Model orders selected by different information criteria for different numbers of trials.

The state-space log-linear models of different orders () are applied to samples of the spike sequences of repeated trials generated from a time-dependent full log-linear model (indicated by the dashed lines in Figure 4C). Three data-driven information criteria, AIC, BIC, and PDIO, are computed for the fitted state-space models of the different orders, . The count for the model order that minimizes the respective criteria is determined by repeating the process for repetitions as in Table 3. The most frequently selected model order, , is displayed for each of the information criteria and for the different numbers of trials, . For comparison, we also show the order of interactions that minimizes the KL risk function (KL-risk) and mean squared error (MSE). We approximated the KL-risk and MSE as follows. At each bin, we compute the KL-divergence (Eq. 21), between a full underlying log-linear model of neurons and the estimated log-linear model whose parameters are given by the MAP estimates of the th-order model. The total sum of the all KL-divergences from bins is used as the distance between the two (time-dependent) models: i.e., , where the function is given in Eq. 21. represents the underlying log-linear parameters used to generate the data. is its estimate from one sample composed of trials. The parameters higher than the th-order that are not included in the model are set to zero. The KL-risk function is estimated as the average of the KL-divergences of realizations of the spike data, each composed of trials. To obtain the MSE, we first computed the sum of the squared errors (SE) : , using one sample composed of trials. The MSE is then estimated as the average of the SEs over 100 samples.