A theory for self-sustained balanced states in absence of strong external currents

TL;DR

An alternative mechanism based on short-term synaptic depression acting on excitatory-excitatory synapses, which dynamically balances the network activity without the need of external inputs is investigated, extending the balanced network paradigm and suggesting that internal synaptic adaptation may play a key role in shaping neural activity.

Abstract

Recurrent neural networks with balanced excitation and inhibition exhibit irregular asynchronous dynamics, which is fundamental for cortical computations. Classical balance mechanisms require strong external inputs to sustain finite firing rates, raising concerns about their biological plausibility. Here, we investigate an alternative mechanism based on short-term synaptic depression (STD) acting on excitatory-excitatory synapses, which dynamically balances the network activity without the need of external inputs. By employing numerical simulations and theoretical investigations we characterize the dynamics of a massively coupled network made up of $N$ rate-neuron models. Depending on the synaptic strength $J_0$, the network exhibits two distinct regimes: at sufficiently small $J_0$, it converges to a homogeneous fixed point, while for sufficiently large $J_0$, it exhibits Rate Chaos. For finite networks, we observe several different routes to chaos depending on the network realization. The width of the transition region separating the homogeneous stable solution from Rate Chaos appears to shrink for increasing $N$ and eventually to vanish in the thermodynamic limit. The characterization of the Rate Chaos regime performed by employing Dynamical Mean Field approaches allow us on one side to confirm that this novel balancing mechanism is able to sustain finite irregular activity even in the thermodynamic limit, and on the other side to reveal that the balancing occurs via dynamic cancellation of the input correlations generated by the massive coupling. Our findings show that STD provides an intrinsic self-regulating mechanism for balanced networks, sustaining irregular yet stable activity without the need of biologically unrealistic inputs. This work extends the balanced network paradigm, offering insights into how cortical circuits could maintain robust dynamics via synaptic adaptation.

Figures (14)

• Figure 1: Dynamical Regimes in Finite Networks. The panels show the time evolution of the neuronal firing rates for four representative excitatory (red) and inhibitory (blue) neurons. (A) Homogeneous fixed-point dynamics for low coupling $J_0 = 0.1$. Neuronal activity rapidly converges to homogeneous steady-state values. (B) Transition regime at intermediate coupling $J_0 = 0.87$. Neuronal activity exhibits heterogenoeus solutions, either stationary or oscillatory. (C) Chaotic dynamics for strong coupling $J_0 = 1.5$. Neuronal rates display broadband irregular fluctuations. Dashed lines in panel (A) indicate the corresponding mean-field predictions for the population-averaged activity \ref{['eq:self_consistent_equations']}. Simulations were performed for network size $N = 10^4$ and no external current ($I_0 = 0$).
• Figure 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 \ref{['eq:self_consistent_equations']}. The solutions obtained in the thermodynmic linit ($\phi_{\infty}^E$, $\phi_{\infty}^I$ and $w_\infty$) are reported as dashed lines. Here we set $J_0 = 0.1$ and $I_0 = 0$. (B) Heatmap showing the combined effect of $(I_0,J_0)$ on the firing rate of the excitatory neurons in a finite network with $N = 10,000$ obtained by using the mean-field predictions \ref{['eq:self_consistent_equations']}. The dashed lines indicate the cuts explored in (C) and (D). (C) Predicted excitatory firing rates at fixed $J_0 = 0.1$ by varying $I_0$ for different network sizes. As $N \to \infty$ the effect of the external current becomes negligible. (D) Same as in (C) by fixing $I_0 = 0.3$ and varying $J_0$, these results show the independence of the asymptotic solution from $J_0$.
• Figure 3: Stability of the Homogeneous Stationary Solutions for Homogeneous Perturbations. (A) Real part of the leading eigenvalue of the Jacobian matrix $\boldsymbol{DF}_{hom}$ as a function of the synaptic strength $J_0$ for different values of the external DC current $I_0$. For this panel $N = 10,000$ (B) Same as in A as a function of $J_0$ for fixed $I_0 = 2$ and increasing network size $N$.
• Figure 4: Stability of the Homogeneous Stationary Solutions for Heterogenous Perturbations. (A) Spectrum of the full Jacobian matrix $\boldsymbol{DF}_{het}$ (blue dots, $\lambda_C$) compared with predictions from the random matrix approximation. The dashed black circle corresponds to the radius $r$\ref{['eq:radius']}, the orange and green markers indicate the predicted outliers $\lambda_{\mathrm{out}}$ and $\lambda_Q$, respectively. In this panel $J_0=0.1$ and $N=5000$. (B) Maximum real part of the eigenvalue spectrum of $\boldsymbol{DF}_{het}$ as a function of $J_0$, compared with the radius $r$ predicted by the random matrix approximation for various values of $I_0$. (C) Critical coupling $J_c$ as a function of network size $N$ for different values of $I_0$.
• Figure 5: 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 $J_0 = 1.0$ and $I_0 = 0$. (D) Theoretical prediction for the average input current (main) and the standard deviation (inset) as a function of the synaptic coupling $J_0$. These correspond to the self-consistent solutions of Eqs. \ref{['eq:media_het']} and \ref{['eq:std_het']}. For all the panels in the figure we have considered $N = 10,000$.