Fig 1.
Connectivity properties synthesized by the generative model.
(A) The generated (dots) and theoretical (solid lines) in-degree distributions with different hybrid parameters q; q = 0 and q = 1 correspond to the pure Poisson and log-normal distributions, respectively. (B) The generated (dots) connection probabilities as a function of inter-neuron distance with different decay constants τD. The decay constants of fitted exponential curves (solid lines) are 5.57 ± 0.05, 11.17 ± 0.04, 19.25 ± 0.15 (95% confidence), respectively. (C) The generated (dots) connection probabilities as a function of shared pre-synaptic common neighbors with different common neighbor coefficients aΓ; the error bars show one SEM of 65 trials. The solid colored lines are the linear fit to the data. For each trial, the results are calculated from 2000 samples; each sample consists of 12 neurons randomly selected from a circular region in the network containing 2000 excitatory neurons to approximate the sampling methods used in [3]. The results are not sensitive to the diameter of the sampling area. The solid black line is the expected curve from the E-R random networks. (D) The mean clustering coefficient (CC) of the network (colored and black dots) increases as the common neighbor coefficient aΓ increases. The solid line is the fitted exponential function (0.22 − 0.089e−aΓ/7.04; R2 = 0.996). The convergence of mean CC for 4 example aΓ values (colored dots) after iterations of the algorithm is shown in the inset. (E) The generated (dots) and theoretical (solid line; log-normal) distributions of unitary EPSP magnitudes. (F) The average incoming connection strength and the in-degree of individual excitatory neurons (dots) follow the inverse square root scaling (solid line fitted; R2 = 0.917); note the plot is on a log-log scale. All the empirical data sets or points in (A-B) and (D-F) show the results of individual realizations of the generative model with default parameter values (unless otherwise stated).
Fig 2.
Emergent higher-order connectivity features.
(A) Triad motif counts. Within each trial, the motif counts are averaged from 1000 randomly sampled quadruplets using the brain connectivity toolbox [28] and 100 control networks are generated to normalize the motif counts. The error bars show one SEM from 10 trials. The mean clustering coefficient is shown above each motif, with the non-zero ones highlighted in red. The asterisks denote the relative motif counts that are statistically significantly different from one (red asterisks p < 0.001, black asterisks p < 0.034; Holm step-down adjusted p-values [29]). (B) Left y-axis (blue): normalized rich-club coefficients Φ(k), which measures the extent to which the neurons with total degrees > k in a network are over-connected to each other. Within each trial, 100 control networks are generated to normalize the coefficients. The coefficient peaks at 1.60, which is significantly higher than 1 as given by the control networks (p < 0.001, one-sample t-test). Right y-axis (red): dynamical importance (averaged across neurons with the same total degree k within each trial), defined as the fractional change (%) in the largest eigenvalue of the adjacency matrix of the network upon removal of the neuron from it. The solid lines show the averages over 10 trials and the shaded areas show one SEM. Data for k < 200 is omitted from the plot because neurons with total degree k < 200 are sparse. (C) The rich-club curves with different common neighbor coefficients aΓ. Inset: the maximum rich-club coefficient between k = 500 and k = 600 are positively correlated with the common neighbor coefficient. Both motif and rich-club results are calculated for a circular region in the network containing 2000 excitatory neurons to avoid artefacts from the periodic boundaries.
Fig 3.
Transition of the circuit activity states induced by changing the I-E ratio ζ.
(A) Mean firing rate of the excitatory population shows a phase transition around ζ = ζc = 3.375. The red solid line and blue dashed line are the two power functions fitted to the data points marked by red squares and blue circles, in the form of and
, respectively. The fitted coefficients are a1 = 25.47 ± 5.17 (0.95% confidence bounds), b1 = −3.535 ± 0.392, c1 = −23.72 ± 6.10, a2 = 2.199 ± 0.045, and b2 = −2.455 ± 0.116. The black triangles are the data points without fitting. The error bars show one SEM. Inset: the same plot but with the y-axis on a linear scale. (B-E) Snapshots of spiking patterns emerging from our local cortical circuit model with different I-E ratio ζ values. Black dots denote that one spike has been emitted from the excitatory neuron during a 5 ms period, and red dots denote two spikes. The cyan circles in (C) and (E) show one standard deviation of the fitted Gaussian firing rate profile and the blue curves show the trajectory of the centers of the pattern in the previous 40 ms. For the state shown in (E), the pattern appears intermittently and exhibits jumping behavior with a variable propagation trajectory.
