Universidad Nacional de San Martín
14:00 - 17:00
18 de Junio de 2026
Living systems are complex in both structure and function: their forms are typically irregular, and they are organized hierarchically across a range of scales. Because these objects lack a single characteristic scale, classic Euclidean geometry struggles to capture them. Fractal analysis offers a more natural way to represent these systems. Although it was first introduced to characterize mathematical monsters - complex structures created by iteratively applying simple rules - it rose to prominence as a tool for the complex forms found in nature. This is because it condenses complex behavior into compact quantities simply and efficiently, providing a quantitative language for objects across domains - coastlines and clouds, the branching of trees and blood vessels, molecules and galaxies.
The brain is a particularly compelling case. It is extremely complex at all levels of observation, characterizing it is clinically relevant, and the development of neuroimaging techniques allows access to rich data on its structure and functioning. Quantities derived from fractal analysis can serve as interpretable indices of brain organization, and as markers of disruption in neurodegenerative disease.
This workshop develops the ideas from the ground up. We begin with the geometric fractals and random fractals used as models of real signals and images, introducing the concepts of self-similarity, power-law scaling, fractal dimension, and the Hurst exponent in an intuitive way. From there, we introduce Detrended Fluctuation Analysis - a robust, widely used method for estimating scaling behavior from data. Next, space-filling curves - continuous paths that thread through every point of an image - let us extend the analysis from one- to two- and three dimensions. We then apply this novel method - Fractal Space-Filling Curve Analysis - to MRI brain scans, showing the impact of aging and dementia on the brain. The workshop closes with an introduction to the multifractal generalization, which allows for the description of scaling heterogeneity.
Learn how to generate and visualize geometric and random fractals.
Build intuition for the concepts of self-similarity, power-law scaling, fractal dimension.
Estimate the scaling properties of time series with Detrended Fluctuation Analysis.
Estimate the scaling properties of images with Fractal Space-Filling Curve Analysis.
Load and visualize MRI brain scans using Python.
Interpret results of fractal analysis of neuroimaging data.
This is a hands-on workshop. We will work in a Python notebook on Google Colab - a link will be shared at the start of the session. To follow along, please bring a laptop and make sure you have a Google account and are signed in.
No prior exposure to fractals, time-series analysis, or neuroimaging is assumed.
We do assume some familiarity with probability and statistics (a working understanding of linear regression, correlation and random walks).
The workshop involves short, guided coding exercises (if you can write a for loop in any programming language, you will be fine).
Marta Lotka is a physics PhD student at the Doctoral School of Exact and Natural Sciences at the Jagiellonian University in Cracow, Poland. In her research she applies tools of complex systems science to neurophysiological data.