research interests

Research Overview

Physiological rhythms are central for life. Prominent examples are the beating of the heart, the activity of neurons, or the release of hormones regulating growth and metabolism. Although many cells in the body display intrinsic, spontaneous rhythmicity, physiological functions derive from interaction of these cells with each other and with external inputs to generate the rhythms essential for life. In general, physiological oscillations can be synchronized to appropriate external or internal stimuli, so it is important to analyse the effects of stimuli on intrinsic physiological rhythms. Even the simplest theoretical models show the enormous complexity that can arise from periodic stimulation of nonlinear oscillations.

One approach to investigate the origin and dynamics of these rhythmic processes is to analyse qualitative aspects of simplified mathematical models of physiological systems. This involves a mathematical analysis of those features of physiological systems that will be preserved by classes of models that are sufficiently close to the real system. For example, periodic stimulation of a nerve axon gives rise to a wide variety of regular and irregular rhythms that can be modeled by simple as well as complex mathematical models.

Dynamical systems theory is the discipline that analyses these complex nonlinear behaviours and is my main research area. Especially, I am interested in the dynamics of relaxation oscillators. The special feature of relaxation oscillators is that their dynamics evolve on multiple time scales, long intervals of quasi steady state interspersed by short intervals of rapid variations like the beating of the heart or the firing of action potentials in neurons. These oscillators can create a lot of complicated patterns, many of them not well understood. The aim of my work is to derive analytic tools to understand the underlying dynamics that are causing some of these patterns. I am particularly interested in interdisciplinary cooperation with bioscientists who perform the related experiments under study.



Relaxation Oscillation Patterns

Dynamical systems with multiple time scales are modeled by singularly perturbed systems. The geometric approach to analyse such systems within the framework of dynamical systems theory has proven to be very useful and has found many applications. However, for slow-fast systems with turning points which arise in many applications, it was for a long time not clear how to apply geometric methods. Recently, with the development of a new technique, called blow-up, geometric singular perturbation theory succeeded to analyse rigorously such singularly perturbed problems where normal hyperbolicity is lost.

In particular, I developed the blow-up technique for singularly perturbed systems in R^3 with folded two dimensional critical manifolds. I classified and analysed such systems according to the reduced flow on the critical manifold. Generically, solutions reaching the fold-line exhibit jumping behaviour which may lead to relaxation oscillations in systems with S-shaped critical manifold. Using this local result on the jumping behaviour near the fold-line we are able to derive global results for systems with S-shaped critical manifold like the existence and bifurcation of periodic relaxation orbits and their stability. In the case of the famous forced van der Pol oscillator this approach can be used to prove existence of an invariant torus and periodic orbits for moderate forcing.

Furthermore, we are able to show how chaotic attractors arise in a generic way in periodically driven relaxation oscillators by connecting the theory of Henon-like families of planar diffeomorphisms with geometric singular perturbation theory. We give conditions on the vector field of periodically forced oscillators such that the singular limit reproduces the setting studied in the theory of Henon-like maps. These conditions destruct the existence of an invariant torus in periodically driven relaxation oscillators and show one way how more complicated dynamics can arise in relaxation oscillators.

Another way to obtain complicated dynamics in relaxation oscillators is due to the so called canard phenomenon. Solutions usually exhibit jumping behaviour reaching the fold-line as described above. But there is another possibility for solutions reaching the fold-line at exceptional points by passing through the fold-line and then following a canard solution near the repelling slow manifold for a considerable amount of time. This behaviour may lead to a delay of repulsion and has a major impact on the global dynamics of relaxation oscillators. For instance, a significant delay of firing in a network of Hodgkin-Huxley neurons with excitatory synaptic coupling can be explained with the canard phenomenon.

These complicated dynamics are also known as mixed mode oscillations (MMO), for which the oscillatory cycle consists of a number of large amplitude oscillations and a number of small amplitude oscillations. These patterns are well known in chemical oscillators but are also of interest in bio-chemical or neural problems.



Neuronal Networks

Respiratory Pacemaker Network

The inspiratory phase of the respiratory rhythm is believed to originate in a group of neurons in a region of the brain stem referred to as the pre-Boetzinger complex (pre-BoetC). Experiments in brain slices have shown that a synaptically coupled network of pre-BoetC pacemaker cells can display synchronous bursting oscillations, another type of pattern observed in relaxation oscillators where bursts of action potentials are followed by a silent phase of near quiescent behaviour.

