Late penalty: 1 mark per day
You can use any programming language you like, but it is recommended that you use a language with a matrix library and graphing capabilities. Two main suggestions are Python and MATLAB.
As you did in previous assignments, make a population of 20 LIF neurons representing a 1-dimensional value, and compute a decoder for them. For parameters, $\tau_{ref}$=0.002s, $\tau_{RC}$=0.02s, the maximum firing rates are chosen randomly from a uniform distribution between 100 and 200Hz, and the x-intercepts are chosen randomly from a uniform distribution between -2 and 2. Remember that the $\alpha$ and $J^{bias}$ terms are computed based on these x-intercepts and maximum firing rates.
It is generally easiest to compute decoders using the original method from Assignment 1, where we use the rate-mode approximation for the neurons to generate the $A$ matrix, then find $\Gamma=A^T A /S + \sigma^2 I$. You can use this approach to find decoders, and these decoders should work even when you simulate the neurons in terms of spikes. The only difference will be that they will need to be scaled by dt, your simulation time step.
You can use this same method for computing decoders for this whole assignment.
Choose a neuron from part 1 that has a firing rate of somewhere between 20-50Hz for $x$=0. Using that neuron's $\alpha$ and $J^{bias}$ value, construct two neurons: both with the same $\alpha$ and $J^{bias}$, but one with $e$=+1 and the other with $e$=-1. With the function from the last assignment, generate a random input $x(t)$ that is 1 second long, with rms=1, dt=0.001, and an upper limit of 5Hz. Feed that signal into the two neurons and generate spikes. Decode the spikes back into $\hat{x}(t)$ using a post-synaptic current $h(t)$ with a time constant of $\tau$=0.005.
Repeat question 2, but with more neurons. Instead of picking particular neurons, randomly generate them with x-intercepts uniformly distributed between -2 and 2 and with maximum firing rates between 100 and 200 Hz. Randomly choose encoder values to be either -1 or +1.
For this question, use two groups of neurons to compute $y = 2x+1$. The first group of neurons will represent $x$ and the second group will represent $y$.
Start by computing decoders. You will need two decoders: one to decode $f(x)=2x+1$ from the first population, and one to decode $f(y)=y$ (the standard decoder) from the second population. Remember that $\Upsilon$ changes depending on what function you want to decode.
Use the same neuron parameters as for previous questions, and use 200 randomly generated neurons in each population.
a) [1 mark] Show the behaviour of the system with an input of $x(t)=t-1$ for 1 second (a linear ramp from -1 to 0). Plot the ideal $x(t)$ and $y(t)$ values, along with $\hat{y}(t)$.
b) [0.5 marks] Repeat part (a) with an input that is ten randomly chosen values between -1 and 0, each one held for 0.1 seconds (a randomly varying step input)
c) [0.5 marks] Repeat part (a) with an input that is $x(t)=0.2sin(6\pi t)$
For this question, use three groups of neurons to compute $z = 2y+0.5x$. Follow the same steps as question 4, but take the decoded outputs from the first two groups of neurons ($f(y)=2y$ and $f(x)=0.5x$), add them together, and feed that into the third group of neurons.
a) [1 mark] Plot $x(t)$, $y(t)$, the ideal $z(t)$, and the decoded $\hat{z}(t)$ for an input of $x(t)=cos(3\pi t)$ and $y(t)=0.5 sin (2 \pi t)$ (over 1.0 seconds)
b) [0.5 marks] Plot $x(t)$, $y(t)$, the ideal $z(t)$, and the decoded $\hat{z}(t)$ for a random input over 1 second. For $x(t)$ use a random signal with a limit of 8 Hz and rms=1. For $y(t)$ use a random signal with a limit of 5 Hz and rms=0.5.
Do the same thing as questions 4 and 5, but with 2-dimensional vectors instead of scalars. Everything else is the same. For your encoders $e$, randomly generate them over the unit circle.
The function to compute is $w = x-3y+2z-2q$. This requires five groups of neurons: $x$, $y$, $z$, $q$, and $w$. Each of them represents a 2-dimensional value. The outputs from $x$, $y$, $z$, and $q$ all feed into $w$.