Analysis of a Spatialized Brain-Body-Environment System

TL;DR

The paper addresses the limitation of ODE-only brain-body-environment models that miss spatial geometry by introducing spatially extended PDE representations for both brain and body in a minimal swinging agent. It reformulates Thorniley-Husbands’ reactive swinging agent into a mixed PDE-ODE framework with $d_t v = K d_{xx} v + u/(1+r^2) + g cos(\theta) + (1+r) \omega^2 - k r - c v$ and $d_t u = K d_{xx} u + \phi(A tanh(\rho v) - u)$, while keeping $d_theta/dt = \omega$, $d_omega/dt = -(g/(1+r)) sin(\theta) - b \omega$, $d_r/dt = v$ as ODEs. Key contributions include showing that diffusion $K$ dampens chaotic dynamics and raises the sensory-coupling threshold $A$, that boundary conditions and sensor-effector geometry bias signal flow, and that approximations like the boundary-distance model can capture essential effects. These findings bridge abstract neural dynamics and embodied constraints, with implications for designing spatially aware embodied systems and informing neuroscience theories about how neural geometry shapes adaptive behavior.

Abstract

The brain-body-environment framework studies adaptive behavior through embodied and situated agents, emphasizing interactions between brains, biomechanics, and environmental dynamics. However, many models often treat the brain as a network of coupled ordinary differential equations (ODEs), neglecting finer spatial properties which can not only increase model complexity but also constrain observable neural dynamics. To address this limitation, we propose a spatially extended approach using partial differential equations (PDEs) for both the brain and body. As a case study, we revisit a previously developed model of a child swinging, now incorporating spatial dynamics. By considering the spatio-temporal properties of the brain and body, we analyze how input location and propagation along a PDE influence behavior. This approach offers new insights into the role of spatial organization in adaptive behavior, bridging the gap between abstract neural models and the physical constraints of embodied systems. Our results highlight the importance of spatial dynamics in understanding brain-body-environment interactions.

Figures (7)

• Figure 1: The neuromechanical model originally presented in ThornileyHusbands2013. The agent is represented as a mass-spring which is attached to then end of a rigid rod. The rod can rotate about the top and $\theta$ denotes the deviation from the rod in its steady resting state. The dynamics of the movement of the rod are governed by the gravitational and torsional forces which are affected by the length of the agent $r$. The agent can then actuate the value of $r$ to move the rod (swing).
• Figure 2: Bifurcation diagram for a swinging agent system, plotting agent state $u$ (vertical axis) at the vertical stable position against sensory sensitivity $\alpha$ (horizontal axis). We used a normalized parameter $\alpha=\frac{A}{300}$ for computational simplicity. $u$ remains stable at zero until $A$ exceeds a threshold, causing the swing to overturn. Key transitions include the Hopf point (onset of oscillations), pd (period-doubling cascade into chaos), and bp (branch points for computational continuation). The coloring and solid arrows in the bifurcation diagram represent the 3 phases of the system: Dampened regime (red), oscillatory regime (yellow), chaotic regime (blue). The diagram also shows that the transition from oscillatory to chaotic is not instantaneous as it requires a cascade of pd bifurcations, the transitory region is tinted green and denoted with a dashed arrow.
• Figure 3: Here we visualize the variables of the agent as a circuit diagram. $\theta$, $\omega$ and $r$ are three state variables. Variables in a circle represent underactuated environmental variables. The variable $r$ is the barrier between the agent and the environment, playing both roles as the agent's sensor, gathering information from $\omega$ into the weighted region $p(x-x_s)$ centered at $x_s$ in the agent $v$, and as the agent's actuator, realizing the agent's response from $v(x_e,t)$ into the environment $\omega$. Note that we assume the distribution $p(x-x_s)$ to follow $N(x_s,\sigma_s)$, we examine both cases $\sigma_s\rightarrow\infty$ (uniform distribution) and finite $\sigma_s$ (localized distribution). Points on the ends of the state variables $x_{\sf min}$ and $x_{\sf max}$ denote where the Dirichlet boundary conditions are specified.
• Figure 4: Frequency power spectrum of the angular velocity $\omega(t)$ as the diffusivity $K$ and the sensory coupling strength $A$ are varied. The effector position is fixed at $x_e=0$ with global presence in the agent $\sigma_s\rightarrow\infty$. The boundary condition is $D=0$, and thus $BC(x)=0$. We observe a variety of phases of the mixed PDE-ODE system in terms of parameters $K$ and $A$. As $A$ increases we see the sequence of transitions from the dampened regime to the oscillatory regime to the chaotic regime. As $K$ increases this process is delayed prolonging the dynamics of the stable regimes even with high sensory feedback.
• Figure 5: Frequency power spectrum of the angular velocity $\omega(t)$ as the effector position $x_e$ and the sensory coupling strength $A$ are varied. The diffusivity is fixed at $K=1$ and the envionment interacts globally in the agent $\sigma_s\rightarrow\infty$. The boundary condition is $D=0$, and thus $BC(x)=0$. We observe a similar variety of phases of the mixed PDE-ODE system but the transition point and the approach to criticality is modified due to the influence of the Dirichlet boundary condition on the effector is stronger when the effector position $x_e$ is closer to the two ends.