Parameter and hidden-state inference in mean-field models from partial observations of finite-size neural networks

TL;DR

This work tackles the challenge of inferring mean-field parameters and reconstructing hidden macroscopic states from partial observations of finite neural networks that admit exact NGNPM reductions. By combining synchronization-based coupling (noninvasive or invasive) with differential evolution, the authors recover mean-field parameters with sub-1% relative error for networks with $N\ge 1000$ and successfully reconstruct unobserved variables such as $R$ and $S$ (or $A$) across periodic and chaotic regimes. The approach explicitly accounts for finite-size fluctuations and transients, showing robustness across network sizes and demonstrating applicability to QIF-IN (periodic) and QIF-AD (chaotic) networks. The results suggest a viable data-driven pathway for extending NGNPMs and motivate future integration with equation-discovery methods like SINDy to learn governing dynamics from limited observations.

Abstract

We study large but finite neural networks that, in the thermodynamic limit, admit an exact low-dimensional mean-field description. We assume that the governing mean-field equations describing macroscopic quantities such as the mean firing rate or mean membrane potential are known, while their parameters are not. Moreover, only a single scalar macroscopic observable from the finite network is assumed to be measurable. Using time-series data of this observable, we infer the unknown parameters of the mean-field equations and reconstruct the dynamics of unobserved (hidden) macroscopic variables. Parameter estimation is carried out using the differential evolution algorithm. To remove the dependence of the loss function on the unknown initial conditions of the hidden variables, we synchronize the mean-field model with the finite network throughout the optimization process. We demonstrate the methodology on two networks of quadratic integrate-and-fire neurons: one exhibiting periodic collective oscillations and another displaying chaotic collective dynamics. In both cases, the parameters are recovered with relative errors below $1\%$ for network sizes exceeding 1000 neurons.

Figures (8)

• Figure 1: Inference of mean-field model parameters from chaotic time series of mean membrane potential observed in QIF-IN networks of different sizes. Box plots visualize the statistics of relative deviations of inferred parameters from their actual values used in the QIF-IN microscopic network model. The statistics are based on $40$ different initial sets of mean-field model parameters, randomly selected from the optimization bounds presented in Tab. \ref{['tab_Devalle']}. Blue and red colors represent the noninvasive and invasive synchronization methods, respectively. For the noninvasive method, the transient time is $t_\mathrm{trans}=831.3$ ms, the loss function is optimized over a training interval of $t_\mathrm{train}=277.1$ ms and the feedback strength is $K=0.5$. For the invasive method, the parameters are: $t_\mathrm{trans}=1400$ ms, $t_\mathrm{train}=560$ ms, $K=-0.45$ and $T_\mathrm{ext}=28$ ms. The remaining parameters are given in the Tab. \ref{['tab_Devalle']}.
• Figure 2: Box plot of the loss function minima for different sizes of the QIF-IN network. Noninvasive and invasive methods are represented in blue and red colors, respectively. The parameter values are given in the caption of Fig. \ref{['fig:QIF_IN_param']} and in Tab. \ref{['tab_Devalle']}.
• Figure 3: Reconstruction of the dynamics of unobservable macroscopic variables $R$ and $S$ of QIF-IN network consisting of 1000 neurons. (a), (b) and (c) The dynamics of the mean firing rate $R$, mean membrane potential $V$ and synaptic activation variable $S$, respectively. The blue curves correspond to the microscopic model, while red dashed curves show the dynamics of the mean-field model, which is driven by the mean membrane potential of the microscopic model via a master-salve control scheme. The parameters of the mean-field model are set to the median values obtained from optimization using the DE method with noninvasive synchronization. (d), (e) and (f) Error dynamics showing the differences between macroscopic variables estimated from a 1000-neuron network and the corresponding mean-field model variables.
• Figure 4: Inference of mean-field model parameters from chaotic time series of mean membrane potential observed in QIF-AD networks of different sizes. Box plots visualize the statistics of relative deviations of inferred parameters from their actual values used in the QIF-AD microscopic network model. The statistics are based on $40$ different initial sets of mean-field model parameters, randomly selected from the optimization bounds presented in Tab. \ref{['tab_Mont']}. For the noninvasive method, the transient time is $t_\mathrm{trans}=1000$ ms, the loss function is optimized over a training interval of $t_\mathrm{train}=500$ ms and the feedback strength is $K=5$. For the invasive method, the parameters are: $t_\mathrm{trans}=2400$ ms, $t_\mathrm{train}=240$ ms, $K=-4$ and $T_\mathrm{ext}=80$ ms. The remaining parameters are given in the Tab. \ref{['tab_Mont']}.
• Figure 5: Box plot of the loss function minima for different sizes of the QIF-AD network. The parameter values are given in the caption of Fig. \ref{['fig:QIF_adapt_param']} and in Tab. \ref{['tab_Mont']}.