Fig 4.
Near critical dynamics around the transition of the circuit activity states.
(A) Susceptibility of the excitatory population at different I-E ratios ζ. 1 ms time bin is used for the calculation; the results are not sensitive to the choice of bin size. The error bars show one SEM of 10 trials. (B) Branching parameter σ* under subsampling as a function of the normalized bin size at ξ/ξc = 1. The shaded area shows one SEM of 10 trials.
Fig 5.
Stability analysis of the homogeneous, asynchronous state of the spatially-extended spiking circuit.
(A) Mean firing rate of the excitatory population in the simulation of our local cortical circuits (dots) and the rate of spatially uniform activity from the analysis (solid line) with the optimal correction factor. The dashed line shows the critical point ζ = ζc. (B) Eigenvalues with the largest real parts of the network’s dynamics in response to spatially periodic perturbations (spatial Fourier mode k = (1, 1)), with different I-E ratios ζ and the optimal correction factor. The eigenvalues in this parameter regime are real, with positive values emerging as the I-E ratio ζ decreases below the critical point ζc (dashed line), indicating a Turing bifurcation. The dominant spatial eigenmode has wave vector k = (1, 1), indicating that a single pattern is formed, consistent with that observed in the full spiking circuit.
Fig 6.
Common neighbor coefficient aΓ can induce the transition from the asynchronous state to the localized propagating wave state.
(A-C) Snapshots of spiking patterns in the local cortical circuit model with different common neighbor coefficient aΓ values. These patterns are visualized in the same way as in Fig 3B–3E. (D) Phase diagram constructed based on the pattern detection rate (the percentage of time frames where a localized wave is detected) and the average firing rate of the excitatory population. The States 1, 2, 3 and 4 correspond to the asynchronous state, the transition state, the localized propagating wave state and the global plane wave state, respectively. The circle denotes the working point of our local cortical circuit model. (E) Phase diagram as in (D) but with the spike frequency adaptation removed, where the different states are shown by red boundary lines. The state I, II, and III correspond to the asynchronous state, the state with localized activity patterns that barely propagate (see S4 Video), and the global plane wave state, respectively. The excitatory neurons without firing a spike during the 10 second simulation time are detected as quiescent neurons.
Fig 7.
Properties of synaptic inputs to individual neurons in the local cortical circuit model.
(A) Time series of total excitatory and inhibitory inputs received by individual excitatory neurons over 1 s in the different dynamic states (with ζ/ζc = 0.8, 1.0, 1.3 corresponding to the propagating wave state, the transition state and the asynchronous state, respectively; see Fig 3). (B) Average cross-correlation (xcorr) between the total E and I currents into excitatory neurons in the different dynamic states. The shaded area shows one SEM of 10 trials. (C) The convex hulls of the mean and standard deviation data points (pooled from 10 trials) of the total input current into individual neurons in the different dynamic states, denoted by the same color-coding as in (B). The region to the left of vertical dashed line is where the mean input current is less than the threshold current. The region above the inclined dotted line is where the mean plus one standard deviation of the input current is above the threshold current. (D) The relative errors of estimating the total standard deviation σI (and mean μI) of the recurrent currents with the sum of individual components as given by the variance decomposition (Variance decomposition analysis), averaged over the excitatory neurons and over a period of 50 ms within each trial. (E) The contributions to the variance in recurrent currents from various sources are shown across circuits with different I-E ratio ζ. The numbers above the bars are the corresponding total variances averaged over excitatory neurons (nA2).
Fig 8.
Emergent, heterogeneous population coupling in the dynamical working regime of the local cortical circuit model.
