21. Simulation

Where Are We

Up to this point, we have developed code that can read a model into memory and write it back out again, detecting a variety of errors in the model. The only point of writing the model out is to verify that the model has indeed been read correctly.

This highway network model could be used for many purposes:

• We could use the model (augmented with map coordinates) to display a map, the way Google Maps does.
• We could use the model to provide driving directions by running a shortest path algorithm over the model.
• We could use the same model to run a simulation of traffic loading to allow testing of the consequences of various proposed road improvements.

The neuron network model we have discussed could also serve many purposes:

• We could use the model (augmented with 3-d coordinates) to display a map of the physical anatomy of the neurons in an organism. A neuroanatomist could use this model to express the results of study of the actual organism.
• We could use the model to run a simulation of the activity of the nervous system of the organism. This would allow us to compare our understanding of how the nervous system works with actual measurements.

Our goal in this course is to build simulations, and the time has come to discuss this in more detail.

Simulation

An old bumper sticker I picked up at a simulation conference said: "Simulationists do it continuously and discretely." The sticker was a joke because, while members of the general public reading the sticker might guess one subject (sex), the actual statement is entirely true when you read "it" as a reference to computer simulation. There are two broad categories of simulation:

• Continuous simulation typically involves use of differential equations to describe the behavior of a physical system, and the simulation itself involves the numerical solution of those equations.

  Continuous simulation models are common in fields as distinct as analog electonics and macro economics. Here at the University of Iowa, the Hydraulics Institute is largely devoted to continuous simulation of fluid flow. Their building was built in the era when their research was largely done using actual tanks of water, but today, much of their work is done on computers, simulating not only water flow but also atmospheric flow.

• Discrete-event simulation typically involves systems where the behavior of the system can be described in terms of a sequence of events. Between events, nothing happens, but at each event, parts of the model interact to make changes in the simulation variables.

  Discrete event simulations are common in fields as distinct as logistics and digital logic simulation.

Almost all simulation models are based on simplifying assumptions. Most physics models assume that air is a vacuum and that the earth is flat. You can build bridges with these assumptions, although for large bridges, it is worth checking how the bridge responds to high winds such as hurricanes.

Our distinction between continuous and discrete models is also oversimplified. There are mixed models, for example, where the set of differential equations that describe the behavior of a system changes at discrete events. At each such event, you need to do continuous simulations to predict the times of the next events.

A Highway Network Model

In a highway network, the events we are concerned with are:

• Arrival at intersection: When a car arrives at an intersection, it may have to wait for other cars in the intersection to clear the intersection (we can assume that cars spend only a fixed short time in the intersection), and it may have to wait for the light to change. If it does not have to wait, it immediately schedules a road-entry event for the road its navigation algorithm selects.

• Entry to road: When a car leaves the intersection, it unblocks the intersection, allowing another car to pass through (if one was waiting); it does this by scheduling an entry-to-road event for that car. It also schedules the arrival at intersection event for this car at the other end of the road this car is entering.

• Stoplight change: Each intersection with a stoplight has a light that changes periodically by scheduling stoplight changes at that intersection. With each change, a different road (and the queue of waiting cars for that road) is unblocked, allowing cars in that queue to proceed by scheduling a road-entry event for the first waiting car.

Of course, the model can vary considerably in complexity. A simple model might have a fixed travel time on each road segment, while a more complex simulation might model congestion by having the travel time get longer if the population of a road segment exceeds some threshold, where that threshold may itself depend on the unpopulated travel time.

In a crude model, each car might make random navigation decisions at each intersection. A more complex model might have each car follow a fixed route through the road network, while a really complex model might include cars with adaptive navigation algorithms so that drivers can take alternate routes when congestion slows them on the path they originally planned.

A Neural Model

In a neural network model, with neurons connected by syapses, we might have the following events:

• Primary synapse fires: The voltage of the neuron at this time is computed, and then it is incremented by the strength of this synapse. If the resulting voltage is above the neuron's threshold, the neuron fires. This schedules synapse-fire events at each outgoing synapse to occur at future times determined by the delay of each synapse.

• Secondary synapse fires: The strength of the destination primary synapse is adjusted by the strength of this secondary synapse. Secondary synapses that operate this way can serve as the basis of long-term memory because they can turn primary synapses on and off.

The key element in the above that needs extra discussion is how the voltage on a neuron changes with time. Between events, the voltage on a neuron decays exponentially, slowly leaking away unless it is pumped up by a synapse firing. So, for each neuron, we record:

• The voltage, v
• The time at which that voltage was known t

Now, if we want to know the voltage at a later time t', we use this formula:

Of course, once you compute the voltage at the new time t', you record the new voltage and the new time so you can work forward from that the next time you need to do this computation. The constant k determines the decay rate of the neuron (it must be negative).

In a simple model, all the neurons might have the same threshold and the same decay rate, and all synapses might have the same strength. More complex models allow these to be varied.

In simpler models, the voltage on a neuron goes to zero when that neuron fires. In more complex models, the threshold has its own decay rate and the threshold goes up when the neuron fires.

In complex models, the strength of a synapse weakens each time it fires because the chemical neurotransmitter is used up by firing. During the time before the next firing, the neurotransmitter can build up toward its resting level. This allows the neural network to get tired if it is too active. (You can actually see this effect at work in your visual pathways. Look at a bright light and then look away, and you will see a negative afterimage that fades as the synapses that were overexcited recharge.)