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Singular networks and ultrasensitive terminal behaviors

Alessio Franci, Bart Besselink, Arjan van der Schaft

TL;DR

This work develops a rigorous framework to characterize and synthesize terminal behavior in signed nonlinear networks that include negative differential conductance. By extending Kron reduction and applying Lyapunov–Schmidt reduction, the authors define singular and externally singular networks, prove the consistency between full and reduced reductions, and uncover ultrasensitive, selectively responsive terminal dynamics near singularities. They derive conditions for network bifurcations (transcritical and pitchfork) in the single-negative-edge setting and illustrate the theory with a numerical example showing predicted bifurcation and ultra-sensitivity patterns. The approach lays a mathematical foundation for neuromorphic and excitable-network designs, with potential extensions to dynamic edges and port-Hamiltonian multi-physics systems.

Abstract

Negative conductance elements are key to shape the input-output behavior at the terminals of a network through localized positive feedback amplification. The balance of positive and negative differential conductances creates singularities at which rich, intrinsically nonlinear, and ultrasensitive terminal behaviors emerge. Motivated by neuromorphic engineering applications, in this note we extend a recently introduced nonlinear network graphical modeling framework to include negative conductance elements. We use this extended framework to define the class of singular networks and to characterize their ultra-sensitive input/output behaviors at given terminals. Our results are grounded in the Lyapunov-Schmidt reduction method, which is shown to fully characterize the singularities and bifurcations of the input-output behavior at the network terminals, including when the underlying input-output relation is not explicitly computable through other reduction methods.

Singular networks and ultrasensitive terminal behaviors

TL;DR

This work develops a rigorous framework to characterize and synthesize terminal behavior in signed nonlinear networks that include negative differential conductance. By extending Kron reduction and applying Lyapunov–Schmidt reduction, the authors define singular and externally singular networks, prove the consistency between full and reduced reductions, and uncover ultrasensitive, selectively responsive terminal dynamics near singularities. They derive conditions for network bifurcations (transcritical and pitchfork) in the single-negative-edge setting and illustrate the theory with a numerical example showing predicted bifurcation and ultra-sensitivity patterns. The approach lays a mathematical foundation for neuromorphic and excitable-network designs, with potential extensions to dynamic edges and port-Hamiltonian multi-physics systems.

Abstract

Negative conductance elements are key to shape the input-output behavior at the terminals of a network through localized positive feedback amplification. The balance of positive and negative differential conductances creates singularities at which rich, intrinsically nonlinear, and ultrasensitive terminal behaviors emerge. Motivated by neuromorphic engineering applications, in this note we extend a recently introduced nonlinear network graphical modeling framework to include negative conductance elements. We use this extended framework to define the class of singular networks and to characterize their ultra-sensitive input/output behaviors at given terminals. Our results are grounded in the Lyapunov-Schmidt reduction method, which is shown to fully characterize the singularities and bifurcations of the input-output behavior at the network terminals, including when the underlying input-output relation is not explicitly computable through other reduction methods.

Paper Structure

This paper contains 21 sections, 6 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on signed Laplacians
  3. Signed graphs, Laplacians, and their Kron reduction
  4. Effective resistance and singular signed Laplacians
  5. Signed nonlinear networks and their terminal behavior
  6. Signed nonlinear networks
  7. Singular networks and externally singular behaviors
  8. Singular networks
  9. Externally singular behaviors
  10. Lyapunov-Schmidt reduction of singular network terminal behavior
  11. Projecting out global shift invariance
  12. Critical eigenvectors of full and Kron-reduced network singularities
  13. Lyapunov-Schmidt reduction of the full network equations
  14. Lyapunov-Schmidt reduction of the reduced network equations
  15. Consistency of the Lyapunov-Schmidt reduction of full and reduced network behaviors
  16. ...and 6 more sections

Key Result

Lemma 1

Assume that $L_{CC}$ is nonsingular. The signed Laplacian $L$ and its Kron reduction $\widehat{L}=L/L_{CC}$ satisfyRecall that, for a matrix $M$, $\mathop{\mathrm{corank}}\nolimits M=\dim\ker M$.

Figures (2)

  • Figure 1: A signed nonlinear network with five nodes, three terminals (encircled vertices), five resistive edges with unitary resistance ($R=1$), and one negative conductance edge (boxed symbol).
  • Figure 2: Bifurcation diagrams (projected on $\hat{v}$) with respect to the negative conductance gain $k$ of the terminal behavior of the network in Figure \ref{['fig: net example']} for different terminal current vectors $u_B$ and $\beta=0.5$. Branches of stable (unstable) equilibria are indicated by thick (thin) lines.

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 1: Network Laplacian and local sensitivity analysis
  • Remark 2: Nonsingular central Laplacian in the signed and unsigned case
  • Remark 3: Non-computability of signed nonlinear Kron-reduced models
  • Remark 4: Connections with Catastrophe Theory
  • Lemma 2
  • proof
  • ...and 6 more