Benchmarking según escenario
4.1.2 Requisitos de Diseño
In this section, we extend the model 5.1 in two aspects: (1). We replace the linear function for the peer firing vk, k = 1, ...n, by a quadratic polynomial; (2). We trace
Figure 5.27: The coverage proportion increases as we double the sample size, and the coverage eventually converges.
Figure 5.28: Density of the standard error. The red line is 0.131, which is the standard error we got from the real data.
the target. These extensions will make the model more flexible to the real situation, and we will show an example later.
By replacing the linear function β1[K−(t−vk)]1t≥vk with a quadratic function
β10(t−vk)1t≥vk+β11(t−vk)
21
t≥vk, we also relaxed the conditions of the preset value
of K, which sometimes could be unexplainable. Then for the single-peer case, the model is α(t|vk) = p X i=1 θiBi(t) +β10(t−vk)1t≥vk+β11(t−vk) 21 t≥vk.
When multiple peer firings are considered, the peer effects are additive as
α(t|vk) = p X i=1 θiBi(t) +β10(t−v(1)k )1t≥v(1) k +β11(t−vk(1))21t≥v(1) k +β20(t−v (2) k )1t≥v(2)k +β21(t−v (2) k ) 21 t≥vk(2) +.... (5.5)
Similar to Data Analysis I, we will discuss the single-peer firing case first. The analysis is done on the same data set as before. From Figure 5.30 we can see that the influence of spk3 on spk1 is large in magnitude but does not last long, while the other peers have weaker but longer effects. We also checked on all the other setups. When each one from spk2 to spk16 is treated as the target, spk3 always shows the strong but short-term effect which is quite different from the rest.
Due to the nature of neuron 3, we may lose some information when we take only the most recent firing of spk3 as the covariate. To illustrate an example by taking
spk1 as the target and spk3 as the peer, we trace back five firings of spk3 which occur within the ISI of the target. The reason we use five firings ofspk3 is out of the sequential selection according to AIC which will be explained in the coming section. If there are fewer than five firings, then some terms will be zero and so have no effect
Figure 5.29: Quadratic functions of peer effects. The red solid curve is for spk3, while the black curves are for the other peers.
spk1 when spk3 fires at 0.046375s, 0.110450s, 0.181725s, 0.221050s and 0.245200s
(values from the real data). We can see that those five firings play different roles from the change of curve. The result provides realtime guidance for how spk3 affects the firing probability of spk1.
Model Selection Criteria for the Number of Peer Firings
For some spike trains such as spk3 in our data set which have much higher firing frequency, we may need to take multiple firings of the spike train into account rather than only the most recent one. So, we use AIC to select the appropriate number of firings from a sequence of choices for those special cases. The idea is that we trace back for a multiple number of firings starting from the most recent one within each ISI of the target, and then we obtain sequential models with AIC =−2(log−likelihood) + 2k, where k is the number ofθ’s and β’s in the model. Let us take spk3 as an example, and here spk1 is its target. From Figure 5.32, we can see that the AIC is rather flat
Figure 5.30: Quadratic functions of peer effects. The red solid curve is forspk3, and the black curves are for the other peers.
Figure 5.31: Fitted function for peer effect. The peer, spk3, fires at 0.046375s, 0.110450s, 0.181725s, 0.221050s and 0.245200s.
in the region from six and above, so the best model we choose includes five firings of
spk3.
We also conducted a simulation study to validate whether this model selection can pick the appropriate number for us.
The way we generate the data indicates a high probability of the target firing after every two close firings of the peer. Without the peer effects, it barely occurs. The model is ultimately as Equation 5.5, with β10 = 7.2, β11 = −6.2, β20 = 4.5, β21 = −2.3. As we discussed earlier, the generation of the spike train is a Bernoulli
process. With the setup, the spike occurrence follows our design in most cases, but there is still a small portion of data generated which may not be associated with two peer firings exactly. It could be more or less as will see from the results. Figure 5.33 shows the average curves from 25 repetitions, and both AIC and BIC are calculated for simulated data. Overall, the correct model is picked by both the criteria. We
Figure 5.32: Sequential selection for number of firings using AIC. The x-axis is the number of peer firings.
Figure 5.33: The average AIC (left panel) and BIC (right panel) curves from 25 repetitions. The x-axis is the number of peer firing.
also noticed that the difference between number 2 and 3 is not significant, and it is actually supported by the generated data. As we explained earlier, a small portion of the generated data may not exactly follow the design as our simulation.