Hard limits on robust control over delayed and quantized communication channels with applications to sensorimotor control

Abstract

The modern view of the nervous system as layering distributed computation and communication for the purpose of sensorimotor control and homeostasis has much experimental evidence but little theoretical foundation, leaving unresolved the connection between diverse components and complex behavior. As a simple starting point, we address a fundamental tradeoff when robust control is done using communication with both delay and quantization error, which are both extremely heterogeneous and highly constrained in human and animal nervous systems. This yields surprisingly simple and tight analytic bounds with clear interpretations and insights regarding hard tradeoffs, optimal coding and control strategies, and their relationship with well known physiology and behavior. These results are similar to reasoning routinely used informally by experimentalists to explain their findings, but very different from those based on information theory and statistical physics (which have dominated theoretical neuroscience). The simple analytic results and their proofs extend to more general models at the expense of less insight and nontrivial (but still scalable) computation. They are also relevant, though less dramatically, to certain cyber-physical systems.

LQ vs. ℓ<inf>∞</inf> in controller design for systems with delay and quantization

“…Combining this with the LQ cost (8), we can observe a tradeoff between signaling speed and accuracy on system performance, analogously to the one in the ∞ system [4]. Larger bandwidth reduces the quantization cost but increases the delay cost.…”

“…In the sensorimotor system, the body corresponds to the plant, the central nervous system to the controller having delay and limited bandwidth. The speed/accuracy tradeoff (4) for spiking neurons is given by R ∝ d in [4], [1] if we assume the space and energy for those neurons are fixed.…”

“…We summarize here the existing robust control theory for ∞ systems with delay and quantization [12] [4]. For disturbance with bounded support w ∞ ≤ L and stabilizing bandwidth R > log 2 |A|, the optimal performance is max w ∞≤L…”

“…This theory provides insights into the interaction between the tradeoffs at component and system level. This paper mainly focuses on the control theory, while its application in neuroscience is discussed in [3] [4].…”

“…Although the work [4] has demonstrated the system level tradeoff between delay cost and quantization cost in the ∞ system, it was unknown whether that tradeoff holds universally for other systems (or norms). In this paper, we derive a performance lower bound as a function of delay and bandwidth in a LQ system, from which the system level tradeoff between delay cost and quantization cost can also be observed.…”