Open Master projects

• Deep learning techniques for deblurring cell images utilizing (Riemannian/Differential) geometry - MSc assignment
• Deblurring and disentangled representations of cell images via deep learning - MSc assignment

Please contact Hil Meijer for more details.

• The energy-deprived brain (Ischemia, Stroke)
  The brain needs a significant amount of energy to function. Most importantly, there are pumps to maintain ion gradients. These pumps and, more specifically, synapses are most vulnerable to energy deprivation. Within a large consortium of biologists, we aim to put all separate experimental traces together in one united computational model. The group has developed biophysical models of cell dynamics in terms of ions rather than potentials to include the pump.
  1) Modelling PID's. Calcium traces in slices show travelling waves emanating from an ischemic core. As such peri-infarct depolarisation causes secondary tissue damage, we need to understand them first to stop them later.
  2) Signatures of Burst Suppression. EEG of patients after stroke show low voltage activity with occasional bursts. Irregular bursts indicate a better chance for recovery. There is also a mathematical classification of bursters based on a slow-fast analysis. Determine the relationship between the clinical bursting patterns and this classification. This may explain why the regularity is a signature of recovery.
  Collaborators: CR Rose, MJAM van Putten
• Rhythms in Epilepsy and Surgery
  About 1% of the population suffers from some form of epilepsy. Initially, treatment involves anti-epileptic drugs. About one-third of patients, however, does not respond to medication. In some cases, epilepsy surgery may be an option. The workup for surgery involves an intensive monitoring period to delineate the area for surgical resection. Several electrodes are implanted subdurally (depth electrodes for SEEG or grids for ECoG) to record brain activity. Stimulating these electrodes allows mapping of brain networks and evoking pathological rhythms. Such information may help in delineating the resection area. Using signal and model analysis, we aim to improve the protocol and a better understanding of the data.
  1) Speed up the protocol. Analysis of existing data shows that some network properties are quite robust and reproducible during the monitoring period. These measures also indicate the epileptic network. So can we sample these network measures more effectively?
  2) Turning the data into a patient-specific computational model. A real surgery can be only once, but a model allows exploring multiple strategies. This may help in delineating the resection area and could uncover important nodes that would go unnoticed otherwise.
  3) Bifurcation analysis of new-generation neural mass models. Epileptiform activity is modelled using neural mass models (NMM). These NMM's show both healthy and pathological rhythms. The models typically involve a sigmoidal activation function based on the mean membrane potential. A recent formulation of the activation function based on population synchrony displays more complex dynamics but is open for systematic exploration.
  Collaborators: FSS Leijten, GJM Huiskamp
• Osteoarthritis; Forming the right tissue
  Within the NWO-project SCI-MAP, we look at models for cell differentiation to specific tissue. This differentiation is a complex process and difficult to control. A swarm exploration of an existing model for bone formation shows bistability. That is, for the same experimental parameters but different initial conditions, the outcome varies. Initialising the experiment smartly may lead to a higher yield of the desired cell type.
  1) Develop numerical methods for bifurcation analysis of high-dimensional models. With such tools, one can then to identify the saddle mentioned above. Its stable manifold forms the basin of attraction to optimise the yield.
  2) Cell differentiation is highly heterogeneous. Parameters for each cell differ, and cells interact. We would like to predict the final distribution of many cells. Using an agent-based model, investigate the effects of cell heterogeneity. Collaborators: JN Post
• Binocular rivalry and intermittent stimulation
  In a typical experiment, subjects see two competing images, either one for each eye or a figure that is multiple interpretable. For some time, one percept will dominate, and then the percept switches spontaneously to the other. A novel setup with real neurons applies current injection to two cells with mutual inhibition to mimic the rivalry. Recordings with this setup reproduce many of the psychophysics, making it an ideal model to increase our understanding of perceptual decision making. These models show a difference between perceptual dominance and onset dominance. The latter is relevant for stabilisation due to intermittent stimulation. While many computational models for binocular rivalry exist, several challenges remain.
  (1) What is the proper way of modelling noise that impacts perceptual switches? Current computational models for rivalry apply random noise for phenomenology only. The task is to implement the various forms of noise properly as motivated by biology, for instance, synaptic noise, variance in synaptic responses, fluctuations of ionic channel activity. Such new models can be used to determine the dominant source of randomness in binocular rivalry.
  (2) Extend to a network to study more neurons. Here models should come first to guide experiments. This extension could aim for two things. The current two neuron setup is a crude approximation of a population, and it requires more neurons to include plasticity or synaptic depression. Secondly, it allows exploring the interaction of low order eye-dominance and higher-order perception-dominance levels in the hierarchical organisation proposed by Blake.
  (3) Explore the generalisation of existing rivalry models to intermittent stimulation. The aim is to investigate the memory effect by incorporating NMDA, plasticity or other channels with long time scales. A large-scale spiking network and population model is available for testing.
  Collaborators: RJA van Wezel, N Kogo (Radboud)

