93916 - Biological System Modeling

Course Unit Page

Academic Year 2022/2023

Learning outcomes

At the end of the course, the student has the basic knowledge and knows how to use the theoretical and practical tools essential for modeling fundamental biophysical phenomena and processes, and for understanding 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 domain, also with reference to regulation and control problems. - analyze the main properties of non-linear mathematical models - study the behavior of a complex biological system by numerical simulation. - understand the role of mathematical models in different physiological contexts. The student is also able to critically examine the function and role of models in different theoretical and applicative fields related to medicine and biology.

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.


The course is supplemented by exercises with the language 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".


Lecture notes provided by the teacher. This material will be uploaded on the platform for the repository of educational material made available by the University.

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.

M. Lutz, "Learning Python", O'Reilly, 2013

Teaching methods

The course is divided into lectures ex-cathedra and exercises with the computer by using the language PYTHON. 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'.

As concerns the teaching methods of this course unit, all students must attend Module 1, 2 on Health and Safety [https://elearning-sicurezza.unibo.it/] with e-learning modality.

Attendance at lectures is strongly recommended for both ex-cathedra lessons and matlab exercises, since all the aspects provided in the lecture notes are deepened and commented in detail by the techer.

Assessment methods

The final exam is based on a written test (about 150 minutes). The written test consists of an exercise on linear models, an exercise on non-linear models and some questions on theoretical aspects and / or physiological models analyzed during the course.

The use of portable computers is allowed to perform the calculations during the written test.

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.

  To achieve a laude it is necessary to have carried out the entire written exam without errors and to have demonstrated an excellent mastery of the theory and of physiological models. The score is progressively scaled based on the number of errors committed and of their conceptual gravity.

In case of sufficient vote, it is possible to refuse the vote only once.


Teaching tools

Blackboard, document camera, 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