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.
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.
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.
II. Probabilities, moments, cumulantsA. Probabilities, observables, and momentsB. Transformation of random variablesC. CumulantsD. Connection between moments and cumulants
III. Gaussian distribution and Wick's theoremA. Gaussian distributionB. Moment and cumulant generating function of a GaussianC. Wick's theoremD. Graphical representation: Feynman diagramsE. Appendix: Self-adjoint operatorsF. Appendix: Normalization of a Gaussian
IV. Perturbation expansionA. General caseB. Special case of a Gaussian solvable theoryC. Example: Example: "phi^3 + phi^4" theoryD. External sourcesE. Cancellation of vacuum diagramsF. Equivalence of graphical rules for n-point correlation and n-th momentG. Example: "phi^3 + phi^4" theoryV. Linked cluster theoremA. General proof of the linked cluster theoremB. Dependence on j - external sources - two complimentary viewsC. Example: Connected diagrams of the "phi^3 + phi^4" theory
VI. Functional preliminariesA. Functional derivative1. Product rule2. Chain rule3. Special case of the chain rule: Fourier transformB. Functional Taylor series
VII. Functional formulation of stochastic differential equationsA. Onsager-Machlup path integral*B. Martin-Siggia-Rose-De Dominicis-Janssen (MSRDJ) path integralC. Moment generating functionalD. Response function in the MSRDJ formalism
VIII. Ornstein-Uhlenbeck process: The free Gaussian theoryA. DefinitionB. Propagators in time domainC. Propagators in Fourier domain
IX. Perturbation theory for stochastic differential equationsA. Vanishing moments of response fieldsB. Vanishing response loopsC. Feynman rules for SDEs in time domain and frequency domainD. Diagrams with more than a single external legE. Appendix: Unitary Fourier transform
X. Dynamic mean-field theory for random networksA. Definition of the model and generating functionalB. Property of self-averagingC. Average over the quenched disorderD. Stationary statistics: Self-consistent autocorrelation of as motion of a particle in a potentialE. Transition to chaosF. Assessing chaos by a pair of identical systemsG. Schrödinger equation for the maximum Lyapunov exponentH. Condition for transition to chaos
XI. Vertex generating functionA. Motivating example for the expansion around a non-vanishing mean valueB. Legendre transform and definition of the vertex generating function GammaC. Perturbation expansion of GammaD. Generalized one-line irreducibilityE. ExampleF. Vertex functions in the Gaussian caseG. Example: Vertex functions of the "phi^3 + phi^4"-theoryH. Appendix: Explicit cancellation until second orderI. Appendix: Convexity of WJ. Appendix: Legendre transform of a Gaussian
XII. Application: TAP approximationInverse problem
XIII. Expansion of cumulants into tree diagrams of vertex functionsA. Self-energy or mass operator Sigma
XIV. Loopwise expansion of the effective action - Tree levelA. Counting the number of loopsB. Loopwise expansion of the effective action - Higher numbers of loopsC. Example: phi^3 + phi^4-theoryD. Appendix: Equivalence of loopwise expansion and infinite resummationE. Appendix: Interpretation of Gamma as effective actionF. Loopwise expansion of self-consistency equation
XV. Loopwise expansion in the MSRDJ formalismA. Intuitive approachB. Loopwise corrections to the effective equation of motionC. Corrections to the self-energy and self-consistencyD. Self-energy correction to the full propagatorE. Self-consistent one-loopF. Appendix: Solution by Fokker-Planck equation