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Book•DOI•

Mathematical foundations of neuroscience

G. Bard Ermentrout, David Terman
01 Jan 2010-Vol. 35
TL;DR: The Hodgkin-Huxley Equations are applied to the model of Neuronal Networks to describe the “spatially distributed” networks.
Abstract: The Hodgkin-Huxley Equations.- Dendrites.- Dynamics.- The Variety of Channels.- Bursting Oscillations.- Propagating Action Potentials.- Synaptic Channels.- Neural Oscillators: Weak Coupling.- Neuronal Networks: Fast/Slow Analysis.- Noise.- Firing Rate Models.- Spatially Distributed Networks.

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G. Bard Ermentrout • David
H.
Terman
Mathematical Foundations
of Neuroscience
43 Springer
Contents
1
The Hodgkin—Huxley Equations
.................................................................
1
1.1
The Resting Potential
...................................................................
1
1.2
The Nernst Equation
.....................................................................
3
1.3
The Goldman—Hodgkin—Katz Equation
.........................................
5
1.4
Equivalent Circuits: The Electrical Analogue
................................
8
1.5
The Membrane Time Constant
.....................................................
11
1.6
The Cable Equation
......................................................................
13
1.7
The Squid Action Potential
...........................................................
16
1.8
Voltage-Gated Channels
...............................................................
18
1.9
Hodgkin—Huxley Model
...............................................................
20
1.10
The Action Potential Revisited
.....................................................
25
1.11
Bibliography
.................................................................................
27
1.12
Exercises
......................................................................................
28
2
Dendrites
........................................................................................................
29
2.1
Multiple Compartments
................................................................
29
2.2
The Cable Equation
......................................................................
33
2.3
The Infinite Cable
.........................................................................
34
2.4
Finite and Semi-infinite Cables
.....................................................
36
2.5
Branching and Equivalent Cylinders
.............................................
38
2.6
An Isolated Junction
.....................................................................
40
2.7
Dendrites with Active Processes
....................................................
42
2.8
Concluding Remarks
.....................................................................
45
2.9
Bibliography
.................................................................................
45
2.10
Exercises
.......................................................................................
45
3
Dynamics
........................................................................................................
49
3.1
Introduction to Dynamical Systems
...............................................
49
3.2
The Morris—Lecar Model
..............................................................
49
3.3
The Phase Plane
............................................................................
51
3.3.1
Stability of Fixed Points
...................................................
52
3.3.2
Excitable Systems
...........................................................
53
3.3.3
Oscillations
.....................................................................
55
ix
x
3.4
3.5
Contents
Bifurcation Analysis
......................................................................
56
3.4.1
The Hopf Bifurcation
........................................................
56
3.4.2
Saddle—Node an a Limit Cycle
........................................
58
3.4.3
Saddle—Homoclinic Bifurcation
.......................................
60
3.4.4
Class
I
and Class II
..........................................................
62
Bifurcation Analysis of the Hodgkin—Huxley Equations
.................
63
3.6
Reduction of the Hodgkin—Huxley Model to a Two-Variable Model 66
3.7
FitzHugh—Nagumo Equations
........................................................
69
3.8
Bibliography
..................................................................................
70
3.9
Exercises
.......................................................................................
70
4
The Variety of Channels
........................................................................
77
4.1
Overview
.......................................................................................
77
4.2
Sodium Channels
...........................................................................
78
4.3
Calcium Channels
..........................................................................
80
4.4
Voltage-Gated Potassium Channels
...............................................
82
4.4.1
A-Current
........................................................................
83
4.4.2
M-Current
........................................................................
85
4.4.3
The Inward Rectifier
.........................................................
86
4.5
Sag
................................................................................................
87
4.6
Currents and lonic Concentrations
.................................................
88
4.7
Calcium-Dependent Channels
........................................................
90
4.7.1
Calcium Dependent Potassium:
The Afterhyperpolarization
..............................................
90
4.7.2
Calcium-Activated Nonspecific Cation Current
.................
93
4.8
Bibliography
..................................................................................
95
4.9
Exercises
.......................................................................................
95
4.10
Projects
.........................................................................................
100
5
Bursting Oscillations
..............................................................................
103
5.1
Introduction to Bursting
.................................................................
103
5.2
Square-Wave Bursters
....................................................................
105
5.3
Elliptic Bursting
............................................................................
111
5.4 Parabolic
Bursting
..........................................................................
114
5.5
Classification of Bursters
...............................................................
117
5.6 Chaotic Dynamics
.........................................................................
118
5.6.1
Chaos in Square-Wave Bursting Models
..........................
118
5.6.2
Symbolic Dynamics
.........................................................
121
5.6.3
Bistability and the Blue-Sky Catastrophe
.........................
123
5.7
Bibliography
125
5.8
Exercises
.......................................................................................
126
Contents
6
