III. Dynamical Systems and Bifurcation Theory

The intention of this page is a short presentation on my research in the field of dynamical systems and bifurcation theory to give a brief overview on this topic.

In general, a dynamical system is a mathematical formulation of the scientific concept of a deterministic process. This model describes the behaviour of the state in time, e.g. in a biological, chemical or physical process. A dynamical system consists of a phase space or state space (set of its possible states) and is defined by an initial value problem. Thus, we consider the system of autonomous ordinary differential equations 

where 

denote the state variables and  the (bifurcation) parameters, respectively. Moreover, we consider a smooth function

 Here, we are interested in the behaviour of the equilibrium/steady state of the autonomous system of ordinary differential equations, i.e. the solution of the system that does not change with time and therefore solves the equation

 Further, we assume that the preceding algebraic equation can be solved for every fixed, such that we have an equilibrium that depends on the parameter, i.e.

 and changes its value by varying this parameter. Notice that a dynamical system does not have to depend on a bifurcation parameter. In general, an equilibrium is a solution of the algebraic equation

 Nevertheless, we will consider problems depending on a bifurcation parameter. In fact, we are interested in the qualitative change in dynamics of the dynamical system produced by varying this parameter. At this stage we derive at the topic of bifurcation theory. Bifurcation theory has proven to be a very helpful and powerful tool in order to investigate dynamical systems and their (complex) dynamics. Furthermore, numerical bifurcation analysis has become a profitable tool in the study of (for instance) climate, neuronal and cardiac models. A bifurcation occurs at some parameter value if there are parameter values arbitrarily close to with dynamics which are topologically not equivalent from those at , e.g. an equilibrium may lose or win stability, or a limit cycle may appear. This large topic we do not want to explain too much in details, since there are so many greats books available, see e.g.

1. Kuznetsov. Elements of Applied Bifurcation Theory. Springer.

2. Shilnikov, Shilnikov, Turaev and Chua. Methods of Qualitative Theory in Nonlinear Dynamics (Part I and II). World Scientific Series on Nonlinear Science.

3. Guckenheimer and Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.

But, we would like to explain this topic with the following example. We consider a certain Hodgkin-Huxley model describing an action potential; see the paper J Physiol. 117(1952), no.4 p.500–544. In physiology, an action potential (AP) is a temporary, characteristic variance of the membrane potential of an excitable biological cell (e.g. neuron or cardiac muscle cell) from their resting potential. The molecular mechanism of an AP is based on the interaction of voltage-sensitive ion channels. The reason for the formation and the special properties of the AP is established in the properties of different groups of ion channels in the plasma membrane. An initial stimulus activates the ion channels as soon as a certain threshold potential is reached. Then, these ion channels break open and/or up such that this interaction allows an ion current, which changes the membrane potential. A normal AP is always uniform and the cardiac muscle cell AP is typically divided in four phases, i.e. the resting phase, the upstroke phase, the (long) plateau phase and the repolarisation phase:

Here, we would like to highlight the Rudy Laboratory, where the MATLAB code for the Luo-Rudy original model, 1991, "A model of the ventricular cardiac action potential - depolarization, repolarization, and their interaction" using a Hodgkin-Huxley-type formalism, is available. Further models of cardiac cell can be found on scholarpedia.

Therefore, the Hodgkin-Huxley model enables the reproduction of (normal) APs and moreover, to analyse and investigate the dynamics in a cardiac muscle cell. If there are depolarising variations of the membrane voltage, then we are speaking about afterdepolarisations (ADs), see e.g. link. These ADs are divided in early afterdepolarisations (EADs) and delayed afterdepolarisations (DADs). This division depends on the timing obtaining of the AP. The EADs occur either in the plateau or in the repolarisation phase of the AP and are benefited by an elongation of the AP, while the DADs occur after the repolarisation phase is completed. Thus, one could use the Hodgkin-Huxley model to investigate cardiac arrhythmia. EADs are additional small amplitude spikes during the plateau or the repolarisation phase of the action potential (AP), i.e. pathological voltage oscillations during one of these phases of a cardiac AP. Furthermore, EADs are caused by ion channel diseases, oxidative stress or drugs and are often associated with deficiencies in potassium currents or enhancements in the calcium or sodium current. The cardiac arrhythmia we are interested are EADs, cf. the following figure:

