Regulation without calibration

TL;DR

This paper revisits the internal model principle and argues that exact trajectory regulation imposes impractical calibration demands. It shifts focus from precise trajectory tracking to reliable event regulation, leveraging excitable generators and synaptic-like feedback to generate and synchronize discrete events despite environmental variability. Through the pendulum and neuromorphic circuit examples, it demonstrates how contraction properties enable reliable event timing and how event-based coupling avoids full trajectory synchronization in heterogeneous networks. The work highlights a path toward a formal theory of event regulation, with potential impact on robust neuromorphic design and bio-inspired control in uncertain environments.

Abstract

This article revisits the importance of the internal model principle in the literature of regulation and synchronization. Trajectory regulation, the task of regulating continuous-time signals generated by differential equations, is contrasted with event regulation, the task of only regulating discrete events associated with the trajectories. In trajectory regulation, the internal model principle requires an exact internal generator of the continuous-time trajectories, which translates into unrealistic calibration requirements. Event regulation is envisioned as a way to relieve calibration of the continuous behavior while ensuring reliability of the discrete events.

Figures (12)

• Figure 1: Reliability of spike timing in response to different input stimuli from mainen1995reliability. The same cortical neuron was stimulated repeatedly over 25 trials using two different current inputs. In panel A (left column), a constant step (DC) current was applied. The middle trace shows the input stimulus, the top plot overlays the resulting voltage responses from the first 10 trials, and the bottom raster plot marks the spike times from all 25 trials. In panel B (right column), the neuron was stimulated using the same realization of Gaussian white noise across all 25 trials. While the step input triggers variable spike timing across trials, the frozen noise input leads to highly reliable spike times.
• Figure 2: Control structure of the pendulum tracking problem. The problem can be understood as the interaction between three pendulums: (i) the reference pendulum (the Exosystem), which defines the desired trajectory; (ii) the Internal Model, which is a copy of the Exosystem and replicates the reference behavior; (iii) the real pendulum (the Plant), which is controlled to follow and track the reference position.
• Figure 3: Illustration of the pendulum tracking problem in the bistable regime. The reference position $\theta_r$ (dashed green) has been reset at times $t=80$ and $t=130$ to induce small and large oscillations behaviors. The controlled position $\theta$ (blue) tracks the reference using control law \ref{['e.ex1.u']} with $(\hat{\theta}_r, \dot{\hat{\theta}}_r)$ and error feedback.
• Figure 4: Circuit representation of the disturbance rejection problem. The FitzHugh-Nagumo circuit (Plant, green) is modelled by an RLC circuit in parallel with a current generator and a nonlinear diode $f(\cdot)$, driven by an external current $I$. The disturbance element (the exosystem) is introduced as an external disturbance $d$, and it is compensated by a current signal $u$, generated by an internal model that mimics the disturbance. These two additional blocks are connected in parallel to the main circuit to achieve disturbance rejection, cancelling out the effect of $d$ on the voltage trajectory $v$.
• Figure 5: Illustration of the disturbance rejection problem with constant step input $I$ in \ref{['ex.eq.fn.con']}, zoomed in the interval $[300, 500]$. The constant applied current causes a limit cycle which is perturbed by extra spikes from the synaptic disturbance $d(t)$ (top). The control law \ref{['e.ex2.u']} compensates the extra spikes in the voltage trajectory (bottom, dashed green), but leaves a residual phase shift (bottom, blue). This residual phase shift illustrates the lack of contraction of the reference trajectory in the absence of error feedback. Error feedback is necessary for regulation whenever the exosystem is not contractive.