Φ -Haar Wavelet Operational Matrix Method for Fractional Relaxation-Oscillation Equations Containing Φ -Caputo Fractional Derivative

This paper proposes a numerical method for solving fractional relaxation-oscillation equations. A relaxation oscillator is a type of oscillator that is based on how a physical system returns to equilibrium after being disrupted. The primary equation of relaxation and oscillation processes is the relaxation-oscillation equation. The fractional derivatives in the relaxation-oscillation equations under consideration are defined in the Φ -Caputo sense. The numerical method relies on a novel type of operational matrix method, namely, the Φ -Haar wavelet operational matrix method. The operational matrix approach has a lower computational complexity. The proposed scheme simplifies the main problem to a set of linear algebraic equations. Numerical examples demonstrate the validity and applicability of the proposed technique.

Phase response analyses support a relaxation oscillator model of locomotor rhythm generation in Caenorhabditis elegans

Neural circuits coordinate with muscles and sensory feedback to generate motor behaviors appropriate to an animal’s environment. In C. elegans, the mechanisms by which the motor circuit generates undulations and modulates them based on the environment are largely unclear. We quantitatively analyzed C. elegans locomotion during free movement and during transient optogenetic muscle inhibition. Undulatory movements were highly asymmetrical with respect to the duration of bending and unbending during each cycle. Phase response curves induced by brief optogenetic inhibition of head muscles showed gradual increases and rapid decreases as a function of phase at which the perturbation was applied. A relaxation oscillator model based on proprioceptive thresholds that switch the active muscle moment was developed and is shown to quantitatively agree with data from free movement, phase responses, and previous results for gait adaptation to mechanical loadings. Our results suggest a neuromuscular mechanism underlying C. elegans motor pattern generation within a compact circuit.

Implementation of types of VCO

A Voltage Controlled Divider (VCO) is a basic building block in most of the electronic systems. Phase-locked loop (PLL), tone synthesizers, Frequency Shift Keying (FSK), frequency synthesizers, etc make use of VCO’s to generate an oscillating frequency that can be decided with the help of components. Voltage Controlled Divider can be implemented for analog applications. The project proposes three types of VCO using Electric tool and LT Spice XVII tool. The three VCO’s that are implemented are CMOS Ring Oscillator, Colpitts Oscillator and Relaxation Oscillator. These circuits generate two oscillating frequencies that is decided by the circult components. Keywords: Voltage Controlled Divider (VCO), CMOS Ring Oscillator, Colpitts Oscillator, Relaxation Oscillator, oscillating frequency.