Mathematical Modelling of Brain

This is an interdisciplinary project aiming at developing mathematical tools for the analysis, simulation, and modelling the behavior of brain. The project involves the University of Leicester (UK), Eindhoven University of Technology (The Netherlands) and RIKEN Brain Science Institute (Japan).

Participants and Investigators

Dr Ivan Tyukin, Professor Alexander Gorban, David Fairhurst (University of Leicester, UK)

Dr Alexey Semyanov, Professor Cees van Leeuwen, Dr Inseon Song (RIKEN BSI, Japan)

Professor Henk Nijmeijer, Erik Steur (Eindhoven University of Technology, The Netherlands)


Mathematical Modelling of Neural Systems (I Tyukin and H Nijmeijer, organisers). Special Session at the 2-d IFAC Symposium, CHAOS'09, 22-24 June 2009

Syncronization, Control and Mathematical Modelling of Neural Systems (I Tyukin, organiser), 3-d IEEE Physics and Control Conference, Potsdam, 3-7 September 2007

Task 1. Modelling of Individual Cells from in-vitro Recordings

We consider the problem of state and parameter reconstruction of typical conductance model neurons from in-vitro measurements of membrane potential. Previous reports available in the literature suggest that the problem is ill-posed mathematically, e.g. distinct parameter values lead to models with indistinguishable behavior.

In order to deal with this problem we propose a novel technique that is based on the concepts of observer design, phase-space analysis of invariant sets of the model and the concept of weakly attracting sets from dynamical systems theory. We show that, subject to mild conditions on sufficient richness of measured signal, the values of model parameters can be reconstructed up to its equivalence class. Success of the method is illustrated with the problem of modelling of membrane conductance variations in neural membrane using simple Hindmarsh-Rose oscillators.

Progress and publications

IYu Tyukin, E Steur, H Nijmeijer and C van Leeuwen. Non-uniform small-gain theorems for systems with unstable invariant sets. SIAM Journal on Control and Optimization, 47(2): 849-882, 2008 (full text pdf, preprint).

E Steur, IYu Tyukin, H Nijmeijer and C van Leeuwen.  Reconstructing Dynamics of Spiking Neurons from Input-Output Measurements in Vitro. In Proceedings of the 3rd IEEE Conference on Physics and Control. Potsdam, Germany, 3-7 September 2007 (full text pdflink to the IPACS open access library)

IYu Tyukin, E Steur, H Nijmeijer and Cees van Leeuwen. Adaptive Observers and Parametric Identification for Systems in Non-canonical Adaptive Observer Form. Submitted to Automatica, preprint available at

D Fairhurst, I Tyukin, H Nijmeijer and Cees van Leeuwen. Non-canonic Observers for Canonic Models of Neural Oscillators. Submitted to Mathematical Modelling of Natural Phenomena. Preprint available at

IYu Tyukin, E Steur, H Nijmeijer and C van Leeuwen. Non-uniform Small Gain Theorems for Systems with Unstable Invariant Sets. In Proceedings of the 47th IEEE Conference on Decision and Control,  5080-5085, Cancun, Mexico, 9-11 December 2008.

IYu Tyukin, E Steur, H Nijmeijer and C van Leeuwen. State and Parameter Estimation for Systems in Non-canonical Adaptive Observer Form. In Proceedings of the 17th IFAC World Congress on Automation Control. Seoul, Korea, 6-1 July 2008 (full text pdf)

IYu Tyukin, H Nijmeijer and C van Leeuwen. Non-uniform Small-gain Theorems for Systems with Critical and Slow Relaxations. In Proceedings of the 17th IFAC World Congress on Automation Control. Seoul, Korea, 6-11 July 2008 (full text pdf)

Task 2. Mathematical Modelling of Extrasynaptic Transmission

Synaptic signal transmission is traditionally believed to be the principal medium for neural interaction. Recent studies show that spillover of neural transmitters from the synaptic clefts may constitute an additional channel for neural interaction.

There is an experimental evidence (H Nishiyama and D Linden, Nature Neuroscience 2007) that extrasynaptic signalling accounts for up to 75% of interneuronal communication. Despite the fact that these empirical findings have attracted substantial interest worldwide, there are few theoretical modeling studies that take these observations into account.

Understanding and proper mathematical modeling of this phenomenon will allow us to further progress in understanding of physical principles behind computations in biological brain. Furthermore, detailed knowledge of how the brain function would change if we modify parameters of diffusion (e.g. changing concentration of the transmitters, or blocking the gates on the membranes) will enable extra degree of controlling the brain. The latter is potentially relevant for medical purposes, for instance as a possible way for manipulating/enhancing of the brain functions.

In order to understand how the extrasynaptic signal transmission affects properties of a single cell we must compare the cell behavior in two different conditions. First condition constitutes the case when the gates (these are located on the membrane of the cell) through which the transmitter enters the cell are open. Second condition corresponds to the situation when the gates are blocked (a special chemical, toxin, was used to block these gates on the membrane).

We have found that the cell in these two conditions responded differently to the same external stimulation. That is, the cell's properties are changed with changes of the concentration of neurotransmitter. The problem, however, is how to characterise these changes physically and mathematically? We must be able to characterise these changes in terms of the properties of the membranes and internal currents of the cell. Simultaneous measurement is largely invasive, and is a difficult problem technically. This is because most of the intrinsic variables of a living cell are not available for direct observation. 

Mathematical modeling of the cell dynamics (Task 1) offers a solution to this problem. Indeed, if trajectories of a realistic mathematical model fit the actual data then internal variables of the model (we can access these variables easily) can be estimated. Thus dynamics of hidden variables of the cell can be restored and analysed.

Progress and Publications

D Rijlaarsdam, IYu Tyukin, H Nijmeijer, A Semyanov and C van Leeuwen. Synchronization of Neural Oscillators with Diffuse Coupling: Does the Leakage of Neurotransmitter Matter? In Proceedings of the 3rd IEEE Conference on Physics and Control. Potsdam, Germany, 3-7 September 2007 (full text pdf, link to the IPACS open access library)

Task 3. Modelling and Analysis of the Dynamics of Neural Populations

In this task we aim to study collective behaviour of coupled (electrically and chemically) neural oscillators. As our main theoretical framework we adopt the theory of passive and semipassive systems. Within this framework we derived sufficient conditions for synchronisation of diffusively coupled Hindmarsh-Rose systems. Recently we have shown that most canonical models of neural osciilators are in fact semipassive, hence the theorems on synchronisation (partial and total) automatically apply to these systems.

Progress and Publications

E Steur, I Tyukin and H Nijmeijer. Semipassivity and synchronization of diffusively coupled neural oscillators. Submitted. Preprint available at

W Oud and IYu Tyukin. Sufficient conditions for synchronization in an ensemble of Hindmarsh and Rose neurons: passivity-based approach. In Proceedings of the 6th. Shtutgart, Germany, 1-3 September 2004 (full text pdf)

Press Releases

Brain power - breakthrough in mathematical modelling , University of Leicester Press Release, September 2007

Alpha Galileo, The world's leading resource for European research news, September 2007.

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