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Biological systems inherently possess much ambiguity or uncertainty. Computational electrophysiology is the one area, from among the vast and rapidly growing discipline of computational and systems biology, in which computational or mathematical models have succeeded. This book provides a practical and quick guide to both computational electrophysiology and numerical bifurcation analysis. Bifurcation...
The challenge for the biosciences in the twenty-first century is to integrate genome sequencing information into a better understanding of biology, physiology, and human pathology. Such attempts at integration are moving the world toward a new generation of bioscience and bioengineering, where biological, physiological, and pathological information from humans and other living animals can be quantitatively...
The theory of dynamical systems has grown up to a very big discipline in mathematics now, and deals mainly with measurable (or ergodic), topological, and differentiable (or smooth) dynamics of systems. The dynamical system theory is related to other areas of mathematics also, for example, to the (analytic or geometric) singular perturbation theory. The singular perturbation theory is very important...
Hodgkin and Huxley (1952) proposed the famous Hodgkin–Huxley (hereinafter referred to as HH) equations which quantitatively describe the generation of action potential of squid giant axon, although there are still arguments against it (Connor et al. 1977; Strassberg and DeFelice 1993; Rush and Rinzel 1995; Clay 1998). The HH equations are important not only in that it is one of the most successful...
What are models? The HH equations (2.3) are often called a physiological model, whereas the models appeared in the following sections are simplified models or abstract models. However, there is no model in which any simplifications or abstractions have not been made. Of course, many features of real neurons are ignored even in the HH equations. All models have their applicability and limits to describe...
In Chap. 2, we have explored the dynamics of the original Hodgkin–Huxley equations of a squid giant axon where only the parameter Iext was changed. The HH equations, however, include various constants or parameters whose values were determined based on physiological experiments, and thus the values inherently possess a certain ambiguity. Also, the “constants” are not really constant but change temporally...
Following the HH formalism introduced in Chap. 2, various kinds of HH-type models of neurons and other excitable cells are proposed (Canavier et al. 1991; Chay and Keizer 1983; Cronin 1987; Gerber and Jakobsson 1993; Hayashi and Ishizuka 1992; Keener and Sneyd 1998; Noble 1995; Rinzel 1990; Traub et al. 1991), and are analyzed (Alexander and Cai 1991; Av-Ron 1994; Bertram 1994; Bertram et al. 1995;...
A major part of biology has become a class of physical and mathematical sciences. We have started to feel, though still a little suspicious yet, that it will become possible to predict biological events that will happen in the future of one’s life and to control some of them if desired so, based upon the understanding of genomic information of individuals and physical and chemical principles governing...
Cells are the basic units of biological structure and functions. They make up tissues and our bodies. A single cell includes organelles and intracellular solutions, and it is separated from outer environment of extracellular liquid surrounding the cell by its cell membrane (plasma membrane), generating differences in concentrations of ions and molecules including enzymes. The differences in charges...
Networks of cells form tissues and organs, where aggregations of cells operate as systems. It is similar to how single cells function as systems of protein networks, where, for example, ion channel currents of a single cell are integrated to produce a whole cell membrane potential. A cell in a network may behave differently from what it does alone. Dynamics of a single cell affect to those of others...
In this chapter, usage of the insilico platform is demonstrated. The insilico platform is composed of three blocks, i.e. insilicoML, insilicoIDE and insilicoDB. InsilicoML (ISML) (Asai et al. 2008) is a language specification based on XML to describe mathematical models of physiological functions. InsilicoIDE (ISIDE) (Kawazu et al. 2007; Suzuki et al. 2008, 2009) is a software program on which users...
Mathematical models, in particular dynamic system models of physiological functions in from molecules and cells to individual organisms play a key role for integration of vast stores of knowledge on physiology since they are capable of describing time evolution of biological system states quantitatively based upon physical and chemical principles or phenomenological logic governing system behavior...
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