Fundamental limits and optimal estimation of the resonance frequency of a linear harmonic oscillator

All physical oscillators are subject to thermodynamic and quantum perturbations, fundamentally limiting measurement of their resonance frequency. Analyses assuming specific ways of estimating frequency can underestimate the available precision and overlook unconventional measurement regimes. Here we derive a general, estimation-method-independent Cramer Rao lower bound for a linear harmonic oscillator resonance frequency measurement uncertainty, seamlessly accounting for the quantum, thermodynamic and instrumental limitations, including Fisher information from quantum backaction- and thermodynamically driven fluctuations. We provide a universal and practical maximum-likelihood frequency estimator reaching the predicted limits in all regimes, and experimentally validate it on a thermodynamically limited nanomechanical oscillator. Low relative frequency uncertainty is obtained for both very high bandwidth measurements (≈10−5 for τ = 30 μs) and measurements using thermal fluctuations alone (<10−6). Beyond nanomechanics, these results advance frequency-based metrology across physical domains.

Spontaneous variability in gamma dynamics described by a linear harmonic oscillator driven by noise

SUMMARYCircuits of excitatory and inhibitory neurons can generate rhythmic activity in the gamma frequency-range (30-80Hz). Individual gamma-cycles show spontaneous variability in amplitude and duration. The mechanisms underlying this variability are not fully understood. We recorded local-field-potentials (LFPs) and spikes from awake macaque V1, and developed a noise-robust method to detect gamma-cycle amplitudes and durations. Amplitudes and durations showed a weak but positive correlation. This correlation, and the joint amplitude-duration distribution, is well reproduced by a dampened harmonic oscillator driven by stochastic noise. We show that this model accurately fits LFP power spectra and is equivalent to a linear PING (Pyramidal Interneuron Network Gamma) circuit. The model recapitulates two additional features of V1 gamma: (1) Amplitude-duration correlations decrease with oscillation strength; (2) Amplitudes and durations exhibit strong and weak autocorrelations, respectively, depending on oscillation strength. Finally, longer gamma-cycles are associated with stronger spike-synchrony, but lower spike-rates in both (putative) excitatory and inhibitory neurons. In sum, V1 gamma-dynamics are well described by the simplest possible model of gamma: A linear harmonic oscillator driven by noise.

Nonlocal transformations of the generalized Liénard type equations and dissipative Ermakov-Milne-Pinney systems

We employ the method of nonlocal generalized Sundman transformations to formulate the linearization problem for equations of the generalized Liénard type and show that they may be mapped to equations of the dissipative Ermakov-Milne-Pinney type. We obtain the corresponding new first integrals of these derived equations, this method yields a natural generalization of the construction of Ermakov–Lewis invariant for a time-dependent oscillator to (coupled) Liénard and Liénard type equations. We also study the linearization problem for the coupled Liénard equation using nonlocal transformations and derive coupled dissipative Ermakov-Milne-Pinney equation. As an offshoot of this nonlocal transformation method when the standard Liénard equation, [Formula: see text], is mapped to that of the linear harmonic oscillator equation, we obtain a relation between the functions [Formula: see text] and [Formula: see text] which is exactly similar to the condition derived in the context of isochronicity of the Liénard equation.