It’s called spherical cow by myself and others. In The Big Bang Theory it became spherical chickens in a vacuum: A simple model that is either used for back-of-the-envelope calculations or one that is actually useless. Let’s forget such jokes and go beyond simple models that are rather calculations, let’s go somewhere really much more interesting. I will tell you the story of so-called toy models, in particular their place in science within what might be seen as three categories of models: simple models, toy models, and ideal models. I don’t see this as a hierarchical categorization but surely a useful one.
To invite confusion between these three categories of models, in particular between toy and ideal models, certainly doesn’t seemed wise to me. And yet, the longer I consider the difference, the less clear I see it. First, an ideal model is at the end of the day just a scientific model. In science, there is always idealization. This puts emphasis on the special role of toy models. They are neither ideal models nor do they have anything to do with spherical cow-jokes or simplified calculations. The problem is, many people believe toy models are more like these simple models, more like spherical cows. They are wrong.
If you do not read on, please take this away: Toy models are extremely valuable scientific tools. If someone takes away from a scientist his or her toy model, or if someone argues these models are a joke, maybe even rejecting a paper because it only considers a toy model, the shocked scientist will not cry but certainly will think it’s cool to fight back. As I did many times and will now.
Let us start with some examples, actually no, not just some examples but the most famous examples out there, one for each type. Let’s start in opposite order.
1. The ideal gas model. Almost no need to say more. And I almost won’t. At this point, I only mention that the model’s right to live is—I am simplifying for clarity—to derive a law: the ideal gas law PV=nRT.
2. The Ising model. It is a so-called toy model for ferromagnetism. The Ising model is certainly less well known. Not unlike the spherical cow or chicken, we rip off any unnecessary details that make ferromagnets more complicated than needed to behave like one.
3. The spherical cow. Suppose you need a quick estimate for the upper limit xmax of the milk production in liters at a dairy farm with n cows. This could be xmax=n(h/2)¾π/1000, with parameter h being the average hight of a cow.1
I will focus mainly on toy models. The question I address is, what is the purpose of such models?
Not much mathematics?
Toy models often remind me at a first glimpse at sophisticated though experiments, looking longer and closer, they simply classify the mechanism of phenomena. But let me start with why I write this post today.
A few days ago, I got review reports of a submitted paper (you may read it on the arxiv). One report remarked:
… the author says „is still simple and explicit enough to be amenable for future mathematical analysis“, but there is virtually no mathematics in the manuscript, except for eqns (1) and (2), which the author characterizes as a „toy model“.
The anonymous reviewer (who wrote overall a really nice review) obviously believes writing „toy model“, as I did, indicates a reduced value of what was accomplished. As if I would have said: „Sorry, not much of mathematics here, so far only a toy model to kid around; you know spherical cows and stuff like that“. That could not be more wrong. I should emphasize, the paper was not rejected, quite the opposite, I got really good reports. But this particular point was criticized. In some of my former papers, the same issue came up and often I had to fight much harder. I always won. Finally, I decided to write this post about what I learned on my way.
‚Toy model‘ is fixed term
The mathematics in toy models can get as complicated as math can be. To get back to the second example above, it needs some dozens of pages full of mathematical derivations, a mathematical tour-de-force to analytically describe the phase transition in the Ising model. It undergoes a phase transition so that beyond a well defined critical temperature a spontaneous magnetization cannot longer be maintained. But this holds true only in a two dimensional (2D) Ising model. In 1D, the model does not undergo a phase transition—to proof this we still need half a page. Moreover, when a phase transition does occur, it is a second-order phase transition, not first order. (Remember above I mentioned classification of phenomena as the purpose?) I could go on with further results obtained from this toy model. Indeed, we can write books and teach courses suitable only for graduate students of physics and mathematics about the Ising model, about a toy model for ferromagnetism.
It’s called a „toy“ because it is not quite a ferromagnet, or model thereof. It only behaves like one „in principle“. As we ripped off any unnecessary details that make ferromagnets more complicated, we took care for not changing its principal behavior. Then we call it a toy model.
This is different to what theoreticians once did when they first considered an ideal gas.
