Fiches de cours 2017-2018

Biological modeling of neural networks

Gerstner Wulfram

English

Summary

In this course we study mathematical models of neurons and neuronal networks in the context of biology and establish links to models of cognition.

Content

I. Models of single neurons 1. Introduction: brain vs computer and a first simple neuron model 2. Models on the level of ion current (Hodgkin-Huxley model) 3./4.  Two-dimensional models and phase space analysis II. Neuronal Dynamics of Cognition  5./6. Associative Memory and Attractor Dynamics (Hopfield Model)   7. Neuronal Populations and networks 8. Continuum models and perception 9. Competition and models of Decision making III. Noise and the neural code 10. Noise and variability of spike trains (point processes, renewal process, interval distribution) 11: Variance of membrane potentials and  Spike Response Models IV. Plasticity and Learning 12.  Synaptic Plasticity and Long-term potentiation and Learning (Hebb rule, mathematical formulation) 13. Summary: Fitting Neural Models to Data

Keywords

neural networks, neuronal dynamics, computational neuroscience, mathematical modeling in biology, applied mathematics, brain, cognition, neurons, memory, learning, plasticity

Learning Prerequisites

Required courses

undergraduate math at the level of electrical engineering or physics majors

Recommended courses

Analysis I-III, linear algebra, probability and statistics
For SSV students: Dynamical Systems Theory for Engineers or "Mathematical and Computational Models in Biology" course, Felix Naef

Important concepts to start the course

Differential equations, stochastic processes,

Learning Outcomes

By the end of the course, the student must be able to:
• Analyze two-dimensional models in the phase plane
• Solve linear one-dimensional differential equations
• Develop a simplified model by separation of time scales
• Analyze connected networks in the mean-field limit
• Formulate stochastic models of biological phenomena
• Formalize biological facts into mathematical models
• Prove stability and convergence
• Apply model concepts in simulations
• Predict outcome of dynamics
• Describe neuronal phenomena

Transversal skills

• Plan and carry out activities in a way which makes optimal use of available time and other resources.
• Collect data.
• Write a scientific or technical report.

Teaching methods

Classroom teaching, exercises and miniproject

miniprojects

Assessment methods

Written exam (67%) & miniproject (33%)

Resources

Bibliography

Gerstner, Kistler, Naud, Pansinski : Neuronal Dynamics, Cambridge Univ. Press 2014

Reference week

MoTuWeThFr
8-9
9-10INM200
10-11
11-12INM200
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22

Lecture
Exercise, TP
Project, other

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• Autumn semester
• Winter sessions
• Spring semester
• Summer sessions
• Lecture in French
• Lecture in English
• Lecture in German