Synaptic shot-noise triggers fast and slow global oscillations in balanced neural networks

TL;DR

This work develops a complete mean-field theory (CMF) that explicitly incorporates synaptic shot-noise in sparse balanced inhibitory networks of quadratic integrate-and-fire neurons driven by external input. By transforming to a phase-oscillator framework and employing Kuramoto–Daido order parameters, the CMF derives macroscopic evolution equations that reveal global oscillations arising through two distinct mechanisms: cluster activation at low in-degree and drift-driven oscillations at high in-degree, with a re-entrant Hopf bifurcation line in the $(K, i_0/g_0^2)$ plane. Comparisons with diffusion (DA) and third-order (D3A) approximations show that DA fails in several regimes, while the CMF accurately captures both the asynchronous and oscillatory states, including hysteresis and coexistence regions. The results connect the slow and fast GO regimes to gamma rhythms observed in cortex and hippocampus and demonstrate how GO frequency can be tuned over a wide range by adjusting the external drive and inhibition, providing a powerful framework for understanding noise-driven coordination in neural circuits.

Abstract

Neural dynamics is determined by the transmission of discrete synaptic pulses (synaptic shot-noise) among neurons. However, the neural responses are usually obtained within the diffusion approximation modeling synaptic inputs as continuous Gaussian noise. Here, we present a rigorous mean-field theory that encompasses synaptic shot-noise for sparse balanced inhibitory neural networks driven by an excitatory drive. Our theory predicts new dynamical regimes, in agreement with numerical simulations, which are not captured by the classical diffusion approximation. Notably, these regimes feature self-sustained global oscillations emerging at low connectivity (in-degree) via either continuous or hysteretic transitions and characterized by irregular neural activity, as expected for balanced dynamics. For sufficiently weak (strong) excitatory drive (inhibitory feedback) the transition line displays a peculiar re-entrant shape revealing the existence of global oscillations at low and high in-degrees, separated by an asynchronous regime at intermediate levels of connectivity. The mechanisms leading to the emergence of these global oscillations are distinct: drift-driven at high connectivity and cluster activation at low connectivity. The frequency of these two kinds of global oscillations can be varied from slow (around 1 Hz) to fast (around 100 Hz), without altering their microscopic and macroscopic features, by adjusting the excitatory drive and the synaptic inhibition strength in a prescribed way. Furthermore, the cluster-activated oscillations at low in-degrees could correspond to the gamma rhythms reported in mammalian cortex and hippocampus and attributed to ensembles of inhibitory neurons sharing few synaptic connections [G. Buzsaki and X.-J. Wang, Annual Review of Neuroscience 35, 203 (2012)].

Figures (14)

• Figure 1: Population firing rates $\nu(t)$ versus time for $i_0=0.00055$: $K=10$ (a) and $K=210$ (b); $i_0=0.00033$ and $K=100$ (c) and $i_0=0.00027$ and $K=60$ (d). The black lines refer to network simulations with $N=80000$ neurons, the violet (red) lines to Langevin simulations with shot-noise (DA) with $N=20000$ (a-b) and $N=80000$ (c-d). The blue solid lines denote the results of the CMF approach (\ref{['eqGP03']},\ref{['eqGP12']}). All data refer to $g_0=1$. Details on the integration of the Langevin equations are reported in Appendix B.
• Figure 2: Population firing rates $\nu(t)$ versus time for $i_0=0.02$, $K=200$ (a) and $i_0=0.006$, $K=400$ (b). The black circles refer to network simulations with $N=80000$ (a) and $N=20000$ (b), the violet (red) solid lines to integration of the continuity equation \ref{['CE']} (FPE \ref{['fpe']}). The green solid lines denote the results of the integration of the GFPE including third order terms \ref{['gfpe']}. All data refer to $g_0=1$, the numerical integration of the partial differential equations has been performed by employing time splitting pseudo-spectral methods with a time step $\Delta t = 10^{-4}$ ms and with 128--512 Fourier modes (for more details see Appendix C).
• Figure 3: Relative error in the estimation of the firing rate $\nu$ for a stationary situation (asynchronous dynamics) computed with (\ref{['eqGP03']})--(\ref{['eqGP12']}) plotted versus the in-degree $K$ for $M=10$ (black line), $20$ (red line), $40$ (green line). The reference firing rate was estimated by employing $M=80$ and it is shown in rescaled units as a blue line. Parameters: $(i_0,g_0)=(0.006,1)$.
• Figure 4: Spectrum of the eigenvalues $\{\lambda_i\}$ for a stationary solution of the system (\ref{['eqGP03']},\ref{['eqGP12']}) for $(i_0/g_0^2,K)=(0.02,400)$ (a), and $(0.00055,10)$ (b). An enlargement is reported in the inset in (b). Black circles refer to the CMF, the red squares to the DA and the green pluses to the D3A.
• Figure 5: (a) Phase diagram for the QIF network in the plane $(i_0/g_0^2, K)$: the black (blue) solid line is the super-critical HB line obtained within the DA (D3A); the orange solid (dashed) line is the super- (sub-) critical HB line given by the CMF; the symbols refer to numerical estimations of the HBs and SNBs. The green (violet) circles denote HBs obtained by performing quasi-adiabatic simulations by varying $K$ ($i_0$) for constant $i_0$ ($K$) values; the cyan stars indicate SNBs. The dimension of symbols takes in account for the error bar in the transition point evaluation. The vertical red dotted line is the critical value defined in noi within the DA for the emergence of fluctuation-driven balanced asynchronous dynamics (for more details see the text). (b) The coefficient $\mathrm{Re}\,C_3$ in rescaled units at the critical points $K=K_{cr}$ versus $i_0/g_0^2$: CM orange solid line), DA (black line) and D3A (blue line). In all the simulation $g_0=1$.