## Advanced Calculus and Dynamical Systems

### Full course description

A brain scientist is able to develop mathematical descriptions of phenomena that evolve in space and time, and can interpret and model high-dimensional data. These abilities rest on a solid understanding of advanced calculus and dynamical systems theory. In this course, the students gain the foundations required to build sophisticated, biophysical models of neural phenomena, and the tools needed for the analysis and the computational modelling of brain and behavioural data.

This course builds on the Calculus and Linear Algebra Courses in periods 1-2. It provides the foundations of multivariate calculus, ordinary and partial differential equations, and the analytical and numerical methods to perform computations in one, two and three dimensions.

The students are subsequently introduced to the basics of dynamical systems theory: the course covers linear systems, stability of equilibria, bifurcation analysis and oscillatory systems, using relevant examples of neuronal systems whenever possible.

The course discusses several models that appear in different courses from a mathematical point of view: it links to Brain Cells, which runs in parallel, and provides the foundations for Biophysical Models.

The final assessment for this course is a numerical grade between 0,0 and 10,0.

### Course objectives

- Perform multivariable differentiation and integration both analytically and numerically.
- Classify and interpret ordinary differential equations (ODEs) and partial differential equations (PDEs).
- Understand a dynamical systems model comprised of coupled ODEs.
- Find and classify the equilibrium point of dynamical system.
- Perform a bifurcation analysis of dynamical systems.
- Collaborate with your peers to solve mathematical problems.

- J. Huys
- O. D'Huys