Activity 3: Exploring Coupled Oscillators

Motivation

In this activity, you will use a program called fireflies+ to study the properties of a system of coupled oscillators. The program is written in StarLogo, a programming language which is designed to implement parallel algorithms. We are interested in coupled oscillators because of their ability to capture the behavior of biological systems which synchronize with each other or with periodic phenomena in their environment. An example of such a biological system is the fireflies described in the Strogatz and Stewart article. These fireflies have the capacity to synchronize their flashes with nearby fireflies.

In this activity we will not be concerned with actually modeling these fireflies. Instead we will be investigating some of the general properties of coupled oscillators. For convenience, however, we will refer to the oscillators as "fireflies." The main behavior we would like to see in our fireflies is synchronization; that is, all of the fireflies should flash at roughly the same time. We will vary the model by changing the values of parameters that control its behavior. In each case, we will observe the effect of the changes and try to explain it.

When you are done with the activity, you should have a better idea of what a computational model is and what it means to experiment with a computational model, as well as how coupled oscillators work and why we are interested in them.

Using StarLogo

When you start up StarLogo, you get 7 windows. Windows which are not visible can be made so by selecting them from the Windows menu. The Interface window contains the various controls to run programs (you won't see anything there until you open a StarLogo program such as fireflies+). The Graphics window shows the graphically displayed output of the program. The title of this window will be the name of the StarLogo program you are running (or "StarLogo" if you aren't running anything). The Output window is where textual output from the program you are running is printed out. The Info window contains information about the program. The Procedures window contains the program itself; you will not need to be concerned about it at this time. We will not be using the Command Center and Toolbar windows. The Toolbar window is the small window containing four images which sits in front of the Interface window. You should close it (by clicking in the upper-left corner) because it will only get in the way.

Start up a particular program, such as fireflies+, in StarLogo by doubling-clicking on its icon or by selecting it after choosing Open... from the StarLogo File menu. It will take a few seconds for the program to load. When it is done, you will see several buttons and sliders in the Interface window. You click on the buttons to set up and run the program. A button with arrows pointing on two directions causes the program to keep doing the same thing over and over again until you click on the button again. The sliders are for setting the values of various variables which control how the program runs.

`fireflies+`

Make sure that you have a copy of the latest version of the program. It is called fireflies+. You can find it using the instructions for obtaining files with FTP which you should already be familiar with. The file is in ftp.cs.indiana.edu/pub/gasser. Do not get the file fireflies, which is an older version of the program.

The fireflies+ program is described in the Info window which you can access after you have started up the program. What follows is a copy of the text from the Info window of the program.

OVERVIEW

This program implements a system of coupled oscillators
rougly analogous to a group of fireflies of the type which
synchronize their flashes.  The oscillators will hereafter
be referred to as "fireflies".  A firefly couples with other
fireflies only when those fireflies are "flashing".  Flashes
happen just before and after a firefly's zero phase angle.

The program has two modes, "spatial" and "non-spatial".
In spatial mode, the position of the fireflies matters.
A firefly can "see", that is, can couple with,  other
fireflies only when it is close enough to them; how close
depends on the "flashing radius" of the fireflies.  In
spatial mode, the fireflies move in random directions around
their world.  You can control how often they move.

In non-spatial mode, a firefly sees (couples with) any other
firefly which is flashing; that is, the fireflies have no
position, and hence they do not move.

Fireflies are represented in the Graphics window by blue
dots which turn white when the fireflies are flashing. 
Beyond this detail, however, the display for the two modes
is completely different.  In spatial mode, the fireflies are
shown in their current locations.  When a firefly flashes,
the flashing radius around it is displayed, and any firefly
within the flashing radius, that is, any firefly which
couples with the flashing firefly, turns temporarily pink. 
In this mode, there is also a display option which indicates
the current phase angle and period of each firefly.  These
are represented by a brown line whose direction indicates
the phase angle (3 o'clock is 0 and the movement is
counter-clockwise), and whose length indicates the period
(a length of 10 is a period of 100).

