Large-Deviation Approach to Random Recurrent Neuronal Networks: Parameter Inference and Fluctuation-Induced Transitions

We here unify the field-theoretical approach to neuronal networks with large deviations theory. For a prototypical random recurrent network model with continuous-valued units, we show that the effective action is identical to the rate function and derive the latter using field theory. This rate function takes the form of a Kullback-Leibler divergence which enables data-driven inference of model parameters and calculation of fluctuations beyond mean-field theory. Lastly, we expose a regime with fluctuation-induced transitions between mean-field solutions.

Figure 1

Maximum likelihood parameter estimation for $\varphi (x)=\mathrm{erf}(\sqrt{\pi }x/2)$, potential $U(x)=\frac{1}{2}{x}^2+s\ln \cosh x$, and external noise $D$. (a) Color-coded sketch of potential and noise. (b)–(d) Activity of three randomly chosen units for coupling strengths $g$ indicated in title. (e) Parameter estimation via non-negative least squares regression (black lines) based on Eq. (12). (f) Power spectra on the left- (dark, solid curves) and right-hand sides (light, dotted curves) of Eq. (12) for the inferred parameters. Further parameters: $\tau =1$, $N=10000$, temporal discretization $dt={10}^{-2}$, simulation time $T=1000$, time span discarded to reach steady state $T_0=100$.

Figure 2

Order parameter fluctuations for $\varphi (x)=\mathrm{erf}(\sqrt{\pi }x/2)$ [(a),(b)] and metastability for $\varphi (x)=\mathrm{clip}[\tan (x),-1,1]$ [(c),(d)]. (a) Temporal order parameter statistics across ten simulations (bars) and theory (solid curve) from Eq. (13). (b) Order parameter variance for 10 realizations of the connectivity with standard error of the mean (symbols) and theory (solid curve) from Eq. (13). (c) Mean order parameter for different initial values $q_0$ from simulations (symbols) and self-consistent theory (solid curves). (d) Fluctuation-induced bistability of the order parameter for $N=750$, $g=0.95$. Further parameters: $T=5000$ in (a),(d); $U(x)=\frac{1}{2}{x}^2$ in (a)–(d); other parameters as in Fig. 1.

Figure 3

Maximum likelihood parameter estimation for two populations with different time constants ${\tau }_1=5$, ${\tau }_2=1$. (a) Output power spectra ${\mathcal{S}}_{\varphi (x)}^{\alpha }(f)$ of two unconnected populations $g_{12}^2=g_{21}^2=0$ with $g_{11}^2=4$ and $g_{22}^2=6$. (b) Estimated (blue) and true (black) parameters corresponding to (a). (c) Output power spectra of two connected populations with $g_{11}^2=0.5$, $g_{12}^2=1.5$, $g_{21}^2=2.5$, and $g_{22}^2=3.5$. (d) Estimated (blue) and true (black) parameters corresponding to (c). Further parameters: $N_1=N_2=5000$, $\varphi (x)=\mathrm{erf}(\sqrt{\pi }x/2)$, $U(x)=\frac{1}{2}{x}^2$, and $D=0$; simulation parameters as in Fig. 1.