Stochastic collocation schemes for Neural Field Equations with random data

TL;DR

The paper develops a two-tier numerical framework for forward uncertainty quantification in neural field equations with random data, combining a generic spatial projection with stochastic collocation to approximate mean and variance of the solution. It establishes well-posedness, $C^0_\sigma$-regularity, and analyticity of the spatially discretised solution with respect to random parameters in the RLNF setting, and provides explicit bounds on derivatives that yield exponential convergence of the stochastic-collocation component. A comprehensive total-error bound is derived, separating spatial-projection error from stochastic-collocation error, with rates depending on whether random inputs are bounded or unbounded and on sub-Gaussian density assumptions. Numerical experiments on linear and nonlinear neural-field problems demonstrate spectral convergence with respect to the stochastic dimension and validate the theoretical predictions, including behavior near bifurcations. The framework is flexible to nonlinearities and scalable to higher parameter counts, with potential extensions to sparse grids and Bayesian inverse problems in neuroscience.

Abstract

We develop and analyse numerical schemes for uncertainty quantification in neural field equations subject to random parametric data in the synaptic kernel, firing rate, external stimulus, and initial conditions. The schemes combine a generic projection method for spatial discretisation to a stochastic collocation scheme for the random variables. We study the problem in operator form, and derive estimates for the total error of the schemes, in terms of the spatial projector. We give conditions on the projected random data which guarantee analyticity of the semi-discrete solution as a Banach-valued function. We illustrate how to verify hypotheses starting from analytic random data and a choice of spatial projection. We provide evidence that the predicted convergence rates are found in various numerical experiments for linear and nonlinear neural field problems.

Figures (4)