The ideal in science
Idealization, which led, among other things, to the ideal gas law, is a process that makes the system not merely easier to understand (or solve mathematically). More importantly, idealizations define the regimes where the model leads to a „perfect“ description (as far as the usual accuracy of measurement is concerned). Such laws are formulated in our natural coordinates and quantities that we use to describe the system, or at least we have some means to get back to such ordinary coordinates and quantities. For a gas, these are the thermodynamic state functions volume, pressure, amount of a substance, and temperature. On the one side, these quantities can be measured and, on the other side, they are represented in the model. Both parts should match accurately. In toy models, nothing really needs to match quantitatively. Actually toy models may not even be formulated in coordinates and quantities that can directly be measured.
Ideal laws are even more easy to understand outside the field of thermodynamics.2 In this case we even often drop the adjective „ideal“. We call them law of motion. Think about Newton’s law of gravitation and it will be clear why. Objects fall only in a vacuum in accordance to Newton. Of course we don’t consider the air and its resistance as part of the system, but it is an idealization nevertheless. We might run into a philosophical discussion when we compare the idealization in Newton’s law of gravitation and, say, Newton’s law of cooling, which again involves thermodynamics and the idealization is more directly related to the system, a heat transferring narrow insulated rod, for example. Let’s not do this, because this is not my point today.
The point is: A toy model is quite different. Although it also might be considered a „law of motion“, the motion is in some abstract space. (When I equal model and law, as done here, it is meant in the sense introduced for the ideal gas model from which we directly derive the law. At some point, we got used to use „law“ and „model“ interchangeably. That is not at all only physicist’s jargon. In psychophysics, for example, psychologist say they „fit data with a model“, which is nothing but a function, or more fancy: a law.)
I lied above
When Wilhelm Lenz proposed what to rip off and how to design a toy model for a ferromagnet, he actually did not take care at all whether or not during this process the principal behavior of the model is still unchanged. Lenz did what every professor would do. He asked a graduate student to do the work for him and proof this for the given model. That is how Ernst Ising came into the game.
Ising investigated whether or not the principal behavior survived in the model that Lenz handed over to him and which eventually would get his name. Moreover, Ising was supposed to gain insight into the fundamental mechanism of the phase transition that ferromagnets obviously show in experiments. But Ising couldn’t do it. He did not fail, but „his“ model did fail. The model did not show any phase transition at all, at least not in the 1D version, which Ising investigated. As I mentioned already above, only in 2D, the Ising model showed the desired effect. To proof this, it took some more years. Finally, Rudolf Peierls, a doctoral student of Werner Heisenberg, proofed the existence of the phase transition in 2D and Lars Onsager solved this problem analytically in the afore mentioned mathematical tour-de-force. Without this result, the Ising model would not be today a famous toy model. In fact, it would not be a toy model at all. It would live it’s life as a spherical cow-joke.
For creating a useful toy model, it does not matter in which order we go through this process3. At the end, you need a model that shows the right behavior and is amenable for mathematical analysis.
Toy model = mechanism
In any case, the road Wilhelm Lenz, Ernst Ising and the others took is the one we also follow today to get a handle on the mechanisms in complex systems like the brain. It is the road I took to study migraine.
In general, we propose a toy model for some kind of brain activity not yet knowing whether it is correct or not. To propose a toy model is nothing but to propose an analogous mechanism to classify this behavior under consideration. Beside migraine, examples are neural activity in the basal ganglia that cause tremor in Parkinson’s disease or seizure activity in epilepsy.
The toy model then predicts further qualitative phenomenological consequences that are shared by all systems that are members of the class the toy model describes. It may sounds a bit like a circular reasoning, but it is not. Usually, we start with only some features, say from the observed brain activity. Once we have a proper toy model for these known patterns, we invest based on the model prediction whether or not also other yet unknown features can be found. Usually that means we wait for clinicians to perform new studies—after trying really, really hard to convince colleagues to do what we think should be done. In iterative steps, we narrow down the „correct“ toy model, that is, the properties of a complex system. A toy model can be considered as a mechanism.
This procedure works particularly well in pathologies of the brain because such dynamics usually take place on low-dimensional manifolds, while normal mental functions might be irreducible.
