By Anne Beuter, Leon Glass, Michael C. Mackey, Michele S. Titcombe
This booklet offers with the applying of arithmetic in modeling and knowing physiological structures, specially these regarding rhythms. it's divided approximately into sections. within the first a part of the publication, the authors introduce principles and methods from nonlinear dynamics which are appropriate to the research of organic rhythms. the second one half comprises 5 in-depth case stories during which the authors use the theoretical instruments built prior to enquire a few physiological methods: the dynamics of excitable nerve and cardiac tissue, resetting and entrainment of organic oscillators, the consequences of noise and time hold up at the scholar mild reflex, pathologies linked to blood mobilephone replication, and Parkinsonian tremor. One novel characteristic of the ebook is the inclusion of classroom-tested machine workouts all through, designed to shape a bridge among the mathematical thought and physiological experiments.
This e-book might be of curiosity to scholars and researchers within the common and actual sciences desirous to find out about the complexities and subtleties of physiological structures from a mathematical perspective.
The authors are participants of the Centre for Nonlinear Dynamics in body structure and medication. the fabric during this e-book used to be built to be used in classes and used to be offered in 3 summer time colleges run by means of the authors in Montreal.
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Additional info for Nonlinear Dynamics in Physiology and Medicine
13. 21). The initial conditions are g(O) =O and g'(O) >O. neighborhood of a fixed point, in which case the associated linear system will have a Jacobian matrix as matrix of coefficients. The classification of the type and stability of fixed points is not as clean and elegant as in the case of two-dimensional systems, and, of course, phase spaces of high dimension are difficult to represent graphically! The matrix approach is necessary in these cases to carry out a systematic investigation. 24) = h(x,y,z), there will be three eigenvalues to characterize the stability of a fixed point, corresponding to three roots of a cubic polynomial, which is given as the determinant of a matrix.
17) is well known in linear algebra as the eigenvalue problem. 18) where det denotes the determinant of the matrix. The determinant of a matrix with 2 rows and 2 columns is defined by det 1 ~ ~ 1= ad - bc. 18) is called the characteristic equation. 18) yields >. 2 - (A+ D)>. 19) which is equivalent to equation (2. 18). 1 = --2- + J(A-D)2+4BC 2 ' >. 2 = A+D _ J(A-D)2+4BC_ 2 2 The roots of the characteristic equation are called the characteristic roots, or eigenvalues. 15) are given by + K 2 e>-.
9) This equation states that the rate of change of the concentration c as a function of time is the difference between the intake i/V and the removal kc. 9) satisfies c* = i/kV. 9) has a derivative (with respect to c) that is the negative constant -k. 9). 10. For an arbitrary nonlinear function f(x), it might be difficult or impossible to determine a solution analytically. 10. Drug distribution evolution in the one-compartment model. namics. For example, any point for which f(x) =O is a fixed point.