Random graphs in the brain: devising a renewed line of research

To make sense of neural connectivity is among the key challenges in neuroscience. This remains fundamentally a neuromathematical challenge, to the extent that the mathematical theory to understand patterns of connectivity is still to be made, but also also a communicational challenge, since it requires the bridging of relevant but still unconnected fields, such as experimental neurophysiology, neuroanatomy, functional imaging, probability, statistics and computer science. These two challenges are the core of the scientific project of FAPESP’s Research, Innovation and Dissemination Center for Neuromathematics (RIDC NeuroMat), which held from November 23 to 27 the workshop “Random Graphs in the Brain,” at the University of São Paulo, Brazil.

On the blackboard, mathematician Remco van der Hofstad, one
of the organizers of the workshop "Random Graphs in the Brain," in November, 2015, at NeuroMat/University of São Paulo.

Mathematical challenges include the devising of models that could give a realistic sense of how neurons interact, taking into consideration neuronal and functional dimensions of the brain, hierarchical properties of connectivity and system criticality. An organizer of the workshop along with Antonio Galves (University of São Paulo), Christophe Pouzat (Paris-Descartes University) and Claudia Vargas (Federal University of Rio de Janeiro), Remco van der Hofstad (Eindhoven University of Technology) gave a series of three lectures on complex networks and the brain and emphasized that a key open question in random graph theory and the brain is: "What is a good network model for brain functionality?” What is at stake, according to Hofstad, is the model one should put in place on how vertices should be associated in order to describe neuronal pairing so that fundamental features of neural activity are not disregarded, especially functional dimensions. His presentation slides are available here.

Most models that are currently being used depend on deterministic methods, yet probabilistic models might lead to more efficient accounts. This has been a core assumption of the work done within NeuroMat, especially the founding paper by Galves and Löcherbach, in 2013, here. In this paper, the authors rely on a critical directed Erdös-Rényi-type random graph, and the discussion of whether or not this graph is appropriate to make sense of neural connectivity has informed working sections during the workshop. In an interview to NeuroMat’s newsletter, here, Hofstad claimed that the Erdös-Rényi model was too “egalitarian" to be a fair depiction of what happens in the brain.

An work path that was discussed within the workshop at NeuroMat is the development of graph theory within the mathematical framework introduced by Galves and Löcherbach. According to mathematician Christophe Pouzat, in reference to his introduction to the event, "Progress in understanding the brain is slowed down by a deficit of clear understanding of how the results of the different disciplines involved are obtained. On the one hand, a mathematician has almost no idea of how recordings from synaptically connected neurons in brain slices are performed and has therefore no intuition for the kind of artifacts they generate and for the precision of the measurements they provide. On the other hand, an experimentalist often fails to grasp that the definiteness on which mathematician are so insistent is the very condition needed to start mathematical work, which is by proving theorems.”

The workshop brought together scientists from 20 universities, who are attempting to cross the language and discipline crossover. During the event, eminent neurobiologist Almut Schüz gave four lectures to introduce neuroanatomy, pinpointing quantitative dimensions in the understanding of the cortex. Her presentation slides are available here.

The workshop was streamed online, with over different 200 watchers, and videos should be released soon. The link to the event call and material is here.

This piece is part of NeuroMat's Newsletter #22. Read more here

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