Effects of Heterogeneity in Two-Cell Feedforward Networks

TL;DR

The paper investigates how heterogeneity in two-cell feedforward networks—comprising pitchfork and Stuart–Landau cells—affects signal amplification and beam steering. It combines model reduction, analytical and computational bifurcation analyses, and singularity theory to quantify the impact of parameter disorders, revealing that excitation inhomogeneity can enhance output growth, while frequency inhomogeneity may reduce amplification but often preserves phase locking with occasional torus dynamics. For pitchfork cells, a two-parameter unfolding governs the equilibrium structure and basins of attraction, enabling macroscopic jumps in response under controlled perturbations. For Stuart–Landau oscillators, the work demonstrates that inhomogeneity in frequency broadens phase-locked regions but generally lowers amplification, and a nonzero cubic imaginary term shifts phase diagrams without altering the primary amplification trend; singularity theory provides a unifying framework to locate bifurcation and hysteresis loci across reduced and original systems, linking theoretical insight to practical sensor design and beam-steering applications.

Abstract

As the need for higher performance from biological and electronic sensors continues to outpace current technologies, new strategies for designing, developing, and implementing novel sensor systems are emerging. A recently introduced feedforward network-based approach can simultaneously enhance a signal while steering a radiating beam in radio frequency communication systems. Furthermore, the approach is also model-independent, thus making it suitable for other applications. In this work, we aim to understand the effects of inhomogeneities in feedforward arrays, which are inevitable in real-world implementations. We investigate a collection of two-cell feedforward networks composed of pitch-fork cells and Stuart-Landau oscillators and quantify the effects of parameter inhomogeneities using system reduction, analytical and computational bifurcation analyses, and a singularity theory approach. Contrary to common intuition, inhomogeneity in the excitation parameter can be exploited to enhance the network output growth rate. While frequency inhomogeneity in Stuart-Landau networks primarily has an adverse effect on signal amplification, phase locking persists over a surprisingly broad range of inhomogeneity.

Figures (25)

• Figure 1: Representative example of a three-cell feedforward network. Arrows indicate coupling, with coupling strength $\lambda$. Each cell represents a dynamical system assumed to be operating near a Hopf bifurcation.
• Figure 2: Signal amplification in a feedforward network (a) with self-coupling of the first cell and (b) without self-coupling. Parameters are: $\mu = (0.5)^6$, $\omega = 1$, $\lambda = 1$. (a): With self-coupling. (b): Without self-coupling.
• Figure 3: Beam steering can be achieved in a feedforward network by varying the phase-locking angle $\theta$. The sequence of snapshots shows the radiating pattern with 20 nonlinear oscillators.
• Figure 4: A schematic of a two-cell feed-forward network with inhomogeneous cells. Each cell represents a system prone to a pitchfork bifurcation.
• Figure 5: A phase diagram and representative bifurcation diagrams for system \ref{['eq:system1']}. Solid and dashed lines represent stable and unstable branches, respectively. The branches of $y$ existing at $x = \sqrt{\mu}$ and $x = -\sqrt{\mu}$ are blue and red, respectively. Black, yellow, and purple branches of $y$ correspond to $x = 0$. The curve separating the one- and three-root regions, plotted in blue, red, and yellow, is defined by Eq. \ref{['eq:poly2roots']}.

Theorems & Definitions (4)

• Proposition 1
• Remark
• Proposition 2
• proof