PMC

The gain in discriminability by using the joint distribution of motif counts rather than the marginal distribution is limited. (A) The outcome of the FCM procedure to find two clusters. Red points indicate counts obtained from a correlated network and blue points are those for the anti-correlated network. The plusses indicate points correctly classified by FCM and the dots represent incorrectly classified network realizations. Each motif-pair ROC curve is obtained by applying the ROC analysis to the FCM generated probability of each network to belong to cluster one (see methods). (B) The resulting AUC as a function of sub-network size for motif 98 (red) is higher than for all possible pairs of motifs (black curves). The AUC shown is for comparing anti-correlated and correlated networks. (C) When the motif count is pooled across N_av = 20 realizations some two-motif curves exceed the single motif curve, which are shown separately in panel (D). This suggests that for a specific size of the sub-network distinguishability can be improved by considering pairs of motif counts.

The basin of attraction of the LFS is larger for anti-correlated networks indicating enhanced stability against fluctuations. (A) Simulations were started from initial states with a different number N_a of active neurons, which is translated into an N_eff value (see text) to allow for a fair comparison of initial conditions. We show the firing rate as a function of time (in units of iterations). For low N_a the anti-correlated network converged to the LFS, whereas for high N_a runs it converged to the HFS. (B) This was reflected in the histogram where green filled bars indicate the number of states with a particular N_eff that converged to the LFS and the open bars indicate the number of states that converged to the HFS. Data for anti-correlated network with J = 25. (C) N_eff,90 as a function of coupling constant J for uncorrelated (green), correlated (red) and anti-correlated (blue) networks together with the number N_a,av of active neurons corresponding to the firing rate of the mean-field solution (cyan dashed line) as a reference. (D) Close-up of panel (C). The data were obtained from a network of N = 2000 neurons, with a baseline rate of 1 Hz. For each coupling strength, and, each network type we used N_r = 1000 initial conditions and averaged across 4 realizations of the network.

Detection can be improved by including past activity and weighting neurons depending on how many inputs they receive from directly stimulated neurons. (A,B) Density representation of the firing rate return map, wherein the probability of obtaining consecutive rate values (r_t, r_t+1) is represented by a color scale, with red indicating the highest probability and blue indicating a near zero probability. The results are shown for (A) spontaneous activity and (B) stimulated activity. The plusses indicate the location of the peak in panel (B). (C) Analysis of factors that contribute to a neuron's weight in detection decision that is outputted by the perceptron procedure. There was significant but small correlation between weight and (top left) in-degree or (top right) out-degree. There was a correlation between the weight and the number of direct inputs from stimulated neurons (bottom left), but only a weak correlation with the number of inputs from cells that received direct input from the stimulated cells (bottom right). The network was comprised of N = 2000 neurons, coupling constant J = 18, baseline firing rate r₀ = 1 Hz. In the stimulated network, n_p = 8 neurons were stimulated for a duration of T_stim = 6 time bins.

Construction of networks with a correlation between out- and in-degree. In panels (A) to (D), we show scatterplots of the out-degree vs. in-degree, whereas the corresponding marginal distributions are shown for (E) the in-degree and (F) the out-degree. We considered four types of networks, each with N = 2000 neurons and a connection probability of p_c = 0.05. (A) The Erdos-Renyi (ER) network in which each connection is chosen at random with a probability p_c = 0.05, for which there is no correlation [ρ = 0.0034 (standard deviation: 0.018)] and the relative variance of in- and out-degree across neurons is small for large networks. In order to examine networks with a higher variance of degree values, we first generated a degree distribution in the form of a truncated, bivariate Gaussian. In (B) the covariance matrix was diagonal, with equal variance for the out- and in-degrees, which yielded uncorrelated in- and out-degrees [ρ = 0.0010 (0.019)]. To generate correlations we started from a covariance matrix with unequal variances and rotated it by 45 degrees anticlockwise to obtain (C) anti-correlated [ρ = 0.821 (0.0085)] and by 45 degrees clockwise to obtain (D) correlated degree distributions [ρ = 0.821 (0.0085)]. In the anti-correlated case, nodes with a high out-degree had a low in-degree and vice versa, whereas in the correlated case, nodes with a high out-degree also had a high in-degree, as illustrated by the insets in (C) and (D), respectively. (E,F) The networks were constructed so that the marginal distributions for the correlated (red), anti-correlated (blue) and uncorrelated (green) case were the same. The ER network (purple) had much tighter marginal distributions.

