I have developed several computer simulations for an undergraduate course I teach at Indiana University called "Complex Adaptive Systems." The course deals with systems that evolve and adapt over time.  Psychology, computer science, economics, biology, and neuroscience depend upon a deeper understanding of the mechanisms that govern adaptive systems.  A common feature of these systems is that organized behavior emerges from the interactions of many simple parts.  Ants organize to build colonies, neurons organize to produce adaptive human behavior, and businesses organize to create economies.  To address the essential question of "What are the properties of complex adaptive systems?," case studies of several systems are explored: chaotic growth in animal populations, human learning, cooperation and competition within social groups, and the evolution of artificial life.  The central thesis is that widely different systems (businesses, ant colonies, brains) share fundamental commonalities.

These topics will be explored by extensive hands-on use of interactive computer simulations.  The basic problem with teaching complex systems is that they are .... complex.   Appreciating how structured, organized behavior can emerge from the interactions of many units is almost impossible unless one can create a working model of these interactions.  Similarly, it is hard for students to understand how simple mathematical equations can explain how the brain is organized, how populations grow and shrink, and fires spread (to take three arbitrary examples), unless they can experiment with these equations, and if they can see the results of these experiments in a graphic, intuitive manner.  By experimenting with computer simulations, students can bridge the gap between mathematical formulae and large-scale behavior.  The philosophical motivation of pedagogy for these simulations is:
 

• Students should gain both an intuitive appreciation for the behavior of complex adaptive systems as well as an understanding of their formal underpinnings.
• An active exploration of complex systems is the only true way to come away with a practical understanding of them.  In many cases, it is quite surprising to see the highly organized, orderly behavior that emerges from systems composed out "agents" behaving in accord with very simple rules.  Many people who have worked with these computer simulations for an extended period of time have been struck by the qualitative increase in their knowledge, relative to simply seeing the equations underlying the systems or a verbal description.
• Simulations should show how agents organize themselves dynamically, over space and time.
• Students should be able to actively change the behavior of the systems by "tweaking" parameters.  All of the simulations that I have written contain a substantial amount of flexibility in them.  Students can feel in control of the simulation, and can discover new phenomena and interrelations by systematically changing parameters.  Students can conduct experiments, and instantly see the results of the experiments.  In addition, the interactivity makes the simulations more appealing to use, hopefully increasing the amount of time students will spend with them.

There is no monetary charge for using these simulations, but I consider them to be Grant-ware. That is, if you use the programs, please send me an email message (rgoldsto@indiana.edu) so that I can document that they are being used.  This can simply be a few lines long, telling me how you used the programs, and perhaps what you liked and didn't like about them.  In this way, perhaps someday I may be able to get a grant to develop new simulations.  All of the programs here were written by Robert Goldstone, Copyright © 1999, 2000.

All of the programs on this page are written for Macintosh computers.  They should work on any powermac, but several of them are computationally demanding, and will run slowly on anything more feeble than an Imac or G3.  The optimal screen size is 1024 X 768.

NEW Two students working with me have ported over some of these Macintosh simulations to be Java applets, with the intention of creating platform-independent simulations.  Stay tuned as more of the simulations are ported over to Java.  For now, check out the simulations by these students:

• Benjamin Ashpole: Competitive Learning, Input/Output Learning, Hebbian Learning, Traveling Salesman
• Yasuaki Sakamoto: Dropping Balls, Path Finder, Resource Cover, Pine Cone, Pattern Formation

If you download software, please take a moment to fill out the following information:

Press here to download Ball Dropper

Press here to download Path Finder

Press here to download Resource Cover

Press here to download Pattern Learner

Press here to download Predators and Prey

Press here to download Hebbian Learning

Press here to download Pine Cone

Press here to download Input/Output Learning

Press here to download Traveling Salesman

Press here to download Apparent Motion

Press here to download The Chaos Game

Press here to download Learning Facilitating Evolution

References and Further Reading for Simulations

What follows is a brief description of the currently available complex system simulations

Ball dropper
 

This is a computer simulation of balls falling on a windy day.  It was designed to teach students about local and global minima, and the effectiveness of a "simulated annealing" schedule for finding global minima.  In this simulation, simulated annealing involves gradually reducing the amount of randomness in the balls' motions.  Each ball follows a few simple rules:
 

1.  The balls will generally fall downwards.
2.  The balls will also be influenced by a random factor.
3.  The balls will not pass through anything painted green.

