34799 - Models of Biological Systems (2nd cycle)

Learning outcomes

At the end of the course, the student has the basic knowledge on how to use the essential theoretical and practical tools for the analysis and modeling of biophysical processes, and to understand the behavior of complex biological systems. In particular, he/she is able to: - describe the main phenomena and biophysical processes using mathematical models - analyze the main properties of linear mathematical models, in the time and frequency domains, also with reference to the problems of regulation and control - analyze the main properties of non-linear mathematical models - study the behaviour of a complex biological system by numerical simulation.

Course contents

Modelling theory

General principles on the construction and validation of mathematical models in physiology and biology.

Linear systems: systems of differential equations. Linear models: the free motion and the forced motion. The transition matrix. The solution in the case of linear time-invariant systems: the transfer matrix . The stability of dynamical systems. Stability bounded-input bounded output, and stability of equilibrium points. The case of time-invariant linear systems: eigenvalues and poles . The classification of the equilibrium point of a second order linear system: focus, node, saddle, degenerate points. The linear feedback systems. Effect of a feedback on the transfer function and on the placement of the poles. The Nyquist criterion with relevant examples. The effect of a pure delay on the stability.

Non-linear systems - Effect of non- linearities and linearization in the neighbourhood of equilibrium points. The Hartman - Grobman theorem. Examples of second order non-linear systems. Main types of bifurcation: saddle-node, transcritical, pitchfork. Second-order nonlinear systems. The conservative systems. Limit cycles and related theorems. The Van der Pol oscillator. The Hopf bifurcation. Some issues on models with order higher than second. The deterministic chaos. The Lorentz equations and the Rossler equations.

Population dynamics. The logistic equation. The dynamics of two populations in antagonism. The prey - predator problem. The Lotka -Volterra equations. Predator-prey equations in the presence of a Michaelis - Mentis kinetics. Considerations on the solutions found in each case.

Discrete dynamical systems - A brief introduction to discrete dynamical systems. The stability of a discrete system. The discrete logistic equation, the flip bifurcation and the transition to chaos.

Physiological systems

Model of cardiovascular dynamics integrated with the baroreceptor control. Effect of the control on systemic arterial pressure and cardiac output.

Model of solute exchange between the intracellular and extracellular fluid. The control of concentration by dialysis. Linear model and nonlinear model (effect of osmosis and of change in volumes) .

Model of mechanical ventilation, with free and forced ventilation. The alveolar ventilation and the dead space. Effect of frequency and amplitude of breathing.

Model of gas exchange in the alveoli and in tissue. The chemoreceptor control of ventilation. The Cheyne - Stokes oscillations of breathing.

Cellular electrophysiology. The membrane potential at equilibrium and the Nernst potential. The electric analogue of the cell membrane.

The excitable cell. Description of "voltage dependent" ionic channels and the experiment of voltage clamp. The Hodgkin-Huxley equations and their parameter assignments. The genesis of the action potential.

Model of propagation along the axon. The telegrapher's equation . The solution in the linear case. Some issues on to the propagation of the action potential along the fibre: the myelinated and unmyelinated fibres.

Exercises

The course is supplemented by exercises with the software package MATLAB. In particular, simulators are built for many of the physiological models described during the course, in order to study their behaviour through analysis "in silico".

Readings/Bibliography

Lecture notes provided by the teacher..

The following texts may be useful to deepen some aspects (well beyond the exam):

S. H. Strogatz, “Nonlinear dynamics and chaos : with applications to physics, biology, chemistry and engineering “, Cambridge (MA) : Westview press, 2000.

J. Keener, J. Sneyd, “Mathematical Biology I: Cellular Physiology”, Springer, 2009.

J. Keener, J. Sneyd, “Mathematical Biology II: Systems Physiology”, Springer, 2009.

M. C. K. Khoo, “Physiological Control Systems: analysis, simulation and estimation”, Wiley, 1999.,

P. Dayan, L.F. Abbott. “Theoretical Neuroscience. Computational and Mathematical Modeling of Neural Systems”. The MIT Press, London, England, 2001.

Teaching methods

The course is divided into lectures ex-cathedra and exercises with the computer by using the software package MATLAB. The lessons are designed to provide the student with a theoretical knowledge about linear and non-linear modelling techniques, and knowledge about important models used in biology and physiology, and to make him/her aware of the advantages and limitations of each technique. The exercises are designed to provide the student with the ability to simulate such models and to analyze their behavior 'in silico'.

Assessment methods

The examination at the end of the course is based on a written test (120-150 minutes) and an oral exam with the student (duration about 20 minutes). The written test consists of a first exercise on linear models and a second exercise on non-linear models. The oral exam will focus on the theoretical aspects and/or on the physiological models analysed during the course.

The overall examination aims to assess the achievement of learning objectives, in particular:

- Knowledge of the main tools for the analysis of linear models;

- Knowledge of the main tools for the analysis of non-linear models;

- The main techniques for controlling a physiological system;

- Knowledge of some important physiological models;

- The capacity to simulate models and analyse the results.

The analytical and synthetic attitudes of the student, and his/her language skills, and clarity of presentation are also part of the final judgment.

Teaching tools

Blackboard, videoprojector.

Notes provided by the Professor. Xeros copies of images on neurosciences and cognitive sciences.

Laboratory equipped with personal computers.

Software package MATLAB, for performing practical exercises on the simulation of models “in silico”.

Office hours

See the website of Mauro Ursino