Stochastic models in biology

proseminars

The subject of the seminar will be: (a) modeling of information processing in biological neural networks, (b) modeling the formation of clusters (swarms) in the systems of interacting individuals (birds, fish, players in evolution games with migration). It will be done using the basic tools of stochastic analysis and the elements of information theory, such as: Markov chains, Birth and Death processes, Fokker-Planck equation, the concepts of entropy and mutual information, elements of statistical physics and machine learning. We will discuss some examples from neuroscience and biology both in micro scale (dynamics and plasticity of synapses), as well as in macro scale (learning and information storing in neural networks, formation of swarms, elements of evolutionary games).

Biological systems such as neural networks process and store information in their activities or structure. Empirical observations suggest that this information processing is very efficient despite of molecular noise (stochasticity). It means that neurons in the brain can very accurately encode the incoming information from the outside world, and write it in synapses in a stable manner, which is related to learning and memory. Learning processes are also present on a macro scale in the creation of moving spatial structures (swarms, clusters of cooperators in spatial evolution games).

The proseminar will be devoted to mathematical modeling of these topics, using probabilistic models with the elements of information theory and machine learning. We do not assume any detailed biological knowledge of students - all the necessary facts will be given during initial lectures (meetings). Also, the necessary mathematical tools will be presented in the course of the seminar.

Mathematical requirements: knowledge of the basics of probability theory, and the ability to analyze and solve simple ordinary differential equations.

Knowledge and skills:

1. Knows basic models of neural networks and swarms, and their functions.

2. Can construct a simple mathematical model of a chosen biological system.

Social competences:

1. Knows how to talk to biologists.

2. Is not afraid to cooperate with biologists.

3. Can give a talk about mathematical modeling of biological systems.

Participation in seminars, giving talks.