The Autistic "Equation of Motion"
Using physics to try to explain the dynamics of the autistic mind...
Introduction and a brief foray into physics
If you saw my previous post regarding my year in the life, you may have noticed I wrote some stuff in LaTeX, which allows me to format mathematical and physics related things very precisely. This got me thinking about my physics background. A lot of my work revolved around working with equations of motion, whether it was applying Newton's Second Law, or using advanced, purpose made computer programs designed to apply the Feynman rules that can be "read off" the Standard Model Lagrangian, which dictate what interactions are allowed.
Can we do the same thing for autistic brains? Can I develop a "Lagrangian" or a "equation of motion" for that?
This post will try to do that.
Before we tackle this question, we should briefly mention what an equation of motion is.
In mechanics, an equation of motion is a mathematical equation that describes the evolution of a system over time. There are several ways in which one can derive the equation of motion for a system. These are:
1. Newtonian mechanics, which involves applying Newton's second law,
2. Lagrangian mechanics, which involves a quantity called the Lagrangian (L), which is equal to the kinetic energy minus the potential energy of the system, and
3. Hamiltonian mechanics, which involves another quantity called the Hamiltonian (H), which is usually the same as the sum of the kinetic and potential energies.
It is the second formulation that we will use for developing the autistic brain's "equation of motion". The procedure to applying this formulation for a physical problem is as follows:
We identify the system in question. While we are doing this, we choose a good coordinate system that will make our math easier.
Determine the kinetic and potential energies.
Construct the Lagrangian based on the kinetic and potential energies we found in the previous step. In equation form
\(\mathcal L = T - V\)where the script L denotes the Lagrangian, T denotes the kinetic energy, and V denotes the potential energy.
Construct the Euler Lagrange equations for each of the coordinates that we have in our problem. This involves setting up equations of the following form1:
\(\frac {\partial \mathcal L}{\partial x} = \frac{d}{dt} (\frac{\partial \mathcal L}{d \dot x})\)
where the fancy d’s denote partial derivatives, and dots over variables denote time derivatives.
Perform the various derivatives. Once completed, the equations of motion are produced.
Okay, great! But how do we apply it to the autistic (or neurodivergent) mind?
With this brief intermezzo into physics done, we can now try to apply this approach for the autistic or neurodivergent mind. We shall follow the procedure that I wrote above.
Our first step is to identify the system in question and determine a good "coordinate" system. This is where we run into some difficulty. The system itself is easy to identify as it is the brain. However, the brain is not a mathematical construct, but rather a tangible object that does a lot of processing. The human brain contains anywhere from 70 to 85 billion neurons2, which have some interesting process, such as synaptic pruning. In neurodevelopmental conditions like autism, these processes proceed differently than in a neuronormative person3. So to try to treat the brain’s neurons like a system of point particles would be impossible and, even if it were possible, would likely miss out on some nuances.
Instead, let's take a different approach. We should look at the variables that cause the brain to think differently, as well as thinking about how to interpret the different elements of the Lagrangian and the Euler-Lagrange equations in this biological system.
To simplify matters, we shall consider one person. The "Lagrangian" that we would construct here would obviously be person dependent, as each person would respond to the same stimulus differently. I might be aloof to some stimulus (say, spicy food), yet somebody else might be overstimulated by the same stimulus. To take a "derivative" in people would not make much sense, as people are discrete, that is, they each take one spot on the vast spectrum of neurodivergence.
This “Lagrangian” would also be time dependent, but in two sorts of ways. There’s a day-to-day component, which depends on what’s going on immediately in that person’s life, and a slower, longer term component which encompasses a bunch of things, which include, but are not limited to:
Changes in “special” interests - whether it’s different ones entering the fray or the levels of intensity of those interests
Changes in the environment - not only in the person’s life (location, who the person hangs out with, work, etc.) but also in regards to the world at large
Changes in the person themselves as they age and acquire skills and knowledge, as well as hanging with others to gain experience in social interactions.
Things that reduce the amount of burden life has, such as using appropriate accommodations (AAC devices, Hidden Disability Sunflower, the ability to work remotely, etc.)
Changes in sensory processing - perhaps things that were once intolerable can be handled, and vice versa, where once tolerable things are now things that we are repulsed by.
