Correlations and functional connections in a population of grid cells

We study the statistics of spike trains of simultaneously recorded grid cells in freely behaving rats. We evaluate pairwise correlations between these cells and, using a generalized linear model (kinetic Ising model), study their functional connectivity. Even when we account for the covariations in firing rates due to overlapping fields, both the pairwise correlations and functional connections decay as a function of the shortest distance between the vertices of the spatial firing pattern of pairs of grid cells, i.e. their phase difference. The functional connectivity takes positive values between cells with nearby phases and approaches zero or negative values for larger phase differences. We also find similar results when, in addition to correlations due to overlapping fields, we account for correlations due to theta oscillations and head directional inputs. The inferred connections between neurons can be both negative and positive regardless of whether the cells share common spatial firing characteristics, that is, whether they belong to the same modules, or not. The mean strength of these inferred connections is close to zero, but the strongest inferred connections are found between cells of the same module. Taken together, our results suggest that grid cells in the same module do indeed form a local network of interconnected neurons with a functional connectivity that supports a role for attractor dynamics in the generation of the grid pattern.Comment: Accepted for publication in PLoS Computational Biolog

Mean spacing and orientation for the 3+1 modules.

<p>Mean spacing and orientation for the 3+1 modules.</p

Couplings versus phase distance.

<p>Inferred couplings are positive for small phase differences while they become negative for larger phase separations, both for data set 1 (A) and data set 2 (B). When we break the population to the three contributing modules of data set 1, this pattern persists for the smaller modules (C,D) while for the largest module (E) the excitatory part is absent. In each plot, the circles represent the inferred values using the full data length. The black lines show the average values of the couplings calculated from 20 random partitions of the data.</p

Noise correlations versus phase distance.

<p>(A) shows all three modules of data set 1 combined, while (B) shows the one module of data set 2. For the smaller modules of data set 1 (C, D) the noise correlations are positive for small phase differences while they approach zero for larger phase separations. No significant pattern can be observed for the cells from the largest module of data set 1 (E). The distance in phase was normalized by the average spacing of the spatial fields in each module. In each plot, the circles represent the inferred values using the full data length. The noise correlations were calculated by binning the environment into 7.5 × 7.5 cm spatial bins. The black lines show the average values of the correlations calculated from 20 random partitions (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004052#sec004" target="_blank">Material and Methods</a>) of the data. The error bars are the standard deviation of the mean values over these 20 random partitions. Note that the normalized maximal phase distance occurs at the minimum overlap between the two commonly oriented hexagonal patterns and is 0.5/cos(30) ≈ 0.6.</p

Trajectory and speed.

<p>In panel A, the trajectories of the rats are shown. Figures in Panel B show the frequencies of different speeds during the recordings for the two data sets.</p

Effect of theta on the couplings.

<p>(A) Adding theta to the Gaussian model has little effect on the couplings (data set 1) with PCC, All = 0.95, PCC, SC = 0.97, PCC, NonSC = 0.94. (B) Mean of couplings from the two theta clusters in the Gaussian model with and without theta included. Black: couplings between cells with similar theta phase preference. Blue: couplings between cells with opposite theta phase preference. Error bars show the standard error of the mean. Without theta taken into account, the connections between cells that fire in the opposite theta phase are on average negative, while they are positive for those that tend to fire in the same theta phase. This difference is suppressed when theta is taken into account.</p

Stability of the inferred couplings.

<p>Stability of the phase-dependent trend in inferred couplings filtered for cell pairs where at least one cell is phase precessing (A), as well as for couplings filtered for cells on the same tetrode (B). The phase dependence of the coupling can be seen to be similar to when all pairs were included. Couplings inferred using one random half of the data plotted against those inferred from the other half, assuming constant external field (C) or Gaussian spatial fields (D). The within module couplings (green triangles) consistently show more stability across partitions of the data than the between module couplings (blue circles), but not as much as the self-couplings (red triangles). A: PCC, within modules = 0.88, PCC, between modules = 0.73, PCC, SC = 0.99. B: PCC, within modules = 0.73, PPC, between modules = 0.51, PCC, SC = 0.94. (E) The effect on between grid cells-couplings from including non-grid cells in the inference for the biggest data set (data set 1, 65 cells) is small.</p

Statistical importance of the parameters.

<p>The negative log-likelihood per cell per time bin for the constant field model (A: data set 1, C: data set 2) and Gaussian field model (B: data set 1, D: data set 2) with different covariates included. Smaller values correspond to better explanatory power. The blue segment of the bar shows the negative log-likelihood. Adding parameters to a model will yield a log-likelihood-value greater than or equal to the model with fewer parameters. To avoid overfitting by including parameters, we performed an Akaike correction on the log-likelihood (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004052#sec004" target="_blank">Material and Methods</a>). The value of the Akaike-correction is shown for each covariate on top of the negative log-likelihood (blue) for each model: head direction (red), theta preference (yellow), and couplings (green). In (C, D), grey is the Akaike-correction due to the Gaussian spatial fields. These two plots show that adding the couplings always increases the explanatory power of the model, e.g. for the model with theta including couplings reduces the negative log-likelihood more than the penalty from the Akaike-correction for the added number of parameters.</p

The couplings of the kinetic Ising model.

<p>We considered different forms of spatial external input to the neurons, boxes of length 37.5 cm (A), 7.5 cm (B) and fields formed as a weighted sum of Gaussian basis functions (C) for data set 1. For each case, we compared the resulting couplings to that of a model with spatially and temporally constant fields. The effect of input with spatial variation is to slightly weaken the couplings. Pearson correlation coefficient (PCC) was calculated for all the couplings together (All), as well as for just the self-couplings (SC) shown by red stars, and the non-self-couplings (NonSC) shown by blue circles. The corresponding values are A: PCC, All = 0.91, PCC, SC = 0.98, PCC, NonSC = 0.86. B: PCC, All = 0.91, PCC, SC = 0.94, PCC, NonSC = 0.90. C: PCC, All = 0.92, PCC, SC = 0.94, PCC, NonSC = 0.91.</p