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. 2008 Jan 9;28(2):505-18.
doi: 10.1523/JNEUROSCI.3359-07.2008.

A Maximum Entropy Model Applied to Spatial and Temporal Correlations From Cortical Networks in Vitro

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Free PMC article

A Maximum Entropy Model Applied to Spatial and Temporal Correlations From Cortical Networks in Vitro

Aonan Tang et al. J Neurosci. .
Free PMC article

Abstract

Multineuron firing patterns are often observed, yet are predicted to be rare by models that assume independent firing. To explain these correlated network states, two groups recently applied a second-order maximum entropy model that used only observed firing rates and pairwise interactions as parameters (Schneidman et al., 2006; Shlens et al., 2006). Interestingly, with these minimal assumptions they predicted 90-99% of network correlations. If generally applicable, this approach could vastly simplify analyses of complex networks. However, this initial work was done largely on retinal tissue, and its applicability to cortical circuits is mostly unknown. This work also did not address the temporal evolution of correlated states. To investigate these issues, we applied the model to multielectrode data containing spontaneous spikes or local field potentials from cortical slices and cultures. The model worked slightly less well in cortex than in retina, accounting for 88 +/- 7% (mean +/- SD) of network correlations. In addition, in 8 of 13 preparations, the observed sequences of correlated states were significantly longer than predicted by concatenating states from the model. This suggested that temporal dependencies are a common feature of cortical network activity, and should be considered in future models. We found a significant relationship between strong pairwise temporal correlations and observed sequence length, suggesting that pairwise temporal correlations may allow the model to be extended into the temporal domain. We conclude that although a second-order maximum entropy model successfully predicts correlated states in cortical networks, it should be extended to account for temporal correlations observed between states.