Propagating Action Potentials
.....................................................
xi
129
6.1
Traveling Waves and Homoclinic Orbits
................................
130
6.2
Scalar Bistable Equations
........................................................
132
6.2.1
Numerical Shooting
..................................................
135
6.3
Singular Construction of Waves
..............................................
136
6.3.1 Wave Trains
................................................................
139
6.4
Dispersion Relations ................................................................
139
6.4.1
Dispersion Kinematics ..............................................
141
6.5
Morris—Lecar Revisited and Shilnikov Dynamics
.................
141
6.5.1
Class II Dynamics ......................................................
142
6.5.2
Class
I
Dynamics
.......................................................
143
6.6 Stability of the Wave
................................................................
145
6.6.1
Linearization
..............................................................
146
6.6.2
The Evans Function
...................................................
147
6.7
Myelinated Axons and Discrete Diffusion
.............................
149
6.8
Bibliography
..............................................................................
151
6.9
Exercises
....................................................................................
152
7
Synaptic Channels
.......................................................................
157
7.1
Synaptic Dynamics
...................................................................
158
7.1.1
Glutamate
...................................................................
161
7.1.2 y-Aminobutyric Acid
.................................................
162
7.1.3
Gap or Electrical Junctions
........................................
164
7.2
Short-Term Plasticity
................................................................
164
7.2.1
Other Models
.............................................................
167
7.3
Long-Term Plasticity
................................................................
168
7.4
Bibliography
..............................................................................
169
7.5
Exercises
....................................................................................
169
8
Neural Oscillators: Weak Coupling
...........................................
171
8.1
Neural Oscillators, Phase, and Isochrons ...............................
172
8.1.1
Phase Resetting and Adjoints
....................................
174
8.1.2
The Adjoint .................................................................
177
8.1.3
Examples of Adjoints
..................................................
178
8.1.4 Bifurcations and Adjoints
..........................................
181
8.1.5
Spike-Time Response Curves
....................................
186
8.2
Who Cares About Adjoints?
.....................................................
187
8.2.1
Relationship of the Adjoint and the Response
to Inputs
.......................................................................
187
8.2.2
Forced Oscillators
......................................................
189
8.2.3
Coupled Oscillators
.....................................................
193
8.2.4
Other Map Models
......................................................
199
8.3
Weak Coupling
..........................................................................
202
8.3.1
Geometrie Idea
.............................................................
203
8.3.2
Applications of Weak Coupling
.................................
205
xii
8.3.3
Synaptic Coupling near Bifurcations
......................
8.3.4
Small Central Pattern Generators
............................
8.3.5
Linear Arrays of Cells
............................................
8.3.6
Two-Dimensional Arrays
........................................
8.3.7
All-to-All Coupling
................................................
Contents
206
208
213
217
219
8.4
Pulse-Coupled Networks: Solitary Waves
............................
223
8.4.1
Integrate-and-Fire Model
........................................
226
8.4.2
Stability
.................................................................
229
8.5
Bibliography
........................................................................
229
8.6
Exercises
.............................................................................
229
8.7
Projects
...............................................................................
238
9
Neuronal Networks: Fast/Slow Analysis
.........................................
241
9.1
Introduction
.........................................................................
241
9.2
Mathematical Models for Neuronal Networks
......................
242
9.2.1
Individual Cells
......................................................
242
9.2.2
Synaptic Connections
............................................
243
9.2.3
Network Architecture
............................................
245
9.3
Examples of Firing Patterns
................................................
246
9.4
Singular Construction of the Action Potential
.....................
249
9.5
Synchrony with Excitatory Synapses
...................................
254
9.6
Postinhibitory Rebound
.......................................................
258
9.6.1
Two Mutually Coupled Cells
..................................
258
9.6.2
Clustering
..............................................................
260
9.6.3 Dynamic Clustering
..............................................
260
9.7
Antiphase Oscillations with Excitatory Synapses
................
262
9.7.1
Existence of Antiphase Oscillations
.......................
263
9.7.2
Stability of Antiphase Oscillations
........................
266
9.8
Almost-Synchronous Solutions
...........................................
269
9.8.1
Almost Synchrony with Inhibitory Synapses
.........
269
9.8.2
Almost Synchrony with Excitatory Synapses
.........
271
9.8.3
Synchrony with Inhibitory Synapses
.....................
274
9.9
Slow Inhibitory Synapses
....................................................
275
9.9.1
Fast/Slow Decomposition
......................................
275
9.9.2
Antiphase Solution
...............................................
276
9.9.3
Suppressed Solutions
.............................................
278
9.10
Propagating Waves
.............................................................
278
9.11
Bibliography
.......................................................................
282
9.12
Exercises
............................................................................
282
10
Noise
......................................................................................................
285
10.1
Stochastic Differential Equations
........................................
287
10.1.1
The Wiener Process
..............................................
288
10.1.2
Stochastic Integrals
...............................................
289
10.1.3
Change of Variables: Itö's Formula
........................
289
Citations
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Journal Article•DOI•
On the nature of seizure dynamics

