Fig. 1.

Observed behaviors and simplified model for the multiagent shepherding task. (A and B) Depiction of successful S&R and COC (Left). The black line trajectory indicates the previous 5 s of behavior. Timeseries during one successful trial of each angular component of the participant’s position (Center) and corresponding power spectra (Right), with a 0.5-Hz frequency boundary used to distinguish between S&R and COC behavioral modes. (C) Simplified shepherding model developed to describe behavior seen in ref. 19) (Materials and Methods has the more detailed model equations used for this study). Eq. F1 is a linear damped mass spring equation that reduces the difference between agent i’s current radial distance, $r_i$, to the radial distance, $r_{sf\left(t\right),i}$, of the farthest sheep on agent i’s one-half of the game field at time t plus a fixed distance, $r_{\min }$, to ensure repulsion toward the center. Eq. F2, excluding, $\beta {\dot{\theta }}_i^3+\gamma {\theta }_i^2\dot{{\theta }_i}$, is identical to Eq. F1 but for the control of agent i’s angular movement, ${\theta }_i$. The inclusion of the terms $\beta {\dot{\theta }}_i^3+\gamma {\theta }_i^2\dot{{\theta }_i}$ convert the linear damped mass spring to a nonlinear system with behavior that can exhibit both point attractor dynamics (akin to Eq. F1) when $b_{\theta i}>0$ and limit cycle dynamics when $b_{\theta i}\le 0$ (the blue agent in C, Right). The dissipative coupling function on the right side of Eq. F2 ensures oscillatory synchronization between agent i and partner j. Eq. F3 is a parameter-dynamic function that determines the value of parameter $b_{\theta i}$, such that $b_{\theta i}$ will be attracted to a value that is less than or equal to zero when the radial distance of agent i’s farthest sheep $r_{sf\left(t\right),i}\le r_{\mathrm{\Delta }}$. Materials and Methods has more details, including human- and task-specific modifications for the experiments presented in this paper.