In non-spatial mode, fireflies trace paths in circles around
the center of the Graphics window.  Their position on the
path indicates their phase angle (again 3:00 is 0 phase,
and movement is counter-clockwise), and their distance from
the center indicates their period.  To make it easier to see
the firefly periods, gray bands on the background indicate
period increments of 20.  The edge of the outermost gray
band represents a period of 200 (twice the average).  When
the period of a firefly goes beyond 200, it is displayed in
red, and it remains at the outer edge of the largest band,
no matter what how great its period, until its period drops
below 200 again.  Where there is more than one firefly on a
particular square, they are displayed in a lighter blue
color.

The phase angle and period of each firefly is updated on
each "time slice".  Each time slice represents a time of
length 2.  The average period of fireflies is 100, that is,
50 time slices.  An option allows the user to have
information printed out about the fireflies on every 10th
time slice.

Both phase angle and period coupling are implemented.  Since
coupling only takes place when a firefly flashes near its 0
phase, the relevant variable is the current phase angle of
the affected firefly.  The coupling function is the negative
sine of this phase angle; the value of this function goes to
zero at a phase angle of .5.  The period of fireflies does
not decay to a resting period.

BUTTONS

setup
This button creates a new set of fireflies and initializes
the display according to the value of the "spatial" slider.

go
This forever button repeatedly updates the phase angles
and periods of the fireflies and redraws the display.

SLIDERS

You can change some of the sliders (clock, mv-prob, flash%,
output?, pa-C, and pd-C) even when the fireflies are
running. To change the other sliders (spatial, howmany,
fl-rad, and pd-rng), you need to first stop the fireflies if
they are running (by clicking on the go button), then move
the slider, and then click on the setup button to create a
new set of fireflies.

spatial
This slider determines whether spatial or non-spatial mode
is to be used.  A value of 0 indicates spatial mode, a
value of 1 non-spatial model.  As discussed above, the two
modes differ in the behavior of the fireflies and in the
type of display which appears.  With this slider, you must
click on setup for its new value to take effect.

clock
This slider is only relevant for spatial mode (spatial = 1).
When it is set to 1, the "clock hands" indicating phase
angles and periods of fireflies are displayed.

output?
When this slider is set to 1, information about the
fireflies (number flashing; phase angle, period, and
resting period of each firefly) is display in the Output
Window on every 10th time slice.

howmany
This value of this slider determines the number of fireflies
in the system.  With this slider, you must click on setup
for its new value to take effect.

fl-rad
This slider is relevant only in spatial mode.  It indicates
the "flashing radius" of the fireflies, that is, the
maximum distance from a flashing firefly at which another
firefly is affected.  With this slider, you must click on
the setup button for its new value to take effect.

mv-prob
This slider is also relevant only in spatial mode.  It
indicates the probability with which each firefly moves on
each time slice.  When it is 0, the fireflies never move;
when it is 100, they always move.

flash%
The value of this slider is the percentage of a firefly's
cycle during which it flashes.  If this is set to 10, for
example, each firefly flashes from a phase angle of .95
until a phase angle of .05.

pd-rng
The value of this slider indicates the range of initial
periods of the fireflies.  The average period is 100.  If
this slider is set to 100 (its maximum), the initial periods
of the fireflies range from 50 to 150.  With this slider,
you must click on the setup button for its new value to take
effect.

pa-C
This slider sets the phase angle coupling factor.  When it
is set to 0, there is no phase angle coupling.

pd-C
This slider sets the period coupling factor.  When it is set
to 0, there is no period coupling.

The Activity Itself

In this activity, you will be asked to describe what happens under certain conditions. You should attempt to single out what is interesting to describe. For example, it is not especially interesting to note that "one firefly in the lower-left corner and another in the center had very similar phase angles." But it might be interesting to say that "about half of the fireflies were synchronized with each other, while the others appeared to be completely unrelated." Also try to explain what happens whenever this seems possible.

