# 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

# 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:

$\begin{split} \varepsilon\,\dot{x} &= T\,x - x^3 - y, \\ \dot{y} &= x - x_d ; \end{split} \qquad\quad x(0) =1, \quad y(0) =0.$
In this model, the variable x represents the length of a muscle fiber in the heart and the variable y represents an electrical control variable that triggers the electro-chemical wave leading to the heart contraction; positive constant T represents a tension of muscle and is related to blood pressure and ε is a small positive constant which plays a role in fast eigenvalues of the system. Parameter xd is the average muscle length in the diastole (the period of time when the heart refills with blood after contraction).

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

$x = x_d , \quad y = x_d^3 - T\, x_d ,$
where xd is a positive parameter. The Jacobian of Zeeman system is
${\bf J}(x,y) = \begin{bmatrix} \frac{1}{\varepsilon} \left( T - 3x^2 \right) & -\frac{1}{\varepsilon} \\ 1& 0 \end{bmatrix} .$
The condition for the real part of the eigenvalue to be negative is $$3\,x^2 -Y > 0 .$$ So the system can be stable if
$x \ge \sqrt{T/3} \qquad\mbox{and} \qquad x \le \sqrt{T/3} .$
These conditions can be satisfied if the value of xd be 1.024. Then the equilibrium point becomes (1.024, -0.0497) in the state space, for parameters T = 1 and ε = 0.2. In this case, all of the trajectories, irrespective to their primary conditions, go to the diastolic equilibrium point.

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:

$\begin{split} \varepsilon\,\dot{x} &= T\,x - x^3 - y, \\ \dot{y} &= \left( x - x_d \right) + \left( x_d - x_s \right) u(t) , \end{split}$
where, when heart is in systolic state, xs is an additional parameter representing a typical fiber length and u(t) shows the cardiac pacemaker controls mechanism that leads heart into diastolic and systolic states. The feedback controller can be found in the form
$u(t) = -k\,x(t- \tau ) ,$
where τ is the time-delay and is nonnegative.    ■

FitzHugh--Nagumo model

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