Let's recall that the Lagrangian depends on the kinetic energy and the potential energy. Hence we can work on each term separately.
The Kinetic Energy Term - Where the Magic Is
In physics, kinetic energy is the energy associated with the motion of a particle.
In the mind, this could be viewed as our ability to think beyond what is minimally necessary to live (i.e. controlling our breathing and heart rate, digestion, basically anything that the body handles automatically without us consciously thinking about it). Any sort of creative thinking, learning about things (whether related to a special interest or not), or planning is a measure of this.
When kinetic energy is high, then we have the ability to get a lot done, provided that there is nothing getting in the way of doing stuff. We feel joy in doing stuff, we’re willing to try things beyond our usual comfort zone, and we feel free to truly express who we are. Likewise, if kinetic energy is low, then we don’t feel quite as adventurous. We would rather stay home, where we are in a safe place and we don’t have to expend as much energy trying to “fit in” with the crowd or deal with things that overstimulate or understimulate us.
The Potential Energy Term - Barriers
In physics, potential energy can be thought of as the ability for a particle to do work, based on its position or its configuration. It is also equal to the work done against any restoring forces.
With regards to the neurodivergent or autistic mind, we can think about barriers that affect our ability to function. These include, but are not limited to:
Materials that are inaccessible due to their format
Sensory issues
“Masking” to fit into the mold that others expect
People not understanding fully the struggles we deal with
Transportation
Being forced to talk with our voice, rather than use methods of communication that we prefer (for example, typing/texting)
Making executive decisions (what am I going to make for dinner? Where would I like to go?)
Constructing “roadblocks” to try to maintain routine, because you are resistant to change4
The list of barriers can go on and on, but broadly speaking there are two types - external (those that appear because of how others operate) and internal (those that come as a result of how our mind and body work). When working alone these barriers can be quite formidable and require a lot of energy to circumnavigate. This saps into the free energy that we have.
We also have to include in this term the amount of energy that activities of daily living and work require. So the potential energy term has four (or three, if you wish to lump both types of barriers together) terms:
Putting it All Together - Our Energy Reserve and Connecting to “Spoons”
When the potential energy is taken away from the kinetic energy, we get the Lagrangian.
When applying this to the mind, we can view this as the amount of “free” energy that we have - the amount of energy that we have remaining to do things. This sort of idea has been described in the disability space before, in particular, with regards to chronic illness. Spoon Theory states that for the day, we start with a limited amount of “spoons”, and each action requires a certain number of these (could be 1 or more). When all the spoons are used up, that’s it5.
The advantage of using this Lagrangian approach is that we can model the amount of “free” energy continuously, instead of in the discrete chunks that “spoons” entail.
When the potential energy exceeds that of the kinetic energy, we become slowly drained. This might not appear outwardly at first but as the deficit becomes bigger, or is sustained for a long period of time, you start to feel a pain in your head, which grows if changes are not made. Eventually, if the deficit is not resolved, burnout can occur, which can drastically impact the amount you can do, possibly permanently, as you navigate the burnout and try to pull yourself out of it. Such a thing occurred to me as I was working through my PhD, and only once I pulled out of it did I feel the weight of my head slowly ease.
What’s the meaning of taking derivatives here?
As we saw in step 4 of the procedure of deriving the equations of motion in Lagrangian mechanics, we have to take derivatives. As we don’t have a mathematical version of the autistic “Lagrangian”, we can’t exactly do that. However, we can envision what the meaning of doing this is.
Perhaps the variable that makes the most sense to take a derivative of here is time. If we take a time derivative of the Lagrangian, we can see if, over time, we are gaining energy or losing energy. If we are gaining energy over time, then we can push ourselves a bit more, or perhaps introduce some additional novelty. On the other hand, if we see that we are losing energy over time, then we may need to consider ways to conserve energy during the day, or find activities/work that require less energy, so that we can try to avoid going into pre-burnout (or worse, burnout).
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Actually solving these is beyond the scope of this blog post.
Technically you could use more spoons than you have for a given day, but this goes against next day’s spoons, which is not desirable, especially because you never know when you may fall ill with something else, or catch a cold, and thus having a reserve of spoons is important!