Figure 1.
Connectivity induces correlations.
A: Activity in a pair of neurons (red) in a network can become correlated due to direct connections (blue) and different types of shared input (cyan). B: For a complete description a large number of indirect interactions (yellow,orange) and indirect common input contributions (green) have to be taken into account. However, not all nodes and connections contribute to correlations (grey).
Table 1.
Used symbols (in order of appearance).
Figure 2.
Hawkes' theory reproduces rates and correlations in a simulated random network.
Network parameters are . A: Top: spike raster plot showing asynchronous irregular activity (mean coefficient of variation 1.03). Inset: Inter-spike intervals of a typical spike train are exponentially distributed (logarithmic scale). Bottom: Population spike counts in bins of length
. Mean
standard deviation (thick red line, shaded area), standard deviation from predicted correlations (green dashed line). B: Fluctuating rates of 50 neurons (grey traces), their average (red line) and distribution across time and neurons (blue). A small part reaches below zero (dashed line). C: Simulated time averaged rates scattered vs. predicted rates (blue). Diagonal (red) plotted for direct comparison. Inset: Distribution of predicted (red) and measured (blue) rates. Broad rate distribution with significant deviations from predictions only for small rates. D: Simulated correlations scattered vs. predicted ones. Larger errors due to finite simulation time. Inset: correlation distributions (green: measured, red: predicted). Although a non-vanishing part of fluctuating rates is below zero, most of the time averaged rates and correlations are predicted accurately.
Figure 3.
Correspondence of motifs and matrix powers.
A: Examples for matrix expressions and symbols. B: Graphical interpretation of paths contributing to elements of the matrix . C: Same for the matrix
.
Figure 4.
Motif contributions to average correlations in random networks.
Top: low connectivity, . Bottom: higher connectivity,
. Other parameters as in Figure 2. A: Spectra of connectivity matrices (fixed out-degree), eigenvalues in the complex plane. Red circle: theoretical radius for bulk spectrum. Red cross: mean input to a neuron. The networks are inhibition dominated (
) and real parts of all eigenvalues are below one (dashed line). B: Contributions of different motifs to average correlation. Comparison between theoretical prediction, random networks with uniform connection probabilities (average across 10 realisations, error bars indicate standard deviation), and networks with fixed out-degree. While in the sparse network only the first few orders contribute, higher orders contribute significantly in the dense network. The analytical expression for the average correlation reproduces the values for networks with fixed out-degrees and approximates the values for random networks. Within one order, chain motifs hardly add to correlations, while common input motifs have a larger contribution. Since inhibition dominates in the network, contributions are positive for even orders
and negative for uneven orders. Refer to Figure 3 for the correspondence of symbols and paths.
Figure 5.
Strongly recurrent networks have broad correlation distributions.
A: Low connectivity, , B: high connectivity,
. Other parameters as in Figure 2. The discrete distribution of direct interactions (blue) is washed out by second order terms (green) to a bimodal distribution for low and a unimodal distribution for high connectivity. higher-order terms (red) contribute significantly only for high connectivity. C: Correlation distributions change from a bimodal to a unimodal distribution for increasing connectivity (grey-scale indicates probability density). Average correlation (blue) increases smoothly and faster than the average interaction (black), which is the sum of excitatory (green) and inhibitory (yellow) interaction, but slower than the average common input (cyan) due to higher-order terms. The analytical prediction from Equation (22) for the average correlation (dashed red) fits the numerical calculation, especially for low connectivities. Vertical dotted lines indicate positions of the distributions in A and B.
Figure 6.
Correlation distributions depend on range of inhibition in ring networks.
A: Distance dependent connectivity in a ring. Nodes are connected with a fixed weight to neurons with a probability depending on their mutual distance. Average interaction is the product of connection probability and weight averaged over populations. The connectivity profile may be different for excitatory neurons (red, positive weight) and inhibitory ones (green, negative weight). Average interaction on a randomly picked neuron at a distance corresponds to the sum (blue). B: Typical spectrum for a connectivity matrix with local inhibition. Parameters: , others as in Figure 2. C,D: Examples for average interaction profiles used in E and F. C: Global inhibition (
) and local excitation,
for small (dashed) and large (dotted)
, hat profile. D: Local inhibition and global excitation, inverted hat profile. Other parameters as in B. E,F: Top: Correlation distributions for fixed
and increasing
(E) and fixed
and increasing
(F), logarithmic colour scale. Values between
(random network) and
(connectivity in boxcar 0.5). Overall connectivity
remains constant. Average correlation (dashed blue: numerical, red: analytical) does not change. Bottom: real parts of eigenvalues for corresponding connectivity matrices. Rings with local excitation tend to be less stable.
Figure 7.
Distance dependence of correlations and population fluctuations.
A,B: Evaluation of for parameters
and
, localised inhibition (A) and
, localised excitation (B). Other parameters as in Figure 2. Contributions of different paths, numerically (full lines) and analytically (dashed lines). Higher orders add up to extreme values for localised excitation but cancel out for localised inhibition. Correlations of individual neurons with distant neighbours vary considerably (grey, 50 traces shown). C: Variance of population spike counts over population size. Comparison between populations of neighbouring neurons in a ring and in a random network with fixed output. Plotted are results from analytical approximation, numerical calculation using the connectivity matrix and direct simulation, averaged over 5 populations in each case. Network Parameters: random network as in Figure 2, ring:
. Simulation parameters: total simulation time:
, bin size for spike counts:
, others as in Figure 2.
Figure 8.
Higher-order contributions to correlations are increased by connected excitatory hubs.
A: Construction. Excitatory neurons are divided into hubs (large out-degree) and non-hubs (small out-degree). The fraction of excitatory outputs from hubs to hubs is varied. B: Average correlations increase with hub-interconnectivity
(average across 10 networks, error bars from standard deviation). Densely connected hubs (
, dashed vertical line) lead to large average correlations. C: Contribution to average correlations of different motifs for random network and networks with broad out-degree distribution and
(disassortative network) and
(assortative network). Average across 20 networks, error bars from standard deviation. Although contributions are different from random networks, in networks with broad degree distribution low order contributions (indirect and common input) are independent of hub interconnectivity, in contrast to certain higher-order contributions.
Figure 9.
Patches in separate populations selectively affect high-order common input motifs.
A: Connection rules for networks with patches. Output connections of a neuron are restricted to a randomly chosen region. B: Average correlations depending on patch size. Comparison between random networks, patchy networks with randomly distributed neuron type and separate populations, average across 5 networks, error bars from standard deviation. Only separate populations lead to increased correlations. Larger increase occurs for smaller patch-size. Total connection probability , other parameters as in Figure 2. C: Contributions of different motifs. Differences to random networks occur only in common input terms of higher order. Patch size
.