# A mathematical framework for modelling the dynamic nature of ADHD symptoms

## TL;DR

This paper introduces mathematical models to capture how ADHD symptoms like inattention and hyperactivity change over time and with context.

## Contribution

The study presents interpretable, theory-based mathematical models for the dynamic nature of ADHD symptoms.

## Key findings

- Inattention was modelled using modulated exponential decay functions.
- Hyperactivity was represented by a modulated sinusoidal function.
- Impulsive choice was modelled using hyperbolic delay discounting and a probabilistic softmax choice rule.

## Abstract

Attention-Deficit/Hyperctivity Disorder (ADHD) is characterized by core symptoms of inattention, hyperactivity, and impulsivity that fluctuate dynamically based on context. Standard diagnostic criteria provide static descriptions, failing to capture this variability, while existing computational models may lack interpretability or flexibility for clinical application. There is a need for dynamic, theory-driven models to represent ADHD.

This study aimed to develop and present a set of interpretable mathematical models representing the dynamic, context-dependent nature of the core symptoms of ADHD, grounded in established neuropsychological principles.

Algebraic equations were formulated to represent symptom dynamics. Inattention was modelled using modulated exponential decay functions. Hyperactivity was represented by a modulated sinusoidal function reflecting its oscillatory pattern. Impulsive choice was modelled using hyperbolic delay discounting combined with a probabilistic softmax choice rule.

The study produced specific mathematical equations that quantify the temporal dynamics and contextual modulation for each core symptom domain. These equations provide a formal representation of how attention decays, hyperactivity fluctuates, and impulsive choices are made, incorporating individual sensitivities and situational factors pertinent to ADHD.

The proposed mathematical models offer a novel, quantitative framework for understanding and representing the dynamic nature of ADHD symptoms. Grounded in neuropsychological theory, these interpretable models provide a potential advance over static descriptions and may facilitate improved clinical assessment, personalized treatment strategies, and targeted research into the mechanisms underlying ADHD. Further empirical validation is warranted to establish their clinical utility. Further empirical validation is warranted to establish their clinical utility.