Book contents
- Frontmatter
- Contents
- Preface
- 1 On a conjecture by A. Durfee
- 2 On normal embedding of complex algebraic surfaces
- 3 Local Euler obstruction, old and new, II
- 4 Branching of periodic orbits in reversible Hamiltonian systems
- 5 Topological invariance of the index of a binary differential equation
- 6 About the existence of Milnor fibrations
- 7 Counting hypersurfaces invariant by one-dimensional complex foliations
- 8 A note on topological contact equivalence
- 9 Bi-Lipschitz equivalence, integral closure and invariants
- 10 Solutions to PDEs and stratification conditions
- 11 Real integral closure and Milnor fibrations
- 12 Surfaces around closed principal curvature lines, an inverse problem
- 13 Euler characteristics and a typical values
- 14 Answer to a question of Zariski
- 15 Projections of timelike surfaces in the de Sitter space
- 16 Spacelike submanifolds of codimension at most two in de Sitter space
- 17 The geometry of Hopf and saddle-node bifurcations for waves of Hodgkin-Huxley type
- 18 Global classifications and graphs
- 19 Real analytic Milnor fibrations and a strong Łojasiewicz inequality
- 20 An estimate of the degree of ℒ-determinacy by the degree of A-determinacy for curve germs
- 21 Regularity of the transverse intersection of two regular stratifications
- 22 Pairs of foliations on surfaces
- 23 Bi-Lipschitz equisingularity
- 24 Gaffney's work on equisingularity
- 25 Singularities in algebraic data acquisition
17 - The geometry of Hopf and saddle-node bifurcations for waves of Hodgkin-Huxley type
Published online by Cambridge University Press: 07 September 2011
- Frontmatter
- Contents
- Preface
- 1 On a conjecture by A. Durfee
- 2 On normal embedding of complex algebraic surfaces
- 3 Local Euler obstruction, old and new, II
- 4 Branching of periodic orbits in reversible Hamiltonian systems
- 5 Topological invariance of the index of a binary differential equation
- 6 About the existence of Milnor fibrations
- 7 Counting hypersurfaces invariant by one-dimensional complex foliations
- 8 A note on topological contact equivalence
- 9 Bi-Lipschitz equivalence, integral closure and invariants
- 10 Solutions to PDEs and stratification conditions
- 11 Real integral closure and Milnor fibrations
- 12 Surfaces around closed principal curvature lines, an inverse problem
- 13 Euler characteristics and a typical values
- 14 Answer to a question of Zariski
- 15 Projections of timelike surfaces in the de Sitter space
- 16 Spacelike submanifolds of codimension at most two in de Sitter space
- 17 The geometry of Hopf and saddle-node bifurcations for waves of Hodgkin-Huxley type
- 18 Global classifications and graphs
- 19 Real analytic Milnor fibrations and a strong Łojasiewicz inequality
- 20 An estimate of the degree of ℒ-determinacy by the degree of A-determinacy for curve germs
- 21 Regularity of the transverse intersection of two regular stratifications
- 22 Pairs of foliations on surfaces
- 23 Bi-Lipschitz equisingularity
- 24 Gaffney's work on equisingularity
- 25 Singularities in algebraic data acquisition
Summary
Abstract
We study a class of ordinary differential equations extending the Hodgkin-Huxley equations for the nerve impulse under a travelling wave condition. We obtain a geometrical description of the subset in parameter space were the equilibria lose stability. This may happen in two ways: first, the linearisation around the equilibrium may have a pair of purely imaginary eigenvalues with multiplicity j. This happens for parameters in the set Nj, the possible sites for Hopf bifurcation, where a branch of periodic solutions is created. Second, a real eigenvalue may change sign at the site for a possible saddle-node bifurcation. This happens at parameter values in the sets Kl where there is a zero eigenvalue of multiplicity l. We show that the sets Nj and Kl are singular ruled submanifolds of the parameter space RM+3 with rulings contained in codimension 1 affine subspaces. The subset of regular points in Nj has codimension 2j - 1 in RM+3 and its rulings have codimension 2j + 1. The subset of regular points in Kl has codimension l in RM+3 with rulings of codimension l + 1. We illustrate the result with a numerical plot of the surface N2 for the Hodgkin-Huxley equations simplified to have M = 2.
Introduction – equations of Hodgkin-Huxley type
The Hodgkin-Huxley model [2] describes the variation of the difference x ∈ R of the electrical potential across a nerve cell membrane, as a function of the distance s ∈ R along an axon and of the time t ∈ R, for an electrical stimulus of intensity I ∈ R.
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- Chapter
- Information
- Real and Complex Singularities , pp. 229 - 245Publisher: Cambridge University PressPrint publication year: 2010