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Class 2019-2020

Modeling, simulations, data analysis in neuroscience and applications to medical predictions using Model-Machine-Learning

David Holcman



WHEN : Oct 2019-Jan 2020

Wednesday. 17h00-20h00. Starting date : beginning of October

WHERE Fifth floor "Salle 513" : ENS 46 rue d’Ulm, 75005 Paris

Class common to PSL-ENS-Sorbonne University-JLL Master in applied mathematics

General description : The class aims at describing modern modeling in neuroscience from the molecular, cellular to the Brain level with medical prediction from the Electroencephalogram. We will start by introducing stochastic models to analyze large amount of single particle trajectories obtained by superresolution microscopy. Models are used to reconstruct and interpret high density domains found at pre and post synaptic terminal. We will discuss vesicular release and calcium nanodomains. The class continues with Neural network bursting modeling, model of recurrent bursting in electrophysiology traces and Up-Down state. Finally, we will introduce Model-Machine-learning method to predict Brain behavior from EEG time series.

The goal of the class is to present modeling approach, stochastics analysis, signal processing and analysis of the model equations. The class is ideal for engineers, mathematicians, physicists, theoretical chemist or computer scientists.

This class (in english) will use the Holcman’s Cambridge lecture and e-class presented in and Youtube
contact : david.holcman chez


Part I

  • Stochastic processes, Fokker-Planck equation
  • Recovering a stochastic process from noisy trajectories
  • Exit problem and boundary layer for linear PDE and Mean First Passage Time Equations.
  • Small hole theory : search for a small target.

Part II and III :

  • Model Up-Down state, distribution of time in the Up-state by studying the non-selfadjoint Fokker-Planck and the full spectrum
  • Diffusion in the cleft+ method of simulations. Calcium dynamics in a dendritic spine.
  • Model of vesicular release and calcium in the pre-synaptic terminal. Diffusion in microdomains : Molecular and vesicular trafficking. Hybrid (Markov and mass-action) model of reaction-diffusion.
  • Model of reconstruction a source for growth-cone from diffusion flux to receptors.
  • Method of SPT Analysis for CaV, calreticulin, AMPAR, NMDA, Gly,..receptors : Model of reconstruction for high density regions, ER-network, potential wells analysis, based on density statistics and vector field reconstruction. Introduction to vector field indices.
  • Calcium signaling and role of extreme statistics.
  • Modeling synaptic transmission and plasticity. Model of the synaptic current.
  • Large-scale model of Neuron-glia interactions.
  • Model of electro-diffusion, asymptotic and singularities, simulations. Electro-neutrality.
  • Deconvolution of time series (voltage dye)
  • EEG analysis. Band spectral analysis,
  • Machine learning classification, feature extractions. Application to Coma, Anesthesia and sleep.
  • Evaluation : small projects.

References :
 D. Holcman Z. Schuss, Stochastic Narrow Escape : theory and applications, Springer 2015
 D. Holcman, Z. Schuss, Asymptotics of Singular Perturbations and Mixed Boundary Value Problems for Elliptic Partial Differential Equations, and their applications, Springer 2018
 Schuss, Z., Theory and Applications of Stochastic Processes (Hardback, 2009) Springer ; 1st Edition. (December 21, 2009)

Basics :
D. Holcman Z. Schuss, 100 years after Smoluchowski : stochastic processes in cell biology, J. Phys. A (2016).
Z. Schuss D. Holcman, The dire strait time, SIAM Multicale Modeling and simulations, 2012.
D. Holcman Z. Schuss, the Narrow Escape Problem, SIAM Rev 56 no. 2, 213–257, 2014.
D. Holcman, Z. Schuss Control of flux by narrow passages and hidden targets in cellular biology, Reports on Progress in Physics 76 (7):074601. (2013).
Z. Schuss, Brownian Dynamics at Boundaries and Interfaces, Springer series on Applied Mathematics Sciences, vol.186 (2013).
Advanced :
• D. Holcman N.Hoze, Statistical methods of short super-resolution stochastic single trajectories analysis, Annual Review of Statistics and Its Application, 4, 1-35 (2017).
• N Rouach, KD Duc, J Sibille, D. Holcman, ionic fluxes regulated neurons and astrocytes. Dynamics of ion fluxes between neurons, astrocytes and the extracellular space during neurotransmission, Opera Medica et Physiologica 4 (1), 1-18, 2018.