Lecture 1 Notes, Thursday 25 August 2005

A physical dynamical system evolves (changes) in time t. This evolution is deterministic, that is, governed by a set of rules that specify the current state of the system and how that state changes with respect to time. In this course we will be concerned with deterministic (non-random) dynamical systems in the form of cells, membranes, ions, and ion channels. We will use the language of nonlinear dynamics to model and then analyze a wide array of behaviors in cellular biophysics.

The concept of a differential equation is the basic form of explanation of physical change, and dynamical systems are often explicitly modelled by differential equations dx/dt=v(x), where x(t)=(x₁(t), ..., x_n(t)) is a vector of state variables, t is time, and v(x)=(v₁(x), ..., v_n(x)) is a vector of functions that encode the dynamics. For example, in a chemical reaction, the state variables represent concentrations. The differential equations represent the kinetic rate laws, which usually involve nonlinear functions of the concentrations.

Such nonlinear equations are typically impossible to solve analytically, but one can gain qualitative insight by imagining an abstract n-dimensional state space with axes x_1, ..., x_n. As the system evolves, x(t) flows through state space, guided by the 'velocity' field dx/dt=v(x) like a speck carried along in a steady, viscous fluid.

Suppose x(t) eventually comes to rest at some point x*. Then the velocity must be zero there, so we call x* a fixed point. It corresponds to an equilibrium state of the physical system being modelled. If all small disturbances away from x* damp out, x* is called a stable fixed point, that is, it acts as an attractor for states in its vicinity.

Another long-term possibility is that x(t) flows towards a closed loop and eventually circulates around it forever. Such a loop is called a limit cycle, and it is also an example of an attractor. It represents a self-sustained oscillation of the physical system.

A third possibility is that x(t) might settle onto a strange attractor, a set of states on which it wanders forever, never stopping or repeating. Such erratic, aperiodic motion is considered chaotic if two nearby states flow away from each other exponentially fast. Long-term prediction is impossible in a real chaotic system because of this exponential amplification of small uncertainties or measurement errors.

To see the full article: (Strogatz, Nature 410: 268, 2001).

Dynamical systems can can be expressed using ordinary differential equations (ODEs) or iterated maps. ODEs describe evolution of systems in continuous time, whereas maps evolve in discrete time steps. ODEs are far more applicable to biological systems at the cellular level, and will be the primary focus in this course.

ODEs have only one independent variable, t, whereas partial differential equations (PDEs) have more than one independent variable, which varies depending on the system. We will examine cell behaviors that change with time, thus most of our problems in Cellular Biophysics and Modeling will be conerned wtih one independent variable and thus ODEs.

A general framework for ODEs is given by (2) in Strogatz p.6:

dx₁/dt = f₁(x₁, ... , x_n)
.
.
.
dx_n/dt = f_n(x₁, ... , x_n).

Each variable x₁, ... , x_n represents a meaningful and measurable aspect of the system, which is essential to describe its state. The functions f₁, ... , f_n are determined by the biophysical aspects of the particular system.

The notation for ODEs can take several equivalent forms including: dx/dt and x′ for the first derivative with respect to time, and d²x/dt² and x″ for second the derivative with respect to time. Note: I will use the notation x′ more often than dx/dt because it is easier to enter and read online.

Complicated looking equations with first and second derivatives such as the damped harmonic oscillator

mx″ + bx′ + kx = 0,

can be re-written in the general framework by introducing a new variable:

x′ = v
v′ = -(b/m)x′ - (k/m)x.

(1) Write out the following general information:

(2) Obtain XPPAUT and install on your computer

(3) Do Fall et al. Chapter 1.5, Exercise 2. Verify that the solutions satisfy the ODEs by differentiation and substitution.

(4) Consider the ODE: x′ = -ax, with inital value x(0) = x₀ for t >= 0.

(5-10) Do exercises 2.2.1, 2.2.3, 2.2.11, 2.4.1, 2.4.6, and 2.4.7 in Strogatz. You might find these resources helpful: Table of integrals, another Table of integrals, and Table of derivatives.

(11) Create an *.ode file for XPPAUT containing this text:

# exponential decay model
#
x'=alpha*x
par alpha=-0.1
init x=4
@xlo=0, xhi=50,ylo=-0.4,yhi=5, total=50
done