As a goal, imagine that you are trying to have all of the balls drop as low as possible.  How can this be done?  Even though the wind is random, you can use it to help you.

Experiment with dropping balls on different landscapes.  Do the balls find the lowest possible spot in each landscape?  If they don't, why not?

Experiment with different ways of adjusting the amount of randomness and amount of movement of the balls.  What technique does the best job of putting the balls at the lowest spot?

How can you avoid having your balls settle in shallow valleys when deeper valleys exist?

Path Finder

This is a computer simulation of balls influencing each other, and as a group, finding pathways around obstacles.  It also demonstrates notions related to local and global minima, and simulated annealing.  In fact, the reason for including both "Ball Dropper" and "Path Finder" is to demonstrate how the same abstract principle may apply in different domains, and to train students in skills related to extracting underlying abstract similarities.  The balls follow the following simple rules:
 

1.  Each ball is assigned two neighbors, making a set of balls arranged from first to last.  One of the fixed blue points is the neighbor of the first ball.  The other blue point is the neighbor of the last ball.
2.  Each ball will move toward of its two neighbors, and will also move with a bit of randomness.
3.  If the location that a ball would move to is painted green, then it does not move.

With these simple rules, and the right setting of parameters, it is possible to have the balls find a pathway that connects the two blue endpoints and avoids the green obstacles.

Experiment with different ways of adjusting the amount of randomness and amount of movement of the balls.  What technique does the best job of usually finding good pathways?

Here are some other questions to ponder as you play with this simulation:

What is the best number of balls for a maze?  Why?

What kinds of mazes can and cannot usually be solved?

If the balls have trouble finding a pathway, what can you do to help them?

 

 
 

Before

After

Press here to download Path Finder

Resource Cover

This is a computer simulation of ants covering food that is placed down, or more generally, any set of agents trying to optimally cover a set of resources (e.g. oil drills covering an oil field, or scientists covering a domain of inquiry).  The following simple steps determine the behavior of the ants:

1.  One piece of food (green pixel) is randomly selected from all the food present.
2.  The ant that is closest to the selected food moves toward the food with one speed.
3.  All of the other ants move toward the food with another speed.

Experiment with different ways of adjusting how quickly the ants move.  What technique does the best job of usually helping the ants cover the food optimally?

Here are some other questions to ponder as you play with this simulation:

Why don't you want to move the ants as quickly as they can go?

If you have 5 ants, and one region of food is 4 times bigger than another, how do the ants distribute themselves?

If you make two food piles and place two ants in one pile, what can you do to make the ants find different piles, and why does this work?

Before

 

After

Press here to download Resource Cover

Pattern Learner

This is an example of a type of artificial intelligence called a "neural network."  It shows how a computer can learn to categorize pictures even without a person telling it what pictures belong together.  For example,  if a person draws pictures of "A"s and "B"s, the computer may create two categories, one for each letter.  As with the first pair of simulations, this simulation is abstractly related to the previous "Resource Cover" simulation.  Both simulations involve finding ways to optimally cover presented information by creating agents that are specialized for subsets of the information.

As a person paints different pictures, the computer will create categories, using the following simple steps:

1.  The person tells the computer how many pictures and categories to allow.
2.  A picture is randomly selected.
3.  The category that is closest to the picture adapts itself to the picture by a certain amount.
4.  All of the other categories adapt themselves to the picture by a different amount.

As you use this neural network, try to figure out how the computer is able to automatically form categories of pictures simply by using the steps just described.  Try situations where the number of categories is the same as the number of pictures, and also situations where there are fewer categories than pictures.

Here are some other questions to ponder as you play with this simulation:

Why aren't good categories found when all of the categories adapt equally quickly to selected pictures?

If there are two categories and two pictures, what can you do to make the two categories each become specialized for a different picture, and why does this work?

Once each category is specialized for one picture, make a change to one of the pictures.   Why does only one category (the correct one) change?

Before

 

After

Press here to download Pattern Learner

Predators and Prey

This is a computer simulation of the changes in populations of predatory and preys.  An historical example is the cyclic variation of lynx and rabbit populations.  However, we will consider populations of rabbits and grass because their interactions are better modeled by the equations used here.  Often it is observed that as the amount of grass grows in a regions, the population of rabbits will increase with their food source.  However, as the rabbit population increases, the grass population will decrease from over-harvesting.  Then, the population of rabbits will decrease because of the dwindling food supply of grass.  With fewer predators, the grass population increases again, perpetuating the cyclic population variation.

