Fig 1.
Hawkes process theory reproduces rates, correlations and third cumulants in a simulated network with Erdős-Rényi type random connectivity.
Network parameters are N = 1000, NE = 800, NI = 200, p = 0.1, gE = 0.015 and gI = −0.075. Top left: Fluctuating firing rates of 50 randomly chosen neurons (gray traces) and their average (red line). The average rate only rarely goes below 0 (dashed line). Top right: Estimated temporal averages of firing rates scattered vs. rates predicted by Hawkes theory. The diagonal (green line) indicates a perfect match. Note that there is a slight discrepancy between theory and simulation for very low rates. Bottom left: Estimated integrated pairwise covariances (of all possible neuron pairs) scattered vs. integrated covariances predicted by Hawkes theory. Bottom right: Estimated integrated joint third cumulants (see the following sections for a definition) of a 100 randomly chosen neurons, scattered vs. integrated joint cumulants computed from Hawkes theory. The larger discrepancies are due to finite simulation time and a relatively small sample size.
Table 1.
Symbols used in text (in order of appearance).
Fig 2.
Pictorial representation of terms contributing to the pairwise covariance density.
Each entry of C(τ) is a weighted sum of integral terms, corresponding to rooted trees with 2 leaves, i and j.
Fig 3.
Pictorial representation of terms contributing to κijk(t1, t2, t3).
Each κijk(t1, t2, t3) is a weighted sum of integral terms, corresponding to rooted trees with leaves i, j and k (see Eq 21). The first term maps to the left tree, while the three remaining terms correspond to three possible ways in which three labeled leaves can be arranged into two groups to form the tree on the right. The first group would represent the daughter nodes of vertex m, and the second group would be a single child of the root node n.
Fig 4.
The pictorial representation of all terms, contributing to κijk.
The tree shapes depicted in this figure were obtained from those in Fig 3 by performing all possible contractions of branches (see text).
Fig 5.
Efficacy of the quadratic approximation to (Eq 47) for different network sizes N.
All four panels: and its quadratic approximation,
, plotted for different values of the connection probability p. Each panel corresponds to a network of a given size.
Fig 6.
Contributions of some tree structures to the average third cumulant in a random network.
Top: Theoretical (narrow out-degree distribution approximation) and sample contributions to the average third cumulant of T6 (see Fig 4) tree topologies with fixed branch lengths. Ticks on the x-axis code for the lengths of the four branches of the tree. The ordering of the indices in the tick labels is done in a top-to-bottom and left-to-right fashion. More precisely, the first number corresponds to the length of the “leftmost” branch emanating from the root node, and so forth. The sample contributions were computed as averages of 3 independent realizations of a random network. Bottom: Theoretical and sample contributions to the average third cumulant of T1 tree topologies with fixed branch lengths.
Fig 7.
Contribution of tree motifs with longer total branch length increases in networks with excitatory hubs.
Top: Theoretical (narrow out-degree distribution approximation for random networks) and sample contributions (in non-regular networks) of the average third cumulant of T6 (see Fig 4) tree topologies with fixed branch lengths. Ticks on the x-axis code for the lengths of the branches of the tree. The ordering of the indices in the tick labels is done in a top-to-bottom and left-to-right fashion. The sample contributions were computed as averages of 3 independent realizations of an assortative network with a geometric out-degree distribution. Bottom: Theoretical and sample contributions to the average third cumulant of T1 tree topologies with fixed branch lengths.