Figure 1.
Network activity in simulation and experiments.
A Bright field image (left panel) of a region of a neuronal culture at day in vitro 12, together with its corresponding fluorescence image (right panel), integrated over 200 frames. Round objects are cell bodies of neurons. B Examples of real (left) and simulated (right) calcium fluorescence time series, vertically shifted for clarity. C Corresponding averages over the whole population of neurons. D Distribution of population-averaged fluorescence amplitude for the complete time series, from a real network (left) and a simulated one (right).
Figure 2.
Independence of network dynamics from clustering coefficient.
A Examples of spike raster plots for three networks with different clustering coefficients (non-local clustering ensemble), showing that their underlying dynamics are similar. B Histograms of the inter-burst intervals (IBIs), with the vertical lines indicating the mean of each distribution. The insets illustrate the amount of clustering by showing the connectivity of simple networks that have the same clustering coefficients as the simulated ones.
Figure 3.
Dependence of the directed functional connectivity on the dynamical state.
A The distribution of averaged fluorescence amplitudes is divided into seven fluorescence amplitude ranges. The functional connectivity associated to different dynamical regimes is then assessed by focusing the analysis on specific amplitude ranges. B Quality of reconstruction as a function of the average fluorescence amplitude of each range. The blue line corresponds to an analysis carried out using the entire data sampled within each interval, while the red line corresponds to an identical number of data points per interval. C Visual representation of the reconstructed network topology (top 10% of the links only), together with the corresponding ROC curves, for the seven dynamical regimes studied. Edges marked in green are present in both the reconstructed and the real topology, while edges marked in red do not match any actual structural link. Reconstructions are based on an equal number of data points in each interval, therefore reflecting the equal sample size performance (red curve) in panel B. Interval I corresponds to a noise-dominated regime; intervals II to IV correspond to inter-burst intervals with intermediate firing rate and provide the best reconstruction; and intervals V–VII correspond to network bursts with highly synchronized neuronal activity. Simulations were carried out on a network with local topology () and light scattering in the fluorescence dynamics. The results were averaged over 6 network realizations, with the error bars in B and the shaded regions in C indicating a 95% confidence interval.
Figure 4.
TE-based network reconstruction of non-locally clustered topologies.
A ROC curve for a network reconstruction with generalized TE of Markov order , and with fluorescence data conditioned at
. The shaded area depicts the 95% confidence intervals based on 6 networks. B Comparison between structural (shown in blue) and reconstructed (red) network properties: clustering coefficients (top), degree distribution (center), and distance of connections (bottom). C Reconstructed clustering coefficients as a function of the structural ones for different reconstruction methods. Non-linear causality measures, namely Mutual Information (MI, red) and generalized Transfer Entropy (TE, yellow), provide the best agreement, while a linear reconstruction method such as cross-correlation (XC, blue) fails, leading invariably to an overestimated level of clustering. The error bars indicate 95% confidence intervals based on 3 networks for each considered clustering level. All network realizations were constructed with a clustering index of 0.5, and simulated with light scattering artifacts in the fluorescence signal.
Figure 5.
TE-based network reconstruction of locally-clustered topologies.
A ROC curve for a network reconstruction with generalized TE Markov order , with fluorescence data conditioned at
. The shaded area depicts the 95% confidence interval based on 6 networks. B Comparison between structural (top) and reconstructed (bottom) connectivity. For the reconstructed network (after thresholding to retain the top 10% of links only) true positives are indicated in green, and false positives in red. C Comparison between structural (blue) and reconstructed (red) network properties: clustering coefficients (top), degree distribution (center) and distance of connections (bottom). D Reconstructed length scales as a function of the structural ones for different reconstruction methods. The non-linear causality measures, Mutual Information (MI, red) and generalized Transfer Entropy (TE, yellow), provide good reconstructions, while the linear cross-correlation (XC, blue) always provides an underestimated length scale. The error bars indicate 95% confidence intervals based on 3 networks per each considered length scale. All network realizations were constructed with a characteristic length scale
, and simulations included light scattering artifacts.
Figure 6.
Dependence of performance level on network clustering, conditioning level and light scattering artifacts.
The color panels show the overall reconstruction performance level, quantified by TP (black, 0% true positives; white, 100% true positives), for different target ground-truth clustering coefficients and as a function of the used conditioning level. Five different reconstruction algorithms are compared: cross-correlation (XC), Granger Causality (GC) with order
, Mutual Information (MI), and Transfer Entropy (TE) with Markov orders
. The top row corresponds to simulations without artifacts, and the bottom row to simulations including light scattering. Reconstructions with linear methods perform well only in the absence of light scattering artifacts. TE reconstruction with
shows the best overall reconstruction performance, even with light scattering and for any target clustering coefficient. An optimal reconstruction is obtained in a narrow range surrounding the conditioning value of
.
Figure 7.
Dependence of reconstruction quality on TE formulations and recording length.
A ROC curves for network reconstructions of non-locally clustered (left panel) and locally-clustered topologies (right), based on three TE formulations: conventional TE (blue), generalized TE with same bin interactions only (red) or also including optimal conditioning (yellow). The vertical lines indicate the performance level at , and provide a visual guide to compare the quality of reconstruction between different formulations. B Decay of the reconstruction quality as measured by
for the two topology ensembles and for generalized TE with conditioning, as a function of the data sampling divisor
. A full simulated recording of 1 h in duration provides a data set of length
, corresponding to a data sampling fraction of
. Shorter recording lengths are obtained by shortening the full length time-series to a shorter length given by
, with
. The insets show the same results but plotted as a function of
in semi-logarithmic scale. For both A and B, the panels in the left column correspond to the non-locally clustered ensembles (cfr. Figure 4), while the panels in the right column correspond to the locally-clustered ensemble (cfr. Figure 5). Shaded regions and error bars indicate 95% confidence intervals based on 6 networks.
Figure 8.
Network reconstruction of an in vitro neuronal culture at DIV 12.
A Example of TE reconstructed connectivity in a subset of 49 neurons (identified by black dots) in a culture with marked neurons (regions of interest) in the field of view, studied at day in vitro 12. Only the top 5% of connections are retained in order to achieve, in the final reconstructed network, an average connection degree of 100 (see Results). B Properties of the network inferred from TE reconstruction method (top panels) compared to a cross-correlation (XC) analysis (bottom panels). The figure shows reconstructed distributions for the in-degree (left column), the connection distance (middle column), and the clustering coefficient (right column). In addition to the actual reconstructed histograms (yellow), distributions for randomized networks are also shown. Blue color refers to complete randomizations that preserves only the total number of connections, and red color to partial randomizations that shuffle only the target connections of each neuron in the reconstructed network.
Table 1.
Synaptic weights used in the simulation.