Around 1926, Lotka and Voltera independently came up with following two equations two describe the dynamics of predator-prey populations:
dG/dt = a*G - b*G*R                dR/dt = d*b*G*R - c*R

where:
a=growth rate of grass without predation
b= rate of consumption of grass by rabbits per encounter
c = natural death rate of rabbits
d = efficiency of turning grass into new rabbits

dG/dt and dR/dt are the rates of change of the grass and rabbit populations from one instant to the next.  As such, the populations of rabbits and grass only depends on the immediately preceding population sizes.

One way of modeling populations is simply to apply the Lotka-Voltera equations given a set of four parameters.  The computer will show this simulation when asked to show the "expected populations" according to this model.  A dynamical, graphical simulation of these equations is also provided.  This simulation proceeds by the following rules:
 

1.  Initial populations of rabbits and grass are set by parameters.  Organisms are randomly positioned in a field with a size set by the user.
2.  For each patch of grass, if a random number between 1 and 100 is less than the "grass grow rate" (parameter "a" of the Lotka-Volera system) then a new piece of grass is created near the grass.
3.  For each rabbit, if a random number between 1 and 100 is less than the "rabbit death rate" (parameter "c") then the rabbit will die.
4. If a rabbit is on top of a patch of grass, and if a random number between 1 and 100 is less than "rate of consumption" (parameter "b"), then the grass will die, and the rabbit will give birth to another rabbit if another number between 1 and 100 is less than the "efficiency rate" (parameter "d").  The new rabbit will be placed next to the old rabbit.
5.  Each rabbit and grass will randomly move to an adjacent square.

There is a limit of 20,000 rabbits or grass, and this limits how many pieces of grass can co-exist on the same patch (capacity = 20,000/number of patches).

You will notice that, for any combination of parameters, the populations predicted by the Lotka-Voltera equations are not the same as the populations predicted by the graphical population model just described.  One of the main reasons for this is that the Lotka-Voltera equations have no notion of space.  All predators and preys are imagined to be on the same patch.  One consequence of this is that predators are much more likely to die out in the graphical/spatial model, because they will be less likely to be standing on a grass patch at any given time.  It may be that the graphical model is a more realistic simulation of populations.  It can be shown that the Lotka-Voltera equations always (see below for one exception) predict cyclic, rather than relatively stable, populations.  Population cycles are found in nature, but not always.

The spatial model of populations predicts many more relatively stable populations.  Why?

Other things to think about:

• Why isn't the population of rabbits always made larger when "rabbit death rate" is decreased?
• If one wants to have the largest population of rabbits as possible, which parameters are important/unimportant and why?
• If one wants to have a lot more grass than rabbits, what should the parameter values be?
• Looking at the Lotka-Voltera equations, it can be shown that the ONLY stable point occurs  if Initial Rabbits = a/b and Initial Grass = c/bd.  Confirm that this is true for the "expected" graphs.  Why is this the case (Hint: populations are stable if their rate of growth is zero.  Thus, set dG/dt and dR/dt to zero, and solve for R and G).

For pure fun, let a crop of grass grow without any rabbits, and set "grass grow rate" to 1.  Now, try to eliminate all grass with the "erase grass" tool.  Can you do it?  What if "grass grow rate" = 2?  Why the big change?

Press here to download Predators and Prey

Hebbian Learning

This is an example of a type of artificial intelligence called a "neural network."  It shows how a computer can be trained with some pictures, and can then later recognize distorted versions of the pictures.  A single memory is used to store information from all of the presented pictures.  The memory is "distributed" in that all parts of the memory are used for representing all of the pictures, rather than having each picture in its own memory.  The memory of the network consists only of connections between each pixel of a 15 X 15 grid to each other pixel (thus, it is made up of 50,625 numbers).  Every pixel in a picture takes on a value between -1 and 1.  Each connection in memory describes the relation between two pixels.  If the pixels tend to be in the same state, then the connection will be positive; If they are in different states, the connection will be negative.  Memories are learned using a modification of the Hebbian learning rule (named after Donald Hebb, a Canadian neurophysiologist).  According to this modified rule:

The change in the link between pixels A and B is proportional to (1) the product of their states, (2) a learning rate, and (3) the difference between the current link and the minimum or maximum value a link can take.

