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Correlations and functional connections in a population of grid cells.

Dunn B, Mørreaunet M, Roudi Y - PLoS Comput. Biol. (2015)

Bottom Line: We find similar results also 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 in the same module and those from different modules can be both negative and positive, with a mean close to zero, but with the strongest inferred connections 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 grid pattern.

View Article: PubMed Central - PubMed

Affiliation: Kavli Institute for Systems Neuroscience and Centre for Neural Computation, NTNU, Trondheim, Norway.

ABSTRACT
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 maximum entropy kinetic pairwise 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. They take positive values between cells with nearby phases and approach zero or negative values for larger phase differences. We find similar results also 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 in the same module and those from different modules can be both negative and positive, with a mean close to zero, but with the strongest inferred connections 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 grid pattern.

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Grid cell spatial firing rate map.The smoothed rate maps from the original spike data (panel A) and synthetic spike data (panel B). Three example grid cells from the three different modules identified in data set 1 are shown here: left column, module 1 (T4C4 is cell identity—tetrode 4 cell 4), middle column, module 2, and right column, module 3. The synthetic data (panel B) was generated using Eq. 1 (with Hi(t) determined by the inferred values for the Gaussian basis functions plus a constant field) and the trajectory of the rat. The rate maps in both panel A and B were generated by first binning the spike data into 3 cm spatial bins, for which the mean rate was calculated and then smoothed using a Gaussian filter (standard deviation = 2 bins).
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pcbi.1004052.g003: Grid cell spatial firing rate map.The smoothed rate maps from the original spike data (panel A) and synthetic spike data (panel B). Three example grid cells from the three different modules identified in data set 1 are shown here: left column, module 1 (T4C4 is cell identity—tetrode 4 cell 4), middle column, module 2, and right column, module 3. The synthetic data (panel B) was generated using Eq. 1 (with Hi(t) determined by the inferred values for the Gaussian basis functions plus a constant field) and the trajectory of the rat. The rate maps in both panel A and B were generated by first binning the spike data into 3 cm spatial bins, for which the mean rate was calculated and then smoothed using a Gaussian filter (standard deviation = 2 bins).

Mentions: Focusing on data set 1, which had the most cells, we first inferred couplings, assuming that each neuron receives an external field which is constant across time and space, hi(t) = hi. Next, we studied how the inferred couplings were affected by increasing the spatial resolution of the external fields, hi(t), to account for the spatial variation in firing rate by dividing the environment into spatial bins, considering the cases of bins of size 37.5 cm and then bins of size 7.5 cm, assigning one external field per box to each cell. We also considered external fields in the form of a sum of Gaussian basis functions. Fig. 2 shows the resulting couplings, plotted against couplings found in the model that assumed spatially and temporally constant external input, hi, for each neuron. As can be seen, increasing the resolution of the external fields made the couplings weaker but not inconsistent with the constant field case, even in the case of Gaussian fields, where the spatial rate maps were well captured by the model, as shown in Fig. 3. In this case, there was a significant weakening of the couplings (the estimated variance of the Gaussian field model couplings () was significantly smaller than that of the constant field model (), (F-test for equal variances, P<0.001)). In each of the models, the total external fields were negative and often strong, as one would expect for data sets with low firing rates (mean firing rate 2.4 Hz).


Correlations and functional connections in a population of grid cells.

Dunn B, Mørreaunet M, Roudi Y - PLoS Comput. Biol. (2015)

Grid cell spatial firing rate map.The smoothed rate maps from the original spike data (panel A) and synthetic spike data (panel B). Three example grid cells from the three different modules identified in data set 1 are shown here: left column, module 1 (T4C4 is cell identity—tetrode 4 cell 4), middle column, module 2, and right column, module 3. The synthetic data (panel B) was generated using Eq. 1 (with Hi(t) determined by the inferred values for the Gaussian basis functions plus a constant field) and the trajectory of the rat. The rate maps in both panel A and B were generated by first binning the spike data into 3 cm spatial bins, for which the mean rate was calculated and then smoothed using a Gaussian filter (standard deviation = 2 bins).
© Copyright Policy
Related In: Results  -  Collection

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Show All Figures
getmorefigures.php?uid=PMC4340907&req=5

pcbi.1004052.g003: Grid cell spatial firing rate map.The smoothed rate maps from the original spike data (panel A) and synthetic spike data (panel B). Three example grid cells from the three different modules identified in data set 1 are shown here: left column, module 1 (T4C4 is cell identity—tetrode 4 cell 4), middle column, module 2, and right column, module 3. The synthetic data (panel B) was generated using Eq. 1 (with Hi(t) determined by the inferred values for the Gaussian basis functions plus a constant field) and the trajectory of the rat. The rate maps in both panel A and B were generated by first binning the spike data into 3 cm spatial bins, for which the mean rate was calculated and then smoothed using a Gaussian filter (standard deviation = 2 bins).
Mentions: Focusing on data set 1, which had the most cells, we first inferred couplings, assuming that each neuron receives an external field which is constant across time and space, hi(t) = hi. Next, we studied how the inferred couplings were affected by increasing the spatial resolution of the external fields, hi(t), to account for the spatial variation in firing rate by dividing the environment into spatial bins, considering the cases of bins of size 37.5 cm and then bins of size 7.5 cm, assigning one external field per box to each cell. We also considered external fields in the form of a sum of Gaussian basis functions. Fig. 2 shows the resulting couplings, plotted against couplings found in the model that assumed spatially and temporally constant external input, hi, for each neuron. As can be seen, increasing the resolution of the external fields made the couplings weaker but not inconsistent with the constant field case, even in the case of Gaussian fields, where the spatial rate maps were well captured by the model, as shown in Fig. 3. In this case, there was a significant weakening of the couplings (the estimated variance of the Gaussian field model couplings () was significantly smaller than that of the constant field model (), (F-test for equal variances, P<0.001)). In each of the models, the total external fields were negative and often strong, as one would expect for data sets with low firing rates (mean firing rate 2.4 Hz).

Bottom Line: We find similar results also 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 in the same module and those from different modules can be both negative and positive, with a mean close to zero, but with the strongest inferred connections 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 grid pattern.

View Article: PubMed Central - PubMed

Affiliation: Kavli Institute for Systems Neuroscience and Centre for Neural Computation, NTNU, Trondheim, Norway.

ABSTRACT
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 maximum entropy kinetic pairwise 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. They take positive values between cells with nearby phases and approach zero or negative values for larger phase differences. We find similar results also 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 in the same module and those from different modules can be both negative and positive, with a mean close to zero, but with the strongest inferred connections 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 grid pattern.

Show MeSH
Related in: MedlinePlus