Biomedical Engineering

ADRIANO BARRA
SANDRA CARILLO

The course provides the Biomedical Engineering student with the basic notions
concerning partial differential equations in mathematical physics.
Specifically, after a brief overview on partial differential equations which
model physical phenomena, first order and higher order p.d.es,
linear and nonlinear equations, some resolution methods are given. Specifically,
initial boundary value problems are studied and the physical interpretation of the
obtained results is discussed.

Moreover, in the case of differential equations (both o.d.es and p.d.es)
non-linear problems in which "small" parameters appear, are considered
on introduction of "perturbative methods". Applications and examples are
provided.

General objectives
The ultimate goal of the course is to make the student aware of the limits of a reductionist representation of reality, with particular attention to Biological Sciences (and related biomedical aspects), and equally to provide the same mathematical methods and models to overcome such description and infer emergent properties in non-linear dynamic systems such as neural or leukocyte networks (example: the single neuron emits at most a simple electrical signal - when it is not quiescent - while a network of billions of highly interconnected neurons is able to learn and recognize a person's face: these learning and recognition capabilities arise spontaneously at the level of the network but are not present in the individual elements that compose it).

Specific objectives
The aim of this course is to provide students with mathematical methods and models for understanding information processing in biological networks, both in direct problems (essential when building a model) and in inverse problems (fundamental when testing a model) and with concrete applications. Specifically, the course, from a formal point of view, is divided into three main modules (related, with increasing complexity), namely (i) Analytically solvable Dynamical Systems, (ii) Dynamical Systems solvable by means of perturbation theories, and (iii) Noisy Dynamical Systems, which are analyzed with Stochastic Processes techniques: each module has its own set of applications.
At the end of the course, the student will master a vast plethora of techniques for the study of dynamical systems, both deterministic and stochastic, and in particular will have understood how to infer the emergent information processing in large assemblies of these systems such as many-body networks, be they neural networks, lymphocyte networks or others.

Knowledge and understanding
At the end of the course, the student is expected to have developed knowledge and skills to collaborate in public and private research activities, also with researchers in the medical and biological areas, both in basic medical research and by applying innovative methodologies and technologies for diagnostics and modeling of biomedical systems. Furthermore, the student is expected to be able to develop innovative and original approaches to technical problems that also require creativity.

Applying knowledge and understanding
The student who masters the mathematical techniques developed during the course should then be able to apply them profitably at least to biological systems covered during the course (neural networks and leukocyte networks), with the hope that - having seen these examples - the same will then be able to generalize when needed.
Furthermore, he should be able to verbally communicate the design choices and the scientific orientations underlying them and be able to draft written technical reports relating to the development of systems of interest to biomedical engineering.
Finally, the student must be able to explain advanced modeling and control techniques of complex systems for the analysis of data and information in the biomedical sector.

Critical and judgment skills
Critical skills will be refined through numerous examples of application of the theory which will give the student the opportunity to test very concretely his maturation during the delivery - and study - of the material that forms the body of the course.

Communication skills
Communication skills will be developed by providing the student with all the appropriate terminology in the sector and ensuring that the semantics of each word embraces its most complete - and correct - domain of meanings.

Learning skills
Learning skills will be continuously tested with exercises carried out in class (by the teacher).

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.

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

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).

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).

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).

0) Solve a given system of ordinary differential equations.
1) What does the Lyapunov stability theorem say?
2) How do you classify the equilibrium points of a dynamical system in R^2?
3) Show the level curves of a conservative dynamical system in the phase space.
4) When does a stochastic Markov process converge to the Gibbs distribution?
5) What is an ergodic Markov chain?
6) Solve the Fourier PDE with a given initial/boundary condition.
7) Describe the stochastic dynamics of a single "integrate&fire" neuron.
8) Describe the emergent collective properties of a network of "integrate&fire" neurons
9) Discuss the stochastic process underlying the V(D)J generation of the antibody repertoire.

• Arguments as they flow have already been inserted with the program.
  • Books: Please see the "textbook" section: therein all the required books have been reported.