Inference of hidden common driver dynamics by anisotropic self-organizing neural networks

TL;DR

This work tackles the challenge of inferring hidden common drivers from observations of driven nonlinear dynamics. It combines time-delay embedding, intrinsic-dimension estimation, and Dimensional Causality to characterize the hidden driver and then trains an anisotropic Self-Organizing Map (ASOM) to decompose the observed attractor into submanifolds for self-dynamics and shared dynamics. The method yields robust reconstruction of the hidden driver, outperforming linear and several nonlinear baselines across coupled logistic and tent-map systems, with correlations reaching up to $\rho = 0.91$ in held-out tests. This dimension-guided, unsupervised approach provides a principled framework for hidden-variable reconstruction with potential applications in neuroscience and other complex systems where latent drivers influence observed dynamics.

Abstract

We are introducing a novel approach to infer the underlying dynamics of hidden common drivers, based on analyzing time series data from two driven dynamical systems. The inference relies on time-delay embedding, estimation of the intrinsic dimension of the observed systems, and their mutual dimension. A key component of our approach is a new anisotropic training technique applied to Kohonen's self-organizing map, which effectively learns the attractor of the driven system and separates it into submanifolds corresponding to the self-dynamics and shared dynamics. To demonstrate the effectiveness of our method, we conducted simulated experiments using different chaotic maps in a setup, where two chaotic maps were driven by a third map with nonlinear coupling. The inferred time series exhibited high correlation with the time series of the actual hidden common driver, in contrast to the observed systems. The quality of our reconstruction were compared and shown to be superior to several other methods that are intended to find the common features behind the observed time series, including linear methods like PCA and ICA as well as nonlinear methods like dynamical component analysis, canonical correlation analysis and even deep canonical correlation analysis.

Figures (6)

• Figure 1: Crossmapping between coupled logistic systems, represented in 3D embedding. A-D: In the case of circular coupling between the systems, the mapping behaves similarly in both directions. A local neighborhood in the state space of system X (A) is mapped into a local neighborhood in the state space of system Y (B). Similarly, a local neighborhood in the state space of system Y (D) is mapped into a local neighborhood in system X (C). However, in unidirectional coupling (E-H), the dimensions of the two systems differ, resulting in different behavior for the projections in the two directions. A small local neighborhood in the cause (E) is mapped onto a one-dimensional submanifold in the state space of the consequence (F), while a small local neighborhood in the consequence (H) is mapped into a small local neighborhood in the cause (G).
• Figure 2: Cross-mapping with a hidden common driver. The Z system is the hidden common driver (C) that drives the two uncoupled systems X and Y on A and B respectively. Cross-mapping of the orange and green local patches forming two local neighborhoods in the state space of $X$ (A), results in two well-localized patches in the state space of the unobserved Z system (C), however mapping these local neighborhoods onto the $Y$ system results in two one-dimensional submanifolds (B).
• Figure 3: Dimensional Causality analysis. Three connected logistic map was simulated and analysed. A: time series of the hidden common driver, z(t), (green). B and F: time series of both observed time series (x(t), red and y(t) blue). C and G: the observed time series are embedded in 3 dimensions. D: Joint embedding (J(t), black). H: Time permuted joint embedding (I(t), yellow). E: Estimated dimensions as a function of the neighborhood size (k). The interval between 10 and 20 was used for the Bayesian estimation. I: Estimated posterior probabilities of the five basic causal relations: $X \rightarrow Y$: $X$ drives $Y$; $X \leftrightarrow Y$: bidirectional coupling; $X \leftarrow Y$: $Y$ drives $X$; $X \mathrel{\hbox{$\curlyveeuparrow$}} Y$: a hidden common cause drives both; $X \perp Y$: $X$ and $Y$ are independent. The dimensional analysis assigns the highest probability to the existence of the hidden common driver.
• Figure 4: Development of the SOM during anisotropic learning. During each time step, a random point in manifold $X$ was chosen, and its local neighbourhood of $K=20$ points were presented to the network, forming a bundle in the manifold $Y$ marked by different colors. The $20 \cdot 40$ nodes of the grid represents the $C_{i,j}$ centers of the receptive fields of the neurons, while the edges corresponds to the predefined neighbourhood structure of the network.
• Figure 5: Readout of the hidden common driver. A: The trained 2D self-organizing map, after $N=10000$ training steps, fits well to the Y manifold and divides it into stripes marked by different colors. The nodes of the grid represent the $C_{i,j}$ centers of the receptive fields of the neurons. The self-dynamics of system Y move the system within the colored stripes on the SOM, while stripes of different colors correspond to the various states of the hidden common driver, which are marked by the same colors in B. C: Comparison of the actual and the reconstructed hidden common driver. Both time series are normalized by standard deviation (SD). D: Correlation between the actual and the reconstructed normalized hidden common driver values ($\rho=0.91$).