This learning rule shows the following useful behaviors:

• Once a picture is learned, noisy versions of the picture become regularized, becoming more similar to the original picture.
• Once a picture is learned, if a version of the picture is shown that has parts deleted, they will be restored.
• Once a picture is learned, if a version of the picture is shown that has extraneous parts added, they will be deleted.
• If individual links of the memory are destroyed, the behavior of the system is not changed very much.

All of these properties can be traced to the simple rule for creating memories, and the simple rule for changing pictures by the learned memories.

While experimenting with the simulation, try the following:

Create a picture, and then see how distorted versions the picture are "cleaned up."

Create two pictures, and see how the memory can retain traces of both at the same time.  If you feed something that looks like one of the pictures, it will become more like whichever stored picture it is more similar to.

Trying limiting the distance that links can reach, and feeding in a small part of picture (try feeding in the picture with both black and neutral green backgrounds).  How is the complete picture reproduced?

Press here to download Hebbian Learning

Pine Cone

This is a computer simulation of how some elegant mathematical relationships commonly found in plants emerge.  Many of the relationships are based on the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
where each number is obtained by adding together the two previous numbers.

Seeds are often arranged in spirals on  flower heads.  Often times there are both noticeable counterclockwise and  clockwise spirals.  The number of spirals going in one direction is usually one of the numbers from the Fibonacci sequence.  The numbers of clockwise and counterclockwise spirals are usually SUCCESSIVE NUMBERS in the Fibonacci sequence.  For example, a sunflower may have 21 clockwise spirals and 34 counter clockwise spirals.  Below, you'll see an actual photograph of a sunflower from http://www.sunflowers.com/.  With a little effort, you can probably see patches where there are counterclockwise and clockwise spirals, and you may even be able to count the number of each type of spiral (I count 34 clockwise spirals).

Another place where the Fibonacci sequence emerges is if we examine each of the spirals more closely.  We can number each plant element (seed head, petals, needle, etc.) by when it was formed.  For example, the first "petal" on a pine cone would be 1, the second is 2, etc.   All of the petals that lie on a spiral are a constant number of petals apart.   These constants are numbers from the Fibonacci sequence.  Furthermore, the constants associated with the counterclockwise and clockwise spirals will be SUCCESSIVE numbers from the Fibonacci sequence.

A final, related mathematical fact is that successive elements of a plant are often separated by 137.5 degrees.  This is an interesting number, because it is the equivalent of the golden section for a circle.  The Golden Section, Phi, a mathematical phenomenon known and used architecturally by the ancient Greeks, is the only number that when squared, is simply equal to itself + 1.  That is,  Phi*Phi = Phi+1.  Plants very often send off leaves, seeds, stems, etc. at angles based on the Golden Section.  What is the relation to the Fibonacci sequence?  The ratio of successive numbers of the Fibonacci sequence becomes a better and better approximation to Phi as the sequence continues.  For example, 13/8 = 1.625, 21/13 = 1.615..., and 34/21 = 1.619..., while Phi=1.618033989...

The current simulation is based on Doady and Couder's (1992) work on modeling plant growth.  The basic idea is that the next plant element to be formed will be formed where there is the most space.  Plant elements will be formed somewhere on the radius around the center of the plant.  They will move away from the center with a certain velocity.  They will be created at a certain rate.  Those three parameters (radius, velocity, creation rate) will determine how the entire plant's growth appears.

The following simulation creates plant forms by the following simple steps:

1.  Place the next plant part at point P on Radius R around the center where energy is at a minimum.

That is, energy will be the sum of an inverse function of the distances between the point P and the other, already existing plant parts A.  Thus, a point P will be selected for  the next plant part if it is, on average, the farthest point on the radius from existing parts.

2.  For plant parts that have already been formed, move them away from the center with a parametrically determined velocity, in the same direction as they were originally placed.

By executing these simple rules with different parameters, many common plant types can be created, from the simple alternating leaf arrangement of corn to the complex seed head spirals of daisies and sunflowers.  Try experimenting with different parameters, and see if you can get parts separated by 137.5 degrees, or spirals related to Fibonacci numbers.
 
