Migraines suck—understatement of the year right there. Migraines are also poorly understood, and it's kind of hard to come up with effective remedies if you don’t understand what you are treating.

One way to understand migraines is through physics. The brain is a network of neurons that are constantly talking to each other. This, in physics-speak, is a dynamical system. Dynamical systems can have more than one stable operating point. A horrible consequence of this particular system might be that migraines represent a stable operating point of the brain.

No going back

A recent paper—one that isn’t especially exciting—has forced me to write about oscillating brains. I’ve talked about chaotic systems, nonlinear dynamics, and dynamical systems before. I won’t repeat everything written previously, but I do want to emphasize the concept of transitions that don’t easily reverse.

If you have something like a weight on a spring, then the stiffness of the spring and mass of the weight are enough to specify the weight’s oscillation. It operates at a fixed frequency, which, if friction did not exist, would continue forever. If you give the spring a shake, the weight will jiggle about randomly for a bit before returning to its steady oscillation. This is called a stable point of operation.

The stiffness of the spring would, if the spring was ever over-stretched, suddenly change. So, the spring's favored frequency only holds as long as the oscillations of the mass stay small. If we excite the oscillator too much, the mass will travel too far and stretch the spring. That will suddenly change the oscillation frequency of the mass. Critically, it will never change back. The spring has a new stiffness, and that dictates the new oscillation frequency.

This analogy is a bit stretched, as we will see below, but it is a good starting point to understand irreversible transitions. The spring’s stiffness is irreversibly modified by large oscillations, which changes the frequency of the oscillation.

Your brain is strangely attractive

Although the brain is not a simple mass on a spring, its network of neurons does create a dynamical system. Each neuron either excites or suppresses the neurons that it is connected to. The sequence of firing, the overall activity, and the measurable average electromagnetic activity all oscillate within certain bounds, even as they respond to external stimuli.

Unlike our mass on a spring, which has a single stable point, the brain operates on a multidimensional surface. That surface has remarkable properties: it is infinite in extent, meaning that the brain has access to an infinite number of possible states. But the infinite surface occupies a finite volume, which means that there are also an infinite number of states that the brain cannot access because they are not on the surface.

The surface is called a strange attractor. The presence, absence, and shape of this surface depends on the physics of the brain. In a strange way, the brain is entirely unpredictable, while still, in certain ways, being predictable.

Migraine studies suggest that some unknown stimulus changes the brain in a way that changes the shape of the surface of available states. The brain then moves to the new surface and travels from state to state along this surface. Unfortunately, these new states all involve pain.

Watching the pain spread

The paper that set me writing modifies an existing model. The model basically links the change in activity of a local part of the network of neurons to the activity of adjacent regions, adding a driving factor from external stimulus and a damping factor that lets activity die out. That means there are a few knobs to be played with: the amount of external stimulus, the sensitivity of the local activity to neighboring activity, and the speed with which activity dies out.

Since previous research shows that migraines spread from one location of the brain outward, the researchers were mainly interested in how the model behavior was influenced by changing the sensitivity of neighboring brain regions to each other. To do this, one region was chosen as the initiating point for the migraine. The sensitivity of that region was modified to see how it changed the behavior of the whole brain.

The researcher’s model shows transitions from a normal brain to a migraine state (called "prodromal") and from there to a pain state as the sensitivity of a single region is gradually changed. Importantly, the prodromal state does not appear to be a strange attractor—neural activity becomes periodic and predictable, rather than wandering across a surface of possible states.

The transition to and from the prodromal state is presaged by greater variability in local peaks in brain activity. The pain state looks like another stable region (and is also a strange attractor), so, once there, it takes a significant change to drive the brain back to its normal state.

Does the model make a difference?

To bring about treatment based on this requires electrodes in the brain to help kick it out of this state. Many experimental systems that involve electrodes being placed in the brain have problems. Most notably, the body recognizes the invading electrodes and encases them in scar tissue. This reduces their ability to sense and modify brain activity.

That leaves drugs, which puts yet another step between the model description of the brain and how the treatment is applied. The more steps we put in, the less likely it is that we can derive insight from the model that can be linked to the treatment.

I guess what I am trying to say is that I really love these models as a way of understanding in a global sense why the brain does what it does. But I don’t believe that this global understanding can be directly used to create treatments.

European Physics Letters, 2018, DOI: 10.1209/0295-5075/123/10006. (About DOIs).