Anti-correlations in the degree distribution improve the stability of the low firing rate state (LFS). We compared the stability of finite-size networks with different degree correlation structure by iterating Equation 9 (which is Equation 1 without taking into account stochastic spiking). (A) The mean-field solution, corresponding to an infinite-size network, is simulated by assuming that the firing rate of each neuron is equal, yielding Equation 6, of which all roots are shown in the graph. (B) Mean firing rate r vs. coupling constant J in the mean-field limit for different values for the baseline firing rate r₀. When the LFS loses stability, the only remaining solution is the HFS. As a result the plotted firing rate suddenly jumps to the maximum possible rate of 100 Hz (corresponding to 1 spike per bin). (C) The range of stable coupling constants, which are between 0 and J_c, decreases with increasing baseline firing rate. (D) The firing rate r_c of the LFS just before it turns unstable increases linearly with r₀. (E) The stability of the LFS depends on system size and approaches the mean-field limit (cyan) gradually as network size N increases (baseline rate r₀ = 1 Hz). The anti-correlated network (blue) is always more stable than the ER (purple), correlated (red), and uncorrelated (green) networks. (F) The difference between the mean field J_c and that of the finite-size networks decreases with baseline firing rate (network size N = 2000).

Network sensitivity, when evaluated using an ROC analysis, depends only on the mean out-degree of the stimulated neurons and not on the degree correlations. (A) Distribution of firing rate across cells in a 10 ms time bin for spontaneous activity (red) and for the stimulated network (blue), in which 8 random cells were stimulated. Note that the stimulated cells were not included in this ROC analysis and we used the binary responses x_i(t) to determine the firing rate. (B) The corresponding ROC curve (blue) quantifies the difference between the distributions, relative to the diagonal (gray), which represents distributions that cannot be distinguished. (C) The area under the curve (AUC) for the ROC curves calculated for different time bins. The AUC before stimulation was close to 0.5 because the distributions were the same apart from fluctuations due to sampling. After the stimulation, which started at t = 10 and ended at t = 15, the AUC rose to around 0.75. (D) The AUC increases with increasing coupling constant and (E) with increasing baseline firing rate. (F) The AUC depended on the mean out-degree of the stimulated neurons. Neurons were divided into ten groups according to their out-degree, with the first group having the highest out-degree. The group index is indicated on the x-axis. The results in (D–F) were not significantly different for correlated (red), anti-correlated (blue) or uncorrelated (green) networks, t-test, p = 0.4479, 0.6279, 0.7421, respectively. The network was comprised of N = 2000 neurons, of which n_p = 8 neurons were stimulated for the duration of T_stim = 6 time units starting on the 10^th bin. In panel (A–C) results for an uncorrelated network are shown. Parameters: (A–C,F) J = 18, r₀ = 1; for (D) r₀ = 1 and (E) J = 20.

Motif 98 is the most sensitive to degree correlations. (A) There are 13 motifs that involve 3 connected nodes. Below the graphical representation we plot the numbering used here, which follows Itzkovitz et al. (). The expected number of motifs depends on network size, hence we normalize the count by N³(k/N)^e, with N the number of nodes, k the expected number of edges per node and e the number of edges in the motif. In addition, we include a numerical factor representing the equivalent permutations [listed in Table 3 in Itzkovitz et al. ()]. (B) The normalized counts, averaged across a thousand realizations, converge to constant values for sub-networks larger than 50–100 nodes, with the precise value depending on the complexity of the motif involved. (C) The standard deviation of the normalized counts fall off as N^−3/2. We illustrate the results for the anti-correlated network, which are typical for the correlated and uncorrelated network also. In addition, we omitted motif 238 because it occurs at such a low probability that it makes the statistics noisy. (D) The normalized counts for each motif for the (red) correlated, (blue) anti-correlated and (green) uncorrelated networks. We used the counts for the full network, rather than sub-networks. Network size in panel (D) and (E) was N = 200. We used a bivariate Gaussian degree distribution with a mean number of nodes equal to 10, a standard deviation along the long axis of σ_y = 3.33 and along the short axis of σ_x = 1.0. (E) The maximum difference in mean count between all three possible comparisons (black bars), relative to the mean standard deviation of these counts across the three network types. The motifs are ordered on the count over standard deviation ratio, starting with the largest. According to this analysis motif 98 should be used to best distinguish between different network correlation structures.

