Neurodegenerative diseases, such as Parkinson’s disease, can be thought of as arising from malfunctions in the network of neuronal agglomerates in the brain. It is therefore often useful to apply insights from a branch of mathematics called network theory when studying the development of these diseases.

A group of European physicists and engineers led by Maria Mannone of the National Research Council of Italy, the University of Potsdam, Germany, and the Potsdam Institute for Climate Impact Research (PIK), Germany, has now taken this further by defining a matrix transforming the brain network of a healthy individual into one affected by Parkinson’s disease.

This has been published in The European Physical Journal Special Topics,

“Our work derives from two historic ideas: that brain functions can be mapped to specific areas, and that connections between them can be mapped non-invasively,” explains Mannone. “These ideas are behind the technique of functional magnetic resonance imaging (fMRI), and we used fMRI images to define our matrices.”

The researchers borrow an idea from theoretical physics, that the brain network can be described as a matrix and any change in that matrix, such as that occurring when an illness develops, can be modeled as a mathematical operator—represented as a matrix—acting on it.

“In classical times, illnesses were seen as demons or deities acting on patients,” says Mannone. “This concept is not dissimilar; we chose to name our ‘demon operator’ K, for the German ‘Krankheit,’ disease.”

They computed K for Parkinson’s disease from analysis of the brains of patients using fMRI data from the Parkinson’s Progression Markers Initiative at the University of Southern California and a healthy volunteer, and observed the parts of the network where most changes occurred.

This essentially theoretical approach can have many applications. It is possible to monitor the course of disease by seeing how K changes over time, and perhaps also to compute an ‘inverse’ operator, simulate disease reversal, and see which parts of the Parkinson’s brain would benefit most from further intervention.

“And, of course, this idea does not only apply to Parkinson’s disease,” concludes Mannone. “We are already looking at Ks for Alzheimer’s disease and schizophrenia.”