Fig 1.
Model architecture and dynamic modes: (A) Diagram of the recurrent excitation–inhibition network. External input rext has a background part r0 and an extra stimulus part r1. (B, C, E, F) Examples of dynamic modes for AS, Cri, SS, and P states. In each case, a period of the local field potential and the corresponding spike raster plot of Exc neurons, the distributions of the Pearson correlation coefficient (PCC) and coefficient of variance (CV) of inter-spike intervals, and the population activity autocorrelation (AC) are shown. (D) Examples of avalanche size distributions for AS, Cri, and SS states. Here, the background input strength is r0 = 0.8/ms, and synaptic parameters are for AS, Cri, SS, and P states, respectively.
Fig 2.
Trial-to-trial variability in ongoing and stimulus states: (A–C) Upper panels: the local field potentials (LFPs) of five single trials (labeled in different colors) and the all-trial-averaged LFP (bold black line). Lower panels: the cross-trial variance of LFP. (D–F) Upper panels: raster plots of 300 Exc neurons in five trials (labeled in different colors). Lower panels: trial-averaged firing rate and Fano factor (FF) (flanking traces are the value of ±std, where the std is due to the different sampling neuron groups used in the FF computation). The insets compare the pre- and post-stimulus FF using boxplots. Dashed black vertical lines indicate the stimulus onset time. The results of subcritical (left), critical (middle), and supercritical (right) dynamics are for parameters , respectively. The background input rate is r0 = 0.55/ms, and the stimulus strength is r1 = 0.2/ms.
Fig 3.
Event-related potential (ERP), gamma frequency modulation, and criticality preservation induced by stimuli with different strengths.
Stimulus starts at t = 0 and ends at t = 600 ms. The blue, green, and red colors in (A–E) represent post-stimulus strengths r1 = 0.1, 0.35, and 0.6/ms, respectively. (A) Raster plots of 500 neurons in a trial. (B) Distributions of avalanche size S, avalanche duration T, and average size 〈S〉 for duration T in the post-stimulus period. The top horizontal purple lines indicate the ranges of estimated power-law distributions for the case of r1 = 0.35/ms. For comparison, we also show the avalanche distributions in the pre-stimulus period (cyan curves) and transient period within 100 ms after the stimulus onset (black curves) for the case of r1 = 0.1/ms. These properties are similar to those for other stimulus strengths r1. (C) ERPs under different input strengths. (D) Transient network firing rates. (E) Power spectrum density (PSD) of post-stimulus local field potential (LFP), measured 100–600 ms after stimulus onset. The inset shows the peak frequencies for different input strengths (dashed-dotted, solid, and dotted lines represent the subcritical, critical, and supercritical cases, with , respectively). (F–H) Time evolution of the powers of different LFP oscillation frequencies for different input strengths. Dots at t = −200 ms indicate the post-stimulus peak frequency. The background input is r0 = 0.3/ms. The synaptic parameter is
for (A, B) and
for (C–H).
Fig 4.
Critical avalanches in networks with different sizes.
We simulate networks subjected to noisy inputs with a constant strength rin that linearly scales with the network size and exhibits critical dynamics (). We show the distributions of avalanche size S, avalanche duration T, and the average size 〈S〉 under duration T. Horizontal purple lines indicate the ranges of estimated power-law distributions. From left to right, the network sizes are N = 2500, 5000, 10,000, 15,000. The input strengths are rin = 0.9, 1.8, 3.6, 5.4/ms, to maintain the scale condition rin~O(N) for E–I balanced networks. Avalanches are measured with adapted time bin Δt = Tm = 0.11, 0.03, 0.02, 0.013 ms respectively.
Fig 5.
Further measurement of critical avalanches using different sizes of time bins.
We further measure avalanches in networks in Fig 4 using different time bins Δt. The distributions of avalanche size S, avalanche duration T, and the average size 〈S〉 under duration T are shown. (A) Measurement of avalanches in networks with sizes N = 2500, 5000, 10,000, 15,000 using fixed bin Δt = 0.02ms. (B) Measurement of avalanches in a network with size N = 15,000 using different time bins Δt = 0.4Tm, 0.8Tm and 1.2Tm, where Tm = 0.013 ms. The horizontal lines on top indicate the ranges of estimated power-law distributions of the corresponding cases.
Table 1.
Comparison of dynamic features at different dynamic states.
Fig 6.
Mean-field theory prediction of the response dynamics: (A) Excitatory firing rates vs. input strength at the AS state, with . The red/blue markers represent network simulation results under noisy/constant inputs. The black curve represents the field model fixed-point estimation under fixed σα parameters indicated at the end of this caption. The purple curve represents the result under σα parameters estimated with different rin values. (B) The real part of the eigenvalue evaluated at equilibrium. The purple curve indicates the zero value (i.e., the deterministic Hopf bifurcation points), and the black curve corresponds to the case where the real part of the eigenvalue is −0.05. (C) The prediction of the population oscillation frequency f = ω/2π, where ω is the imaginary part of the eigenvalue at equilibrium. Only results above the effective critical black line in (B) are shown. The inset shows the frequencies for input strengths rin = 0.4, 0.65, 0.9/ms (solid and dotted lines correspond to the critical and supercritical cases with
), similar to the inset of Fig 3E. (D) The linear noise prediction of Var(VE) for different
and rin values (before the Hopf bifurcation line). The black curve is the critical line in (B). (E) Plot of Var(VE) vs. input strength rin for subcritical and critical states with
. (F) Same as (E) but for quantity
. The field equation parameters are σe = 3.2, σi = 3.8, β = 0.2.