Jacob Duncan
Winona State assistant professor Jacob Duncan has published a mathematical model of addiction. Photo: Paul Scott
 

WINONA, Minn. — What if there was natural law behind the universe of pain?

More to the point, what if addiction, with its dizzying highs and devastating lows, could be explained by an equation on a chalkboard?Almost as if falling off the wagon, that most human of failures, had its very own E= mc²?

That's what Jacob Duncan, a Zumbrota native with a PhD in mathematical biology, set out to do just over two years ago. A newly appointed instructor focused on the study of insect outbreaks and their ability to devastate forests in cyclical fashion, it occurred to Duncan that the cravings and relapse of addiction follow a similar pattern.

"It was kind of always in the back of my mind to develop scientific theory of this thing which has been stigmatized," says Duncan, who is 38 and teaches at Winona State. "The idea of illuminating how this can be thought of mathematically just like gravity, another phenomena in nature we can model scientifically, hopefully taking some of the negative pressure off of addicts who struggle with this."

The result of that work was published this spring in SIAM Journal of Applied Dynamical Systems, a prestigious house organ of the Society for Industrial and Applied Mathematics. The paper, "A Fast-Slow Dynamical System Model of Addiction: Predicting Relapse Frequency," is one of only a handful of studies to ever approach addiction as mathematical law. In the model, Duncan and two co-authors identified a universal fast-slow system, one driving experiences that can too easily resemble ordinary human failure.

"It came down to a couple of differential equations that track craving and mood levels experienced by an addict in time," Duncan says. Duncan describes craving as "like a spring that you're compressing until it breaks." Armed with just those two variables, rising craving rising against dropping mood, "from that we were able to come up with a formula to predict the frequency of relapse. " Which, as depicted as mathematical animation, is alarming in its cold, unwavering efficiency.

"You get to those moments where your mind changes," he says. "They call it a relapse of the mind. It is the change in thought patterns that lead you inevitably to relapse, a point that's like, right about somewhere around there," he said, pointing at a spot in a graph where a black line denoting cravings shoots rapidly upward before descending to return where it started.

"When you simulate the model on the computer, you can see the cycle as a loop," Duncan says, "one in which you have craving levels going up, slowly reach a threshold, relapse happens quickly, and then there's a mood spike followed by withdrawal." During this withdrawal the brain, previously awash in dopamine, undergoes a compensatory depletion of the reward neurotransmitter.

"For a while cravings are satiated in this recovery period," Duncan says. But as the emotional low of withdrawal wears off, the brain can take on a false sense of security, with cravings re-emerging. "Your brain has recovered, everything feels fine, and you start to use that as a reasoning that well, maybe I'm OK."

Over time, Duncan says relapses will begin happening quicker, according to the model, until the system tips into a nonstop bender. "There's a bifurcation point where suddenly the dynamics change drastically," he says. "It will become this constant where the person is perpetually using, always intoxicated, always high."

That's the bad scenario. The good scenario to his model, one following the introduction of some sort of treatment, medication or therapy intervention, can cause this same cycle of addiction to collapse. Duncan has also developed an animation to show what that outcome looks like.

"A lot of it is just bewilderment," Duncan says of morning-after regrets. "It's like, how did it happen again when you were so set on staying clean? It’s almost as if your volition has been taken over by something else. But it’s not something else. It's the fact that your dopamine pathways, neurotransmitters and reward system is stuck out of whack. That is the disease."

If that sounds like insight drawn from experience, it's because Duncan says he is a recovering alcoholic. "Family members have struggled with it. Friends. Myself in the past. This was sort of a personal endeavor to unite the two sides of my brain. One is mathematics and the other is the addiction that has been ever-present in my life, whether in me or other people in my life."

Duncan says the model applies to addiction of all kinds.

"All that stuff affects dopamine pathways, opiates, alcohol, food, sex, gambling ....The dopamine system is there for a reason. It's evolutionarily advantageous to feel a reward when you find a food and eat it. But drugs can fool your brain into thinking you're getting something you need, something good, when you're not."

Though the model holds up mathematically, Duncan still needs to test it with real-world data drawn from the daily moods and cravings of people struggling with addiction. He's confident the results will tell him the truth, whatever that is.

"Mathematics," he explains, "is the law of the universe. "

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