Dynamics of Relaxation Oscillations

Abstract

Relaxation oscillations are characteristic of periodic processes consisting of segments which differ greatly in time: a long-time span when the system is moving slowly and a relatively short time span when the system is moving rapidly. The period of oscillation, the sum of these contributions, is usually treated by singular perturbation theory which starts from the premise that the long span is asymptotically extended and the short span shrinks asymptotically to a single instant. Application of the theory involves the analysis of adjacent dynamical regions and multiple time scales. The relaxation oscillations of the Stoker-Haag piecewiselinear discontinuous equation and the van der Pol equation are investigated using a simpler analytical method requiring only the connection at a point of the two dynamical fast and slow regions. Compared to the results of singular perturbation theory, the quantitative results of the present method are more accurate in the Stoker-Haag case and marginally less accurate in the van der Pol case. The relative simplicity of the formulation suggests extension to threedimensional systems where relaxation oscillations can become unstable leading to bistability, multiple periodicity and chaos. *

An Analytic Picture of Neuron Oscillations

“…For the symmetric case, the period according to the asymptotic Eq. (5") is 9.75, which is corrected to 13.56 including the 7.0143 (k ) 1 3 term [Dorodnitsyn, 1947;Phillipson & Schuster, 2001], in good agreement with the computer value of 13.08. Equation (5'…”

“…Since p 1 p 2 = 1, from Eq. (A.3), T 1 ≈ p 2 ≈ 1 k → 0 and to the same approximation v s1 → 0 [Phillipson & Schuster, 2001]…”

“…(2,3,4) include both I(τ ) and dI dτ for which analytic solutions for arbitrary I can be simply arrived out for the Reduced-Broken Linear Model. In particular, a periodic I(τ ) can produce multiple periodicity, bistability, entrainment and chaos for the same dynamical reasons that they arise in the harmonically forced van der Pol equation or its Broken-Linear reduction of the harmonically driven Stoker-Haag equation [Phillipson & Schuster, 2001]. Linearization of the F-N equations around fixed points in its phase plane representation with an imposed time dependent input has demonstrated some of these phenomena [Di Garbo et al, 2001].…”

“…This expression for the period demonstrates logarithmic approach to steady state behavior as I approaches the barriers I 1,2 = q 1,2 [T → ∞]. Furthermore, the period is a minimum at I min = q 1 +q 2 2 , at which point T min = T (I min = 2k ln 3 which is identical to the asymptotic period of the symmetric Stoker-Haag oscillator [Stoker, 1950;Phillipson & Schuster, 2001].…”

“…(4) for V (τ ) (black curve) and v(τ ) ≡ dV dτ = dx dτ (red curve). Following a procedure identical to that applied to calculation of the period for the symmetric oscillations of the Stoker-Haag oscillator [Phillipson & Schuster, 2001] we consider the system initially at point a at the lower boundary:…”