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Viktor K. Jirsa1, Viktor K. Jirsa2, William C. Stacey3, Pascale P. Quilichini2, Pascale P. Quilichini1, Anton Ivanov2, Anton Ivanov1, Christophe Bernard2, Christophe Bernard1 •
French Institute of Health and Medical Research1, Aix-Marseille University2, University of Michigan3
01 Aug 2014-Brain
TL;DR: A taxonomy of seizures based on first principles is established and only five state variables linked by integral-differential equations are sufficient to describe the onset, time course and offset of ictal-like discharges as well as their recurrence.
Abstract: Seizures can occur spontaneously and in a recurrent manner, which defines epilepsy; or they can be induced in a normal brain under a variety of conditions in most neuronal networks and species from flies to humans. Such universality raises the possibility that invariant properties exist that characterize seizures under different physiological and pathological conditions. Here, we analysed seizure dynamics mathematically and established a taxonomy of seizures based on first principles. For the predominant seizure class we developed a generic model called Epileptor. As an experimental model system, we used ictal-like discharges induced in vitro in mouse hippocampi. We show that only five state variables linked by integral-differential equations are sufficient to describe the onset, time course and offset of ictal-like discharges as well as their recurrence. Two state variables are responsible for generating rapid discharges (fast time scale), two for spike and wave events (intermediate time scale) and one for the control of time course, including the alternation between ‘normal’ and ictal periods (slow time scale). We propose that normal and ictal activities coexist: a separatrix acts as a barrier (or seizure threshold) between these states. Seizure onset is reached upon the collision of normal brain trajectories with the separatrix. We show theoretically and experimentally how a system can be pushed toward seizure under a wide variety of conditions. Within our experimental model, the onset and offset of ictal-like discharges are well-defined mathematical events: a saddle-node and homoclinic bifurcation, respectively. These bifurcations necessitate a baseline shift at onset and a logarithmic scaling of interspike intervals at offset. These predictions were not only confirmed in our in vitro experiments, but also for focal seizures recorded in different syndromes, brain regions and species (humans and zebrafish). Finally, we identified several possible biophysical parameters contributing to the five state variables in our model system. We show that these parameters apply to specific experimental conditions and propose that there exists a wide array of possible biophysical mechanisms for seizure genesis, while preserving central invariant properties. Epileptor and the seizure taxonomy will guide future modeling and translational research by identifying universal rules governing the initiation and termination of seizures and predicting the conditions necessary for those transitions.