Problem A: Phase-angle coupling
When you start up the program, it will be in non-spatial mode. That is, each firefly will respond to any other firefly which flashes, and a firefly's position in the Graphics window indicates its phase and period. You should set the sliders to the following positions if they are not already there
```
spatial   0
howmany   20
flash%    10
pd-rng    0
pa-C      20
pd-C      0
```
The settings of the other sliders don't matter, but if you set output? to 0, the program will run much faster. These settings create a set of 20 fireflies all with the same initial period (100). Period coupling is turned off, but phase angle coupling is turned on, with the relatively low value of 20. This value controls the amount of influence each flashing firefly exerts on the other fireflies. The flash% setting of 10 means that fireflies flash for 10% of each cycle (5% before and 5% after their 0 phase).
Click on the setup button to initialize the fireflies. Notice that their phase angles are random but that their periods are the same.
1. Run the fireflies by clicking on the go button. Notice that the program is slow enough that you can actually see the individual time slices. Let the program run until its behavior is relatively stable. Describe what has happened and what the final state of the system is.
2. Now stop the fireflies by clicking on the go button again. Move the howmany slider to 60 to create a larger population; don't change of the other settings. Click on the setup button to reinitialize the fireflies. Run the fireflies again by clicking on the go button. Describe how the fireflies' behavior differs this time.
3. Now stop the fireflies and reset the howmany slider to 20. Set the pa-C slider to 50, increasing the degree of coupling between the fireflies. Set up the system and run it again. Describe the behavior that you observe this time, and compare it to what you saw in 1 and 2 above.
4. Some oscillator models implement a refractory period, a short time in which an oscillator which has just "fired" (= "flashed" in our system) cannot fire again. How might this improve the behavior of the systems you observed in 2 and 3 above?
Problem B: Period coupling
Set the sliders to the following values.
```
spatial   0
howmany   20
flash%    10
pd-rng    50
pa-C      20
pd-C      0
```
This situation is just like that in Problem A.1., except that fireflies with different initial periods will be created because of the pd-rng setting.
1. Set up the fireflies and run them as long as you feel you need to in order to observe what is happening. (The system may not stabilize.) Describe how the behavior differs from the situation in Problem A.1. when all fireflies had the same period.
2. Stop the fireflies and set the pd-C slider to 20. This starts up period coupling (a flashing firefly influences the periods of the other fireflies). Set up the system again and run it long enough to see what is going on. Describe how this differs from what happened in part 1.
3. Now set the sliders to the following values:
```
spatial   0
howmany   10
flash%    10
pd-rng    100
pa-C      24
pd-C      16
```
  This creates a system with 10 fireflies that have very different initial periods and that interact with both period and phase-angle coupling. Set up and run the fireflies 5 or so times, each time waiting until the system seems to have stabilized. (This may take 5 minutes or so.)
  1. Describe the stable behavior(s) you observe.
  2. Does the system seem to be very sensitive to initial conditions (the initial phase angles of the fireflies)?
Problem C: Fireflies in physical space
Set the sliders to the following values:
```
spatial   1
clock     1
howmany   40
fl-rad    10
mv-prob   0
flash%    10
pd-rng    0
pa-C      100
pd-C      0
```
This creates a set of 40 fireflies whose locations matter for coupling but which do not move at all (because of the mv-prob setting). The initial periods are all the same, and there is very strong phase-angle coupling, but no period coupling.
1. Run the fireflies until the situation is fairly stable. This may take five minutes or so. (To speed things up, you can set clock to 0 for part of the time.) What sort of behavior do you observe?
2. Stop the fireflies, and set fl-rad to 16. This makes the flashing radius of each firefly (the area this is visible to other fireflies) somewhat larger. (It also makes the program much slower, so be patient.) Also set pa-C to 50, lowering the coupling rate. Setup a new set of fireflies, start them up again, and wait for things to be relatively stable. How is the behavior here different from what it was in part 1 of this problem?
3. Stop the fireflies, and reset fl-rad to 10. Then set mv-prob to 100 so that the fireflies move on every time slice. Leave the value of pa-C at 50. Set up a new group of fireflies, start them up once more, and wait for a stable situation. How does the behavior differ from that in parts 1 and 2 of this problem?

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URL: http://www.indiana.edu/~gasser/activity3.html
Last updated: 26 September 1995
Comments: gasser@salsa.indiana.edu
Copyright 1995, The Trustees of Indiana University