Probabilistic Spiking Neuronal Nets Neuromathematics for the Computer Era

This book provides a self-contained introduction to a new class of stochastic models for systems of spiking neurons. These systems have a large number of interacting components, each one evolving as a stochastic process with a memory of variable length. Several mathematical tools are put to use, such as Markov chains, stochastic chains having memory of variable length, point processes having stochastic intensity, Hawkes processes, random graphs, mean field limits, perfect sampling algorithms, the Context algorithm, and statistical model selection.

The book’s focus on mathematically tractable objects distinguishes it from other texts on theoretical neuroscience. The biological complexity of neurons is not ignored, but reduced to some of its main features, such as the intrinsic randomness of neuronal dynamics. This reduction in complexity aims at explaining and reproducing statistical regularities and collective phenomena that are observed in experimental data, an approach that leads to mathematically rigorous results.

With an emphasis on a constructive and algorithmic point of view, this book is directed towards mathematicians interested in learning about stochastic network models and their neurobiological underpinning, and neuroscientists interested in learning how to build and prove results with mathematical models that relate to actual experimental settings.

A Neurophysiology Primer for Mathematicians.- A Discrete Time Stochastic Neural Network Model.- Mean Field Limits for Discrete Time Stochastic Neural Network Models.- But Time is Continuous!.- Models without Reset: Hawkes Processes.- What is a Stationary State in a Potentially Infinite System?.- Statistical Estimation of the Interaction Graph.- Mean Field Limits and Short-Term Synaptic Facilitation in Continuous Time Models.- A Non-Exhaustive List of Some Open Questions.- Appendix A.- Appendix B.- Appendix C.- Appendix D.- Appendix E.- Appendix F.- References.- Index.

Antonio Galves was a Senior Professor at the Institute of Mathematics and Statistics at the University of S. Paulo. His later research focused on statistical model selection as a conceptual framework to model neurocognition and on modeling networks of spiking neurons as systems of interacting point processes with memory of variable length. From 2011 until 2023, he was the coordinator of the Research, Innovation and Dissemination Center for Neuromathematics (NeuroMat). He was a member of the Brazilian Academy of Sciences and he was awarded the Brazilian National Order of Scientific Merit. He passed away on September 5, 2023.

Eva Löcherbach is presently Professor at the University Paris 1 Panthéon Sorbonne. Previously she was a professor at CY Cergy Paris Université, where she started to work on the stochastic modeling of neurons, going from models for single neurons (the stochastic Hodgkin–Huxley model and its longtime behavior) to systems of interacting and spiking neurons described by point processes, their mean field limits and the emergence of collective behavior. She was a plenary speaker at ECMTB in Lisbon, 2018.

Christophe Pouzat is a researcher at the Centre National de la Recherche Scientifique (CNRS) and at the University of Strasbourg. His PhD and postdoctoral work was in experimental neurophysiology (adapting multi-electrode arrays extracellular recordings to the first olfactory relay of insects). He was then recruited by the CNRS in 2001, where he combined experimental work with methods development for 10 years. He then joined the maths department of his University (then Paris-Descartes) and has since been working on neurophysiological data analysis, as well as on stochastic models of neuronal networks and their simulation.