Participants have to present group work from the afternoon exercise sessions

Course workload

Course workload category

Hours

Preparation / Self-Study

15.00

Course hours

36.00

Evaluation / reporting

15.00

Sum

66.00

Content

The objective of the proposed summer school is to bring together 25-35 PhD students in quantitative life sciences and teach them a range of mathematical approaches to analysis and modeling in neuroscience. The workshop will be jointly organized with the European Society for Mathematical and Theoretical Biology (ESMTB), and fits into a series of annual workshops on mathematical biology organized with the EMS (http://www.esmtb.org/schools).

The lecturers cover a broad range of approaches in mathematical and computational neuroscience, and both single cell and network models will be covered, allowing participants to get a good overview of what is possible and which approaches will be useful for their projects. In order to expose the participants to a wide range of approaches, we have invited Professor of Applied Mathematics Stephen Coombes from University of Nottingham, expert in application of principles from nonlinear dynamics and statistical physics to the study of neural systems, Principal Investigator Alex Roxin from Centre Recerca Matematica, Barcelona, expert in Theoretical Neurobiology of Cortical Circuits, Professor in Probability Eva Löcherbach from Université Cergy-Pontoise, expert in stochastic processes and interacting graphs, and Professor Patricia Reynaud-Bouret from University of Nice, expert in point process models and statistics for spike trains and their dependence structure. All four have agreed to teach on the school.

The students will be presented to the following topics.

Dynamics of neural networks The tools of dynamical systems theory are having an increasing impact on our understanding of patterns of neural activity. First some of the more popular single neuron models will be introduced and their behaviour in terms of bifurcation diagrams, phase-planes and phase-response curves will be explained. For limit cycle oscillators, the coupled oscillator approach that has provided a framework for understanding behaviour in neural networks with weak synaptic and gap junction coupling will be reviewed. Then it will be shown how results for strong coupling can be obtained by focusing on a specific class of spiking neural models, namely (non-smooth) planar integrate-and-fire models. Next it will be described how to build tractable tissue level models that maintain a strong link with biophysical reality. These models typically take the form of nonlinear integro-differential equations. Their non-local nature has led to the development of a set of analytical and numerical tools for the study of waves, bumps and patterns, based around natural extensions of those used for local differential equation models. The relevance of neural field models for describing the brain at the large scales necessary for interpreting EEG and MEG data will be discussed. Recent results on next generation neural field models obtained via a meanfield reduction from networks of nonlinear integrate-and-fire neurons will also be presented.

Meanfield theory for networks of spiking neurons If functional units in the brain are large networks then we may not need to know the detailed dynamics of each neuron, but rather just a measure of the mean activity. Can we develop such a meanfield theory for networks of model spiking neurons? The by now classical meanfield theory for irregularly spiking neurons will be discussed, which is based on two main assumptions: 1 - neurons receive large numbers of weak, uncorrelated inputs and 2 - neurons are Poisson processes. This is a very general theory which works for many spiking neuron models, but which only allows us to calculate stationary states, their stability, and the response to weak inputs. A complementary theory will also be discussed, which only works for quadratic integrate-and-fire neurons with quenched variability, but which allows us to derive exact equations for the mean-field. These equations capture the full nonlinear response of the network and is simple enough to allow for extensive analysis.

Modeling networks of interacting neurons as systems of processes with variable length A class of recently introduced models to describe networks of neurons as stochastic processes with memory of variable length will be presented. These are non-Markovian processes in high or infinite dimension in which the past dependence of transition probabilities or intensities has a range that is finite but depends on the particular history. Starting from existence results, we study related mean-field models in continuous time and their large population limits, and discuss the relation with associated Piecewise Deterministic Markov Processes (PDMP’s) and state results concerning their longtime behavior. Finally, we touch two important problems of statistical inference in such models. Firstly, estimation of the spiking rate function, and secondly, estimation of the neuronal interaction graph.

Detection of dependence between neurons and synchronisation Synchronisation is an important phenomenon in neuroscience, and this kind of dependency between neuronal activity may play an important role for how the brain encodes information. An essential mathematical/statistical question is therefore how to detect such phenomena in a set-up where the observations are scarce, noisy and always fluctuating. The Unitary Events methods will be described (first introduced by Grün in the 90's) which makes rigorous the distinction between synchronization and pure coincidence and how statistical testing procedures can distinguish between them. Then the notion of local independence between the spiking activity of neurons and how estimating a graph of local independence can infer functional connectivity will be explained.

Literature

The course is based on recent results in the field of mathematical neuroscience. The literature will be relevant publications and lecture notes written for the course.

Teaching and learning methods

The mornings will start with plenary lectures by the invited experts. These lectures will showcase exemplary stories that have combined mathematical modeling and experimental neuroscience, as well as discuss a number of mathematical methods in-depth. The lectures will cover a range of topics, including, but not limited to: dynamics of neural networks; neural field models; piecewise deterministic Markov processes and related statistical questions, such as estimation of spiking rate or the interaction graph; detection of dependence between neurons and synchronisation; signal processing of experimental data in neuroscience. In the afternoons, the workshop will focus on hands-on group work on selected problems raised in the morning lectures.

Group work

Signal processing of experimental data in Neuroscience Topics will cover Filters for processing experimental data, Time-frequency analysis, and Spike rate - theory and estimation. Introductory lectures will give a general introduction to analysis of electrical signals from the nervous system. We alternate between lectures on the different topics and then problem based learning by the students via the computer language MATLAB. After this introduction we will use matlab to analyse and process data. There will be an assignment, which the students can work on either in groups or by themselves, as a part of the course. The teachers will be there to assist the students at all times.

Numerical experiments and simulation of stochastic models The students will play with methods to determine if synchronisation among spike trains is present or not. This will be done through a free package interfaced with R to perform all statistical analysis. Also some Matlab based tutorials/worksheets for numerically simulating coupled oscillator networks and neural field models will be introduced in the handson exercise sessions.

Lecturers

Guest Lecturers:

Professor of Applied Mathematics Stephen Coombes from University of Nottingham, expert in application of principles from nonlinear dynamics and statistical physics to the study of neural systems.

Principal Investigator Alex Roxin from Centre Recerca Matematica, Barcelona, expert in Theoretical Neurobiology of Cortical Circuits.

Professor in Probability Eva Löcherbach from Université Cergy-Pontoise, expert in stochastic processes and interacting graphs.

Professor Patricia Reynaud-Bouret from University of Nice, expert in point process models and statistics for spike trains and their dependence structure.

Remarks

The summer school takes place on Bornholm. We ask you to be present at the hotel the night before the course start, Sunday 25 June 2017.

Price for lodging including breakfast, lunch, coffee breaks and conference dinner: When registering before 1 March: EUR 260 (DKK 1.900). When registering after: EUR 310 (DKK 2.300). Deadline for registration: 15 April 2017.

Read about the payment on the summer school website. Your registration for the course is not accepted until we have received the payment!

In addition to signing up on this site (click Enroll on the top of the page), please also fill out the form you'll find at the summer school website.

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