Autonomous learning of nonlocal stochastic neuron dynamics

Abstract

Neuronal dynamics is driven by externally imposed or internally generated random excitations/noise, and is often described by systems of random or stochastic ordinary differential equations. Such systems admit a distribution of solutions, which is (partially) characterized by the single-time joint probability density function (PDF) of system states. It can be used to calculate such information-theoretic quantities as the mutual information between the stochastic stimulus and various internal states of the neuron (e.g., membrane potential), as well as various spiking statistics. When random excitations are modeled as Gaussian white noise, the joint PDF of neuron states satisfies exactly a Fokker-Planck equation. However, most biologically plausible noise sources are correlated (colored). In this case, the resulting PDF equations require a closure approximation. We propose two methods for closing such equations: a modified nonlocal large-eddy-diffusivity closure and a data-driven closure relying on sparse regression to learn relevant features. The closures are tested for the stochastic non-spiking leaky integrate-and-fire and FitzHugh-Nagumo (FHN) neurons driven by sine-Wiener noise. Mutual information and total correlation between the random stimulus and the internal states of the neuron are calculated for the FHN neuron.

Keywords: Colored noise; Equation learning; Method of distributions; Nonlocal; Stochastic neuron model.

Fig. 1

(a) PDF $f_x(X;t)$ of the membrane voltage x(t) in (24), computed via the PDF equation with the semi-local closure, for several combinations of the strength ($\sigma$) and correlation length ($\tau$) of colored noise $\xi (t)$. (b) Direct comparison of the PDF $f_x(X;t)$ computed with the semi-local closure and the yardstick PDF $f_{\text{MC}}(X;t)$ for $\sigma =0.133$ and $\tau =0.20$ at times $t=1$ and 3

Fig. 2

Temporal evolution of the KL divergence, $D_{\text{KL}}(f_{\text{MC}}||f_x)$, between the membrane voltage PDF computed with the semi-local closure, $f_x(X;t)$, and its Monte Carlo counterpart, $f_{\text{MC}}(X;t)$, for several combinations of the strength ($\sigma$) and correlation length ($\tau$) of colored noise $\xi (t).$

Fig. 3

Diffusion coefficients associated with the semi-local closure, $D_2(t)$, the data-driven closure with constant coefficients, ${\widehat{\beta }}_2$, and the data-driven closure with a temporal Legendre basis expansion, ${\widehat{\beta }}_2(t)$. We present the cases for which the KL divergence $D_{\text{KL}}(f_{\text{MC}}\Vert f_x)$ of the data-driven closure with constant $\widehat{\mathit{\beta }}$ is highest. In all cases, the data-driven coefficients approximated with a Legendre basis expansion agree well with the other closures

Fig. 4

Temporal snapshots (at times $t=0.5$, 5, 12.5, and 25; for $\sigma =0.133$ and $\tau =0.20$) of the membrane voltage PDFs $f_x(X;t)$ alternatively computed with the semi-local closure (SL), the data-driven closure with constant coefficients (${\text{DD}}_{\text{C}}$), and the data-driven closure with a temporal Legendre basis expansion (${\text{DD}}_{\text{P}}$)

Fig. 5

Temporal evolution of the KL divergence, $D_{\text{KL}}(f_{\text{MC}}||f_x)$, between the membrane voltage PDF computed with the data-driven closure, $f_x(X;t)$, and its Monte Carlo counterpart, $f_{\text{MC}}(X;t)$, for selected combinations of the strength ($\sigma$) and correlation length ($\tau$) of colored noise $\xi (t)$. The data array ${\widehat{f}}_x$ used in the data-driven closure is computed with $N_{\text{MC}}^{\text{tr}}=3\times {10}^4$ Monte Carlo realizations

Fig. 6

Dependence of the KL divergence $D_{\text{KL}}(f_{\text{MC}}\Vert f_x)$, averaged over all temporal grid nodes, on the number of Monte Carlo runs used in the sparse regression, $N_{\text{MC}}^{\text{tr}}.$

Fig. 7

The coefficients of the semi-local closure, $D_4$ and $D_5$, and the data-driven closure, ${\widehat{\beta }}_3$ and ${\widehat{\beta }}_4$. For $\sigma =0.05$ and $\tau =0.1$, the optimization algorithm sets ${\widehat{\beta }}_3=0$ while $D_4=\mathcal{O}({10}^{-7})$, which is near zero when compared with ${\widehat{\beta }}_4$ and $D_5.$

Fig. 8

Temporal snapshots of the joint PDF $f_{\mathbf{x}}(\mathbf{X};t)$ of the random state variables $x_1(t)$ and $x_2(t)$ in the FHN model (33). The dynamics of $f_{\mathbf{x}}(\mathbf{X};t)$ is governed by the PDF equation (37), for $\sigma =0.2$ and $\tau =0.1.$

Fig. 9

Temporal evolution of the KL divergence $D_{\text{KL}}(f_{\text{MC}}\Vert f_{\mathbf{x}})$ between the yardstick Monte Carlo solution $f_{\text{MC}}(\mathbf{X};t)$ of (33) and the PDF $f_{\mathbf{x}}(\mathbf{X};t)$ computed, alternatively, via (37) and (4) with the data-driven closure (38)

Fig. 10

Snapshots (at times $t=20$ and 65) of the marginal PDFs $f_{x_1}(X_1;t)$ (left column) and $f_{x_2}(X_2;t)$ (right column) alternatively computed with the local (L), semi-local (SL) and data-driven (DD) closures and with Monte Carlo simulations (MC), for the stochastic FHN neuron with $\sigma =0.05$ and $\tau =0.1.$

Fig. 11

Dependence of the KL divergence $D_{\text{KL}}(f_{\text{MC}}\Vert f_x)$, averaged over all temporal grid nodes, on the number of Monte Carlo runs used in the sparse regression, $N_{\text{MC}}^{\text{tr}}.$

Fig. 12

Temporal evolution of MI, I, between the different FHN neuron states and of the total correlation, C, between all three states. The PDF equation (4) is closed with the data-driven closure, and the parameter values are set to $\sigma =0.05$ and $\tau =0.01.$