Fig 1.
Network construction and search.
A: Our networks were constructed with 4000 clustered excitatory and 1000 unclustered inhibitory units. Probabilities of connection (those from excitatory to excitatory units, and from inhibitory to inhibitory units) were inferred from experimental literature and determined via grid search (those from excitatory to inhibitory units and vice versa). Simulation runs began with 30ms of 20Hz Poisson input onto a subset of 500 units. B: Synaptic weights followed a log-normal (heavy-tailed) distribution. Synapses were conductance-based, so weights are in units of nanosiemens. Connections originating from inhibitory units were 10x stronger than those from excitatory units. C: For each network, we defined 50 clusters in total and randomly assigned each excitatory unit to two of these clusters. The wiring probability between units within the same cluster is twice that of units in different clusters. This resulted in heterogeneously-sized clusters. Here we show the cluster size distribution (in counts) for 500 networks. D: Visualization of a subset of 300 clustered excitatory units in our network.
Fig 2.
Grid search yields networks with dynamics resembling neocortex.
A: We performed two rounds of grid search for the topological parameters that yielded consistent low-rate, critical, and asynchronous dynamics. The first search was at a lower resolution to narrow down our region of interest, and the second was at a finer resolution. B: One scoring metric we used was branching. The branching parameter [13] is a proxy for criticality. It measures the ratio of active descendant units to active ancestor units. A branching value of 1 indicates a balanced (or critical) network, which is the value we optimized for. C: Distribution of the interspike interval coefficient of variation for individual neurons. D: Distribution of the interspike interval coefficient of variation for networks. E: L2 error between autocorrelation and normalized autocovariance.
Fig 3.
Simulations on the same network topologies yield sustained or truncated runs.
A: A raster plot of a single complete 1000ms simulation on one of our networks. Excitatory units are numbered 1-4000 on the y axis, and inhibitory units are 4001-5000. B: A raster plot of a single truncated simulation (700ms) on the same network with the same input. C: Distribution of truncation times in ms for all truncated runs. D: Instantaneous rate across time for the simulations in rasters A and B, binned at 10 ms. Blue: sustained run; orange: truncated run; solid line: excitatory units; dashed line: inhibitory units. E: Instantaneous branching across time for the simulations in rasters A and B, binned at 10 ms. Same legend conventions as in D. F: Instantaneous Van Rossum distance across time for the simulations in rasters A and B, binned at 50 ms. Same legend conventions as in D.
Fig 4.
Score distributions for sustained and truncated runs.
A: Distributions of spike rate scores for completed (blue), late-truncated (> 500 ms duration, green), and early-truncated (< 100 ms duration, orange) simulations. B: Distributions of criticality scores (branching parameter) for completed (blue) and truncated (orange) simulations. C: Distributions of asynchrony scores (Van Rossum spike distance) for completed and truncated simulations. D: Same asynchrony score data as in C, with completed and truncated simulations now separated along the y axis by their total durations. Each dot indicates an individual simulation.
Fig 5.
A: The synaptic graph is the ground-truth topology of our networks. Based on spiking activity during each simulation, we construct a series of active synaptic subgraphs—one for each time bin. These are graphs made of units which spiked in that bin, connected via the same edges as in the synaptic graph. We infer a single functional graph from whole-trial spiking activity using confluent mutual information—these graphs represent the functional connectivity of the network for each simulation trial. The intersection of the functional graph with the active subgraph for a given time bin yields the recruitment graph for that time bin. B: The three triangle motifs we examine—fan-in, fan-out, and middleman—are isomorphic by rotation. When calculating motif clustering, the choice of reference node is key. C: Calculation of the clustering coefficients of the different triangle motifs on weighted directed graphs, as defined in Fagiolo 2007 [49]. The clustering coefficient is defined as the ratio of the actual to the possible motif counts.
Fig 6.