The main aim is the study of EADs and to understand the mechanisms which induce EADs. Here, we are interested in the reason why different oscillations occurs for different values of a certain parameter. To this end, we use several approaches. One approach is the bifurcation analysis. In the following, we will explain (not too much in details) some ideas and illustrate this utilising MATLAB and the continuation toolbox MATCONT, see the following link. Here, we have to mention that the visualisation of a three-dimensional system with one bifurcation parameter is (could be) already very difficult. Here, we try to give the best possible impression of this problem. Therefore, we utilise several points of view and we also created some movie hoping to make the comprehension much easier. We start to determine the equilibria or the equilibrium curve, respectively. This means that we have an equilibrium depending on a fixed bifurcation parameter. If we now vary this parameter, we get for each value of the parameter an equilibrium and therefore, an equilibrium curve. We are able to determine one condition for the equilibrium, but because of the structure of the Hodgkin-Huxley-type formalism, we are not able to solve this condition explicitly. To this end, we need a numerical approach, or we could use for example MATCONT, see also the dsweb page for further toolboxes. Utilising MATCONT provides simultaneously the stability of the equilibrium curve and certain bifurcation points. This yields the following equilibrium curve:

The preceding figure shows the equilibrium curve in the phase space (x,f,V) and the equilibrium curve in a bifurcation diagram. Here, we see that on the one hand there are stable equilibria/steady states depending on the bifurcation parameter and on the other hand unstable ones. Moreover, the considered system exhibits two Andronov-Hopf bifurcations - a sub- and a supercritical one - in a reasonable range. Close to the subcritical Andronov-Hopf bifurcation the Jacobian evaluated at the equilibrium has the following eigenvalues: 

Here, the equilibrium changes stability at the subcritical Andronov-Hopf bifurcation and moreover, an unstable limit cycle bifurcates from it, while from the supercritical Andronov-Hopf bifurcation a stable limit cycle bifurcates. This we will illustrate in the following movies. In the first movie, one can see the equilibrium curve, where unstable limit cycles bifurcate from the subcritical Andronov-Hopf bifurcation (yellow, non-transparent) disappearing at the supercritical Andronov-Hopf bifurcation. Additionally, there occur period doubling bifurcations (red cycles) and limit point (fold) bifurcation of cycles (stable - black cycles, unstable - blue cycles). Please note that the limit cycle branch becomes stable via the limit point bifurcation of cycle. From the period doubling bifurcations there also bifurcates limit cycles.

Here, we used MATCONT to start the continuation of limit cycles from the subcritical Andronov-Hopf bifurcation. This yields these cycles and a closed curve (using accurate MATCONT parameters). For a clearer understanding we show here the previous video in a cross section:

Finally, we present the parameter dependence of the equilibrium, where we replace the gating variable x by the bifurcation parameter and consider the corresponding bifurcation diagram. Here, we show only a cross section:

One could see the limit cycles bifurcating from the Andronov-Hopf bifurcations (yellow). Moreover, there are limit cycles bifurcating from period doubling bifurcation (blue, transparent), see also:

As a last point we show here a final movie that illustrates several trajectories with fixed bifurcation parameter and how they coincide with the bifurcation diagram:

By the previous presentation it is obviously not easy to analyse the complete behaviour of the full system. If we consider the same dynamical system describing cardiac AP with a small change of two parameters it could become more complicate, cf.

