Using mathematics to translate the brain

Collaborators with NEUROMAT, a research center supported by FAPESP, discuss the use of random graphs to study neuronal and brain region connectivity

By Karina Toledo  |  Agência FAPESP – The human nervous system and social media websites such as Facebook have at least two things in common: they are complex networks made up of countless interacting components and their future behavior depends on their history.

The creation of mathematical models to represent how these complex networks work and to predict their behavior is a major challenge for scientists in several fields. Generally speaking, such models can be classified as part of random graph theory. 

“Scientists around the world are using random graph models to study how the brain works, but the mathematical basis for those models isn’t as sound as it could be. We aim to develop a new mathematics language to address the problems in neurobiology,” said Antonio Galves, a professor at the University of São Paulo’s Mathematics & Statistics Institute (IME-USP) in Brazil and head of the Neuromathematics Research, Innovation & Dissemination Center (NEUROMAT), one of the Research, Innovation & Dissemination Centers (RIDCs) supported by FAPESP.

Galves coordinated a workshop on “Random Graphs in the Brain”, held at the University of São Paulo (USP) on November 23-27. The interdisciplinary workshop was attended by NEUROMAT’s researchers from Brazil and several other countries, including specialists and students in experimental neurophysiology, neuroanatomy, functional imaging, probability, statistics, and computer science. The main aim was to discuss the mathematical, statistical and computational challenges that need to be addressed to achieve a deeper understanding of the brain.

Galves explained that in mathematical theory a graph comprises a set of vertices and edges linking pairs of vertices. In neuroscience applications, the vertices can represent neurons or brain regions.

“For example, we can use graphs to analyze time series of brain data recorded by electroencephalography (EEG),” he said. “Electrodes attached to a volunteer’s scalp record the brain waves generated while the subject receives visual or auditory stimuli. Graphs are created as global descriptions of the data collected.”

According to Remco van der Hofstad, a professor at Eindhoven University of Technology in the Netherlands and co-organizer of the workshop, it would be impossible to describe the interactions among our brain’s 100 billion neurons deterministically, showing precisely which neurons are connected to which neurons and how they interact.

“A random graph is a probabilistic way of describing neuronal connectivity,” he said. “We can assume that each neuron is connected to a thousand others on average. So we try to build models with this property based on probabilistic principles.”

For Hofstad, the use of graphs to analyze brain data could one day enable more cost-effective, efficient and early diagnoses of Alzheimer’s disease and other brain disorders simply by means of an EEG exam.

“But what I’d like to see in the long term is for us to be able to build the simplest possible model to understand how neurons are connected. And the next questions would be: Can we model functionality? Can we describe not just what’s happening in the brain but also how it responds to stimuli?” Hofstad said in an interview with Agência FAPESP.

If the model is sufficiently robust, he went on, it could be used to predict behavior or effects, paving the way for a mathematical theory that would help understand the plasticity of the brain.

“For example, it would be possible to predict the probability for a certain brain region to take over a function previously performed by a region damaged in a stroke. Of course, all this is far in the future, but it may be possible one day,” Hofstad said.

Constant change

According to NEUROMAT’s principal investigator, Claudia Domingues Vargas, a professor at the Federal University of Rio de Janeiro (UFRJ) and another co-organizer of the workshop, one of the key challenges of creating models that describe how the brain works is that the brain is constantly being changed by new experiences as it develops.

“In neuroscience, we study how the interactions of billions of neurons in different brain regions generate properties such as language, thought, emotions, movement, and everything that represents our being in the world. Formalizing the links between these interactions and behavior proper is one of NEUROMAT’s missions,” Vargas said.

Sidarta Ribeiro, Head of the Brain Institute at the Federal University of Rio Grande do Norte (UFRN) and a speaker at the workshop, uses graphs to understand patients’ and volunteers’ thought patterns in experiments.

“We record their speech and convert reports of their dreams into pathways of words,” he said. “Each node is a word, and each edge is a sequence of words. We do this to plot thought patterns and determine, for example, whether the individual suffers from bipolar disorder or schizophrenia. What psychiatrists learn to recognize through subjective training, we can measure and represent in a graph. Then, we can analyze its properties.”

Another participant in the workshop was Wojciech Szpankowski, a professor of computer science at Purdue University in the United States and Head of the Science of Information Center, an organization funded by the National Science Foundation along similar lines to FAPESP’s RDICs.

“Our mission is to study information in every form, including the information flows in the brain. We’re followers of Claude Shannon, who created information theory,” said Szpankowski, a member of NEUROMAT's International Committee. Information theory is a branch of probability theory and statistical mathematics that addresses communication systems, data transmission and compression, cryptography, coding, noise, and error correction, among other things.

In his presentation on “Emerging Frontiers of Information Science”, Szpankowski spoke about new challenges, such as network security, and described theoretical information models pertinent to the study of biological systems.

Additional speakers included Almut Schütz from the Max Planck Institute for Biological Cybernetics (Germany), Audrey Mercer from University College London (UK), Christophe Pouzat from Paris Descartes University (Paris V, France), and Anderson Winkler from Oxford University (UK). The presentation slides are available at http://neuromat.numec.prp.usp.br/rgbrain, and videos of the presentations will also be uploaded onto the website soon.