Figure 1.
Flow diagram of the maximum entropy test.
Table 1.
Step procedure of the Monte Carlo Maximum Entropy Test.
Figure 2.
Evaluation of the maximum entropy test on artificial data.
(A) Probability mass functions of the maximum entropy distribution (left) and the Gaussian copula based distribution
(right) of the mixture distribution
with a linear correlation coefficient of
. The Poisson marginals (
) are plotted along the axes. (B) Same as A but for the mixture distribution
with a linear correlation coefficient of
. (C) Percent rejections of the null hypothesis using the entropy difference as the divergence measure. Significance level was
. Rejection rates were estimated over 100 tests. Different lines correspond to different numbers of samples drawn from the candidate distribution: 10 (red dotted line), 50 (green dash-dotted line), 100 (blue dashed line), and 200 (black solid line). (Left) Results for the
family for varying correlation coefficient
. (Center) Results for distributions from the
family (
) for varying mixture parameter
(cf. Figure 2 A). (Right) Same for
(
, cf. Figure 2 B)). Poisson rate was
for all candidate distributions (corresponding to 30 Hz and 100 ms bins). Simulated annealing [45] was applied to maximize the p-value (cf. Text S1). Number
of Monte Carlo samples was 1000.
Figure 3.
Effect of sample size on the Monte Carlo maximum entropy test results (solid black line) and on the maximum likelihood ratio test results (dashed blue line) with Poisson rate .
The entropy difference was used as a divergence measure. Significance level was . Rates were estimated over 100 trials. ( A) Percent rejections of the maximum entropy hypothesis. Data were sampled from maximum entropy distributions with random correlation strengths. ( B) Percent rejections of the null hypothesis. Data were sampled from a copula-based mixture model with uniformly random mixture parameter (cf.
, Equation 21).
Figure 4.
Effect of autocorrelations on the Monte Carlo maximum entropy test results (blue) and on the discrete Kolmogorov-Smirnov test results (red).
Interspike intervals of two concurrent spike trains were sampled from a gamma distribution with constant rate and gamma parameter
. Spike counts were calculated over subsequent 100 ms bins. The entropy difference was used as a divergence measure. Significance level was
. Rates were estimated over 100 trials. ( A) 50 spike count pairs were sampled for each test trial. ( B) 100 spike count pairs were sampled for each test trial.
Figure 5.
Illustration of the control and adaptation protocols.
Figure 6.
Results of the maximum entropy test for data recorded from area V1 of anesthetized cat.
The evaluation was performed separately for the control and adaptation conditions. ( A) Fraction of neuron pairs rejected by the Monte Carlo maximum entropy test with the entropy difference as the divergence measure () and for different bin sizes. ( B) Same as in A but using the mutual information difference. Rejection rates were averaged over all neuron pairs and all time bins. Simulated annealing [45] was applied to maximize the p-value (cf. Text S1). Number
of Monte Carlo samples was 1000. The false discovery rate of the rejections was corrected using the Benjamini-Hochberg procedure [35].
Figure 7.
Subpopulation analysis of the data that are presented in Figure 6 C.
(A) Overall firing rates of the 11 neurons in the data set from Figure 6 for the control and adaptation conditions. The rates were averaged over all stimuli. ( B) Fraction of neuronal pairs rejected by the maximum entropy test with the mutual information difference as the divergence measure () for the high firing rate (
, cf. A) population of neurons. ( C) Same as in B but for the low firing rate population (
). Rejection rates were averaged over all neuron pairs and all time bins. Simulated annealing [45] was applied to maximize the p-value (cf. Text S1). Number
of Monte Carlo samples was 1000. The false discovery rate of the rejections was corrected using the Benjamini-Hochberg procedure [35].