Shinji Doi & Junko Inoue 
Computational Electrophysiology 

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 textbook provides a practical and quick guide to both computational electrophysiology and numerical bifurcation analysis. Bifurcation analysis is a very powerful tool for the analysis of such highly nonlinear biological systems. Bifurcation theory provides a way to analyze the effect of a parameter change on a system and to detect a critical parameter value when the qualitative nature of the system changes. Included in this work are many examples of numerical computations of bifurcation analysis of various models as well as mathematical models with different abstraction levels from neuroscience and electrophysiology. This volume will benefit graduate and undergraduate students as well as researchers in diverse fields of science.

Table of Content

1 A Very Short Trip on Dynamical Systems

1.1 Difference Equations, Maps, and Linear Algebra

1.2 Differential Equations, Vector Fields, and Phase Planes

1.3 Linearization, Stabilities, Coordinate Transformation

1.4 Nonlinear Dynamical Systems and Bifurcations

1.5 Computational Bifurcation Analysis
2 The Hodgkin–Huxley Theory of Neuronal Excitation

2.1 What is a Neuron? Neuron is a Signal Converter

2.2 The Hodgkin–Huxley Formulation of Excitable Cell Membranes

2.3 Nonlinear Dynamical Analysis of the Original HH Equations
3 Computational and Mathematical Models of Neurons

3.1 Phase Plane Dynamics in the Context of Spiking Neuron

3.2 Simple Models of Neurons and Neuronal Oscillators

3.3 A Variant of the BVP Neuron Model

3.4 Stochastic Neuron Models

3.5 Stochastic Phase-Lockings and Bifurcations
4 Whole System Analysis of Hodgkin–Huxley Systems

4.1 Changing the Parameters: Sensitivity and Robustness

4.2 Bifurcations of the Hodgkin–Huxley Neurons

4.3 Two-Parameter Bifurcation Analysis of the HH Equations

4.4 Numerical Bifurcation Analysis by XPPAUT
5 Hodgkin–Huxley-Type Models of Cardiac Muscle Cells

5.1 Action Potentials in a Heart

5.2 Pacemaker Cell Model

5.3 Ventricular Cell Model

5.4 Other HH-Type Models of Cardiac Cells