Spiking Nonlinear Opinion Dynamics (S-NOD) for Agile Decision-Making

TL;DR

The model, Spiking Nonlinear Opinion Dynamics (S-NOD), provides superior agility, characterized by fast, flexible, and adaptive response to rapid and unpredictable changes in context, environment, or information received about available options.

Abstract

We present, analyze, and illustrate a first-of-its-kind model of two-dimensional excitable (spiking) dynamics for decision-making over two options. The model, Spiking Nonlinear Opinion Dynamics (S-NOD), provides superior agility, characterized by fast, flexible, and adaptive response to rapid and unpredictable changes in context, environment, or information received about available options. S-NOD derives through the introduction of a single extra term to the previously presented Nonlinear Opinion Dynamics (NOD) for fast and flexible multi-agent decision-making behavior. The extra term is inspired by the fast-positive, slow-negative mixed-feedback structure of excitable systems. The agile behaviors brought about by the new excitable nature of decision-making driven by S-NOD are analyzed in a general setting and illustrated in an application to multi-robot navigation around human movers.

Figures (6)

• Figure 2: (a): Trajectory of a robot controlled with NOD (S-NOD) is shown with a blue (pink) line as it navigates towards a goal (star) in the presence of an oncoming human mover (black). The NOD robot experiences a collision; the S-NOD robot does not. (b): Opinion $z$ of the robot over time $t$. Circles denote matching time points along trajectories and opinions. Figures \ref{['fig:intro']}, \ref{['fig:phase plane']}, \ref{['fig:multagent']}, and \ref{['fig:robot_trajectories']} animated at https://spikingNOD.github.io.
• Figure 3: The effect of $K_u$ on the bifurcation diagram of \ref{['eq:NOD']} and the cubic and quintic terms of \ref{['eq:z_dot_taylorExpans1']}. (a): Bifurcation diagrams of NOD \ref{['eq:NOD']} with $K_u$ values corresponding to the vertical dashed lines in (b). Stable (unstable) solutions are shown with solid (dotted) lines. The bifurcation point is $(u_0^*, 0)$. (b): Coefficient $p$ ($q$) as a function of $K_u$ shown as a solid (dashed) black line.
• Figure 4: Opinion solutions of NOD and S-NOD over time and associated bifurcation diagrams. (A): Trajectories of NOD \ref{['eq:NOD']} and S-NOD \ref{['eq:ENOD']} for initial condition $(z, u_0)|_{t=0} = (0.01,0.9)$ for two $K_u$ values. Input signal $b$ is also shown over time. (B): Bifurcation diagrams of \ref{['eq:NOD']} for the two values of $K_u$, showing the solutions in A. moving from initial state to steady state.
• Figure 5: The system solutions and $(u_s,z)$ phase portrait as the basal attention $u_0$ increases. For all, $d \!= \!1, \alpha \!=\! 2,$ (thus $u_0^*\!=\! 0.5$)$, K_u \!=\! 2, K_{u_s} \!=\! 6, \tau_{u_s}/\tau_z \!=\! 10$. (Top): Example solutions of $u_s$ and $z$ over time, with initial condition as $(u_s, z)|_{t=0} = (0.01, 0.01)$ and additive Gaussian distributed white noise. (Bottom): The $u_s$-nullcline ($z$-nullcline) is shown as a blue (pink) line. Solid (dotted) lines indicate stable (unstable) branches of the $z$-nullcline with respect to \ref{['eq:z_dot_ENOD']}. Gray arrows denote the vector field. Black filled circles show stable equilibria, unfilled are unstable equilibria, partially filled is a saddle-node bifurcations. Crosses show saddle equilibria. Saddle-node-homoclinic cycles in (b), and limit cycles in (c) and (d) are in yellow.
• Figure 6: Response of three agents with the same communication network but for opposite signs on graph edge weights and an input $b_1$ applied only to agent $1$. For $i\neq k$, (a) $a_{ik}= + 0.1$ and (b) $a_{ik}= - 0.1$. Parameters are $a_{ii}=1$, $d=1$, $K_u=2.3$, $K_{u_s}=16$, $u_0=0.9$, $\tau_{u_s}/\tau_{z}=20$.

Theorems & Definitions (6)

• Lemma 1
• proof
• Proposition 1
• proof
• Proposition 2
• proof