Ensemble inhibition and excitation in the human cortex: An Ising-model analysis with uncertainties

The pairwise maximum entropy model, also known as the Ising model, has been widely used to analyze the collective activity of neurons. However, controversy persists in the literature about seemingly inconsistent findings, whose significance is unclear due to lack of reliable error estimates. We therefore develop a method for accurately estimating parameter uncertainty based on random walks in parameter space using adaptive Markov-chain Monte Carlo after the convergence of the main optimization algorithm. We apply our method to the activity patterns of excitatory and inhibitory neurons recorded with multielectrode arrays in the human temporal cortex during the wake-sleep cycle. Our analysis shows that the Ising model captures neuronal collective behavior much better than the independent model during wakefulness, light sleep, and deep sleep when both excitatory (E) and inhibitory (I) neurons are modeled; ignoring the inhibitory effects of I neurons dramatically overestimates synchrony among E neurons. Furthermore, information-theoretic measures reveal that the Ising model explains about 80–95% of the correlations, depending on sleep state and neuron type. Thermodynamic measures show signatures of criticality, although we take this with a grain of salt as it may be merely a reflection of long-range neural correlations.

Figure 1

Schematic representation of experimental data. (a) The spike train response of a set of 92 neurons in the human temporal cortex. (b) Discretization of the region delimited by the red rectangle into time bins of width $\mathrm{\Delta }t=50\text{ms}$. (c) The raster corresponding to the discretization in panel (b), where red (${\sigma }_i=+1$) denotes spiking of neuron $i$, and blue (${\sigma }_i=-1$) represents silence.

Figure 2

Maximum entropy model for the population of 92 neurons in the awake state. (a) The mean spiking activity $m_i=\langle {\sigma }_i\rangle$. (b) The bias terms $h_i$ of the inferred model, with a mean error of 0.01. Note that in the presence of interactions, the relationship between $h_i$ and $m_i$ no longer follows the closed form given in Eq. (2.6). (c) The correlation coefficients between pairs of neurons $C_{ij}=\langle {\sigma }_i{\sigma }_j\rangle -\langle {\sigma }_i\rangle \langle {\sigma }_j\rangle$. The inset shows the population distribution of the correlation coefficients. (d) The pairwise coupling terms $J_{ij}$ of the inferred model, with a mean error of 0.004. The inset shows the population distribution of the pairwise couplings. Note that panels (c) and (d) only show the lower triangle of the symmetric matrices $C$ and $J$. In all panels, neurons are ordered by decreasing spiking frequency.

Figure 3

Cosine similarity between all pairs of 10 parameter vectors $\mathbf{\theta }$ estimated from 10 independent runs of our algorithm, when keeping only the $N^{\prime }$ most active neurons. Higher values of the cosine similarity indicate a smaller variance in parameters from run to run, so we see that the parameters of more active neurons can be more reliably measured. Our threshold for reliable neurons (horizontal line) is set at 0.25.

Figure 4

Model predictions for the network of 19 reliable neurons during the awake state. (a) The probability of occurrence of each activity pattern predicted by the maximum entropy model [$P^{\left(1\right)}(\mathbf{\sigma },\mathbf{h})$ or $P^{\left(2\right)}(\mathbf{\sigma },\mathbf{\theta })$] is plotted against the observed pattern frequency from measured data $\left[P^{\left(N\right)}\left(\mathbf{\sigma }\right)\right]$. The black line corresponds to prediction matching observation. The apparent stratification on the lower end of the spectrum is due to the finite size of the data set. (b) The predicted and observed distribution of the number of simultaneously spiking neurons in each time bin. Error bars are asymmetrical because of the logarithmic scale. Similar results hold across all states, timescales, and data sets.

Figure 5

The pairwise coupling matrix $J$ inferred from the population of reliable neurons in the deep sleep state. Neurons are sorted by type, with (a) inhibitory neurons at positions 1–13 and excitatory neurons at positions 14–23 for data set 1, (b) inhibitory neurons at positions 1–15 and excitatory neurons at positions 16–27 for data set 2. The couplings $J_{ij}$ among inhibitory neurons, as well as the couplings among excitatory neurons (diagonal blocks) are almost entirely positive. The couplings $J_{ij}$ between inhibitory and excitatory neurons (off-diagonal blocks) display a mix of both positive and negative values. Similar results hold across all states, timescales, and data sets.

Figure 6

Difference between E and I neurons. The probability of occurrence of each activity pattern predicted by the maximum entropy model [$P^{\left(1\right)}(\mathbf{\sigma },\mathbf{h})$ or $P^{\left(2\right)}(\mathbf{\sigma },\mathbf{\theta })$] is plotted against the observed pattern frequency from measured data $\left[P^{\left(N\right)}\left(\mathbf{\sigma }\right)\right]$ for a network of (a) 14 E neurons and (b) 23 I neurons. The black line corresponds to prediction matching observation. The predicted and observed distribution of the number of simultaneously spiking neurons in each time bin for a network of (c) 14 E neurons and (d) 23 I neurons. Error bars are not symmetrical because of the logarithmic scale. All plots are for the deep sleep state, but similar results hold across all states, timescales, and data sets.

Figure 7

Pearson correlation coefficients $R_{ij}$ between the same set of (a) 14 (data set 1) and (b) 18 (data set 2) reliable neurons during awake and deep sleep states. The coefficients $R_{ij}$ are labeled based on the type of neurons $i$ and $j$. The correlations are different during the two states, with I-I correlations being predominantly larger during deep sleep.

Figure 8

Heat capacity and entropy in maximum entropy models. (a) Heat capacity per neuron $C\left(T\right)/N$ as a function of temperature for different types of neurons in the awake and SWS states. The heat capacity peaks around $T=1$ (vertical dotted line). (b) Entropy per neuron $S\left(T\right)/N$ as a function of temperature for different types of neurons in the awake and SWS states. The entropy experiences a jump around $T=1$ and saturates to its maximum value $S\left(T\right)/N=1$ (horizontal dotted line) at high temperatures. Both panels indicate signatures of criticality. Errors are displayed as shaded regions around the main lines. Similar results hold across all time bins and data sets.