A Fast-Slow Dynamical System Model to Study Drug Addiction

By Karthika Swamy Cohen

Editor's Note: This article originally appeared in SIAM News on July 17, 2019 (https://sinews.siam.org/Details-Page/a-fast-slow-dynamical-system-model-to-study-drug-addiction-1).

Addictive disorders affect billions of people globally. Nearly 50,000 drug overdose deaths were reported in the U.S. alone in 2017. 

Studying the underlying science of addiction is complicated because of the dynamics of the condition: the periodicity, and the interplay between relapse and recovery making it hard to study. 

Addictive disorders affect billions of people worldwide, with nearly 50,000 drug overdose deaths reported in the U.S. alone in 2017.

This is where mathematical models come in: math models—such as continuous dynamical systems—have been used to understand a variety of cyclical processes in natural and manmade systems alike.

Addictive disorders manifest as relapse-recovery cycles, where relapse is a rapid event while recovery is slow and prolonged. Fast-slow dynamical system modelling techniques can best represent this fluctuation between fast and slow phases. For this reason, such systems have been used to describe a range of biological process characterized by periodic abrupt changes while operating on multiple timescales.

The craving level of a patient is one fundamental behavioral aspect of the relapse-recovery cycle that changes periodically throughout.

A relapse occurs when cravings reach and pass a critical threshold. Thus a craving determines when relapse happens. Mood also plays an important role in the cycle since a patient feels gratification upon satisfaction of a craving and depression or anxiety during episodes of withdrawal. Hence, the relationship between craving and mood must be accounted for to adequately describe the relapse-recovery cycle. 

In a paper published recently in the SIAM Journal on Applied Dynamical Systems, Jacob Duncan, Teresa Aubele-Futch, and Monica McGrath use a fast-slow dynamical system model to do exactly this: the model analyzes the interplay between an addictive disorder patient’s levels of mood and craving. 

The abruptness of relapse can be attributed to dopamine surges in the brain’s reward system triggered by the fast action of the drug. The authors use their model to analyze examine the macroscopic effects of this highly complex chemical process.

All drugs of abuse increase levels of dopamine in the brain's pleasure center. Researchers studied this complex chemical process through a math model.

The model utilizes relaxation oscillations with excited states to depict relapse and mood crash, and relaxed states to illustrate recovery and craving satiation. The researchers separate the time scales by neglecting fast states of the relaxation oscillation and linearizing the slow branches of the limit cycle; this decouples the system, allowing an analytic solution of the differential equation governing craving levels.

The solution is then used to determine the durations of craving build-up and satiation – these time periods together approximate to the full cycle period, which allows one to predict relapse frequency. 

As a parameter that is responsive to treatment is varied, its value changes, and the system moves from a state of periodic relapse-recovery to one that is relapse-free through a reverse Hopf bifurcation. The Hopf bifurcation thus portrays how treatment can lead to a break in this cycle, which denotes the point at which an addictive disorder patient can achieve and maintain abstention.

Since the model is based on psychological theories of addiction mechanisms, model-based prediction of the relaxation oscillation period renders a good estimation of the relationship between treatment and relapse frequency. The authors show that treatment efforts, such as craving management, can increase the treatment parameter, leading to less frequent relapses until the cycle is broken completely and results in total remission. 

Read the full paper. 

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