This text is a guide to computational electrophysiology, the one area of computational and systems biology where computational and mathematical models have succeeded. It also covers bifurcation analysis, a tool to analyze highly nonlinear biological systems.

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.

1 A Very Short Trip on Dynamical Systems1.1 Difference Equations,Maps, and Linear Algebra1.2 Differential Equations, Vector Fields, and Phase Planes1.3 Linearization, Stabilities, Coordinate Transformation1.4 Nonlinear Dynamical Systems and Bifurcations1.5 Computational Bifurcation Analysis2 The Hodgkin–Huxley Theory of Neuronal Excitation2.1 What is a Neuron? Neuron is a Signal Converter2.2 The Hodgkin–Huxley Formulation of Excitable Cell Membranes2.3 Nonlinear Dynamical Analysis of the Original HH Equations3 Computational and Mathematical Models of Neurons3.1 Phase Plane Dynamics in the Context of Spiking Neuron3.2 Simple Models of Neurons and Neuronal Oscillators3.3 A Variant of the BVP Neuron Model3.4 Stochastic NeuronModels3.5 Stochastic Phase-Lockings and Bifurcations4 Whole System Analysis of Hodgkin–Huxley Systems4.1 Changing the Parameters: Sensitivity and Robustness4.2 Bifurcations of the Hodgkin–Huxley Neurons4.3 Two-Parameter Bifurcation Analysis of the HH Equations4.4 Numerical Bifurcation Analysis by XPPAUT5 Hodgkin–Huxley-Type Models of Cardiac Muscle Cells5.1 Action Potentials in a Heart5.2 Pacemaker Cell Model5.3 Ventricular Cell Model5.4 Other HH-TypeModels of Cardiac Cells

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.