Dynamics of Biological Systems

From the spontaneous rapid firing of cortical neurons to the spatial diffusion of disease epidemics, biological systems exhibit rich dynamic behaviour over a vast range of time and space scales. Unifying many of these diverse phenomena, Dynamics of Biological Systems provides the computational and mathematical platform from which to understand the underlying processes of the phenomena.

Through an extensive tour of various biological systems, the text introduces computational methods for simulating spatial diffusion processes in excitable media, such as the human heart, as well as mathematical tools for dealing with systems of nonlinear ordinary and partial differential equations, such as neuronal activation and disease diffusion. The mathematical models and computer simulations offer insight into the dynamics of temporal and spatial biological systems, including cardiac pacemakers, artificial electrical defibrillation, pandemics, pattern formation, flocking behaviour, the interaction of autonomous agents, and hierarchical and structured network topologies. Tools from complex systems and complex networks are also presented for dealing with real phenomenological systems.

With exercises and projects in each chapter, this classroom-tested text shows students how to apply a variety of mathematical and computational techniques to model and analyze the temporal and spatial phenomena of biological systems. MATLAB® implementations of algorithms and case studies are available on the author’s website.

Michael Small is a professor of mathematical modelling and director of the Phenomics and Bioinformatics Research Centre in the School of Mathematics and Statistics at the University of South Australia (as of October 2011). He was previously an associate professor in the Department of Electronic and Information Engineering at Hong Kong Polytechnic University. His research interests include nonlinear time series, chaos, and complex systems.

Biological Systems and Dynamics In the Beginning The Hemodynamic System Cheyne-Stokes Respiration

Population Dynamics of a Single Species Fibonacci, Malthus and Nicholsons Blowflies Fixed Points and Stability of a One-Dimensional First Order Difference Equation The Cobweb Diagram An Example: Hormone Secretion Higher Dimensional Maps Period Doubling Bifurcation in Infant Respiration

Observability of Dynamic Variables Bioelectric Phenomena Measurement ECG, EEG, EMG, EOG and All That Measuring MovementMeasuring Temperature Measuring Oxygen Concentration Biomedical Imaging The Importance of Measurement

Biomedical Signal Processing Segmentation Error Measure and Automatic Analysis of EEGsECG Signal Processing Vector Cardiography Embedology and State Space Representation Fractals, Chaos and Nonlinear DynamicsPrediction

Computational Neurophysiology The Cell Action Potentials and Ion Channels Ficks Law, Ohms Law and the Einstein Relation Cellular Equilibrium: Nernst and Goldman Equivalent Circuits Dendrites

Mathematical Neurodynamics Hodgkin, Huxley and the Squid Giant Axon FitzHugh-Nagumo Model Fixed Points and Stability of a One-Dimensional Differential Equation Nullclines and Phase Plane Pitchfork and Hopf Bifurcations in Two Dimensions Excitability

Population Dynamics Predator-Prey Interactions Fixed Points and Stability of Two-Dimensional Differential EquationsDisease Models: SIS, SIR and SEIR SARS in Hong Kong

Action, Reaction and DiffusionBlack Death and Spatial Disease TransmissionReaction-DiffusionCardiac Dynamics

Autonomous AgentsFlockingCelluloid Penguins and Roosting StarlingsEvaluating Crowd Simulations

Complex Networks Human Networks: Growing Complex Networks Small World Networks of Spread of SARSGlobal Spread of Avian InfluenzaComplex Disease Transmission and Immunisation Complex Networks Constructed from Musical CompositionInteraction of Grazing Herbivores Neuronal Networks and Complex Networks

ConclusionModels Are a Reflection of Reality

References

A Summary appears at the end of each chapter.