In a previous post I discussed a paper of Lev Bar-Or about the dynamics of the interactions between T cells and macrophages via the cytokines they produce. The model of that paper is a four-dimensional dynamical system with eighteen parameters. In that post I also discussed a paper of mine where I proved some mathematical results about this system. The features I captured were that for certain values of the parameters there is one stationary solution which is a global attractor while for other special values of the parameters there are three stationary solutions, two of which are stable. Both these cases were already seen in phase portraits included in the original paper. Some numerical simulations I did at that time did not reveal any more complicated behaviour.
More recently experiments I did using Mathematica led me to discover some new phenomena. For certain values of the parameters it is possible to extract a two-dimensional model system with just two parameters and and this already displays interesting dynamical phenomena. The parameter encodes the effect of antigen presentation. A two-dimensional system has the advantage that its qualitative properties can be displayed effectively in the plane. I found the command ‘Manipulate’ in Mathematica to be very helpful in searching through parameter space. First I used it on a phase portrait. This did not get me very far for the following reasons. I found it quite hard to recognize the important dynamical features in the phase portrait and I found that searching in parameter space quickly became tiring. Next I tried displaying nullclines as a method to look for stationary solutions. There it was easier to see what was going on since it is only necessary to see where two curves intersect. My first attempt with this approach also gave nothing new but the fact that it was less tiring meant that I was encouraged to try again. The second attempt revealed parameter values where there are seven stationary solutions. Once I had found these parameter values I could go back and compute the corresponding phase portrait. It shows that four of the seven steady states are sinks while the other two are saddle points.
Having seen these pictures I had the ambition to prove the existence of seven stationary solutions, four of which are stable, under conditions on the parameters which are as general as possible. I succeeded in doing this. In addition I was able to obtain generalizations of these results to the original four-dimensional system. These theorems and their proofs are contained in a new paper. I will now describe a key technique which was used. I first thought that this method was quite different from anything I had used before but I later saw that it does bear some resemblance to the method of Fuchsian equations. I leave it to the interested reader to think about what the relations are. The idea for proving the existence of stationary solutions is as follows. Often there is an obvious way of defining a mapping whose fixed points are in one-to-one correspondence with stationary solutions of the dynamical system. Suppose now that for a given system it is possible to guess that there is a stationary solution near some point with coordinates . Consider a small box of size containing this point. If it is possible to show that this box is invariant under then the Brouwer fixed point theorem implies the existence of a stationary solution inside that box. If it can further be shown that the restriction of to the box is a contraction the stationary solution is the only one in the box. In the example the necessary estimates are obtained with the help of a parameter which has the property that the stationary solution tends to as it gets large. The procedure is then to choose the parameter large and the size of the box to be neither too small nor too large. With the obvious mapping this is enough to capture the four stable stationary solutions. The other three are more difficult and it is necessary to define a different mapping which has the same fixed points as but has other good mapping properties in addition. This more refined technique can be used to get the other three stationary solutions, at least in certain cases.
On the biological side these results suggest a scenario where the alternative states of the immune system are not just distinguished by the dominance of Th1 or Th2 cytokines but also different discrete possibilities for the strength of the dominance. Changing from one of these steady states to the other leads to a 50% increase in the total cytokine concentration and so it is by no means a small effect. I am not aware that this kind of phenomenon has been observed experimentally. One thing I will take away from this project is that in the future I plan to try to combine analytical and numerical approaches in an increasingly sophisticated way when trying to investigate the properties of dynamical systems.