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Approaches to Neuroscience from Complex Systems and Mathematics

by  Luis F. Seoane,

Departamento de Biología de Sistemas, Centro Nacional de Biotecnología (CSIC),

C/ Darwin 3, 28049 Madrid, (SPAIN)




Neuroscience deals with understanding one of the most complex objects that we know of in the universe -- the brain, and cognitive systems in general. This includes challenging topics such as the evolution and development of successful computational structures, understanding of how behavior comes about and is implemented by physical matter, and the emergence of consciousness and free will. It is a field well grounded as a branch of biology, ultimately relying on audacious experiments; but its huge complexity, and the challenges posed by its questions, have made neuroscience a branch ripe for mathematical investigations. Actually, several key developments during the 20th century were possible thanks to a very fruitful coming-together of advanced mathematical tools and empirical data. As computational power in our laptops, data, and our understanding of cognition grow, this interdisciplinary collaboration promises an even brighter future. 

This course will discuss a series of topics in neuroscience from a mathematical and complex systems perspective. The scope is very big, so the lectures will necessarily fall short, and the content is biased towards examples that the lecturer has found more illuminating and interesting. The aim is not to provide an exhaustive view, but rather to offer a catalog of open possibilities and raise enthusiasm about a topic at the very frontier of modern science. While mathematics are an important part of this course, it has been designed with an interdisciplinary audience in mind, emphasizing concepts over technical calculations. 

While this description seems very theoretically focused, throughout the course we will also review some of the most modern sources of data in the neurosciences. Data collection is making impressive strides, and might soon allow us to test the most esoteric hypotheses. A hands-on session will allow us to see in first person the level of detail that we are reaching. 


  1.  The brain 101. Due precisely to the school's interdisciplinary focus, it is very convenient to revisit some basic facts about the brain. This will also help us provide historical context, establish relevant hypotheses that have guided the field, and outline open problems. 
  2.  Classical mathematical models in neuroscience. Some huge advances relied on the development of mathematical models and other computational strategies -- even decades ago before the personal computer took on a most prominent role. While conceptual understandings about the brain reach a general public, the mathematical details are not so popular, and the perspective that math has helped so much in the development of neuroscience is lost. In this lecture we review some of these classic examples. 
  3.  Biological versus artificial brains -- neuroscience and AI. Whether through Darwinian evolution (in biology) or human design (in the artificial case), brains have evolved to deal with a series of computational tasks. Do those tasks impose universal constraints that imply that all brain-like structures must converge to a same design? Or is there a range of different strategies available? In the later case, biological brains would likely be biased by their evolutionary history, while artificial brains are freer to explore alternatives. In this lecture we explore convergences and divergences between biological and artificial brains, noting how certain AI approaches seem to rediscover strategies that Darwinism had found first. 
  4.  Symmetry, asymmetry, and symmetry breaking in the brain. Symmetry is a central theme in mathematics. Mirror symmetry is also a defining trait of bilaterian brains, while other symmetries might lurk within the folds of the cortex. In this lecture, we look at recent advances concerning symmetry (its absence or its falling apart) in neural systems and how it affects certain situations in healthy and pathological brains, as well as in development and evolution. 
  5.  Looking further beyond -- taking mathematical neuroscience seriously. In this lecture we will indulge in wild speculation, while we will also study recent advances in mathematical neuroscience. We will see how one of the most captivating and difficult branches of mathematics, algebraic topology, is repeatedly showing up in theoretical and empirical neuroscience studies. With this, we will allow ourselves to entertain a very outlandish possibility that links together brain, mind, geometry, topology and, in essence, takes the idea of mathematical neuroscience seriously.