Biomedical Engineering

Introduction to the course (= presentation of the syllabus and its calibration on the current course) APPLIED MATHEMATICS (TO BIOLOGICAL PROBLEMS): THEORY Part One: Dynamical Systems (basic tool: Mathematical Analysis): -systems of differential equations: resolution -systems of differential equations: stability -the simple Lotka-Volterra case in detail -the case of the Logistic Map and the genesis of deterministic chaos -the Lyapunov exponent, the small Poincaré denominators and the Pesin relation Part Two: Stochastic Processes (basic tool: Probability Theory): -stochastic processes: detailed balance, ergodicity, irreducibility and Markov theorem -the Bernoulli case: temperature as a random walk and the limit of the Fourier PDE -Wiener processes (with and without drift) and Brownian motions --entropy as a measure of information and the Yajnes inferential principle -the Erhenfest model: statics and dynamics, understanding statistical reductionism -equivalence of Boltzmann entropy with Gibbs and Shannon entropy in the canonical -Kulback-Leibler entropy and mutual information: use in the selection of optimal models APPLIED MATHEMATICS (TO BIOLOGICAL PROBLEMS): APPLICATIONS Part Three: Mathematics applied to neurology: from the dynamics of the neuron to information processing in neural networks -elements of neurobiology: morphology of neurons, electrostatic analysis, famous experiments and historical context of the major discoveries. -classical membrane potential theory and Nerst's law -Lapique's "integrate and fire" neuron -Rall's linear cable theory and the emission of the Hodking-Huxley spike: channel dynamics and sodium-potassium pump -Stein's stochastic neuron and its formal equivalence with Rosenblatt's perceptron: additive functionals and Poisson processes -Gaussian theory of elementary network models: Guerra's interpolation for mean (magnetization) and variance (susceptibility) -Lagrangian theory of elementary network models: Hamilton-Jacobi and Burgers equation: phase transitions & Hopf bifurcations -SK: vademecum (replicas, overlaps, etc.), Guerra's interpolation & Hamilton-Jacobi technique -Hebb's learning rule: synaptic dynamics in the Hopfield neural network model at low load -the network model Hopfield neural network in high load: High Dimensional Statistics and signal-2-noise -multilayer networks (example of retinal neurons) and contrastive divergence: real Hebbian learning -a look at the experiments: Pavlov's conditioned reflex module -a look at the experiments: patch clamp and multi-electrode array at maximum entropy Part Four: Mathematics applied to immunology: coordination of the response, from the single lymphocyte to lymphocyte networks -elements of immunology: primary and secondary response, coordinating branch, effector branch, clonal expansion, Burnet's theory of clonal selection, two signal model for the activation of B and T-killer cells, anergy in self-directed lymphocytes, lymphocytosis and autoimmunity (the case of ALPS). -SiR models for epidemics and their generalization à la Valesini -the concepts of quasi-species and fitness, mutation frequency and estimation of the length of the viral genome -the stochastic processes underlying V(D)J recombination in the formation of the epitopal repertoire of lymphocytes -major histocompatibility complex (class-1 and class-2) interaction with BCR and TCR -host-pathogen-immune system dynamics: free evolution, chronic infection or eradication -host-pathogen-immune system dynamics: evolution with drugs (protease inhibitors and reverse transcriptase inhibitors) -simple antigenic variation (one epitope per pathogen), advanced antigenic variation (multiple epitopes scenario) -a look at the experiments: V(D)J recombination and genesis of the epitopal repertoire of lymphocytes -a look at the experiments from the eyes of a modeler: lymphocyte dynamics on a LabOnChip plate

No one in particular, although a knowledge of the basic courses (both in Mathematics and Physics) experienced during the three-year degree is obviously a conditio sine qua non.

For the Dynamical Systems part: Pure Mathematics: 1) D. Andreucci & E. Cirillo, Lecture notes on systems of differential equations (the authors are Professors of Sapienza). Mathematics applied to Biology: 2) B. May & M. Novak, virus dynamics: differential equations for theoretical immunology and virology. https://academic.oup.com/book/54401 For the part of stochastic processes: Pure Mathematics: 3) E. Marinari & G. Parisi, Trattatello di probabili (the authors are Professors of Sapienza). Mathematics applied to Biology: 4) A.Coolen, R. Kuhn, P. Sollich, Theory of neural information processing systems. https://academic.oup.com/book/53058 For the PDE part: please follow the suggestions by Professor Sandra Carillo (who teaches the 30 hours -out of 90- of PDE Theory).

Attendance is not mandatory but is strongly recommended.

The assessment of knowledge consists of an oral interview with the teacher. If there is the possibility of awarding honors (i.e. "lode"), this will be given only if - having obtained the maximum in the oral exam - the student also demonstrates that he or she is able to understand a technical scientific article in the field (decided in agreement with the teacher).

The lessons will be held on the blackboard or by projector depending on the topics (the theory and exercises are done on the blackboard while showing the biological applications involves the use of the projector to analyze how the experiments are set up and conducted in the laboratory with their related data analysis).