Here it is much more difficult to see what is going on. Therefore, one maybe need same new approach. At this stage, we would like to mention a further way to investigate dynamical system, i.e. the approach of geometric singular perturbation theory of slow-fast systems including mixed-mode oscillations and multiple time scales dynamics, since the Hodgkin-Huxley model exhibits different time scales, see for example

1. Desroches, Guckenheimer, Krauskopf, Kuehn, Osinga, Wechselberger. Mixed-mode oscillations with multiple time scales. SIAM Review, 54(2012).

2. Kuehn. Multiple Time Scale Dynamics. Springer.

In paper [4], we show that early afterdepolarisations are canard- and Hopf-induced depending on the system parameters. Some details on canards are provided here Canards.

The paper [2] and [4] is focused on EADs induced by an enhancement in the calcium current. Using the numerical bifurcation analysis and bifurcation theory, we can find regions, where EADs occur via an enhanced calcium current. Even more, we study in this manuscript a multiple bifurcation parameter problem, i.e. we use the conductance of the calcium current and the potassium current as the two main bifurcation parameters. Here, it turns out that one can balance the influence of an enhanced calcium current via an increased potassium current, i.e. 

Moreover, we used several approaches to control the EADs. To this aim we introduce a control parameter, which we use as bifurcation parameter in our controlled system. A good approach we studied in this paper is to control the EADs by increasing the potassium current, where we simultaneous increase the conductance of the potassium current and decrease the relaxation time constant of the gating variable of the potassium current, please see the preprint version below. This means that we "shift" the "bad" region to the relaxation time constant to get a smaller "bad" region with respect to the conductance, please cf. the last and the following figure:

This we achieve not only with the control parameter, we also add a weight to the relaxation time constant. If we decrease this weight, i.e. we decrease relaxation time constant with a weight equal to 0.1, then we get the following multiple bifurcation diagram, which is not mentioned in this paper:

In this situation, we have only periodic spiking and some oscillations before the supercritical Andronov-Hopf bifurcation, which are neither EADs nor normal APs, or the voltage reaches immediately a stable equilibrium. Therefore, a smaller weight seems to be a too radical measure, please see the previous figure.

Related manuscripts of myself are:

1. Period Doubling Cascades of Limit Cycles in Cardiac Action Potential Models as Precursors to Chaotic Early Afterdepolarizations. BMC System Biology, (2017) 11:42, DOI 10.1186/s12918-017-0422-4, with Mak Bulelzai and Philipp Kügler.

2. Bifurcation analysis of a certain Hodgkin-Huxley model depending on multiple bifurcation parameters. Mathematics, 6(2018), no.6(103), p.1–15, DOI 10.3390/math6060103. preprint version(first version)

3. Early afterdepolarizations in cardiac action potentials as mixed mode oscillations due to a folded node singularity. PLoS ONE, 12(2018), no.13, DOI 10.1371/journal.pone.0209498 with Mak Bulelzai and Philipp Kügler

4. Early afterdepolarisations induced by an enhancement in the calcium current. Processes, 7(2019), no.1(20), p.1–16, DOI 10.3390/pr7010020

5. On complex dynamics in a Purkinje and a ventricular cardiac cell model. Commun. Nonlinear Sci. Numer. Simul., 93(2021), DOI 10.1016/j.cnsns.2020.105511 with Susanne Solem

6. Analysis and simulation of a modified cardiac cell model gives accurate predictions of the dynamics of the original one,  arXiv:2105.05544 with Susanne Solem

Furthermore, an introductory presentation of myself yielding an overview on this research you can find here:

• Talk and presentation from the 13.11.2017 - based on my preprint Canard- and Hopf-induced Early Afterdepolarizations -  at the PDE seminar at UiO, see the slides.

• Talk and presentation from the 26.02.2018 - based on my preprint Bifurcation analysis of a certain Hodgkin-Huxley model depending on multiple bifurcation parameters -  at the Simula Research Laboratory, see the slides.

• Talk and presentation from the 01.06.2021 - at the Applied Analysis Group Seminar, University of Bremen, see the slides.