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## NetLogo User Community Models

## WHAT IS IT?

This simulation investigates the fixation of neurones in an indigenous, neurone-less population.

Animals (large circles, red and black) eat food in their proximity (small circles) and accumulate energy (1 unit per food eaten). Animals expend a basal energetic maintenance cost for living on every time step (parameter maint-cost) and lose energy for attempting to consume food (capture-cost). They split into two identical offspring upon achieving energy Vrep.

Animals come in two breeds: deciders and idiots. Deciders have a neurone that acts as rudimentary food proximity detectors; neurones make deciders more likely to capture food when there is food nearby. Idiots have no such neurone; they simply attempt to consume food on every time step.

A decider is introduced to a stable idiot population. We are interested in the probability that that decider will drive the idiots to extinction (i.e. fixate).

To see the code that produces this simulation, click the procedures tab.

Depending on parameter values, 50 trials can take from 5 minutes (for very successful or very unsuccessful deciders) to 72 hours (for deciders that do slightly better than the idiots in a large idiot population).

## HOW IT WORKS

After clicking the 'begin-invasion' button, this is what happens.

1) Food reproduce until they reach their population cap (fixed at 1000). They reproduce after every time step with a fixed probability of .07 (the actual number affects population sizes, but not the results).

On a time step, the number of food that are introduced is binomially distributed as binom(1000 - (whatever current food population is), .07). The lower the food population, the faster they are introduced, and vice versa. This reproduction rule forces animal and food population sizes to a stable equilibrium.

2) Idiots are introduced. Assuming they make an expected energetic profit per time step, they reproduce and drive the food density down to the point where the idiots receive an expected energetic return of 0 per time step.

3) After the ecosystem achieves stable population sizes, a single decider is introduced.

The decider has an idealised sensor; food send signals out that fall as distance cubed. The decider's sensor multiplies the sum of the signals from the food by the gain, adds uniform noise, and spikes with some probability, given the food's distances from the decider. If food are far away, their signals are weaker, and the decider is less likely to attempt consumption. Though the decider saves energy on average (by not attempting food consumption when food isn't close), the neurone incurs a cost (parameter spike-cost) when it spikes.

4) When the decider population has reached 0 (neurone went extinct) or the idiot population has reached 0 (neurone fixated), we count what happened (fixation or extinction) and reset the simulation.

## HOW TO USE IT

Select the desired parameter settings, and choose the number of times you wish to run the simulation until invasion or extinction. Click 'setup', and then click 'begin-invasion.'

## THINGS TO NOTICE

The idiots always drive the food density down to a stable equilibrium value of
density = (maint-cost + capture-cost) / benefit = maint-cost + capture-cost. If the food population were not constant, then the fixation probability for the decider becomes intractable.

We can obtain a tractable expression for the fixation probability of the decider given two central assumptions:
1) The expected energetic return for the deciders is very close to 0
2) The decider attempts food consumption on almost every time step

There are two kinds of neurone: inhibitory and excitatory. If a decider's neurone is excitatory, the decider attempts food consumption when its neurone spikes. If a decider's neurone is inhibitory, the decider does not attempt food consumption when its neurone spikes. So excitatory neurones are more likely to spike when food is close, whereas inhibitory neurones are more likely to spike when food is distant. Obviously, if a decider attempts food consumption almost all the time, an inhibitory neurone (which doesn't spike very often) is a much more energetically efficient way of transmitting the same amount of energy as the excitatory neurone (which almost always spikes).

## THINGS TO TRY

Given the two central assumptions above, deciders are more likely to fixate when:
1) Capture-cost is large compared to maint-cost
2) When the food density is low (alter capture-cost and maint-cost to drive food density down)
3) When the neurone is either inhibitory or energetically cheap (spike-cost is low)
4) When the noise isn't dominating the the food's signals (when noise = 7 or so, it is very difficult for deciders to fixate)

## CREDITS AND REFERENCES

For a tractable mathematical analysis of this problem, see the manuscript.