A: The left-hand panel shows the reciprocity (ratio of reciprocal connections to total connections) of functional graphs plotted as a function of their density (ratio of existing to possible connections). Data points for sustained runs are plotted in black and form a tight cluster, whereas those for truncated runs are varied. Truncated runs are colored by the ratio of inhibitory to excitatory spike rates. The right-hand panel shows the minimum distance (in reciprocity vs. density coordinate space for functional graphs) between each truncated run and a sustained run as a function of truncation time. Truncated runs which have greater than 200ms duration level off in their minimum distance. Thus, past a certain threshold, the difference between truncated and sustained runs’ density and reciprocity is not related to the run duration. B: The left-hand panel shows the reciprocity (ratio of reciprocal connections to total connections) of recruitment graphs plotted as a function of their density (ratio of existing to possible connections). Sustained runs are plotted in black and form a neat relationship between density and reciprocity and occur within a limited range of values. As in panel A, truncated runs are more diffuse. The color of each point indicates the ratio of inhibitory to excitatory spike rates. And also as in panel A, the right-hand panel shows the minimum distance between each truncated run and a sustained run (in reciprocity vs. density coordinate space) as a function of truncation time, this time for recruitment graphs. Truncated runs which have greater than 200ms duration level off in their minimum distance.
Fig 7.
Standard network triplet motifs.
A: Comparison of triangle motif clustering propensities of the three isomorphic motifs on sustained and truncated runs across all networks. B: Trajectories of all runs on a sample network in 3-dimensional isomorphic motif space. Truncated runs have a larger spread of trajectories along with variation in the ratios of inhibitory to excitatory spike rates. However, sustained runs are consistent in their spike rate ratios. C: Trajectories of all sustained runs alone, on axes of the identical scale as in panel B. D: Example trajectory of a single run on the same network, now enlarged (from inset in panel C). The network begins away from the area of its eventual cyclic trajectory, and the 30ms of Poisson input at the beginning of the run drives it towards this region. E: Example trajectory from panel D shown as fan-out propensity vs fan-in propensity. F: Example trajectory from panel D shown as fan-out propensity vs middleman propensity. G: Example trajectory from panel D shown as fan-in propensity vs middleman propensity.
Fig 8.
Markov comparisons between sustained and truncated runs on standard networks.
A: Probabilities of state dominance of a triplet motif in sustained (left) and truncated (right) runs. B: Second order state probabilities for sustained (left) and truncated (right) runs. C: Second order conditional state probabilities for sustained (left) and truncated (right) runs. D: Expectation of hitting time for Markov model of state dominance transitions in sustained (left) and truncated (right) runs. E: Visualization of Markov matrix for state dominance in complete (left) and truncated (right) runs.
Fig 9.
Networks with increased weights.
Networks have the same structure as those seen in Figs 4 and 5, but all edge weights have been increased by 1.5 times their original values. A: Left, density (ratio of existing to possible connections) for synaptic, functional, and recruitment graphs. Right, reciprocity (ratio of reciprocal to all existing connections) for synaptic, functional, and recruitment graphs. B: Clustering propensity for isomorphic triangle motifs on increased-weight-graph simulations. The y-axis is scaled to match that of Fig 7A (clustering propensities on original graphs) and Fig 10B (clustering propensities on unclustered ER graphs). C: Probabilities of dominance of each triangle motif. The dominant motif at a time point is given by the maximum of mean middleman, mean fan-in, and mean fan-out across units. D: Second order motif state probabilities for progression of temporal recruitment graphs. E: Probabilities for each motif to follow a given second order motif. F: Hitting times for each state for the Markov process defined by motif transition probabilities. G: Trajectories of all complete runs on a sample network in 3-dimensional isomorphic motif space. In blue are the runs on the network with its original weights, in orange are the runs on the same network with weights increased. H: Markov Matrix for transition probabilities between motifs.
Fig 10.
Unclustered (Erdős-Renyi) networks.
A: Left, density (ratio of existing to possible connections) for synaptic, functional, and recruitment ER graphs. Right, reciprocity (ratio of reciprocal to all existing connections) for synaptic, functional, and recruitment ER graphs. B: Clustering propensity for isomorphic triangle motifs on ER graph simulations. The y-axis is scaled to match that of Fig 7A (clustering propensities on original graphs) and Fig 9B (clustering propensities on graphs with 1.5 times increased weights). C: Probabilities of dominance of each triangle motif. The dominant motif at a time point is given by the maximum of mean middleman, mean fan-in, and mean fan-out across units. D: Second order motif state probabilities for progression of temporal recruitment graphs. E: Probabilities for each motif to follow a given second order motif. F: Hitting times for each state for the Markov process defined by motif transition probabilities. G: Trajectories of all runs on a sample ER network in 3-dimensional isomorphic motif space. All runs reached completion. H: Markov Matrix for transition probabilities between motifs.