# Coursebooks 2016-2017

## 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. Cognition, Learning, and Synaptic Plasticity  5. Associative Memory and Attractor Dynamics (Hopfield Model) 6:  Synaptic Plasticity and Long-term potentiation (Hebb rule, mathematical formulation)  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 12.  Population dynamics and membrane potential distribution (Fokker-Planck equation) 13. Dynamics in Plastic networks 14.  Neural Code: Generalized Linear Models and Reverse Correlations

#### 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

### legend

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