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Statistical Field Theory for Neural Networks 1st ed. 2020 [Minkštas viršelis]

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  • Formatas: Paperback / softback, 203 pages, aukštis x plotis: 235x155 mm, weight: 454 g, 5 Illustrations, color; 122 Illustrations, black and white; XVII, 203 p. 127 illus., 5 illus. in color., 1 Paperback / softback
  • Serija: Lecture Notes in Physics 970
  • Išleidimo metai: 21-Aug-2020
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030464431
  • ISBN-13: 9783030464431
Kitos knygos pagal šią temą:
Statistical Field Theory for Neural Networks 1st ed. 2020Didesnis paveikslėlis
  • Formatas: Paperback / softback, 203 pages, aukštis x plotis: 235x155 mm, weight: 454 g, 5 Illustrations, color; 122 Illustrations, black and white; XVII, 203 p. 127 illus., 5 illus. in color., 1 Paperback / softback
  • Serija: Lecture Notes in Physics 970
  • Išleidimo metai: 21-Aug-2020
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030464431
  • ISBN-13: 9783030464431
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783030464431.html
This book presents a self-contained introduction to techniques from field theory applied to stochastic and collective dynamics in neuronal networks. These powerful analytical techniques, which are well established in other fields of physics, are the basis of current developments and offer solutions to pressing open problems in theoretical neuroscience and also machine learning. They enable a systematic and quantitative understanding of the dynamics in recurrent and stochastic neuronal networks.





This book is intended for physicists, mathematicians, and computer scientists and it is designed for self-study by researchers who want to enter the field or as the main text for a one semester course at advanced undergraduate or graduate level. The theoretical concepts presented in this book are systematically developed from the very beginning, which only requires basic knowledge of analysis and linear algebra.
1 Introduction
1(4)
1.1 Code, Numerics, Figures
3(2)
References
4(1)
2 Probabilities, Moments, Cumulants
5(10)
2.1 Probabilities, Observables, and Moments
5(3)
2.2 Transformation of Random Variables
8(1)
2.3 Cumulants
8(2)
2.4 Connection Between Moments and Cumulants
10(3)
2.5 Problems
13(2)
References
14(1)
3 Gaussian Distribution and Wick's Theorem
15(6)
3.1 Gaussian Distribution
15(1)
3.2 Moment and Cumulant-Generating Function of a Gaussian
16(1)
3.3 Wick's Theorem
17(1)
3.4 Graphical Representation: Feynman Diagrams
18(1)
3.5 Appendix: Self-Adjoint Operators
19(1)
3.6 Appendix: Normalization of a Gaussian
19(2)
References
20(1)
4 Perturbation Expansion
21(18)
4.1 Solvable Theories with Small Perturbations
21(2)
4.2 Special Case of a Gaussian Solvable Theory
23(3)
4.3 Example: Example: "φ3 + φ4" Theory
26(1)
4.4 External Sources
27(1)
4.5 Cancelation of Vacuum Diagrams
28(3)
4.6 Equivalence of Graphical Rules for n-Point Correlation and n-th Moment
31(1)
4.7 Example: "φ3 + φ4" Theory
31(2)
4.8 Problems
33(6)
References
38(1)
5 Linked Cluster Theorem
39(14)
5.1 Introduction
39(1)
5.2 General Proof of the Linked Cluster Theorem
40(5)
5.3 External Sources--Two Complimentary Views
45(3)
5.4 Example: Connected Diagrams of the "φ + φ4" Theory
48(2)
5.5 Problems
50(3)
Reference
52(1)
6 Functional Preliminaries
53(4)
6.1 Functional Derivative
53(2)
6.1.1 Product Rule
53(1)
6.1.2 Chain Rule
54(1)
6.1.3 Special Case of the Chain Rule: Fourier Transform
55(1)
6.2 Functional Taylor Series
55(2)
7 Functional Formulation of Stochastic Differential Equations
57(12)
7.1 Stochastic Differential Equations
57(3)
7.2 Onsager-Machlup Path Integral
60(1)
7.3 Martin-Siggia-Rose-De Dominicis-Janssen (MSRDJ) Path Integral
61(1)
7.4 Moment-Generating Functional
62(2)
7.5 Response Function in the MSRDJ Formalism
64(5)
References
67(2)
8 Ornstein-Uhlenbeck Process: The Free Gaussian Theory
69(8)
8.1 Definition
69(1)
8.2 Propagators in Time Domain
70(2)
8.3 Propagators in Fourier Domain
72(5)
References
75(2)
9 Perturbation Theory for Stochastic Differential Equations
77(18)
9.1 Vanishing Moments of Response Fields
77(1)
9.2 Feynman Rules for SDEs in Time Domain and Frequency Domain
78(4)
9.3 Diagrams with More Than a Single External Leg
82(2)
9.4 Appendix: Unitary Fourier Transform
84(2)
9.5 Appendix: Vanishing Response Loops
86(2)
9.6 Problems
88(7)
References
93(2)
10 Dynamic Mean-Field Theory for Random Networks
95(32)
10.1 The Notion of a Mean-Field Theory
95(1)
10.2 Definition of the Model and Generating Functional
96(1)
10.3 Self-averaging Observables
97(3)
10.4 Average over the Quenched Disorder
100(7)
10.5 Stationary Statistics: Self-consistent Autocorrelation as a Particle in a Potential
107(3)
10.6 Transition to Chaos
110(1)
10.7 Assessing Chaos by a Pair of Identical Systems
111(6)
10.8 Schrodinger Equation for the Maximum Lyapunov Exponent
117(1)
10.9 Condition for Transition to Chaos
118(4)
10.10 Problems
122(5)
References
125(2)
11 Vertex-Generating Function
127(30)
11.1 Motivating Example for the Expansion Around a Non-vanishing Mean Value
127(3)
11.2 Legendre Transform and Definition of the Vertex-Generating Function T
130(4)
11.3 Perturbation Expansion of T
134(3)
11.4 Generalized One-line Irreducibility
137(5)
11.5 Example
142(1)
11.6 Vertex Functions in the Gaussian Case
143(2)
11.7 Example: Vertex Functions of the "φ3 + φ4"-Theory
145(1)
11.8 Appendix: Explicit Cancelation Until Second Order
146(2)
11.9 Appendix: Convexity of W
148(1)
11.10 Appendix: Legendre Transform of a Gaussian
149(1)
11.11 Problems
149(8)
References
154(3)
12 Expansion of Cumulants into Tree Diagrams of Vertex Functions
157(8)
12.1 Definition of Vertex Functions
157(5)
12.2 Self-energy or Mass Operator E
162(3)
13 Loop wise Expansion of the Effective Action
165(24)
13.1 Motivation and Tree-Level Approximation
165(2)
13.2 Counting the Number of Loops
167(3)
13.3 Loopwise Expansion of the Effective Action: Higher Number of Loops
170(5)
13.4 Example: φ3 + φ4-Theory
175(2)
13.5 Appendix: Equivalence of Loopwise Expansion and Infinite Resummation
177(3)
13.6 Appendix: Interpretation of T as Effective Action
180(1)
13.7 Appendix: Loopwise Expansion of Self-consistency Equation
181(5)
13.8 Problems
186(3)
References
187(2)
14 Loopwise Expansion in the MSRDJ Formalism
189(14)
14.1 Intuitive Approach
189(3)
14.2 Loopwise Corrections to the Effective Equation of Motion
192(6)
14.3 Corrections to the Self-energy and Self-consistency
198(1)
14.4 Self-energy Correction to the Full Propagator
199(2)
14.5 Self-consistent One-Loop
201(1)
14.6 Appendix: Solution by Fokker-Planck Equation
201(2)
References
202(1)
Nomenclature 203
Moritz Helias is group leader at the Jülich Research Centre and assistant professor in the department of physics of the RWTH Aachen University, Germany. He obtained his diploma in theoretical solid state physics at the University of Hamburg and his PhD in computational neuroscience at the University of Freiburg, Germany. Post-doctoral positions in RIKEN Wako-Shi, Japan and Jülich Research Center followed. His main research interests are neuronal network dynamics and function, and their quantitative analysis with tools from statistical physics and field theory.

David Dahmen is a post-doctoral researcher in the Institute of Neuroscience and Medicine at the Jülich Research Centre, Germany. He obtained his Master's degree in physics from RWTH Aachen University, Germany, working on effective field theory approaches to particle physics. Afterwards he moved to the field of computational neuroscience, where he received his PhD in 2017. His research comprises modeling, analysis and simulation of recurrent neuronal networks with special focus on development and knowledge transfer of mathematical tools and simulation concepts. His main interests are field-theoretic methods for random neural networks, correlations in recurrent networks, and modeling of the local field potential.