Efficient, simulation-free estimators of firing rates with Markovian surrogates

TL;DR

The paper addresses the challenge of estimating firing rates in finite-size spiking neural networks (SNNs) without heavy simulations. It introduces a Markovian approximation that discretizes neuron states into a Markov process and derives two simulation-free estimators, Type I (fast, ignores synchrony) and Type II (includes synchrony via first-moment synaptic-drive dynamics). Using Kolmogorov forward dynamics for population states $\rho^X$ and mean-drive equations $\bar{H}^{XY}$, the approach yields accurate firing-rate predictions across regimes and outperforms standard mean-field and MF+v methods, while preserving spike-reset discontinuities. The results demonstrate that accounting for spiking synchrony significantly improves accuracy in finite networks and provides a practical, parameter-to-rate mapping for neuroscience modeling.

Abstract

Spiking neural networks (SNNs) are powerful mathematical models that integrate the biological details of neural systems, but their complexity often makes them computationally expensive and analytically untractable. The firing rate of an SNN is a crucial first-order statistic to characterize network activity. However, estimating firing rates analytically from even simplified SNN models is challenging due to 1) the intricate dependence between the nonlinear network dynamics and parameters, and 2) the singularity and irreversibility of spikes. In this Letter, we propose a class of computationally efficient, simulation-free estimators of firing rates. This is based on a hierarchy of Markovian approximations that reduces the complexity of SNN dynamics. We show that while considering firing rates alone is insufficient for accurate estimations of themselves, the information of spiking synchrony dramatically improves the estimator's accuracy. This approach provides a practical tool for brain modelers, directly mapping biological parameters to firing rate.

Figures (5)

• Figure 1: Approximation of a single neuron which receives upstream Poisson inputs with constant rates. Left: detailed comparison of predicted steady-state voltage distribution and long-time average from simulation, with two sets of parameters; Right: relative predictive error versus long-time average firing rate from simulation on log-log scale, with 3000 sets of parameters, in which recurrent input rates, refractory periods and external input rates are varied.
• Figure 2: Predictions of the simplified method versus ground truth from simulations. Colors of dots reflect the spike synchrony index of LIF simulations of SNN dynamics. A: comparison of predicted steady-state firing rate and long-time average from simulation, with 5000 sets of parameters; altered parameter from left to right and top to bottom: recurrent connectivities $S^{XY}$, synaptic timescale $\tau^{XY}$, synaptic coupling probabilities $P^{XY}$, refractory period $\tau^{X,\text{ref}}$ and external input rate $\lambda^{X,\text{ext}}$. B: absolute prediction error versus network synchrony, quantified by SSI.
• Figure 3: Predictions of the full method versus ground truth from simulations. A: comparison of predicted steady-state firing rate and long-time average from simulation, tested on the same sets of parameters as in Fig. \ref{['fig:2']}. B: absolute prediction error versus network synchrony, quantified by SSI.
• Figure 4: Comparison of the Type II estimator, the mean-driven limit, and the MF+v algorithm. A: comparison in predicted firing rate with both methods, tested on 5000 sets of parameters, where the refractory period $\tau^{X,\text{ref}}$ is enforced to be $0$ due to limitations of the mean driven limit; altered parameter from left to right and top to bottom: recurrent connectivities $S^{XY}$, synaptic timescale $\tau^{XY}$, synaptic coupling probabilities $P^{XY}$, refractory period $\tau^{X,\text{ref}}$ and external input rate $\lambda^{X,\text{ext}}$. Note that the fourth subplot is not applicable to the method of mean-driven limit as explained above. B: comparison in absolute predictive error with both methods.
• Figure 5: Type II estimator captures the branching phenomena in LIF networks with different sizes. Top row: results in networks with 30 E-neurons and 10 I-neurons; Middle row: networks with 300 E-neurons and 100 I-neurons; Bottom row: networks with 3000 E-neurons and 1000 I-neurons. Each panel compares the steady-state firing rate predicted by Type-I and Type-II estimators and the long-time average ($10$ seconds) from simulation with different initial conditions for each set of parameters. Altered parameters are synaptic coupling probabilities $P^{XY}$. Different branches of LIF simulations and estimators come from different initial conditions.