Statistical parameter identification of mixed-mode patterns from a single experimental snapshot

TL;DR

The paper advances parameter identification for pattern formation by extending Correlation Integral Likelihood (CIL) to handle a single experimental snapshot with mixed-mode patterns. It integrates a multi-feature, Gaussian-approximate CIL framework and two data regimes (SCIL and mixed-mode SCIL) to cope with limited data and heterogeneity, applying them to the Lengyel-Epstein CIMA system in an open spatial reactor. Through synthetic and experimental Pattern data, it achieves MAP parameter estimates and credible intervals that are broadly consistent with chemical measurements while effectively distinguishing pattern types. The approach offers a robust, data-efficient pathway for parameter inference in heterogeneous spatial outputs, with potential applications in developmental biology and chemical patterning.

Abstract

Parameter identification in pattern formation models from a single experimental snapshot is challenging, as traditional methods often require knowledge of initial conditions or transient dynamics -- data that are frequently unavailable in experimental settings. In this study, we extend the recently developed statistical approach, Correlation Integral Likelihood (CIL) method to enable robust parameter identification from a single snapshot of an experimental pattern. Using the chlorite-iodite-malonic acid (CIMA) reaction -- a well-studied system that produces Turing patterns -- as a test case, we address key experimental challenges such as measurement noise, model-data discrepancies, and the presence of mixed-mode patterns, where different spatial structures (e.g., coexisting stripes and dots) emerge under the same conditions. Numerical experiments demonstrate that our method accurately estimates model parameters, even with incomplete or noisy data. This approach lays the groundwork for future applications in developmental biology, chemical reaction modelling, and other systems with heterogeneous output.

Figures (7)

• Figure 1: The main idea of the Correlation Integral Likelihood. To characterise a family of high-dimensional patterns, coming from an abstract pattern formation model for fixed values of parameters, a scalar feature (a distance $\rho$ between two arbitrary patterns) is used to map a family of random patterns to scalar values. A number of eCDF vectors are computed, each for a sufficiently large sample of scalars. The mean and covariance of the vectors are estimated, and the multidimensional Gaussianity is numerically verified. This distribution is used to quantify the statistical distance between a pattern formation model and pattern data.
• Figure 2: The observation of a chemical Turing pattern in the chlorite-iodite-malonic acid (CIMA) reaction performed in a two-sided open spatial reactor as described in Rudovics1996. The reactor consists of two reservoirs A and B (CSTRs), separated by a hydrogel block, that suppresses convection effects. Each reservoir contains chemical mixtures that are inert individually but react when combined. The hydrogel is loaded with a species of reduced mobility, which slows down the diffusion of the activator and changes colour depending on its concentration, making the pattern visible. A camera captures a digital image of the chemical pattern through a glass window on one side of the reactor.
• Figure 3: Statistical comparison between a pattern formation model and pattern data using the CIL approach. In the case of basic CIL (top part), the multidimensional Gaussian distribution of $\bm{y}(\bm{\theta}_0)$ is derived from the experimental data. When a new parameter vector $\bm{\theta}$ is proposed, one realisation of $\bm{y}(\bm{\theta})$ is created using model-generated (synthetic) data and compared with the previously derived Gaussian distribution. In the case of SCIL (bottom part) the roles of synthetic and experimental data are reversed: the Gaussian distribution of $\bm{y}(\bm{\theta})$ is created for each proposed parameter vector $\bm{\theta}$ and next compared with a single realisation of $\bm{y}(\bm{\theta}_0)$ derived from the experimental data.
• Figure 4: [I]: the comparison of the distribution of distances obtained by basic CIL for different subset size $N$ with the distribution of $D_{\bm{\theta}}$ estimated by computing $10^6$ independent realisations of the random variable. [II]: the distribution of eCDF vectors in the case of SCIL and regular pattern formation. [III]: the distribution of eCDF vectors in the case of SCIL and mixed mode patterns.
• Figure 5: Statistical comparison between a pattern formation model and data using the mixed mode CIL approach. Here, an ensemble of Gaussian distributions $\bm{y}(\bm{\theta})$ defined by fixing small numbers of patterns from the family $\bm{S}_{\bm{\theta}}$ is created for a proposed parameter vector $\bm{\theta}$. Each of these distributions is then compared with a single realisation of $\bm{y}(\bm{\theta}_0)$ derived from the experimental data and the results are processed by summary statistics to produce a single scalar output.

Theorems & Definitions (1)

• Remark 1