We provide a mathematical analysis of the mechanisms underlying the transitions from quiescence to bursting and from bursting to spiking in the network, under parameter changes. Within the bursting regime, we examine how synaptic coupling and excitatory inputs combine to influence the silent and active phase durations, and hence the period, of bursting. Furthermore, our results will contribute to the study of the biology of respiration by generating experimentally testable predictions relevant to pacemaker network dynamics in the pre-BoetC.



Hypothalamic Neurons

Thermoregulation is controlled by a continuum of neuronal structures and connections extending from the hypothalamus and limbic system to the lower brainstem and reticular formation to spinal cord and the sympathetic ganglia. One extremely important thermoregulatory area is located at the preoptic area and anterior hypothalamus (PO/AH). This area is thermosensitive and has partial control over all of the physiological and behavioral thermoregulatory responses. Hypothalamic neurons sense changes in deep body temperature and integrate this information with afferent sensory information from thermoreceptors in the skin. In response to peripheral temperature changes, hypothalamic neurons initiate appropriate thermoregulatory responses to maintain constant core temperature.

The goal of this project is to design a mathematical model for these PO/AH neurons. We want to identify the ionic channels which are responsible for thermo-sensitivity and build them into a conductance based Hodgkin-Huxley like model. We are focusing on sodium and potassium channels which may be expressed in PO/AH neurons. We obtain all different kinds of thermo-sensitive behaviours by only changing maximal conductances of the ionic channels in our model. This suggests that the thermo-sensitive properties of PO/AH neurons are a result of different expression levels of certain ionic channels. We will generate experimentally testable predictions to identify all our model ionic channels and their thermo-sensitive properties. Thus the validation of the model has to be done through experiments.



Temperature Regulation

The basic neuronal model explaining temperature regulation, the Hammel model, suggests that in the PO/AH region populations of warm-sensitive and insensitive neurons synaptically control separate populations of effector neurons for each of the thermoregulatory responses. It is the mutually antagonist synaptic input to effector neurons which triggers the thermoregulatory response. At a set-point temperature (usually 37 degrees Celcius for humans) the synaptic input from warm-sensitive neurons is 'equal' and opposite the synaptic input from insensitive neurons, i.e. no response is caused by synaptic input. Shifting the temperature from the set-point will produce a net synaptic input to the effector neurons which will cause a thermoregulatory response to obtain the set-point temperature back. Various drugs and endogenous substances affect temperature regulation by altering the activity of hypothalamic neurons. Perhaps the best examples are pyrogens that cause fever by elevating the regulated set-point temperature.

A quantitative network model of PO/AH neurons which simulates thermoregulation and initiation of fever accurately would be of great interest and is the long-term goal of this project.



Calcium Signaling

Calcium Signaling in Non-Excitable Cells

Calcium acts as an intracellular messenger, relaying information within cells to regulate their activity such as the fertilization of eggs and the secretion of hormones and peptides. In non-excitable cells calcium is supplied mainly by the intracellular calcium stores (e.g. endoplasmic reticulum). The second messenger inositol trisphosphate (IP3) is generated by the action of the enzyme phospholipase C (PLC) on phosphotidylinositol biphosphate (PIP2) at the plasma membrane, in response to the action of growth factors, hormones or neurotransmitters at receptors. The second messanger IP3 acts on so called IP3 receptors in the endoplasmic reticulum, which cause release of calcium from the store.

The main way of calcium signaling is oscillatory but with widely different periods from a few seconds to hours. Models to explain these oscillations are broadly based on the self-propagating regulatory properties of calcium on the IP3 receptor known as calcium induced calcium release, or on the PLC/IP3 signaling pathway. Distinction between the two schemes relies on whether IP3 oscillations, from repetitively switching PLC on and off, are the driving force, or whether calcium alone controls this process by enhancing or inhibiting its own release from internal stores at low or high concentration, respectively. There is first experimental evidence that calcium concentration changes may be induced by IP3 oscillations. The goal of our work is to design a quantitative model which supports this hypothesis and suggest further experiments for validation.