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Section7.3Periodic Models

Subsection7.3.1Overview

This section discusses periodic functions, specifically using the trigonometric functions of sine and cosine as models for periodic functions. A periodic function captures the idea of cycles that repeat perfectly and indefinitely. Oscillatory waves can be modeled with periodic functions, including sound, light and radio waves. The spiking patterns of a neuron can be modeled with periodic functions. Cyclic patterns in populations can even be modeled with periodic functions.

Periodic functions are characterized by basic features. One of the most fundamental features is called the period, which is how much the input of the function needs to increase before the function exactly repeats its output. When a wave is observered spatially, the period of the function corresponds to the wavelength. When a wave is observed temporally (in time), the period of the function corresponds to the physical period, or the time between repeating cycles.

Subsection7.3.2A Barebones Review of Sine and Cosine

The appendix has a much more detailed discussion of trigonometric functions. For our purposes, all that I really want to review is the oscillatory behavior of the sine and cosine functions as prototypes of periodic functions. One dynamical perspective of the graphs results from considering a point that moves around a unit circle at a steady speed.

The point starts at the position \((1,0)\). We consider counter-clockwise rotation as positive motion and clockwise rotation as negative motion, and we measure the distance \(x\) the point travels along the circumference. (This is what the radian measure of an angle measures: the distance traveled on a unit circle.) The sine function gives from the vertical position of the point after traveling some distance along the circumference of the unit circle. The cosine function gives the horizontal position of the point. Thus, based on our starting point \((1,0)\), we have \begin{equation*}\sin(0) = 0 \qquad \cos(0) = 1.\end{equation*} In addition, because both the horizontal and vertical coordinates of the unit circle remain between -1 and 1, we know that the sine and cosine oscillate between a minimum at -1 and a maximum at +1, \begin{equation*}-1 \le \sin(x) \le 1 \qquad -1 \le \cos(x) \le 1. \end{equation*}

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As soon as the point travels a distance of \(2\pi\), it has completed a full revolution of the circle and it is as though it is revisiting the same points. This is why the sine and cosine functions are periodic with a period \(2 \pi\).

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As a consequence of their unit circle definitions, the sine and cosine functions are especially symmetric. For example, the cosine function is an even function, which means that if you reflect the graph horizontally over the \(y\)-axis, the graph is unchanged. This graphical view is expressed in an equation as \begin{equation*} \cos(-x) = \cos(x). \end{equation*} That is, if the graph \(y=\cos(x)\) includes some point \((x,y)=(a,b)\), then the opposite point \((x,y)=(-a,b)\) (changing only the \(x\)-coordinate) is also in the graph.

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How do you read or interpret that equation? On the left, we see the expression \(\cos(-x)\). In the language of transformations, we have an original graph \(y=\cos(x)\). This expression corresponds to a new relation \(v=\cos(-u)\) so that our transformed variables are \(v=y\) and \(x=-u\). That is, \(\cos(-x)\) is the function that corresponds to changing all \(x\)-coordinates by multiplying by -1 (or switching positive/negative sides on the number line). So the equation is saying, on the left, if you take the graph of \(\cos(x)\) and flip all points from left-to-right, then this is exactly equal to the original graph. Again, note that changing variables to interpret the transformation is a convenience and not necessary. You should practice to where this extra step becomes unnecessary.

While the cosine function is even, the sine function is an odd function. This means that if you do a horizontal reflection, instead of getting the original function, you instead get what would be equivalent to a vertical reflection. This is expressed mathematically by the equation \begin{equation*} \sin(-x) = - \sin(x). \end{equation*} Do you see how each side of the equation represents a different transformation, and the equation says that the two different transformations result in the same thing? Graphically, this is the same as saying the graph has a rotational symmetry of 180 degrees or that the graph is symmetric across the origin. That is, if the graph of \(y=\sin(x)\) includes a point \((x,y)=(a,b)\), then the opposite point \((x,y)=(-a,-b)\) (both change because going across the origin) is also on the graph.

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Because a circle is rotationally symmetric, the sine and cosine graphs have the same shapes although they are shifted with respect to one another. If we took the graph of the cosine and shifted it to the right by the arc length corresponding to a quarter rotation, \(\frac{1}{4}(2\pi) = \frac{\pi}{2}\), then we get the sine graph. This is given by either of the following equations. \begin{gather*} \cos(x-\frac{\pi}{2}) = \sin(x)\\ \sin(x+\frac{\pi}{2}) = \cos(x) \end{gather*}

Subsection7.3.3Periodic Functions

Definition7.3.1Periodic Function

A function \(f\) is periodic if there is a number \(p\) so that \(f(x+p)=f(x)\) for all \(x\). The smallest such number \(p\) is called the period of the function.

