Springer, 2014. — 378 p. This book tells the story of an adventure, a scientific adventure moved by the desire to understand the geometrical structures of the visual brain. For geometry we attend here not the anatomical geometry of the brain shape, but the differential geometry of the connectivity between neural cells. This connectivity builds and shapes the hidden brain structures underlying visual perception. The story of the problem runs over the last 30 years, since the discovery of Hubel and Wiesel of the modular structure of the primary visual cortex, and slowly cams towards a theoretical understanding of the experimental data on what we now know as functional architecture of the primary visual cortex. Experimental data comes from several domains. The two classical sources of data, neurophysiology and phenomenology of perception, are nowadays more and more sustained by neurocognitive imaging. Imaging techniques like functional MRI and diffusion tensor MRI allow us to deepen the study of cortical structures at the mesoscale, completing the scale range already well covered by neurophysiology at the microscale and by psychophysics at the global scale. Due to the variety of sources of experimental data, neuromathematematics deals not just with modelling of cortical structures but also with modelling of perceptual spaces, in the Spinozist tradition where res cogitans and res extensa are just the two sides of the same sheet of paper. From themathematical point of view, neuromathematical structures are forged by classical differential geometry and Lie groups, but are more and more demanding for new instruments to pure mathematicians: research in sub-Riemannian geometry is important to model horizontal connectivity, harmonic analysis in non commutative groups is fundamental to understanding the pinwheels structure, as well as non-linear dimensionality reduction is at the base of many neural morphologies and possibly of the emergence of perceptual units. But what is at the center of the neurogeometrical problem is overall the art to harmonize contemporary mathematical instruments with neurophysiological findings and phenomenological experiments towards the construction of a theoretical model of vision. The constributions to this book are devoted to this task and come from the very founders of the discipline.
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