Simulation Results

Suppose that we have a population of $N=40$ neurons, and suppose that the neurons are firing according to Poisson model with dead time $\triangle=2$ ms (this can be taught of as absolute refractory period, i.e. the time interval following a spike in which neuron is unable to fire an action potential). Let us assume that the 40 cells within the population have their mean firing rates as shown by Fig. 2.

Figure 2: Tuning curves.

Figure 3: Raster plots.

These curves could be experimentally obtained by averaging the response of individual neurons to a sequence of constant stimuli of different magnitude. In particular, the tuning curves from Fig. 2 are obtained as raised cosine functions

$$\Lambda_j(x)=k_j\,\cos(x-x_j)+c_j \qquad j=1,2,\cdots,40$$ (11)

where $x\in[0,\,2\pi]$, $x_j$ is the preferred direction of $j-$th neuron and $k_j$ and $c_j$ are constants that determine the height and the width of the tuning curves. For simplicity we take $k_j=c_j$ and we draw $x_j$ from a uniform distribution over $[0,\,2\pi]$. Note that our maximum firing rate in the population does not exceed 24 spikes/s, which should make decoding process more challenging.

In the spirit of detection problem defined by equation (2), let us assume that $x$ takes its values from the set $\mathcal{X}=\{0,\,\pi/4,\,\pi/2,\cdots,\,7\,\pi/4 \}$ ($M=8$). For each $x\in\mathcal{X}$ and for each neuron from the population, the mean firing rate $\lambda$ is calculated according to (11), and sequences of spikes are generated using Poisson generator with dead time $\triangle=2$ ms. The realization of one such collection of random processes is given by Fig. 3. The raster plots from Fig. 3 correspond to one of the eight inputs from $\mathcal{X}$. Given a conditional response, such as the one shown by Fig. 3, our goal is to estimate the most likely value of the input that elicited such response. Since we know what inputs are indeed behind every response, we can use this knowledge for cross-validation of our results. The decoding is performed according to (7).

The results of the decoding procedure across 8 inputs are shown in Fig. 4. The top plot in each subfigure shows the relative frequency of decoded directions. It is not surprising that the decoded input changes in time, despite the fact that the encoded input is constant in time. However, the decoded value stabilizes after some time, and does not change any more. The settling time is different for different inputs, e.g. 300 ms for input 4 ( $x_4=3\,\pi/4$). The middle plot in each subfigure shows the traces of decoded inputs as a function of time. Input 7 is decoded with a 100$\%$ accuracy, i.e. this input emmerges as dominant (most likely) for all times. The bottom image in each subfigure shows the color coded likelihood of each input. The colorbar indicates that the likelihoods are normalized between 0 and 1.

Figure 4: Detection results across 8 inputs.