For more information contact Matthias Schlottbom

• Numerical and analytical methods for data-driven gradient flows
  Topic: Partial differential equations (PDEs) are used widely in modeling natural phenomena, such as weather forecasting.
  In very controlled situations, such as laboratory experiments, it is possible to completely specify all model parameters accurately. In such situations, established approaches from mathematics can be used to simulate the model numerically.
  In many practical situations, the model can, however, be specified only partially. For instance, in weather prediction, neither the initial conditions are known nor all scales can be resolved properly with currently available numerical models. The mathematical model is then not well-posed or inaccurate in general, and additional observational data has to be used to properly specify the model or to render coarse models sufficiently accurate. This is the realm of data assimilation.
  This project considers a particular class of time-dependent PDEs, termed gradient flows, which minimize a corresponding energy functional over time. Based on a reformulation of the time-dependent PDE as a minimization problem to advance from one model state to the next one, it is possible to integrate additional data from measurements.
  A possible final project can be developed along the following two lines:
  • (i) Analysis of the properties of the minimization problems (like convergence in case of observation noise vanishes).
  • (ii) Development of numerical methods for a fast and reliable numerical minimization.

For more information contact: Tom Tyranowski,

• Numerical integration of nonlinear wave equations with nondifferentiable field potentials
  Topological field theories play an important role in modeling physical phenomena in cosmology or condensed matter physics. The nonlinear wave equations underlying such theories are often characterized by solutions called topological defects, e.g., solitons, kinks, vortices, breathers, oscillons. Solutions describing the interaction of such topological defects typically cannot be computed analytically, and therefore have to be approximated by numerical simulations. The goal of this project is to design, implement, and analyze numerical methods for solving a class of nonlinear wave equations characterized by field potentials which are nondifferentiable at their minima. Topological defects in such theories are known to be compactly supported, and their interaction gives rise to solutions which are only piecewise smooth, which poses a challenge to classical numerical techniques such as finite differences.
  
  
• Topics in geometric integration and stochastic geometric integration of Hamiltonian and Lagrangian systems
  Hamiltonian and Lagrangian systems are (ordinary or partial) differential equations whose solutions possess certain geometric properties (e.g., symplecticity, variational structures). Geometric integrators are numerical methods designed to preserve these geometric properties also in computer simulations, which results in numerical solutions demonstrating better stability and accuracy over very long integration times. Stochastic Hamiltonian and Lagrangian systems are extensions of their deterministic counterparts that can account for uncertainties underlying the considered system, such as uncertain parameters, external noise, unresolved physical processes, etc. Such systems can be numerically integrated using stochastic geometric methods. The overarching goal of this project is to design and analyze new geometric integration strategies for Hamiltonian and Lagrangian systems arising in various applications in physics and engineering. These strategies may also be aided by data-driven machine learning techniques.
  A more detailed topic will be tailored to the interested student.