This book is about the dynamics of neural systems and should be suitable for those with a background in mathematics, physics, or engineering who want to see how their knowledge and skill sets can be applied in a neurobiological context. No prior knowledge of neuroscience is assumed, nor is advanced understanding of all aspects of applied mathematics! Rather, models and methods are introduced in the context of a typical neural phenomenon and a narrative developed that will allow the reader to test their understanding by tackling a set of mathematical problems at the end of each chapter. The emphasis is on mathematical- as opposed to computational-neuroscience, though stresses calculation above theorem and proof. The book presents necessary mathematical material in a digestible and compact form when required for specific topics. The book has nine chapters, progressing from the cell to the tissue, and an extensive set of references. It includes Markov chain models for ions, differential equations for single neuron models, idealised phenomenological models, phase oscillator networks, spiking networks, and integro-differential equations for large scale brain activity, with delays and stochasticity thrown in for good measure. One common methodological element that arises throughout the book is the use of techniques from nonsmooth dynamical systems to form tractable models and make explicit progress in calculating solutions for rhythmic neural behaviour, synchrony, waves, patterns, and their stability. This book was written for those with an interest in applied mathematics seeking to expand their horizons to cover the dynamics of neural systems. It is suitable for a Masters level course or for postgraduate researchers starting in the field of mathematical neuroscience.
Author(s): Stephen Coombes, Kyle C. A. Wedgwood
Series: Texts in Applied Mathematics, 75
Publisher: Springer
Year: 2023
Language: English
Pages: 512
City: Cham
Preface
Acknowledgements
Contents
List of Boxes
1 Overview
1.1 The brain and a first look at neurodynamics
1.2 Tools of the (mathematical) trade
Remarks
2 Single neuron models
2.1 Introduction
2.2 Neuronal membranes
2.3 The Hodgkin–Huxley model
2.3.1 Batteries and the Nernst potential
2.3.2 Voltage-gated ion channels
2.3.3 Mathematical formulation
2.4 Reduction of the Hodgkin–Huxley model
2.5 The Morris–Lecar model
2.5.1 Hopf instability of a steady state
2.5.2 Saddle-node bifurcations
2.6 Other single neuron models
2.6.1 A plethora of conductance-based models
2.7 Quasi-active membranes
2.8 Channel models
2.8.1 A two-state channel
2.8.2 Multiple two-state channels
2.8.3 Large numbers of channels
2.8.4 Channels with more than two states
Remarks
Problems
3 Phenomenological models and their analysis
3.1 Introduction
3.2 The FitzHugh–Nagumo model
3.2.1 The mirrored FitzHugh–Nagumo model
3.3 Threshold models
3.4 Integrate-and-fire neurons
3.4.1 The leaky integrate-and-fire model
3.4.2 The quadratic integrate-and-fire model
3.4.3 Other nonlinear integrate-and-fire models
3.4.4 Spike response models
3.4.5 Dynamic thresholds
3.4.6 Planar integrate-and-fire models
3.4.7 Analysis of a piecewise linear integrate-and-fire model
3.5 Non-smooth Floquet theory
3.5.1 Poincaré maps
3.6 Lyapunov exponents
3.7 McKean models
3.7.1 A recipe for duck
Remarks
4 Axons, dendrites, and synapses
4.1 Introduction
4.2 Axons
4.2.1 Smooth nerve fibre models
4.2.2 A kinematic analysis of spike train propagation
4.2.3 Myelinated nerve fibre models
4.2.4 A Fire-Diffuse-Fire model
4.3 Dendrites
4.3.1 Cable modelling
4.3.2 Sum-over-trips
4.3.3 Compartmental modelling
4.4 Synapses
4.4.1 Chemical synapses
4.4.2 Electrical synapses
4.5 Plasticity
4.5.1 Short-term plasticity
4.5.2 Long-term plasticity
Remarks
Problems
5 Response properties of single neurons
5.1 Introduction
5.2 Mode-locking
5.3 Isochrons
5.4 Phase response curves
5.4.1 Characterising PRCs
5.5 The infinitesimal phase response curve
5.6 Characterising iPRCs
5.7 Phase response curves for non-smooth models
5.7.1 PRCs for integrate-and-fire neurons
5.7.2 iPRC for piecewise linear systems
5.8 Phase and amplitude response
5.8.1 Excitable systems
5.9 Stochastically forced oscillators
5.9.1 Phase equations for general noise
5.10 Noise-induced transitions
Remarks
Problems
6 Weakly coupled oscillator networks
6.1 Introduction
6.2 Phase equations for networks of oscillators
6.2.1 Two synaptically coupled nodes
6.2.2 Gap-junction coupled integrate-and-fire neurons
6.3 Stability of network phase-locked states
6.3.1 Synchrony
6.3.2 Coupled piecewise linear oscillators
6.3.3 The splay state
6.4 Small networks
6.5 Clustered states
6.5.1 Balanced (M,q) states
6.5.2 The unbalanced (N,q) cluster state
6.6 Remote synchronisation
6.7 Central pattern generators
6.8 Solutions in networks with permutation symmetry
6.8.1 A biharmonic example
6.8.2 Canonical invariant regions
6.9 Phase waves
6.10 Beyond weak coupling
Remarks
Problems
7 Strongly coupled spiking networks
7.1 Introduction
7.2 Simple neuron network models
7.3 A network of binary neurons
7.3.1 Release generated rhythms
7.4 The master stability function
7.4.1 MSF for synaptically interacting LIF networks
7.5 Analysis of the asynchronous state
7.6 Rate-based reduction of a spiking network
7.7 Synaptic travelling waves
7.7.1 Travelling wave analysis
7.7.2 Wave stability
Remarks
8 Population models
8.1 Introduction
8.2 Neural mass modelling: phenomenology
8.3 The Wilson–Cowan model
8.3.1 A piecewise linear Wilson–Cowan model
8.3.2 The Wilson–Cowan model with delays
8.3.3 The Curtu–Ermentrout model
8.4 The Jansen–Rit model
8.5 The Liley model
8.6 The Phenomenor model
8.7 The Tabak–Rinzel model
8.8 A spike density model
8.9 A next-generation neural mass model
8.9.1 Mapping between phase and voltage descriptions
8.10 Neural mass networks
8.10.1 Functional connectivity in a Wilson–Cowan network
Remarks
Problems
9 Firing rate tissue models
9.1 Introduction
9.2 Neural field models
9.3 The continuum Wilson–Cowan model
9.3.1 Power spectrum
9.3.2 Single effective population model
9.4 The brain wave equation
9.5 Travelling waves
9.5.1 Front construction
9.5.2 Front stability (Evans function)
9.6 Hallucinations
9.7 Amplitude equations
9.8 Interface dynamics
9.8.1 One spatial dimension
9.8.2 Two spatial dimensions
Remarks
Problems
Appendix A Stochastic calculus
A.1 Modelling noise
A.2 Random processes and sample paths
A.3 The Wiener process
A.4 Langevin equations
A.5 Stochastic integrals
A.6 Comparison of the Itô and Stratonovich integrals
A.7 Itô's formula
A.8 Coloured noise
A.9 Simulating stochastic processes
A.10 The Fokker–Planck equation
A.10.1 The backward Kolmogorov equation
A.11 Transforming probability distributions
Appendix B Model details
B.1 The Connor–Stevens model
B.2 The Wang–Buzsáki model
B.3 The Golomb–Amitai model
B.4 The Wang thalamic relay neuron model
B.5 The Pinsky–Rinzel model
Appendix References
Index
