Preface
This section studies some first order nonlinear ordinary differential equations describing the time evolution (or “motion”) of those hamiltonian systems provided with a first integral linking implicitly both variables to a motion constant. An application has been performed on the Lotka--Volterra predator-prey system, turning to a strongly nonlinear differential equation in the phase variables.
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Introduction to Linear Algebra with Mathematica
Glossary
Neuroscience
Zeeman heartbeat model
Example: The mathematical modeling of biological systems has proven to be a valuable tool by allowing experiments which would otherwise be unfeasible in a real situation. The electrocardiogram (ECG) signal is one of the most obvious effects of the human heart operation. The oscillation between systole and diastole states of the heart is reflected in the heart rate. The surface ECG is the recorded potential difference between two electrodes placed on the surface of the skin at pre-defined points. The largest amplitude of a single cycle of the normal ECG is referred to as the R-wave manifesting the depolarization process of the ventricle. The time between successive R-waves has been widely used as a measure of the heart function, and this helps to identify patients at risk for a cardiovascular event or death. Analysis of variations in this time series is known as heart rate variability (HRV) analysis.
The development of a dynamical model will provide a useful tool to analyse the effects of various physiological conditions on the profiles of the ECG. The model-generated ECG signals with various characteristics can also be used as signal sources for the assessment of diagnostic ECG signal processing devices. The dynamic response of the cardiovascular control system to physiological changes is reflected in HRV and blood pressure. Mathematical models are extremely important for understanding biological processes. Now catastrophe theory is one branch of applied mathematics that was developed in order to describe certain biological processes and has been applied by researchers, especially by Christopher Zeeman, who in 1972 resented an important set of nonlinear dynamical equations for heartbeat modelling:
The following properties of this model are considered as fundamental: (i) The existence of an equilibrium state (fixed point) corresponding to the diastole (relaxed state of the heart). (ii) There must be a threshold for triggering the process whereby the heart contracts from a diastole to a systole (fully contracted state and another equilibrium state). (iii) The model should quickly return to the original equilibrium state after the systole.
First, we find equilibrium solution
The system stays at the stable equilibrium point endlessly, until the equilibrium point is stable, except there is an exterior excitation that forces the system move to a new equilibrium point. Based on this new stable system, a control input is added to the Zeeman system as shown below:
FitzHugh--Nagumo model
- Jafarnia-Dabanloo,N., D.C. McLernon, D.C., Zhang, H., Ayatollahi, A., Johari-Majd, V., , Journal of Theoretical Biology, 2007, 244, pp. 180--189.
- Jardóon-Kojakhmetova, H. and Broer, H.W., Polynomial normal forms of constrained differential equations withthree parameters, 2014.
- Jones, D.S., Sleeman, B.D., Differential Equations and Mathematical Biology, Chapman & Hall. London. (2003).
- McSharry, P.E., Clifford, G.D., Tarassenko, L., Smith, L.A., A Dynamical Model for Generating Synthetic Electrocardiogram Signals, IEEE Transactions on Biomedical Engineering, 2003, 50, pp. 289--294.
- Pandiselvi, A. Saravanakumar, R., Sivasankari, M.K., Numerical Solution of Zeeman Heartbeat Systems: A Nonlinear Model, International Journal of Scientific Research and Review,
- Zeeman, E.C., Differential Equations and Nerve Impulse, Towards a Theoretical Biology, 1972, 4, pp. 8-67.
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