Low dimensional dynamics of a sparse balanced synaptic network of quadratic integrate-and-fire neurons

TL;DR

The paper tackles the kinetics of a sparse balanced network of $N$ pulse-coupled quadratic integrate-and-fire neurons driven by an effective shot noise, where the diffusion approximation fails in biologically relevant regimes. It develops a complete mean-field framework and a two-circular-cumulant (2CC) reduction in the genuine phase, yielding a closed 2D system for $(\varkappa_1,\varkappa_2)$ with a rescaled time $\tau=K^{1/4}\sqrt{i_0}\,t$ and firing-rate coupling $\nu$. The 2CC model accurately reproduces time-independent states and dynamic responses to periodic modulation beyond the diffusion-approximation domain, highlighting a low embedding dimensionality of the macroscopic dynamics and offering a tractable tool for theoretical studies of inhibitory–excitatory balanced networks. The work also compares with Ott–Antonsen and derives a diffusion-approximation boundary, clarifying when each approach is applicable and illustrating the limitations of diffusion-based reductions for shot-noise-driven networks.

Abstract

Kinetics of a balanced network of neurons with a sparse grid of synaptic links is well representable by the stochastic dynamics of a generic neuron subject to an effective shot noise. The rate of delta-pulses of the noise is determined self-consistently from the probability density of the neuron states. Importantly, the most sophisticated (but robust) collective regimes of the network do not allow for the diffusion approximation, which is routinely adopted for a shot noise in mathematical neuroscience. These regimes can be expected to be biologically relevant. For the kinetics equations of the complete mean field theory of a homogeneous inhibitory network of quadratic integrate-and-fire neurons, we introduce circular cumulants of the genuine phase variable and derive a rigorous two cumulant reduction for both time-independent conditions and modulation of the excitatory current. The low dimensional model is examined with numerical simulations and found to be accurate for time-independent states and dynamic response to a periodic modulation deep into the parameter domain where the diffusion approximation is not applicable. The accuracy of a low dimensional model indicates and explains a low embedding dimensionality of the macroscopic collective dynamics of the network. The reduced model can be instrumental for theoretical studies of inhibitory-excitatory balanced neural networks.

Figures (5)

• Figure 1: (a): The diagram of the macroscopic regimes (GOs: global oscillations, AS: asynchronous dynamics). The boundary between the GO and AS regimes is the Hopf bifurcation curve plotted with the black solid line for the original complete mean field system (\ref{['eqGP03']}) and (\ref{['eqGP12zn']}) and with the red dashed line for the diffusion approximation (Section \ref{['sec7']}). The cyan line: $K_\mathrm{D}(i_0/g_0^2)$, the diffusion approximation is accurate for in- degree $K$ 1--2 orders of magnitude larger than $K_\mathrm{D}$ (Section \ref{['sec71']}). The gray dash-dotted line: the Hopf bifurcation curve for model reduction (\ref{['eqGPnutilda']})--(\ref{['eqGP14']}). (b): Circular cumulants for the "exact" time-independent solution of (\ref{['eqGP03']}) and (\ref{['eqGP12zn']}) at the Hopf bifurcation line for $i_0/g_0^2=1.78$ (diamonds), $0.178$ (squares), $0.0316$ (triangles), $0.0007$ (black/blue circles for the upper/lower branch). (c): The Lyapunov exponent $\lambda$ of the time-independent solution within the Ott--Antonsen Ansatz [given by Eqs. (\ref{['eqGPnutilda']})--(\ref{['eqGP13']}) with $\varkappa_2=0$] is presented with the shadowgraph of $2\pi\sqrt{i_0}K^{1/4}\lambda/g_0^2$.
• Figure 2: The firing rate for time-independent network states is plotted for $i_0/g_0^2=0.01$ (a), $0.03$ (b), $0.06$ (c), and $0.4$ (d). Black squares: the "exact" solution of the infinite equation chain (\ref{['eqGP03']}) with firing rate (\ref{['eqGP12zn']}); red diamonds: 2CC reduction (\ref{['eqGPnutilda']})--(\ref{['eqGP14']}); blue circles: Ott--Antonsen Ansatz given by Eqs. (\ref{['eqGPnutilda']})--(\ref{['eqGP13']}) with $\varkappa_2=0$. The values of $i_0$ and $K$ for these plots are shown with magenta lines in Figure \ref{['fig1']}a.
• Figure 3: Response of the network to the periodic modulation of excitatory current $\eta=0.4\cos{2\tau}$ is plotted with filled symbols for $i_0/g_0^2=0.2$ (a), $0.4$ (b), $0.8$ (c); black squares: the "exact" solution, red diamonds: 2CC reduction, blue circles: OA Ansatz. In panel (b), the OA solution numerically explodes for $K<200$. In panel (c), open symbols present the response for $\eta=0.4\cos{3\tau}$. The values of $i_0$ and $K$ for the plots are shown with magenta lines in Figure \ref{['fig1']}a.
• Figure 4: Population firing rate versus time for $i_0$ and color coding as in Figure \ref{['fig3']}, in- degree $K=60$, $\eta=0.4\cos{2\tau}$ (solid lines) and $0.4\cos{3\tau}$ (dashed lines).
• Figure 5: For the driving-free system, the firing rate is plotted versus $K$ at $i_0/g_0^2=0.2$ (a) and $0.4$ (b) for the "exact" solution (black squares), 2CC reduction (red diamonds), and OA Ansatz (blue circles). In panel (c), the firing rate oscillations are plotted for $K=1000$, $i_0/g_0^2=0.2$ (dashed lines) and $0.4$ (solid); black: the "exact" solution, red: 2CC model.