 

Input/Output Learning

This is an example of a type of artificial intelligence called a "neural network."  It shows how a computer can learn to associate several pairs of input and output pictures to each other.  A single memory is used to store information from all of the presented pictures.  The memory is "distributed" in that all parts of the memory are used for representing all of the input/output pairs.  The memory of the network consists only of connections between each pixel of a 15 X 15 grid to each other pixel (thus, it is made up of 50,625 numbers).  Every pixel in a picture takes on a value between -1 and 1.  Each connection in memory describes the relation between two pixels.  If the pixels tend to be in the same state, then the connection will be positive; If they are in different states, the connection will be negative.  Memories are learned using the Delta Rule (McClelland & Rumelhart, 1986).  This net input is compared to the desired output - the actual value of the pixel in the output picture.  The bigger the difference between the desired output and the net input to the pixel, the greater the change in connections will be.  According to this rule, if the net input to a pixel is greater than the pixel's actual state of activation, then the change in the weight from i to j will be negative.  This means that if Pixel i is active, then its connection to j will decrease, which is desireable because it means that the next time i is active, it will not cause j to become so active.  Conversely, if the net input to Pixel j is too small compared to the actual state of j, then the change of weight will be positive, making an active unit i more potent in activating j.  After weights between units have been learned, they affect the later behavior of the neural network.  When given a certain set of input activities, the behavior of the network  will be equal to the net input to each output unit.

The delta rule of neural network learning has a number of useful properties:

• Graceful degradation: as parts of the memory for  network (the entire set of input-to-output links) are erased or distorted, the network does not suddenly lose its ability to give the proper output when presented with an input.  Performance gets worse as the memory is disrupted, but surprisingly large disruptions have little influence because the learning is distributed over the entire network.
• Noise tolerance: if a distorted version of an input is presented, the network is typically still able to produce a reasonable fascimile of the proper output.
• Prototype formation: if two similar inputs have similar outputs, and if the shared parts from the two inputs are presented, then the strongest outputs will be for the shared parts from the two outputs. That is, the commonalities between two input/output pairs are selectively emphasized.
• Content addressability: in finding the output associated with an input, the computer does not go through a list of associations one by one.  Instead, it directly accesses the proper output.

Press here to download Input/Output Learning

Traveling Salesman

This computer simulation solves the classic "Traveling Salesman" problem.  The goal of this problem is to find the shortest possible route between N cities, visiting each city once and returning to the starting city.  Finding the route which is definitely the shortest possible route is known to be an NP-Complete problem, meaning that as the number of cities increases LINEARLY, any computer algorithm requires an EXPONENTIALLY increasing number of steps.  NP-complete problems are thus considered to be intractable, but that does not mean that reasonable solutions for them cannot usually be found.  There is no guarantee that these "reasonable" solutions will be the very best possible, but typically they will be very close to the optimal solution if reasonable parameters are chosen.  In this simulation, there is a population of solutions, each of which offers a particular route through a set of cities.  The entire system can develop a good solution for a given map of cities by selectively "breeding" solutions.  Thus, the set of solutions evolve over time.  Each new generation will be based on the previous generation, but with a bit of variation thrown in.  The computer will show the best (most "fit" in evolutionary terms) solution among those available in a particular generation.  Unlike typical computer searches for a good solution, here an entire POPULATION searches for a solution, meaning that the system tends not to get stuck in dead-ends as easily as systems with only a single agent that is searching.

Each solution within a population consists of a simple representation of a route among cities.  Namely, it is an ordered set of N numbers where each number identifies a city.  If an agent's set is:   5 7 2 3 1 6 4
then the route would go from City 5 to City 7, and then to Cities 2, 3, 1, 6, 4, in order, and then would finally return to City 5.  By changing ("mutating" to use the evolutionary term) its solutions, a system can explore different solutions to a map.  Here are the particular types of mutation allowed in this program:

• Move: A randomly selected city is moved to a randomly determined location.  By this method, if an agent was originally 5723164 and City 3 was chosen for moving, its child in the next generation might appear as 5721634.
• Invert: A starting and ending city are randomly selected, and the order of cities within this set is reversed.  By this method, 5723164 would become 5746132 if the set of cities "23164" was selected for inverting.
• Swap: Two cities are randomly selected and swapped.  By this method, 5723164 would become 5726134 if Cities 3 and 6 were selected.

