Figure 1.
Laminar population recordings in response to natural movies.
(a) Example of one of the natural movie stimuli, depicting ducks on a lawn, presented full-field at 150 frames per second. (b) Example data from 22 cells (session B4), binned in 20 ms windows, 2 s of data. Columns of this matrix form the training data for our algorithm. For the spatiotemporal version of the model, several adjacent columns are concatenated. (c) Pairwise correlations in the raw data, and noise correlations computed from 60 repetitions of a 30 s stimulus, binned at 20 ms. Both show small, positive correlations. Shuffling spikes for each of the cells shows that correlations expected due to shared firing rate modulations time-locked to the stimulus are much smaller.
Figure 2.
Functional connectivity patterns of the three models estimated for recording session B4.
The horizontal lines indicate approximate boundaries between cortical layers II/III, layer IV and layers V/VI. (a) Ising model coupling matrix. Each row/column of the matrix corresponds to a neuron, bias terms are shown on the diagonal. The model has many small coupling terms that encode positive correlations. (b) The RBM coupling weights are shown as a bar chart for each hidden unit, ordered from left to right from largest to smallest activity. The first bar chart is the bias for all the visible units, and the blue bar at the top of each plot corresponds to the bias of that hidden unit. Blue bars indicate negative values (the bias terms are predominantly negative, but plotted with flipped sign to fit on the scale of the remaining terms). (c) The sRBM weights are shown in the same way, with the pairwise couplings on the left and hidden units on the right. The pairwise connections are qualitatively very different from those of the Ising model, as most of the structure is better captured by hidden units. (d) Pairwise coupling terms for the Ising model with stimulus terms. Much of the structure, in particular the bias terms, have been explained away by the stimulus terms. (e) Hidden unit couplings for the RBM with stimulus terms. The structure of the hidden units remains largely unchanged, indicating higher-order couplings are due to network interactions and not stimulus correlations.
Figure 3.
Model comparison using likelihood gain over the independent model.
Likelihoods are normalized to bits/spike to account for different population size as well as firing rate. (a) The change in performance with dataset size (22 cells for session B4, 26 cells for MU, and 36 cells for T6) is thus due to additional structure captured from larger populations. B4 and T6 are spike sorted, B4M is a multiunit dataset. All three models outperform the independent model by 0.4–0.8 bits/spike. The higher-order models with hidden units give a small (0.03–0.04 bits/spike, about 10%) improvement over the Ising model for the small datasets, growing to 0.18 bits/spike, about 28%, for the dataset with the large population size. (b) Including stimulus terms provides a large gain in likelihood, even the stimulus PSTH term alone outperforms the network models by a large margin for this dataset. There is still a significant gain by including coupling terms. The difference between the second order Ising model and higher-order RBM is particularly visible in the right hand plot which shows the gain relative to the PSTH only model. All error bars indicate one standard deviation over repeated estimation on different random subsets of the data for training and validation and random initializations of AIS estimation.
Figure 4.
Scatter plot showing empirical probabilities against model probabilities on T6 test dataset.
Different models are distinguished by color, the number of simultaneously spiking cells in each pattern by different symbols. (a) shows the independent model compared to the Ising model, (b) shows the Ising model compared to the RBM. The sRBM is omitted as it is very similar to the RBM. The RBM significantly outperforms simpler models. Inserts in both models show the distribution of synchrony, , where
is the number of cells simultaneously firing in one time bin. The synchrony in the empirical distribution (red line) is greatly underestimated by the independent model (yellow), but well captured by both the pairwise (blue) and higher-order model (green).
Figure 5.
Likelihood comparison as a function of model size.
Spatiotemporal models with 10 cells and a varying number of concatenated time steps. The log-likelihood per spike increases as each neuron is modeled as part of a longer time sequence. This effect holds both for Ising and higher-order models. Since the Ising model cannot capture many of the relevant dependencies, the increase in likelihood saturates after about 3 timesteps for the Ising model, but continues to increase for the higher-order models. Insert: Comparison of the entropy per time slice for Ising and RBM models as a function of model size. As the RBM is better able to model spatiotemporal dependencies, the additional entropy for extra frames is smaller than for the Ising models. The RBM does not reach the point of extensivity, where additional frames add constant entropy. Multiple lines of the same color indicate repeated runs with different random initialization.
Figure 6.
Spatiotemporal network models.
(a) Hidden units of spatiotemporal sRBM model. For each hidden unit, the horizontal axis is time and the vertical axis cells with horizontal bars separating the subgranular, granular and supergranular cortical layers. (b) Log-likelihood gain for each of the 10 cells (ordered by firing rate) conditioned on the remainder of the network state for all three models. (c) Spike prediction from network history. For one of the cells, we show 1 s of predicted activity given the history of the network state. In each case when a spike occurs in the data (gray bars), there is an elevated probability under the models (colored bars).
Figure 7.
(a) Example recording site with lines indicating layer boundaries and a schematic drawing of the 32-channel probe. (b) Current source density (the second spatial derivative of the LFP) in response to a full-field flash stimulus, showing a strong current sink in layer IV and to a lesser degree in layer VI. CSD was used to assign layer boundaries. (c) Cross-correlation for one pair of cells across time, binned at 6.7 ms. Correlations fall off quickly as the time lag increases with a central peak extending indicated by dashed lines.