There are only two parameters, but playing around with them yields some interesting results:

Again, the code to implement this is pretty simple, so I’ll post it here as opposed to GitHub (though I’ll probably post a version on GitHub which draws the system state after each iteration):

a = 1.4;

b = 0.3;

iterations = 10000;

x = zeros(1, iterations);

y = zeros(1, iterations);

% simulation

for i=1:iterations

x(1,i+1) = 1-(a*x(1,i)^2) + y(1,i);

y(1,i+1) = b*x(1,i);

end

% plotting

plot(x(5:iterations),y(5:iterations), '.k','MarkerSize',3)

line1 = sprintf('Henon Map with %.0f iterations', iterations);

line2 = sprintf('a = %.2f, b = %.5f', a, b);

title({line1, line2});

xlabel('x')

ylabel('y')

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“… in which case it is a dissipative system possessing a Liapunov function … thus, released from an initial (non-equilibrium) state, typical solutions approach asymptotically stable fixed points (sinks).”

(Brown & Holmes (2001), *Modelling a simple choice task: Stochastic dynamics of mutually inhibitory neural groups*, Stochastics and Dynamics, 1:2 (2001), 159-191))

So I’ve been doing a bit of background reading on system stability (in linear systems) and Liapunov functions, which can be used to prove the stability of an equilibrium of an ordinary differential equation. I plotted a simple one in MATLAB:

There isn’t really a lot of code behind it, so I’ll just post it here (I also might update this later with slightly more complicated functions):

`x = [-1:.05:1];`

y = x;

[x,y] = meshgrid(x,y);

z = x.^2 + y.^2;

figure(1)

colormap(bone)

surf(x,y,z)

title('Liapunov equation: f(x) = x^2 + y^2')

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Small dt example (populations oscillate in a stable manner):

Plotting the two populations in the phase plane yields this:

Large dt example, where the population growth is without bounds:

Plotting the two populations in the phase plane yields something quite different:

As always, the code is over on my GitHub.

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As always, the code that produced this is available over at my GitHub. I’ll be attempting the pooled inhibition model shortly, as well as investigating some more focused behavioural experiments with honeybees and *drosophila. *

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This is the result:

The code that produced this is over at my GitHub.

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The graph shows the membrane potential of a neuron over time. There’s an inhibitory current injection at 0.04 seconds, followed by an excitatory injection at 0.05 and 0.09 seconds. The second current injection causes the voltage to exceed a set threshold (shown by the red line), and so an activation potential is generated (the neuron spikes). A final excitatory injection is given at 0.11 seconds, however this is within the time of the absolute refractory period: although the voltage crosses the threshold again, the neuron doesn’t spike.

The code is over on my GitHub.

Feng, J. (2001). Is the integrate-and-fire model good enough?–a review. Neural networks : the official journal of the International Neural Network Society, 14(6-7), 955–75.

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I’ll spend a bit more time soaking up the sun before my PhD starts; and reading papers of course.

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However, upon testing his grasp today it seems as though his right hand isn’t as healthy as it should be. For the time being I’ll be getting him to pick things up with his left hand.

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I only have one more week left to work with Zeno before I head home for the Summer; this intern has flown by really quickly but I’ve had lots of fun.

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