Figure 1.
Modeling a spike-count distribution.
(A) Normalized empirical distributions of spike-counts from a pair of neurons recorded in macaque prefrontal cortex (see Section “Materials and Methods”). The bin size was . Gray values of the squares denote the number of occurrences of pairs of spike-counts (dark to bright corresponding to low to high, see scale bar). The corresponding marginals are plotted below and left of the coordinate axes. The distribution is based on
occurrences. (B) Joint distribution and marginals of the discretized and rectified multivariate normal distribution with the mean and covariance matrix set to the sample mean and sample covariance matrix. (C) Joint distribution and marginals of the best fitting Clayton copula (see Section “Multivariate Spike-Count Distributions Based on Copulas”, parameter:
) and negative binomial marginals (parameters:
).
Figure 2.
Bivariate copula probability densities of commonly used families.
(A) Clayton copula (). (B) Gumbel-Hougaard copula (
). (C) Frank copula (
). (D) Ali-Mikhail-Haq copula (
). (E) Farlie-Gumbel-Morgenstern copula (
).
Table 1.
Five commonly used Copula families.
Figure 3.
Probability densities of four different orthant dependencies generated by the flashlight transformation.
The original distribution was the bivariate Clayton copula (parameter ). The transformation takes a set
as a parameter which contains the indices of the elements that are transformed. (A) Original Clayton copula, which is also recovered for
. (B) Element
is transformed (
). (C) Element
is transformed (
). (D) Both elements are transformed (
).
Figure 4.
Deviation of the estimated likelihood from the likelihood for different dependence strengths.
The deviation is given in percent of the likelihood. Samples were drawn from a Clayton-copula model with negative binomial marginals. The marginals were parametrized by maximum likelihood estimates obtained on the entire data that is described in Section “Multi-Tetrode Recordings”. The vertical axis indicates the number of samples in the training set. The evaluation took place on a separate set of samples. Above the black line the deviation is smaller than
. (A) Correlation coefficient
. (B) Correlation coefficient
. (C) Correlation coefficient
.
Figure 5.
Copula-based analysis of bivariate spike-count data.
(A1–A4) Neural network models used to generate the synthetic spike-count data. Two leaky integrate-and-fire neurons (“LIF1” and “LIF2”, see Section “Materials and Methods”) receive spike inputs from three separate populations of neurons (rectangular boxes and circles), but only one population sends input to both of the neurons. All input spike trains were Poisson-distributed. Each neuron had a total inhibitory input rate of . We had three times as many excitatory spikes as inhibitory spikes. We increased the absolute correlation between the spike-counts by shifting the rate of the left and right populations to the center population. The center population was active in half the simulation time. The total simulation time amounted to
. Spike-counts were calculated for
bins. (B) Empirical distribution for the model with an inhibitory input population (see A3) obtained for
bins and a correlation coefficient of
. (C1–C4) Log likelihoods of the best fitting Clayton copulas with negative binomial marginals as a function of the strength of the input correlation. Plots shown (C1
C4) correspond to the four different network models (A1
A4). Dotted, dashed, solid, and dashed-dotted lines correspond to the best fitting Clayton copula with lower, lower-right, upper-left, and upper orthant dependence (see Figure 3). Copulas were fitted using the IFM estimators.
Figure 6.
Log likelihoods of the best fitting MVN, Poisson latent variables, and copula-based models for the validation set.
(A) Log likelihoods for the discretized multivariate normal distribution (circles), the multivariate Poisson latent variables distribution (crosses), the best fitting copula-based model with Poisson (squares), and with negative binomial marginals (diamonds). The figure shows the log likelihoods averaged over all different stimuli, but separately for the pre-stimulus, sample stimulus, delay, and test stimulus phase of the memory task. For the best fitting copula, we considered all the copula families shown in B. AMH denotes the Ali-Mikhail-Haq family, FGM the Farlie-Gumbel-Morgenstern family (see Table 1). For the
order model of the FGM family we set all but the first
parameters to zero, therefore leaving only parameters for pairwise interactions. In contrast, for the
order model we set all but the first
parameters to zero. (B) Difference between the log likelihood of a model with independent spike-counts and negative binomial marginals (“ind. model”) and the log likelihoods of the best fitting representatives of the different copula-based models shown in the legend. Negative binomial marginals were used. Data was again averaged over the
different stimuli.
Figure 7.
Log likelihoods of different Clayton-copula models transformed using the flashlight transformation.
(A) Cartoon indicating the labeling of orthants for the six dimensional space. Each number indicates the orthant, into which the originally lower tail dependence was transformed. (B) Mean log likelihoods on the test interval validation set for all possible flashlight transformed Clayton copulas and negative binomial marginals. The bars mark the standard errors. (C) Mean log likelihoods on the test interval validation set for a mixture of the Clayton copula with all possible flashlight transformed Clayton copulas and negative binomial marginals.
Figure 8.
Distribution of log likelihoods from models fitted to data from the sample stimulus phase.
The Clayton-copula model with different marginals was used. A histogram of samples is shown where each sample represents an average over
spike-count vectors. The solid line corresponds to the log likelihood of the training set whereas the dashed line corresponds to the log likelihood of the validation set. (A) Model with Poisson marginals. (B) Model with negative binomial marginals. (C) Model with empirical marginals.
Figure 9.
Monte Carlo estimates of the mutual information between stimuli and responses.
The estimation is based on the Clayton-copula model with negative binomial marginals. The Monte Carlo method was terminated when the standard error was below . The sample stimulus was presented in phase two, whereas the test stimulus was presented in phase four. For the test stimulus phase, the estimation was performed twice: for the sample stimulus that was previously presented (dashed line) and for the test stimulus (dotted line). (A) Estimated mutual information based on IFM parameters determined on the training set for each of the task phases (pre-stimulus, sample stimulus, delay, and test stimulus). (B) Estimated information increase that is due to the dependence structure. The mutual information
of the model with independent spike-counts and negative binomial marginals was subtracted from and normalized to the mutual information
of the Clayton-copula model with negative binomial marginals.