Fig 1.
Dynamical regimes in finite networks.
The panels show the time evolution of the neuronal firing rates for four representative excitatory (orange) and inhibitory (blue) neurons. (A) Homogeneous fixed-point dynamics for low coupling J0 = 0.1. Neuronal activity rapidly converges to homogeneous steady-state values. (B) Transition regime at intermediate coupling J0 = 0.87. Neuronal activity exhibits heterogeneous solutions, either stationary or oscillatory. (C) Chaotic dynamics for strong coupling J0 = 1.5. Neuronal rates display broadband irregular fluctuations. Dashed lines in panel (A) indicate the corresponding mean-field predictions for the population-averaged activity (5). Simulations were performed for network size N = 104 and no external current (I0 = 0).
Fig 2.
Finite-size characterization of the homogeneous stationary solutions.
(A) Stationary firing rates and synaptic efficacy as a function of the system size N. Symbols correspond to numerical simulations while the solid line shows the self-consistent mean field prediction (5). The solutions obtained in the thermodynamic limit (,
and
) are reported as dashed lines. Here we set J0 = 0.1 and I0 = 0. (B) Heatmap showing the combined effect of
on the firing rate of the excitatory neurons in a finite network with
obtained by using the mean-field predictions (5). The dashed lines indicate the cuts explored in (C) and (D). (C) Predicted excitatory firing rates at fixed J0 = 0.1 by varying I0 for different network sizes, in the inset are reported the corresponding stationary excitatory input currents
. As
the effect of the external current becomes negligible. (D) Same as in (C) by fixing I0 = 0.3 and varying J0, these results show the independence of the asymptotic solution from J0.
Fig 3.
Stability of the homogeneous stationary solutions for homogeneous perturbations.
(A) Real part of the leading eigenvalue of the Jacobian matrix as a function of the synaptic strength J0 for different values of the external DC current I0. For this panel
(B) Same as in A as a function of J0 for fixed I0 = 2 and increasing network size N.
Fig 4.
Stability of the homogeneous stationary solutions for heterogeneous perturbations.
(A) Spectrum of the full Jacobian matrix (blue dots,
) compared with predictions from the random matrix approximation. The dashed black circle corresponds to the radius r (25), the orange and green markers indicate the predicted outliers
and
, respectively. In this panel J0 = 0.1 and
. (B) Maximum real part of the eigenvalue spectrum of
as a function of J0, compared with the radius r predicted by the random matrix approximation for various values of I0. (C) Critical coupling Jc as a function of network size N for different values of I0.
Fig 5.
DMF characterization of the chaotic regime.
(A) Time traces of firing rates (orange) and
(blue) for a subset of neurons in a network of size
at J0 = 1.5 and I0 = 0. The inset shows the time traces of the synaptic efficacies for the same excitatory neurons of the main panel. (B) Distribution of total input currents
across neurons (main panel) for excitatory (orange) and inhibitory (blue) populations. The inset shows the corresponding firing rate distributions
. (C) Autocorrelation function (ACF) of the total input currents for excitatory and inhibitory populations. Solid lines correspond to DMF predictions and symbols to direct simulations. The inset illustrates a fit of the ACF with the function
(black dashed lines). (D) Average firing rates, (E) Total input currents, (F) Input variances and (G) Decorrelation times as a function of J0 for different values of the external current I0 (color-coded). In panels (D-G) inhibitory (excitatory) populations are depicted with solid (dashed) lines.
Fig 6.
Rate chaos approaching the thermodynamic limit.
(A) Average firing rates and synaptic efficacy [w] as a function of network size N. Symbols refer to direct network simulations, while solid lines represent DMF predictions. (B) Variances
of the total inputs for increasing N predicted by DMF theory. (C) Autocorrelation function of the excitatory synaptic input for different N. Inset: Decorrelation times
versus N. (D) |AE|,|AI| versus N. The magenta dashed line indicates a power law decay N−1/2. For this figure all simulations were performed by averaging a time interval
, with I0 = 0 and J0 = 1.5.
Fig 7.
Correlations in the balanced regime.
(A) Population averaged Pearson correlation coefficients among the partial input currents versus N. (B) Population averaged Pearson coefficients
of the total input currents. Inset: Correlation coefficients RE and RI of the excitatory and inhibitory firing rates versus the system size N. The red dashed line refer to a power law decay 1/N. The data are averaged over 10 different realizations of the random network, and over a time interval t = 300, after discarding a transient of duration 500. For this figure I0 = 0 and J0 = 1.5.
Fig 8.
Two distinct bifurcation mechanisms driving the instability of the homogeneous fixed point.
(A) Hopf bifurcation: two complex conjugate eigenvalues crosses the imaginary axis. (B) Zero-frequency bifurcation: One real eigenvalue crosses zero, leading to a stationary heterogeneous solution. In the insets in (A-B) are reported the firing activity of few excitatory (inhibitory) neurons above the corresponding transition displayed in orange (blue). (C) Distribution of the input currents for the excitatory population in a network simulation with heterogeneous fixed point (blue shaded histogram) and the Gaussian theoretical prediction (red line). Inset: Corresponding distribution of the synaptic efficacies. For panels A-C) we have used J0 = 1.0 and I0 = 0. (D) Theoretical prediction for the average input current (main) and the standard deviation (inset) as a function of the synaptic coupling J0. These correspond to the self-consistent solutions of Eqs (53) and (54). For all the panels in the figure we have considered .
Fig 9.
Lyapunov characterization of the routes to chaos.
(A–D) Main Figures: Maximal LE as function of J0 with I0 = 0. After the fixed point loses stability at
a transition regime emerges -shaded area- where, depending on the network realization different routes to chaos can be identified. In the inset of each figure we show the first and second largest LEs calculated with a higher resolution in J0 with the aim of characterizing the transition region towards chaos.
Fig 10.
Fixed point regime in the spiking network.
(A) Raster plot of a subset of excitatory (orange) and inhibitory (blue) neurons showing tonic firing patterns. (B) Sample synaptic input to two neurons in the excitatory and inhibitory populations (solid line) compared with the mean-field rate model prediction (dashed line). (C) Maximum real eigenvalue of the heterogeneous perturbation Jacobian (symbols) and the random matrix theory prediction (solid lines) for varying values of external input I0. (D) Critical coupling strength Jc estimated from the generalization of the Girko’s law as a function of network size N. For this figure J0 = 0.1, I0 = 1.2,
,
.
Fig 11.
Chaotic regime in the spiking network.
(A) Raster plot of a subset of excitatory (orange) and inhibitory (blue) neurons showing bursting firing patterns. (B) Distribution of firing rates (main panel) in the excitatory (orange) and inhibitory (blue) populations. the inset displays the distribution of the coefficient of variation CV. (C) Two samples of neuronal input from the spiking network (solid lines) and average calculated using rate model (dashed lines). (D) Input autocorrelation function for excitatory and inhibitory populations. Squares correspond to spiking network simulations, circles to rate model and solid lines to DMF predictions. For this figure J0 = 0.4, I0 = 1.2,
,
.
Table 1.
Values of the employed parameters.