My migraine toy model
So in particular, I developed in several iterative steps a toy model for migraine with aura. And yes, I also had students who investigated much of the behavior in simulations. This toy model—which is published here
will be published soon but note that it is not the paper mentioned above, yet also available on arxiv—will hopefully help us to understand the activity patterns we have observed with fMRI and patient’s symptom reports during migraine [Dahlem & Hadjikhani, 2009]. Much like the Ising model helped us to understand phase transition in ferromagnets, this model is developed to understand phase transition in the brain causing migraine.
A toy model with a ghost, who would not love it?
We undertook a very detailed investigation with this toy model and addressed particular clinical problems. At this stage, the model makes predictions about how these problems might be resolved. Of course, predictions are the critical test for any model, for toy models, predictions are often even the foremost reason to develop them. For example, we predict the mechanism that leads to the prevalence of migraine subtypes „migraine without aura“ (~70%) and „migraine with aura“ (~30%) based on statistical properties of a phenomenon called critical slowing down. Some of the seemingly conflicting evidence with regard to the prevalence and a concept called „silent aura“ is resolved when our predictions are correct. This topic is currently very controversially discussed in the migraine literature.
It takes another post to go into this. In brief: according to my toy model, the cortical hyperactivity patterns during migraine with aura are formed by ghost behavior near a saddle-node bifurcation. These fancy words, which probably mean nothing to you, are words for phenomena related to a particular nonequilibrium phase transition that we design in the toy model. (Look, for example, into the textbook by Steven Strogatz, „Nonlinear Dynamics And Chaos“ and search for the word ghost.) We then examined this model and looked for the 70/30 distribution that could explain the prevalence data. We found it. This led to a particular new prediction about the distinct and characteristic spatio-temporal activity pattern evolving during the two subtypes. We need to test this in functional magnetic resonance imaging study to close the circle. When we do not find the expected results, the mechanism must be wrong. If we do find it, it still could be by chance but much more likely this that we nailed it from top to bottom.
The toy model’s right to live long and prosper
We (again a talented student involved) are also about to derive ideal models of the predicted cortical hyperactivity during migraine to further support our toy model. Often toy models do not stand alone (although they could) and it takes many years to completely understand their behavior and to successfully apply them to the real system under consideration.
I had to justify the ansatz of developing and using a toy model in migraine research almost in every single event I presented it. Although many scientist perfectly well understand toy models, there is always someone who doesn’t. Maybe also because I continue to call them „toy models“. When you hear someone talking about a toy model, don’t even think he or she is a spoiled and pampered scientist far form any real world problems. By and large, we are right on focus. Please don’t take away his or her scientific toy—a special note for reviewers. It perplexes me enough that even people working with toy models, sometimes do not consider to publish this. Lars Onsager did not publish his tour-de-force analytical solution of the 2D Ising model. Hodgkin and Huxley did not publish their first migraine toy model in the late 1950ties, early 1960ties, a mechanism on which I build mine. That makes me look really modest, I publish toy models.
1You may not call a back-of-the-envelope calculation a model. They are an estimate, like for the upper limit xmax of the milk production in litres at a dairy farm with n cows with an average hight of h. But such estimates are in a wife sense also models. And again, I am not talking about jokes at all, the story is funny enough without them.
2 I invite you to ponder about the difference between both coordinates and quantities in thermodynamics and in classical mechanics. Anyway, there might be some deeper reason to restrict the suggested categorization in toy models and ideal models to thermodynamics far from equilibrium—although I am not sure about it. So I hide this thought in a footnote because any move in this direction would blow up the post beyond reason.
3Nonetheless, there are ways to derive toy models instead of designing them and subsequently proofing their applicability. This derivation, however, runs under a different name, it is called „normal form“ model. Henri Poincaré developed this in his PhD thesis. An algorithm for reducing a system based on changes of coordinates, for instance, from Cartesian to polar coordinates but also more complicated ones. These transformations are chosen to eliminate the oversized description by finding symmetries but also other tricks (resonant modes …), but this would lead us to far. If the toy model is derived in such a way, that is, if it is a normal form, we may be able to say something about asymptotic consequences of this result. Usually we cannot. We always can, however, say something about the nature of the phenomenon. We can classify phenomena according to their normal form models.