Figures

Figure 1.
Figure 1.
Data collection and representation. A, Left, Dissociated rat cortical neurons cultured on 60-channel microelectrode array. Several neurons can be seen around an electrode tip, which appears as a black circle. Overall array is similar to the grid shown in C below and has an interelectrode distance of 200 μm. Right, Spike waveform recorded from dissociated culture. Signals that crossed a threshold were cut out and later sorted. B, Left, Organotypic slice culture of rat cortex on high-density array with 60 μm interelectrode distance. Electrodes are visible as small black dots in a hexagonal lattice. Right, Spike waveform from organotypic culture. Full waveform was stored and later sorted. C, Left, Acute slice of human cortex removed from peritumoral region, placed on 60-channel microelectrode array. Right, LFP waveform from acute slice. All signals that crossed a threshold were recorded as events. D, Raster plot of data show gray boxes for the three time bins used: 20, 4, and 1.2 ms. All data were binned at 20 ms, and then also at one finer resolution (4 ms for 200 μm array; 1.2 ms for 60 μm array). Dots represent spikes from individual neurons or LFP negative peaks from individual electrodes. A correlated pattern occurs when events from several neurons/electrodes appear in one time bin.
Figure 2
Figure 2
The Ising model of neural interactions. A, Harmony between three neurons (σa, σb, σc), their local fields (−h, +h), and their interactions (−Jab, −Jbc, +Jac). Each neuron is represented by a circle, and the direction of the arrow inside depicts whether the neuron is firing a spike (up) or not (down). The gray region surrounding neuron σb has many arrows pointing up, indicating the positive local field +h. The white region surrounding neurons σb and σc has many downward pointing arrows, indicating the negative local field −h. When the arrow of a neuron is pointing in the same direction as its local field, it is in harmony with that field. Neurons in A are also in harmony with their interactions, represented by the solid black lines joining the neurons. A positive interaction like +Jac tends to make neurons share the same state (both up, or both down), whereas a negative interaction like −Jab tends to make neurons have different states (one up, one down). B, When neuron σa changes its state and fires a spike, it introduces frustration. Now σa has an arrow pointing against its local field and against its interactions with neurons σb and σc. To denote this frustration, the parameters −h, −Jab, and +Jac are displayed in a different font. C, Harmonious states in the Ising model are lower in energy and more probable than frustrated states. The exponential probability distribution maximizes entropy, given the constraints of the local fields and the interactions. The Z in the equation just ensures that the probability distribution sums to one.
Figure 3.
Figure 3.
The maximum entropy model successfully predicts correlated states for many in vitro preparations. Figures compare the abundance of correlated states from data and models. The predicted rate is plotted against the observed rate. Correlated states from actual data would lie along the diagonal line. Red dots represent rates for 250 ensembles of 10 cells, each predicted by an independent Poisson model. Blue dots represent rates predicted by the second-order maximum entropy model. Note how blue dots in general lie much closer to the diagonal line, indicating superior prediction over the Poisson model. Horizontal bands of red dots represent multievent states predicted by the Poisson model, and often lie over 10 orders of magnitude below diagonal line. A–E show results on a single representative preparation of each type. For each preparation, 250 ensembles each containing 10 randomly chosen cells were run. Results are shown for data binned at 20 ms. Data binned at finer resolutions fit less well.
Figure 4.
Figure 4.
Quantifying performance in the maximum entropy model. A, The fraction of network correlations captured by the model, I(2)/IN, is plotted against the full multi-information IN, for data binned at 20 ms. Each dot represents one ensemble of 10 cells, randomly chosen from one preparation. Each of the 13 preparations had 250 ensembles, giving 3250 dots. The general trend is for fractions to approach 0.95 as multi-information increases. B, Average fraction of network correlations for each preparation type, for data binned at 20 ms. Each symbol is an average from three preparations of that type, except for the human slice, where there was only one preparation. C, The fraction of network correlations captured by the model for data binned at 1.2 and 4 ms. D, Average fraction of network correlations for each preparation type, for data binned at 1.2 and 4 ms. Error bars show SDs. Note that model performance was better for LFP data and better for 20 ms data.
Figure 5.
Figure 5.
Interactions and local fields for representative preparations with spikes and with LFPs. A, Left, Interaction strengths Jij plotted against correlation coefficients Cij for spike data from an organotypic culture. Each dot represents one neuron pair from one ensemble; 2500 pairs are shown. Right, Interaction strengths plotted against correlation coefficients for LFPs from an organotypic culture. Note much larger correlation coefficients and wider range of interaction strengths in LFP data. B, Left, Distribution of local field strength hi for organotypic spikes. The distribution mean is negative, indicating that most cells are influenced to not spike by other cells outside the ensemble. Right, Distribution of field strength for organotypic LFPs shows similar negative mean. C, Distribution of interaction strengths for organotypic spikes, left, and for organotypic LFPs, right. Positive and negative interactions exist in both preparations, suggesting that frustration is common (Table 1). Results are shown for data binned at 20 ms.
Figure 6.
Figure 6.
Interactions, Jij, and local fields, hi, change with ensemble size. A, Left, Interaction strengths for cell pairs chosen from a 10-cell ensemble, Jij(10), plotted against interaction strengths for the same cell pairs chosen from a 4-cell ensemble, Jij(4), for representative organotypic spike data. Right, Same plot now with representative organotypic LFP data. Note that asymmetry in spike plot is not present in LFP plot, indicating that in this case interaction strength decreases with ensemble size more for spikes than for LFPs. B, Plots of local field strength caused by interactions, hiint, against local field strength caused by intrinsic activity, hi. Density cloud moves across the diagonal line as ensemble size is increased from four to 10, indicating that hiint dominates over hi in larger ensembles. Higher density is coded by darker pixels. C, Left, Average of Jij(10), hi, and hiint for all preparations as ensemble size is increased. Significant changes with ensemble size occurred for Jij (F = 5.91, Fcrit = 2.85, α = 0.05, df = 3, 39) and hi (F = 7.65), but not for hiint (F = 2.54). Right, Individual plots of Jij(10), hi, and hiint for each preparation as ensemble size is increased, showing that average trends were followed in each individual case. Error bars give SDs. Results shown are for data binned at 20 ms. Full results are given in Tables 2 and 3.
Figure 7.
Figure 7.
Estimation of neuronal clique sizes. Information rate for independent model (S1, gray dots) and for actual data (IN, black dots) are plotted in log–log space as a function of number of cells in ensemble. The point at which the S1 and the IN extrapolated lines would intersect gives an estimate of the ensemble size at which entropy would cease to increase. A–E, Representative plots for dissociated culture data, organotypic spike data, organotypic LFP data, acute slice LFP data, and human slice LFP data binned at 20 ms. F, Histogram showing average clique sizes for each data type, for both short (1.2 or 4 ms, gray) and long (20 ms, black) bin widths. Error bars give SDs. Results suggest that entropy will no longer increase with ensemble size in relatively small cliques.
Figure 8.
Figure 8.
Sequence lengths and sizes predicted by concatenation are often significantly smaller than those found in actual data. Concatenated sequences were constructed by randomly drawing states from the model probability distribution. The left column shows sequence length distributions for concatenated sequences (gray) and data (black); the right column shows sequence size distributions. A–E show representative plots for data from dissociated spikes, organotypic spikes, organotypic LFPs, acute slice LFPs, and human slice LFPs. Error bars show SDs. Results shown are for data binned at 20 ms.
Figure 9.
Figure 9.
Comparing sequence length distributions. A, Two sample sequence length distributions from the data and from concatenations, plotted together in log–log coordinates. Sample distributions from the data are plotted in black; those from concatenations are in gray. Sample distributions from the data were taken from a half-length segment of the recording with a randomly chosen start time. To construct distributions from concatenation, correlated states were chosen in proportion to their probability of occurrence in the model and then concatenated. Note the long tail in the data distributions. B, Sum of squares differences between length distributions are sorted in ascending order and plotted. The light gray dashed curve is for differences between pairs of concatenated length distributions; the dark gray dashed curve is for differences between pairs of data length distributions; the solid black curve is for differences between pairs of data and concatenated length distributions. For each curve, 125 pairs of sample distributions like those shown in A were used. Note that differences between concatenations and data are larger than differences within each group. C, Cumulative distributions of differences. The curves in B were converted to cumulative probability distributions, and the maximum difference between the (data–concatenated) curve and the (data–data) curve was found. This difference was called D and was used to test for significance in the Kolmogorov–Smirnov test. Thus, D was a measure of the temporal mismatch between concatenations and data.
Figure 10.
Figure 10.
Candidate factor for a temporal model. A, Schematic showing temporal relationship of activity in neurons/electrodes i and j across time. The appropriate relationship might allow the maximum entropy model to be extended to account for transitions between correlated states. B, The fraction of significant temporal correlations is significantly correlated with maximum D value for 20 ms data (r = 0.627; p = 0.022). C, The fraction of significant temporal correlations is also significantly correlated with maximum D value in 1.2 or 4 ms data (r = 0.735; p = 0.004). To calculate the fraction of significant correlations, all correlations between neuron/electrode pairs across one time step were compared with correlations expected by chance.

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