Degree correlations can be distinguished by pooling fifty measurements of networks at least thirty neurons in size. (A) Motif count varies across network realizations, but degree correlations can be distinguished when the corresponding distribution show little overlap. We show the distribution of the normalized counts of motif 98 for (red) correlated, (blue) anti-correlated, and (green) uncorrelated networks with 200 neurons. (B) For smaller sub-networks (N_sub = 30), the distributions overlap. Furthermore, these distributions are not Gaussian as they are skewed because counts are always positive. Hence, a more general procedure, such as the ROC analysis needs to be used instead of looking at the differences in mean count relative to the standard deviation. (C,D) The area under the ROC curve (AUC) as a function of sub-network size N_sub for the comparison (C) between correlated and anti-correlated networks and (D) between anti-correlated and uncorrelated networks. Each motif is labeled with a line style and color as indicated in the legend. Motif 98 is most sensitive in both cases (as well as for the correlated vs. uncorrelated comparison that is not shown). It is more difficult to distinguish an anti-correlated network from an uncorrelated one than to distinguish it from a correlated network. The average AUC values were determined based on the AUC value for each of twenty different motif distributions of 500 network realizations, which were sampled randomly with replacement out of 1000 realizations. (E) The motif distribution for N_sub = 30 can be pooled across N_av = 50 network realizations in order to shrink the width of the distribution, so that the differences in mean counts become clearer [(compare to panel (B)]. (F) The AUC for larger N_av values reaches unity (distributions are perfectly distinguishable) for smaller sub-network sizes. We show (green) no pooling, (blue) pooling across N_av = 5 realizations and (red) pooling across N_av = 50 realizations. The AUC goes from 0.7 to 1.0 between N_sub = 30 and 50 when pooled across N_av = 50 realizations, indicating that networks of size 30 can be used to determine degree correlation structure.

The anti-correlated network is more stable against fluctuations. (A) The firing rate vs. coupling strength for the mean-field solution (cyan) and networks with uncorrelated (green), correlated (red) or anti-correlated (blue) degree distributions (r₀ = 1 Hz, N = 2000). The anti-correlated degree distribution leads to the most stable network. The dashed box approximately indicates the interval of coupling strengths highlighted in panels (B) and (C). (B) Despite the existence of a stable LFS for a particular coupling strength, fluctuations in network activity may perturb the network away from it and the network ends up in the co-existing stable HFS state. The fraction of states that end up in the HFS state is close to zero far below J_c and increases to unity above J_c. The LFS state is more stable for the anti-correlated (blue) network than for the uncorrelated network (green), which in turn is more stable than the correlated network (red). The dashed lines are fits to the sigmoidal function in Equation 10. (C) The stability depends on the strength of the correlation. When the width (dispersion) corresponding to the small axis in the bivariate Gaussian degree distribution is increased, which means lower correlation, the stability is reduced. Data are for an anti-correlated network. (D) A neuron's firing rate is correlated with its in-degree, but the degree of correlation is reduced to 0.519 (0.014) due to jitter in this relation for Equation 1 (blue dots) from 0.997 (0.002) for Equation 9 (green dots). Data for anti-correlated network, J = 30.96. (E,F) The degree of stability can be qualified by J_gap, the distance of the J_c for the finite-size network from that for the mean-field network, shorter distances meaning more stable networks. J_gap decreases with the (E) baseline firing rate r₀ and with (F) network size. In both panels the anti-correlated network (blue line) corresponds to the lowest curve indicating higher stability compared to ER (purple), uncorrelated (green) and correlated (red), an advantage that increases with network size. The network had N = 2000 neurons, for each coupling strength N_t = 100 simulations were performed, with a length of 500 time steps, of which the first 100 were discarded as a transient.