# 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

### Semaine de référence

LuMaMeJeVe
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

Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand