A four-state adaptive Hopf oscillator

Adaptive oscillators (AOs) are nonlinear oscillators with plastic states that encode information. Here, an analog implementation of a four-state adaptive oscillator, including design, fabrication, and verification through hardware measurement, is presented. The result is an oscillator that can learn the frequency and amplitude of an external stimulus over a large range. Notably, the adaptive oscillator learns parameters of external stimuli through its ability to completely synchronize without using any pre- or post-processing methods. Previously, Hopf oscillators have been built as two-state (a regular Hopf oscillator) and three-state (a Hopf oscillator with adaptive frequency) systems via VLSI and FPGA designs. Building on these important implementations, a continuous-time, analog circuit implementation of a Hopf oscillator with adaptive frequency and amplitude is achieved. The hardware measurements and SPICE simulation show good agreement. To demonstrate some of its functionality, the circuit’s response to several complex waveforms, including the response of a square wave, a sawtooth wave, strain gauge data of an impact of a nonlinear beam, and audio data of a noisy microphone recording, are reported. By learning both the frequency and amplitude, this circuit could be used to enhance applications of AOs for robotic gait, clock oscillators, analog frequency analyzers, and energy harvesting.

A free-running oscillator times and executes centriole biogenesis

The accurate timing of organelle biogenesis and the precise regulation of organelle size are crucial for cell physiology. Centriole biogenesis initiates exclusively in S-phase, when a daughter centriole emerges from the side of a pre-existing mother and grows until it reaches its mother’s size. This process is regulated by Polo-like kinase 4 (Plk4), which is recruited to centrioles in oscillatory waves in flies and human cells 1,2. The nature and function of Plk4 oscillations is, however, unknown. Here we discover that Plk4 forms an adaptive oscillator at the base of the growing centriole, whose function is to time and set the duration of centriole biogenesis in Drosophila embryos. We demonstrate that the Plk4 oscillator is free-running of, but is entrained and calibrated by, the core Cdk/Cyclin cell-cycle oscillator, explaining how centrioles can duplicate independently of the cell cycle 3–5. Mathematical modelling and further experiments indicate that the Plk4 oscillator is generated by a time-delayed negative-feedback loop in which Plk4 recruitment to, and dissociation from, the centriole is monitored via changes in the affinity-state of its centriolar receptor, Asterless. We postulate that such organelle-specific autonomous oscillators could regulate the timing and execution of organelle biogenesis more generally.