Redundant and synergistic interactions in a complex network of single-transistor electronic chaotic oscillators and in neurophysiological recordings

Abstract

Complex networks often exhibit emergent behaviors, where simple dyadic interactions yield collective dynamics that cannot be explained by examining the system's units individually or in pairs. Understanding how redundant and synergistic interaction emerges from elementary connectivity patterns is important in characterizing the behavior of physical, biological, and engineering systems. In this study, the information-theoretic framework of Partial Information Decomposition (PID) is employed to investigate how pairs of signals measured at the nodes of large network systems contribute individually and in cooperation with each other to determine the overall state of the network. The analyzed systems are networks of numerically simulated Roessler oscillators and physical single-transistor electronic chaotic oscillators, reproducing purely pairwise and symmetric links in biological neuronal cultures, and cortical electroencephalographic networks assessed in humans. In the two settings, PID was extensively applied to decompose the information brought by pairs of signals to the overall state of the system, assessed respectively as the dynamic regime (from asynchronous chaos to synchronization) and the subject condition (open vs. closed eyes). Our approach highlights the coexistence of redundant and synergistic interplay in determining the effect of the pairwise dynamics on the system's state. Specifically, we demonstrate how, in a highly synchronized system, where units act following overlapped redundant behaviors, their joint consideration helps determine the system's state though their synergistic interactions. Our findings reveal the emergence of non-trivial behaviors in networked system based on pairwise connections and straightforward neural states, highlighting the need for using appropriate analytical tools to capture complex phenomena.

Figures (10)

• Figure 1: Schematic representation of the pipeline of analysis. In panel ($a$), the network system composed of $M$ units is depicted in multiple configurations dependent on different units' characteristics (represented by varying shades of gray) and connections (represented by lines of varying thickness) collectively determining the state of the overall system's states indicated by the outcomes of the random variable $Y$ depicted by the color of the external circle. In panel ($b$), the state-PID framework applied to decompose the information shared by the target variable $Y$ and two source variables representative of two units of the network, e.g., $X_1$ and $X_2$, is graphically represented through Venn diagrams and the redundancy lattice. Specifically, the Venn diagram representing the decomposition of the joint MI $I(Y;X_1,X_2)$ is reported considering the unique contributions $U(Y;X_1)$ and $U(Y;X_2)$ in orange, the redundant contribution $R(Y;X_1,X_2)$ in blue, and the synergistic contribution $S(Y;X_1,X_2)$ in red. Moreover, the diagram representing the net balance between the redundant and synergistic contributions is shown with the purple area, highlighting how this quantity obscures the effective contribution of the two terms and can assume both positive and negative values, depending on whether redundancy or synergy prevails. The redundancy lattice representing the ordering of the atoms in which the joint MI can be decomposed is also reported. In panel ($c$), a schematic representation of the nearest-neighbor estimation approach used to compute the specific MIs in Eqs. \ref{['specMI_1']} and \ref{['specMI_2']} is reported considering a generic sample $\{x_1,x_2\}$ associated to the specific outcome $y$ of the variable $Y$. Part of the figure adapted from Ref. [bara2025partial].
• Figure 2: Graphical representation of the experimental study from the in-vitro cultured neurons of Wistar rats to the identification of the connectivity matrix representing brain-like network topology (panel a) and 200-length windows of simulated $y$-variable time series for a representative Rössler unit (see Eq. \ref{['Rossler_eq']}) at varying the setting of the parameters $\epsilon$ and $a$ (panel b). Part of the figure adapted from Ref. [minati2025spontaneous].
• Figure 3: High-order interactions between pairwise oscillator activity and system state for the network of coupled Rössler systems. Upper triangular heatmaps depict the average values across the realizations of the information shared by each pair of oscillators and the state of the system (panel a), as well as their unique (panel b), synergistic (panel c) and redundant (panel d) contributions; the balance between the redundant and synergistic decomposition terms with positive values for net redundancy and negative values for net synergy is also depicted (panel e). In each panel, boxplots depict the distribution of the average across realizations of the measures obtained for pairs of oscillators that are physically connected (1) or not (0). The line in the middle of the box represents the median of the distribution, while the bottom and upper limits of the box represent the 25$^{th}$ and the 75$^{th}$ percentile, respectively; the whiskers extend till the minimum and the maximum values of the distribution that are not considered as outliers. Statistically significant differences for each measure: $p<0.05$, 0 vs. 1, Student's t-test.
• Figure 4: State-specific high-order interactions on simulated data. Upper triangular heatmaps depict the average values across the realizations of the information shared by each pair of oscillators and each specific state of the system (panel a), i.e., by varying coupling strength $\epsilon$ in row and the bifurcation parameter $a$ in column, as well as their unique (panel b), synergistic (panel c), and redundant (panel d) contributions; the balance between the redundant and synergistic decomposition terms is also depicted (panel e). The percentage of realizations of the simulation for which the measure was deemed as significant (average over all pairs) is reported in the lower triangular heatmaps for all the possible thirty-six states of the system.
• Figure 5: Single (left) and coupled (right) Minati-Frasca chaotic oscillator (panel a), prototype of the circuit board (panel b), and representative experimentally recorded time series at varying the setting parameters $V_{\textrm{cpl}}$ and $V_\textrm{s}$ (panel c). Part of the figure adapted from Ref. [minati2025spontaneous].