essay #1: your brain forgets things the same way your body eliminates drugs
the decay curve
In the 1880s, Hermann Ebbinghaus sat alone in a room memorizing nonsense syllables and testing himself at intervals, producing the first quantitative measurements of human forgetting. What he found was an exponential decay curve: memory retrievability drops off rapidly at first, then more slowly, following something like
R(t) = e^(−t/S)
where R is the probability you can recall something at time t, and S is a stability parameter — how resistant that particular memory is to fading.
Now open a pharmacology textbook to chapter one. After a single dose of most drugs, plasma concentration follows
C(t) = C₀ · e^(−k·t)
where C₀ is the initial concentration, k is the elimination rate constant, and the whole thing decays exponentially as your liver and kidneys clear the drug from your system.
These are roughly the same equation with different variable names.
half-lives
Pharmacologists talk about elimination half-life: the time it takes for drug concentration to drop by 50%. Aspirin has a half-life of about 4 hours. Diazepam, around 40 hours.
FSRS (the spaced repetition algorithm that emerged from optimizing millions of Anki review histories) uses an analogous concept: memory stability. While FSRS doesn't explicitly frame it as a half-life, the stability parameter S plays exactly the same role. A memory with S = 10 days decays at a rate that means you have roughly a 90% chance of recall at day 10. Double the stability, and that same 90% threshold pushes out to day 20.
In both systems, the half-life (or its equivalent) is the fundamental parameter that governs everything downstream. And in both systems, it changes with each intervention instead of being fixed.
the dosing problem
A pharmacologist's core challenge is maintaining drug concentration within the therapeutic window (high enough to be effective, low enough to avoid toxicity). You do this by giving discrete doses at calculated intervals, each one boosting the concentration back up before it decays too far.
A spaced repetition algorithm's core challenge is maintaining memory retrievability above a target threshold (high enough that you can actually recall the material, but you don't want to review so frequently that you waste time on things you already know). You do this by scheduling discrete reviews at calculated intervals, each one boosting retrievability back up before it decays too far.
Both are solving the same optimization problem, which is minimizing the cost of intervention while keeping a decaying process variable within an acceptable range.
The pharmacologist asks: what's the minimum dosing frequency that keeps plasma concentration above the minimum effective concentration?
The spaced repetition algorithm asks: what's the minimum review frequency that keeps recall probability above the desired retention rate?
They're the same question!
compartments
Pharmacokineticists model drug distribution using compartment models. The simplest is one-compartment: the drug enters the bloodstream and is eliminated at a single rate. But many drugs need two compartments: a central compartment (blood and highly perfused organs) and a peripheral compartment (muscle, fat, other tissues). The drug equilibrates between compartments at different rates, creating the characteristic biphasic elimination curve: a rapid distribution phase followed by a slower elimination phase.
Memory researchers have independently arrived at strikingly similar models. The "new theory of disuse" (Bjork & Bjork, 1992) proposes that every memory has two strengths: storage strength (how deeply encoded it is, analogous to the peripheral compartment) and retrieval strength (how accessible it is right now, analogous to the central compartment). Storage strength accumulates slowly and decays slowly. Retrieval strength fluctuates rapidly with use and disuse.
The mathematical behavior is the same: a fast-decaying accessible component coupled with a slow-decaying deep component. The biphasic curves even look the same on a graph.
FSRS captures this implicitly. When you review a card and get it right, FSRS increases the stability parameter. The memory doesn't just get boosted back to full retrievability, it becomes more resistant to future decay. Each successful review at the right moment of difficulty doesn't just refill the tank; it makes the tank bigger.
This is analogous to how repeated drug dosing at steady state can lead to accumulation in the peripheral compartment: the deep reservoir fills up.
steady state
Give a drug at regular intervals and eventually you reach steady state: the amount absorbed per dose equals the amount eliminated between doses, and plasma concentration oscillates within a predictable range. It takes roughly 4–5 half-lives to reach steady state.
Review flashcards on a consistent schedule and you reach the learning equivalent: memories stabilize at a retrievability that oscillates within a predictable range around your target retention. The interval between reviews lengthens as stability grows, but the pattern of boost-and-decay reaches a kind of dynamic equilibrium.
In both cases, steady state is a sustained oscillation. Concentration goes up with each dose and decays between them. Retrievability goes up with each review and decays between them. The system is never truly flat. But it's stable, and it's predictable, and that's what lets you plan things ahead.
why this matters
I could stop here and this would be a neat observation about mathematical isomorphism. Two fields independently discovering the same structure. But I think the reason these models look the same isn't coincidence or metaphor. It's because they're both instances of a general class of problems: maintaining a quantity above a threshold through periodic discrete interventions against continuous exponential decay, where each intervention also modifies future decay dynamics.
You can see this abstract structure everywhere. Vaccination schedules (immune memory decays; boosters restore it). Retraining machine learning models on shifting data (performance decays; retraining restores it). Maintaining relationships (closeness decays with absence; contact restores it). Practicing a musical instrument.
I think it's pretty neat!
— Cerces
