Maximum-entropy models reveal the excitatory and inhibitory correlation structures in cortical neuronal activity

Maximum entropy models can be inferred from large datasets to uncover how collective dynamics emerge from local interactions. Here, such models are employed to investigate neurons recorded by multi-electrode arrays in the human and monkey cortex. Taking advantage of the separation of excitatory and inhibitory neuron types, we construct a model including this distinction. This approach allows us to shed light on differences between excitatory and inhibitory activity across different brain states such as wakefulness and deep sleep, in agreement with previous findings. Additionally, maximum entropy models can also unveil novel features of neuronal interactions, which are found to be dominated by pairwise interactions during wakefulness, but are population-wide during deep sleep. Overall, we demonstrate that maximum entropy models can be useful to analyze datasets with classified neuron types and to reveal the respective roles of excitatory and inhibitory neurons in organizing coherent dynamics in the cerebral cortex.

Figure 1

Multi-electrode (Utah) array recordings. (a) Utah array position in human temporal cortex (top) and monkey prefrontal cortex (bottom). Figure is adapted from Ref. [13]. (b) Raster plots of spikes recorded for human (top) and monkey (bottom) in wakefulness (left) and SWS (right). Neurons are ordered to separate excitatory (E) from inhibitory (I) cells.

Figure 2

Pairwise Ising model fails to predict SWS synchronous activity, especially for inhibitory neurons. (a) Model schematic diagram. Model parameters are each neuron's bias toward firing, and symmetric functional couplings between each pair of neurons. (b) Empirical and predicted probability distributions of the population activity $K={\sum }_i{\sigma }_i$ for the neuronal population. The Ising model more successfully captures the population statistics during wakefulness than SWS, especially for medium and large $K$ values. (c) Empirical and predicted population activities for E (lower curves, in green or dark gray) and I (upper curves, in red or light gray) neurons. The model particularly fails at reproducing the statistics of I population activity. These results are consistent with the presence of transients of high activity and strong synchrony between I neurons during SWS. Insets show an enlarged view on the region of low population activity, region within which the system spends the vast majority of the time (on a linear scale).

Figure 3

Neural firing is tuned to the neural population's activity, particularly during SWS. (a) Tuning curves of ten example neurons (see text and Appendix pp3) showing that neurons are tuned to the rest of the population's activity. (b) Scatter-plot of the excitatory (green triangles) and inhibitory (red circles) neuron sensitivity to the population activity (see Appendix pp3). Neurons are very consistently more sensitive during SWS ($p<0.001$, Wilcoxon sing-ranked test).

Figure 4

Single-population model shows better performance during SWS than wakefulness. (a) Model schematic diagram. (b) Pairwise covariances, empirical against predicted, for wakefulness (left) and SWS (right) states. Consistently with Fig. 3, the success for SWS, most noticeably for I-I pairs (red circles), suggests that these neurons are most responsive to whole-population activity. Inset shows an enlargement of the small-correlation region.

Figure 5

I neurons are more specifically tuned to the I population during SWS. (a) Example tuning curves from ten neurons of each type to each type of population during SWS, and similarly for the I population. (b) Scatter-plot of neuron sensitivity to E versus I population, during SWS. I neuron (red circles) are more tuned to I population than the E population ($p<{10}^{-3}$, Wilcoxon sign-ranked test). E neurons (green triangles), instead, are weakly sensitive to both populations.

Figure 6

Two-population model shows significant improvement in prediction for all types of neurons. (a) Schematic diagram of the two-population model. Parameters $h_{i{K}^E}^E,h_{i{K}^I}^I$ are the couplings between each neuron $i$ and the E population activity $K^E={\sum }_{i\in E}{\sigma }_i$ and the I population activity $K^I={\sum }_{i\in I}{\sigma }_i$. (b) Pairwise covariances, empirical against predicted, for the two-population model, during SWS. Improvement compared with the whole-population model is confirmed by the Pearson correlations. (c) Deterioration of prediction by shuffling neuron types for the human and monkey datasets. This effect demonstrates that knowledge of neuron types significantly contributes to improving model prediction. This is confirmed by the Mann–Whitney $U$ test $p$ values. Inset shows an enlargement of the small-correlation region.