Hénon Map MATLAB

In my studying of dynamical systems I stumbled upon the Hénon Map today, a two-dimensional dynamical system with a strange attractor which demonstrates chaotic behaviour. The system is described by a coupled pair of equations. Here’s what the classical version looks like:

Classic Hénon

The classical Hénon Map.

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

Hénon

A variation of the Hénon Map

Hénon Map

Another variation of the Hénon Map

Another variation of the Hénon Map

Yet another variation of the Hénon Map

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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Liapunov plot in MATLAB

I seem to do a lot of MATLAB coding… In a paper I’m studying I came across the following:

“… 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:

Liapunov plot

Plot of a simple Liapunov function

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')

Lotka-Volterra MATLAB model

I’m starting to play with dynamical systems so I figured I’d post a baby model. It essentially shows the growth of two populations co-existing together, one being the prey, the other the predators. A small time step (dt) shows that the system is stable; a larger one leads to instability and thus highlights the importance of parameter choice.

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

Population growth with small changes in time.

Population growth with small changes in time.

Plotting the two populations in the phase plane yields this:

Phase plane for the system.

Phase plane for the system.

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

Population growth with large changes in time.

Population growth with large changes in time.

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

Phase plane for the system.

Phase plane for the system.

As always, the code is over on my GitHub.

Drift Diffusion MATLAB Model

Just spent a good eight/nine hours redesigning my website from the ground up. I’m happy with the result (for a few months at least), so figured I’d take a little break and make a blog entry. I finally got round to implementing the drift diffusion model a few weeks back, one of the classical models of decision making. This is essentially the random walker with a bias, or “drift” towards one of the choices (in a two alternative forced choice task). This is one of the results from a simulation:

Drift diffusion model implemented in MATLAB.

Drift diffusion model implemented in MATLAB.

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. 

leaky integrate and fire MATLAB model

LIF is considered as one of the simpler spiking neuron models, and I’ve been doing a bit of work with it in MATLAB:

Leaky intergrate-and-fire graph

Leaky intergrate-and-fire graph

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.

the placement ends

Today was the final day of my placement working with Zeno. I’m really looking forward to seeing how he’ll progress in the coming months; and I will most definitely visit him again when I return to Sheffield. My intern has ended with beautiful skies, being treated to lunch and finding out that I’ll be graduating this year with a first class degree. A lovely day to say the least!

Picture of Sheffield

Sheffield in the sun.

working with robots

I’ve had a really good few days at work this week; there’s been lots of progress with setting up our humanoid robot, ‘Zeno’ , and he performed very well for the department visitors yesterday. So, I figured that today I’d buy him some stacking cups! Hopefully others from the department will donate some toys as well. I also had the chance to pet Paro, the robot seal – who may or may not know his name yet (we couldn’t really tell).

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.