1. The solutions are rank ordered in terms of how good they are (good = small round-trip distance).  The top N% are selected for reproduction, where N is determined by you.
2. Each solution to be reproduced is mutated by the three methods above.  The frequency with which each method is applied is determined by you.  Each of the top N% solutions produces the same number of offspring in the next generation.

While exploring this simulation, here are some questions to consider:

• Why is it not a good idea to make mutations occur very frequently?  It is true that this allows many different types of solution to be considered, but at a definite cost.
• You might notice that larger mutation rates become increasingly effective as the population size increases.  Why?
• What types of mutation are useful in correcting different types of solution difficulties?
• In what situations are your solutions better than the computer's?  When is the computer's solutions better than yours?

Press here to download Traveling Salesman

When static pictures are rapidly alternated, people often spontaneously perceive illusory motion.  By studying apparent motion phenomena, we can learn about how people subjectively structure the external world , the assumptions that people make about stimuli when perceiving, top-down effects in vision, and the specific mechanisms that people use to see motion.  Apparent motion is also indispensible to the motion picture industry because movies are nothing more than a series of still frames that are rapidly displayed.

This program does two things.  First, it allows you to create and display apparent motion phenomena.  You can design an apparent motion display, and then present it to yourself and determine the motion that you spontaneously see.  Second, it provides a computational model of how people see apparent motion.  Typically, you will use this program by presenting the same apparent motion display to yourself and to the computational model, and observing whether the computer "sees" the same kind of motion that you do.
 

The "Motion correspondence problem"

When people see motion between two successively presented frames, their minds  unconsciousnly determine which parts from the first frame correspond to which parts from the second frame.  When people see two parts as corresponding to each other, this means that they interpret the two parts as coming from the same object.  If the two parts are separated from each other, then people will see a single object moving over time.

Determining what parts correspond to each other tells us what parts come from the same object at different times.  However, this task is intrinsically ambiguous.  One part in one frame could correspond to many different parts in the other frame.  Simple experiments will show that people sometimes see two parts as showing the same object moving over time even if the parts do not have the same color, size, location, or even shape!  Your mind spontaneously figures out what parts correspond to each other, but underlying is seemingly effortless feat are sophisticated computations.  Your mind uses several constraints in order to try to determine what parts correspond to each other.
 
 

Assumptions That Solve the "Motion Correspondence Problem"

• Proximity:  Your mind assumes that objects appearing at two different times come from the same object if they are close to each other.
• Shape, size, and color similarity:  Your mind assumes that objects appearing at two different times come from the same object if they are similar in their shape, size, and color.
• Motion similarity: When two objects are placed in correspondence, a direction of motion is specified for them.  Your mind assumes that all of the objects in a display move in similar directions.
• One-to-one mapping constraint:  All else being equal, your mind tries to match up objects so that each object in one frame has ONE AND ONLY ONE corresponding partner in the other frame.

Computational  Modeling of Apparent motion

• Location, color, shape, and size similarity: Correspondence units will tend to be active if the objects they connect have similar positions, colors, shapes, and sizes.  The importance of each of these kinds of similarity can be controlled by the user.
• One-to-one mapping constraint:  Correspondence units will tend to become deactivated if they are inconsistent with other highly active correspondence units.  Correspondences are inconsistent if they would place TWO objects from one frame in correspondence with ONE object from the other frame.
• Match: If an object has no correspondending partner, then it will appear to blink on and off.  Correspondences will tend to develop so that every object has at least one corresponding partner.
• Motion similarity:  Correspondence units will tend to become active if the direction of motion that they indicate is similar to the directions of motion indicated by other highly active correspondences.

Press here to download Apparent Motion

This is a computer simulation of "The Chaos Game."  This simulation obeys the following rules:

1. A set of endpoints are designated, and an initial location for a moving point is selected.
2. One of the endpoints is randomly selected.  Usually, each endpoint is equally likely to be chosen.
3. The NEW location for the moving point will (usually) be halfway between its OLD location and the chosen endpoint.  This NEW location is plotted on the screen as a point, and then becomes the OLD location.

The main puzzle to try to solve with this simulation is:  Why does the random procedure for moving points from time to time produce Sierpinski's gasket?  With a bit of exploration and thought, you can probably figure out the reason.  To help you, we have provided a feature that allows you to plot a point, and click on endpoints to simulate how the point would move if the endpoint was chosen.  Once you understand the behavior of the system, a more challenging question is to try to determine the AREA of the plotted region as the number of iterations goes to infinity.
 