(A) Spike-triggered population rate (stPR) for five representative neurons in a single trial. (B) stPR for the same five neurons but with spikes shuffled. Inset: distributions of stPR coefficients pooled from 10 trials before (black) and after (red) spike shuffling. The coefficients are defined as the stPR values at zero time lag normalized by the median value of the shuffled data. For each trial, 10 samples of 66 excitatory neurons are randomly chosen to match the sample size used in experimental studies [12]. Neurons are sampled from a circular area with a radius of 25 grid units (about 125 ¼m) for approximating the spike detection range of a single shank of silicon electrode array used in [12]. The results are not sensitive to the size of the sampling area or the number of sampled neurons. (C-D) stPRs for ζ = 1.15 and ζ = 0.7, respectively. Insets: shuffled stPRs. For ζ = 1.15, the model is shifted away from the working regime into the regular propagating wave state; while for ζ = 0.7, it is shifted into the irregular, asynchronous state.
Fig 9.
Rich-club connectivity is related to the heterogeneous population coupling.
(A) An example of the rich-club connectivity among high total degree (k > 600) excitatory neurons (red dots). Only a 10% random subset of the synaptic connections are plotted (lines) for visualization purpose, with a probability of 1/Φ(k = 600) colored in blue and otherwise in yellow to illustrate the proportion between the number of connections expected from their total degrees alone (blue) and the number of extra connections due to the rich-club connectivity (green). The circle shows the range within which both the rich-club and stPR coefficients are calculated as used in Figs 2B and 8. (B) The rich-club neurons have high stPR coefficients. Left y-axis (blue): the blue dashed line shows the stPR coefficient z scores for the neurons with a total degree > k averaged over 10 trials and the shaded area shows one SEM. Within each trial, 10 random 66-neuron samples (as in Fig 8) are collected to calculate the k-dependent stPR coefficient z scores. Right y-axis (red): the red dashed line shows the normalized rich-club coefficient curve from Fig 2B and the shaded area shows one SEM. Inset: normalized rich-club coefficient Φ(k) (y-axis) versus k-dependent stPR coefficient z score (x-axis) from 10 realizations of the network (shown with 10 different colors). (C) Distributions of stPR coefficients pooled from 10 trials before (black) and after (red) spike shuffling from the local cortical circuits with different common neighbour factors aΓ. For comparison, the total count (area under curve) and bin size are the same for each distribution. (D) stPR coefficients before shuffling, normalized by the SD of stPR coefficients after shuffling (i.e., coefficients of control case) and plotted in the same k-dependent manner as in (B). Each curve shows the average of 10 trials.
Fig 10.
Dynamic properties of propagating wave patterns emerging in the dynamical working regime of the local cortical circuit model.
(A) Raster plots of 50 neurons randomly sampled from a 10-by-50 grid-unit region, sorted according to their y-coordinates, showing two travelling patterns moving in opposite directions at different time points. The solid blue line is the linear fit to the first spikes at each y-position (y = 0 denoting the y-center of the sampling region). (B) The distribution of the pattern propagation speed (dots) with the log-normal fit (blue solid line). The error bars and shaded area show one SEM.
Fig 11.
Emergent precise spiking structures.
(A) Schematic of a spike triplet that is described by two inter-spike intervals. (B) The probability distribution of two inter-spike intervals from a representative neuron trio. The black square denotes the precisely repeating triplets occurring around ± 10 ms of the mode. (C) Occurrence of precisely repeating triplets peaks shortly after the start of activated states. The time from activated state (AS) onset is calculated based on the first spike of the triplet. The black, red, and green lines are the trial-averaged normalized probability density functions (PDFs) for the original data, shuffled data [14], and data dithered with different window sizes (method L in [56]), respectively. For each trial, 50 samples of 50 excitatory neurons are randomly chosen from a 10-by-50 grid-unit region (about 50-by-250 μm) to approximate the spike detection region of two neighboring shanks of silicon electrode array used in the experimental study by [14]. The results are not sensitive to the width of the rectangular sampling area. Within each sample, 50 neuron trios are randomly chosen to get the original and shuffled/dithered spike triplet counts.
Table 1.
Summary of model definition.
Table 2.
Summary of neuron parameters (in a consistent system of units; see Eq 4).