• Figure 1: Expectation and variance of a solution $u$ to the nonlinear neural field \ref{['eq:nonlinRNF']} with random forcing. The problem is posed on a 1-dimensional ring $D = \mathbb{R}/L \mathbb{Z}$ of width $L = 22$, and time interval $J = [0,200]$. Deterministic data is specified through an excitatory-inhibitory kernel $w(x,y) = W(x-y)$ with $W(z) = (2-z^2)\exp(-z^2)$, a sigmoidal firing rate $f(u) = [ 1+ \exp( -20(u-10) ) ]^{-1}$, and initial condition $v(x) = 2.5 + 0.5/\cosh^2(0.5 x)$. The external input is an oscillating pulse with random instantaneous speed $c$ depending on $6$ random variables: the forcing is given by $g(x,t,Y(\omega_g))$, where $g(x,t,y) = 1.4 \exp(-(x-c(t,y))^2)$, $y = (c_1,c_2,c_3,f_1,f_2,f_3)$, $c(t,y) = \sum_{k=1}^3 c_k \sin(2\pi t /f_k)$. The random variables $Y_i$ are independently distributed with $Y_i \sim \mathcal{U}[\alpha_i,\beta_i]$. The model is discretised in space using a spectrally convergent Fourier collocation scheme with $n = 100$ gridpoints. Expectation and variance are computed with a stochastic collocation scheme, with a dense, tensor-product grid of 10 Gauss-Legendre points in each of the intervals $[\alpha_i,\beta_i]$, for a total $10^6$ points. Parameters $[\alpha_1,\beta_1] = [0,4]$, $[\alpha_2,\beta_2] = [1/6,2/3]$, $[\alpha_3,\beta_3] = [1/10,4/5]$, $[\alpha_4,\beta_4] = [40,60]$, $[\alpha_5,\beta_5] = [10,50/3]$, $[\alpha_6,\beta_6] = [100,200]$.
• Figure 1: Sketch of the domains of analyticity of various functions used in the analycitity proof. The mapping $\psi_t$ of \ref{['thm:r-analyticity']} is defined on $\Gamma_i$ (grey horizontal line) and is $\mathbb{R}$-analytic on the interval $U_{\tau_i} \subset \Gamma_i$ of the real line (red). We use partial derivatives $\partial^k_y u$ defined on $\Gamma_i \subset \mathbb{R}$, and the bounds of \ref{['thm:AnalytLRNFAlt']} to define Fréchet derivatives in an appropriate Banach space and obtain analyticity of $\psi_t$ on $U_{\tau_i}$. The function $\tilde{\psi}_t$ in the proof of \ref{['cor:Sigma']} is defined on $\mathbb{C}$ and is $\mathbb{C}$-analytic in $\tilde{U}_{\tau_i}$ (dark grey). Without upgrading our definition of derivatives we prove that $\tilde{\psi}_t$ is the analytic extension of $\psi_t$ on $\tilde{U}_{\tau_i}$. Finally, we prove the existence of a function $\tilde{u}_n$ that extends $\psi_t$ analytically in the whole $\Sigma(\Gamma_i,\tau_i)$ (light grey).
• Figure 1: Spectral convergence of the stochastic colllocation method paired to Chebyshev collocation in space, for the RLNF problem with data eq:prob1DataDeteq:prob1DataRand. (a) Error $E_{n,k}$, defined in \ref{['eq:numError']} as a function of $q$, for various values of $n$, showing that the error decays expoentially until an $n$-dependent plateau, whose value is lower when $n$ increases. (b) Error $E_{n,k}$, as a function of $n$, for various values of $q$. (c) For fixed $n = 40$ (which gives machine accuracy in the spatial error), the error $E_{40,q}$ is exponentially convergent and increases when the variance of the distribution $\mathcal{U}[\alpha,\beta]$ is larger. Parameters $T = 1$, $L=1$ and, in (a,b), $\alpha = -2$, $\beta = 0.5$.
• Figure 2: Convergence of stochastic collocation error $\tilde{E}_q$\ref{['eq:errorq']}, for the NRNF problem in eq:prob2DataDeteq:prob2DataRand with $2$- and $4$-dimensional random noise. (a) We fix the number of collocation points in the $y_2$-direction and refine the tensor grid along $y_1$-direction. We set $q^*_2=10$ for the normal distribution and $q^*_2=64$ for the uniform ones. The number $q^*_1$ is taken as large as possible to test convergence, but differs for each curve. In (b), we fix $q^*_1=16$ in both normal and uniform settings, and increase $q_2$. In (c), we increase $q_1=q_2=q$ simultaneously, and we plot $q$. In (a--c) we note that the error in $y_2$ dominates over the one in $y_1$, and slower convergence is observed when the variance increases. In (d), the noise in the external input and the initial condition depends on $y_1\sim \mathcal{U}[1.25,1.75]$ and $y_2\sim \mathcal{U}[0.5,1.5]$ and we show an experiment in which $q^*=(15,15)$ is kept fixed for all simulations. $(e)$ We show nine solution profiles at final time $T$ of the NRNF, for $y_1=1$ and ten different values of the parameter $y_2\in[-1, 0.8]$, giving evidence of a bifurcation in this parameter range, and explaining the lower convergence rates seen in (a--c) for normally-distributed parameters (see main text). In (f) we refine one grid direction at a time for the NRNF with 4 parameters, as given in \ref{['subsec:NumericalEx3']}. Other parameter values are: $\sigma_{\omega}=1$, $F_0=1$, $\mu=10$, $h=0.3$, $\omega_g=1$ and $\sigma_g=0.4$, $n^* = 40$, $T = 1$.

Theorems & Definitions (29)

• Theorem 3.5: Abridginng avitabile2024NeuralFields on solutions to neural fields with random data
• Definition 3.6: $m$-dimensional, $p$th-order, $\mathbb{B}$-valued noise
• Example 3.8: Affine parameter dependence
• Example 3.9: Non-affine parameter dependence
• Corollary 3.11: Abridged avitabile2024NeuralFields on $L^p_\rho$-regularity with finite-dimensional noise
• Proposition 4.1: Adapted from avitabile2024NeuralFields
• Remark 4.2
• Lemma 5.2: $C^0_\sigma$-regularity for linear problems with finite-dimensional noise
• Proof 1
• Remark 5.3: Analyticity radius