 

Press here to download The Chaos Game

This is a computer simulation of the "Baldwin Effect."  By this effect, evolution proceeds at a faster rate for species if they are made up of organisms that can learn.  This statement of the Baldwin effect may seem suspiciously similar to the Lamarckian theory of acquired characteristics, by which adaptations within an organism's lifetime are passed down to its children.  For example, Lamarck argued that giraffes who continually reach for high leafs on trees stretch their necks over time.  The neck's new length will be passed down as a physical characteristic to the giraffe's children.  It is now known that characteristics are not acquired in this way.  Physical changes to an organism that occur in its lifetime do not affect its genetic code, and this is what is passed down to the next generation.  However, the Baldwin effect HAS been observed.  What an organisms learns during its lifetime does influence the course of evolution for the species.  This goal of this simulation is to show how this is possible.
 

A Simple Graphical Notion of Fitness

Continuing to simplify matters, we will also represent organisms by points in space.  Their genetic code will define their starting locations, and in their lifetime, they may move their location if they are endowed with an ability to learn.  If they can learn, they will generally move toward the closest fitness peak to them.  For example, an organism may start off life unable to scratch its head to clear off fleas, but it may gradually get better and better at doing this.
 

Adapation Within and Across Lifetimes

1. Organisms are given random genetic codes (starting locations).
2. During each organism's lifetime, it will learn to be closer to its closest fitness peak.  It may also change in random ways.
3. After 30 learning steps (i.e. years), the organisms of a generation will die, but will produce offspring.  The more fit the organism, the more offspring it will leave behind.  Fitness is determined by how close the organism was to the fitness peak.  Specifically, Fitness = e^(-distance*(101-spread)).  By controlling spread, you control how "pointed" the fitness peak is.  If spread=0, then an organism has to be right on top of the peak to be fit.  For example, perhaps being able to partially scratch one's head doesn't help an organism because the remaining fleas can all congregate toward the unreachable spots.  On the other hand, if spread=100, then the fitness peak is still the most fit point, but neighboring points are also reasonably fit.  For example, it may be that the more areas an organism can scratch, the lower it can keep flea populations, and any flea reduction prevents blood loss.  An organism's fitness may also be influenced by how early in its lifetime it was able to find the fitness peak.  This is appropriate in situations where the organism can produce and raise more organisms as soon as it finds the peak.
4. The new generation of organisms will be based on the fittest organisms of the previous generation, but will also include a specifiable amount of "mutation."  Mutations simply move the starting genetic code of an organism in a random direction.

As you explore the simulation, remember that the initial location for an organism before it starts learning can be considered to be its genetic code.  These locations, shown in blue,  may change over the course of generations. This would be evidence for evolution of the entire species.  Thus, there are two types of change to observe.  There is learning within a single organism's lifetime, indicated by moving red dots.  There is also evolution of the genetic code of the entire population, indicated by distributions of blue dots.
 

Questions to Consider

Under what circumstances does learning facilitate evolutionary change?  One factor that matters considerably is whether the fitness peak is more like a vertical spike or a gradual hill.  In one of these situations, learning can allow genetic improvement to occur much more quickly than it would without learning.  Which situation, and why?

There is also a situation where learning can initially cause genetic improvement, but then can later actually interfere with further genetic improvement.  Why does this occur, and what can be done to prevent this interference?

How does the mutation rate influence the Baldwin effect?  What are the costs and benefits of a high mutation rate?

Once a population has evolved well to a fitness peak, try moving the fitness peak far away. What factors allow the population to quickly adapt to the new peak, and why do they work this way?

Compare changing the learning rate to changing the random jitter in organisms.  Do BOTH types of variation produce a Baldwin Effect? Why or why not?

Assuming that mutation rates can be modified in evolutionary time, what does the simulation predict about the relative mutation rates for populations that are close to a fitness peak versus far away?

Before

 
 

After

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Ball Drop

Path Finder

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Traveling Salesman

Hebbian Learning

Pattern Learner

I/O Learner

Predator-Prey Populations

The Chaos Game

Apparent Motion

Pine Cone

The Baldwin Effect

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Date last modified: 9/27/2001