General teaching duties
I have taught various courses on Calculus, Linear Algebra, Differential Equations,
Numerical Analysis, Mathematical Modelling and Neuroscience for the programs Applied Mathematics,
Biomedical Engineering and Technical Medicine.
I am chairman of the Program Committee for Applied Mathematics since January 2017.
Open Master/Bachelor Assignments
I list a few interesting topics. Please contact me 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)
Supervised students
- Wouter van Harten (AM, 2021) Numerical continuation schemes for 1D patterns in neural fields
- Lucas Jansen Klomp (AM, 2021) The influence of feedforward inhibition on spontaneous and evoked activity in coupled neural mass models
- Sophie Ligtenstein (AM/TG, 2020) Modelling the effect of cardiac arreston neural activity using an energy-dependent neural mass model basedon ion concentrations
- Wei Zhong (BME, 2017), Beta oscillatory source localization: Modeling analysis of a new 32-contact DBS lead
- Manu Kalia (AM, 2017) Homoclinic saddle to saddle-focus transitions in 4D systems
- Marij Thijsen (TG, 2016) Single Pulse Electrical Stimulation: automatic detection of delayed responses
- Dorien van Blooijs (TG, 2015) Improving the SPES protocol by automating ER and DR detection and evaluation of the relation between ERs and DRs
- Bert Kiewiet (TW, 2014) On the effect of a Gaussian firing rate function and propagation delays on the dynamics of a network of Wilson-Cowan populations
- Inger Heerts (TG, 2013) Efficacy and mode of action of pulsed radiofrequency
- Albert Bakker (EE, 2011) Cortical activity of near-threshold tactile stimuli using pulse train modulation
- Wessel Woldman (TW, 2011) Computational model on neuronal stabilization in perceptual choice dynamics