gradient systems where there is no dominant linear part.
Designed for a one-semester course for students in applied
mathematics, it is the purpose of this book to present sufficient
rigorous methods and examples to position the reader to investigate
singular perturbation problems in such equations. Titles in this
series are co-published with the Courant Institute of Mathematical
Sciences at New York University.
In studying the dynamics of populations, whether of animals, plants
or cells, it is crucial to allow for intrinsic delays, due to such
things as gestation, maturation or transport. This book is
concerned with one of the fundamental questions in the analysis of
the effect of delays, namely determining whether they effect the
stability of steady states. The analysis is presented for one or
two such delays treated both as discrete, where an event which
occurred at a precise time in the past has an effect now, and
distributed, where the delay is averaged over the population's
history. Both of these types occur in biological contexts. The
method used to tackle these questions is linear stability analysis
which leads to an understanding of the local stability. By avoiding
global questions, the author has kept the mathematical
prerequisites to a minimum, essentially advanced calculus and
ordinary differential equations.
This book describes the signal processing aspects of neural
networks. It begins with a presentation of the necessary background
material in electronic circuits, mathematical modeling and
analysis, signal processing, and neurosciences, and then proceeds
to applications. These applications include small networks of
neurons, such as those used in control of warm-up and flight in
moths and control of respiration during exercise in humans. Next, a
theory of mnemonic surfaces is developed and studied and material
on pattern formation and cellular automata is presented. Finally,
large networks are studied, such as the thalamus-reticular complex
circuit, believed to be involved in focusing attention, and the
development of connections in the visual cortex. Additional
material is also provided about nonlinear wave propagation in
networks. This book will serve as an excellent text for advanced
undergraduates and graduates in the physical sciences, mathematics,
engineering, medicine and life sciences.
In studying the dynamics of populations, whether of animals, plants
or cells, it is crucial to allow for intrinsic delays, due to such
things as gestation, maturation or transport. This book is
concerned with one of the fundamental questions in the analysis of
the effect of delays, namely determining whether they effect the
stability of steady states. The analysis is presented for one or
two such delays treated both as discrete, where an event which
occurred at a precise time in the past has an effect now, and
distributed, where the delay is averaged over the population's
history. Both of these types occur in biological contexts. The
method used to tackle these questions is linear stability analysis
which leads to an understanding of the local stability. By avoiding
global questions, the author has kept the mathematical
prerequisites to a minimum, essentially advanced calculus and
ordinary differential equations.
This book describes signal processing aspects of neural networks, how we receive and assess information. Beginning with a presentation of the necessary background material in electronic circuits, mathematical modeling and analysis, signal processing, and neurosciences, it proceeds to applications. These applications include small networks of neurons, such as those used in control of warm-up and flight in moths and control of respiration during exercise in humans. Next, Hoppensteadt develops a theory of mnemonic surfaces and presents material on pattern formation and cellular automata. Finally, the text addresses the large networks, such as the thalamus-reticular complex circuit, that may be involved in focusing attention, and the development of connections in the visual cortex. This book will serve as an excellent text for advanced undergraduates and graduates in the physical sciences, mathematics, engineering, medicine and life sciences.
An introduction to mathematical methods used in the study of population phenomena including models of total population and population age structure, models of random population events presented in terms of Markov chains, and methods used to uncover qualitative behavior of more complicated difference equations.
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