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Journal Article•DOI•
Spatiotemporal dynamics of continuum neural fields

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Paul C. Bressloff1•
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27 Jan 2012-Journal of Physics A
TL;DR: This work surveys recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields, an important example of spatially extended excitable systems with nonlocal interactions.
Abstract: We survey recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields. Neural fields model the large-scale dynamics of spatially structured biological neural networks in terms of nonlinear integrodifferential equations whose associated integral kernels represent the spatial distribution of neuronal synaptic connections. They provide an important example of spatially extended excitable systems with nonlocal interactions and exhibit a wide range of spatially coherent dynamics including traveling waves oscillations and Turing-like patterns.

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Cites background from "Mathematical foundations of neurosc..."

  • ...We then highlight a sequence of approximations that can be made to reduce a network of spiking neurons to an effective rate-based model, distinguishing between voltage-based and activity-based versions along the lines of Ermentrout [15, 31]....

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The stabilized supralinear network: a unifying circuit motif underlying multi-input integration in sensory cortex.

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Cites background from "Mathematical foundations of neurosc..."

  • ...Wemodel neuronal firing rates, rather than action potential (‘‘spike’’) generation, which suffice to understand many aspects of network behavior when spikes are fired irregularly and asynchronously (Ermentrout and Terman, 2010; Murphy and Miller, 2009)....

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Peter Ashwin1, Stephen Coombes2, Rachel Nicks3•
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06 Jan 2016-The Journal of Mathematical Neuroscience
TL;DR: In this article, a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understand network dynamics in neuroscience.
Abstract: The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.

259 citations

Journal Article•DOI•
Macroscopic Description for Networks of Spiking Neurons

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Ernest Montbrió1, Diego Pazó2, Alex Roxin•
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19 Jun 2015-Physical Review X
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Abstract: Understanding memory and decision making in the human brain requires generating models of how neurons fire. Using ordinary differential equations, researchers formulate an exact firing rate description for an ensemble of spiking neurons.

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Abstract: This paper concerned with basic concepts and some results on (idempotent) semigroup satisfying the identities of three variables. The motivation of taking three for the number of variables has come from the fact that many important identities on idempotent semigroups are written by three or fewer independent variables. We consider the semigroup satisfying the property abc = ac and prove that it is left semi- normal and right quasi-normal. Again an idempotent semigroup with an identity aba = ab and aba = ba (ab = a, ab = b) is always a semilattices and normal. An idempotent semigroup is normal if and only if it is both left quasi-normal and right quasi-normal. If a semigroup is rectangular then it is left and right semi-regular. I. PRELIMINARIES AND BASIC PROPERTIES OF REGULAR SEMIGROUPS In this section we present some basic concepts of semigroups and other definitions needed for the study of this chapter and the subsequent chapters. 1.1 Definition: A semigroup (S, .) is said to be left(right) singular if it satisfies the identity ab = a (ab = b) for all a,b in S 1.2 Definition: A semigroup (S, .) is rectangular if it satisfies the identity aba = a for all a,b in S. 1.3 Definition: A semigroup (S, .) is called left(right) regular if it satisfies the identity aba = ab (aba = ba) for all a,b in S. 1.4 Definition: A semigroup (S, .) is called regular if it satisfies the identity abca = abaca for all a,b,c in S 1.5 Definition: A semigroup (S, .) is said to be total if every element of Scan be written as the product of two elements of S. i.e, S 2

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15 Aug 1991-Journal of Algebra
TL;DR: In this article, the authors investigated the relationship between certain classes of ordered small categories, introduced by Charles Ehresmann in the course of his work on local structures, and the class of U -semiabundant semigroups, first studied by El-Qallali and by de Barros.

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