This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology "Dynamical Systems in Neuroscience" presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties The book introduces dynamical systems starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems Each chapter proceeds from the simple to the complex, and provides sample problems at the end The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum - or taught by math or physics department in a way that is suitable for students of biology This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience

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Dynamical Systems in Neuroscience

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Introduction to the Modern Theory of Dynamical Systems

Existence of 'breathers', that is, time-periodic, spatially localized solutions, is proved for a broad range of time-reversible or Hamiltonian networks of weakly coupled oscillators. Some of their properties are discussed, some generalizations suggested, and several open questions raised.

https://iopscience.iop.org/article/10.1088/0951-7715/7/6/006/pdf

Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators

Discrete breathers — Advances in theory and applications

Breathers in nonlinear lattices: existence, linear stability and quantization

Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos

Equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for Frenkel-Kontorova models

Slow sweep through a period-doubling cascade: delayed bifurcations and renormalisation

Complete proofs are given for some claims of Middleton and of Floria and Mazo about the asymptotic behaviour of chains of balls and springs in a tilted periodic potential and generalizations, under gradient dynamics. AMS classification number: 58F22

Gradient dynamics of tilted Frenkel-Kontorova models

Numerical work of many people on the bifurcations of uniformly travelling water waves (two-dimensional irrotational gravity waves on inviscid fluid of infinite depth) suggests that uniformly travelling water waves have a reversible Hamiltonian formulation, where the role of time is played by horizontal position in the wave frame. In this paper such a formulation is presented. Based on this viewpoint, some insights are given into bifurcations from Stokes’ family of periodic waves. It is demonstrated numerically that there is a ‘fold point’ at amplitude A0 ≈ 0.40222. Assuming non-degeneracy of the fold and existence of an associated centre manifold, this explains why a sequence of p/q-bifurcations occurs on one side of A0, with 0 < p/q [les ] ½, in the order of the rationals. Secondly, it explains why no symmetry-breaking bifurcation is observed at A0, contrary to the expectations of some. Thirdly, it explains why the bifurcation tree for periodic uniformly travelling waves looks so much like that for the area-preserving Henon map. Fourthly, it leads to predictions of a rich variety of spatially quasi-periodic, heteroclinic and chaotic waves.

C. Baesens

Papers

Three coupled oscillators: mode-locking, global bifurcations and toroidal chaos

Equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for Frenkel-Kontorova models

Slow sweep through a period-doubling cascade: delayed bifurcations and renormalisation

Gradient dynamics of tilted Frenkel-Kontorova models

Uniformly travelling water waves from a dynamical systems viewpoint : some insights into bifurcations from Stokes' family