Waves are a good physical example of periodic behavior. (I'm thinking about steady rolling waves or ripples, not breaking waves that do not pass the vertical line test but are much more fun.) If we took a snapshot (single moment of time) and looked at the height of the water along a straight line, we would see peaks and valleys that repeat their shape over and over again. This would give us a periodic function with an independent variable of position. The period would correspond physically to the wavelength, which is the distance at which the shape starts to repeat (usually imagined as distance from peak-to-peak).

On the other hand, if we had a bobber at a single location and measured its height as it bobbed up and down, then we again have a periodic function with an independent variable of time. The period now corresponds to a time interval over which the pattern of height begins to repeat, which equals the time between the bobber reaching its highest point.

In reality, physical waves exist in both space and time and should be considered as functions of both independent variables. In such a case, the physical period is called the wavelength and the temporal (time) period is called simply the period. For the purposes of our discussion, we will only consider a single independent variable and use the idea of period and wavelength interchangeably.

The 1963 Nobel prize in Physiology or Medicine was awarded to the scientists Alan Hodgkin and Andrew Huxley for their development in 1952 of a model of how ion channels in a neuron control the neuron's firing patterns. One prediction of their model was that if the stimulus on the neuron was sufficiently high then the neuron would fall into a steady rhythm of firing or spiking. The resulting graphs illustrate a periodic function that is more complex than a simple sine or cosine function. The period can be estimated by measuring the interval (spacing) between any repeating point on the graph.

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Figure7.3.2 Neuron spiking given by the Hodgkin--Huxley model of ion channels.

Finding the period from such a graph is challenging because we can only estimate the coordinates of the points. Using the actual data as portrayed in a detailed table would be better. In the above figure, it looks like over the course of 100 ms the pattern repeats almost exactly 7 times. So the period should be close to \(100/7 \approx 14.3\) ms. The data (table, not shown) suggest that the moment when the voltage first crosses \(V=0\) occurs at \(t=1.81\) ms. The voltage crosses this again coming down at \(t=10.16\) ms, but the moment when the pattern begins again is when it crosses going up at \(t=16.45\) ms. So our best estimate of the period is \(p=16.45-1.81=14.64\) ms.

For symmetric waves like the sine and cosine functions, the amplitude of the wave describes the height of the wave away from the center, which would be the value in the absence of a wave. For this reason, the amplitude of a periodic function is usually defined as half the distance between the maximum and minimum values. It is harder to justify this definition for examples like the Hodgkin–Huxley spiking model where the center line should be 0 V (neutral) so that there is an asymmetry between the distance to maximum and distance to minimum.

Subsection7.3.4Transformations and Periodic Functions

Suppose we are given data representing some periodic behavior and we want to establish a model for those data using a known periodic function. It is almost certain that the known periodic function will have the wrong amplitude, the wrong period, and the wrong starting point. We use the idea of transformations to create a new function that will do a better job of representing our data. We will use the sine and cosine functions as examples of how this is done.

One way to do this is to imagine enclosing exactly one cycle of the elementary function in a rectangle and then enclose exactly one cycle of the data as the transformed function. The width of each rectangle corresponds to the period. The top of the rectangle corresponds to the maximum value (peak) and the bottom corresponds to the minimum value (valley). Then use the method of function transformation to find the new function.

Example7.3.3
Find the formula for the sinusoidal function shown in the following graph.

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Solution

The previous example demonstrated how knowing basic information about a periodic function allows us to model that function using an elementary trigonometric function. In the example, we discovered that the fundamental attributes that we needed were the centerline and amplitude (to determine the vertical transformation) and the period and a starting point (to determine the horizontal transformation). Once these features are known, it is straightforward to develop the model.

The reverse process also works. If we start with a formula of a sinusoidal function, we can interpret the parameters of the formula to determine the period, the amplitude, the center line (vertical shift), and the phase shift (horizontal shift).

Example7.3.5
Interpret the formula \(y=3 \sin(6x+\pi) - 3\) and use this to sketch the graph.
Solution