The choice of a doctoral project in Neuroscience is aimed at combining my mathematical and physical interests with the study of the brain, in particular at the local level of the neuro-glial vascular unit and subsequently at the network level by exploring the connectomics. The philosophy behind this work is to seek a bridge between mathematical models deriving from Dynamical Systems, Statistical Mechanics, Neural Networks, numerical solution of PDEs, and the Physiology of the system.

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During the second year of my master studies I developed a particular interest in mathematical physics topics related to the Nonequilibrium Statistical Mechanics, as

• Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics - D. Ruelle
• Nonequilibrium and Fluctuation Relation - G.Gallavotti
• A dynamical-systems interpretation of the dissipation function, T-mixing and their relation to thermodynamic relaxation - L.Rondoni, O. G. Jepps

and Ising-like models, e.g. Spin-glass models of neural networks which are deeply connected and based on the works of Parisi et al., inter alia the Replica Symmetry Breaking (RSB), and their mathematization carried out by Talagrand; see:

• Spin-glass models of neural networks - D. J. Amit, H. Gutfreund, and H. Sompolinsky
• Mean Field Models for Spin Glasses, Volume I: Basic Examples - M. Talagrand
• Introduction To The Theory Of Neural Computation - John A. Hertz
• Theory of Simple Glasses: Exact Solutions in Infinite Dimensions - G. Parisi, P. Urbani, F. Zamponi