Forced Oscillations and Resonance

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Journal of Mathematical Neuroscience (2011) 1:11 DOI 10.1186/2190-8567-1-11 RESEARCH Open Access Spontaneous voltage oscillations and response dynamics of a Hodgkin-Huxley type model of sensory hair cells Alexander B Neiman · Kai Dierkes · Benjamin Lindner · Lijuan Han · Andrey L Shilnikov Received: 26 May 2011 / Accepted: 31 October 2011 / Published online: 31 October 2011 © 2011 Neiman et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License Abstract We employ a Hodgkin-Huxley-type model of basolateral ionic currents in bullfrog saccular hair cells for studying the genesis of spontaneous voltage oscilla- tions and their role in shaping the response of the hair cell to external mechanical stimuli. Consistent with recent experimental reports, we find that the spontaneous dynamics of the model can be categorized using conductance parameters of calcium- activated potassium, inward rectifier potassium, and mechano-electrical transduction (MET) ionic currents. The model is demonstrated for exhibiting a broad spectrum AB Neiman (  ) · LHan Department of Physics and Astronomy, Neuroscience Program, Ohio University, Athens, OH 45701, USA e-mail: neimana@ohio.edu LHan School of Science, Beijing Institute of Technology, 100081 Beijing, People’s Republic of China e-mail: hanljbit@gmail.com KDierkes· B Lindner Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany KDierkes e-mail: kai@pks.mpg.de B Lindner e-mail: benji@pks.mpg.de B Lindner Bernstein Center for Computational Neuroscience, Physics Department Humboldt University Berlin, Philippstr. 13, Haus 2, 10115 Berlin, Germany AL Shilnikov Neuroscience Institute and Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, USA e-mail: ashilnikov@gsu.edu Page2of24 Neimanetal. of autonomous rhythmic activity, including periodic and quasi-periodic oscillations with two independent frequencies as well as various regular and chaotic bursting pat- terns. Complex patterns of spontaneous oscillations in the model emerge at small values of the conductance of Ca 2+ -activated potassium currents. These patterns are significantly affected by thermal fluctuations of the MET current. We show that self- sustained regular voltage oscillations lead to enhanced and sharply tuned sensitivity of the hair cell to w eak mechanical periodic stimuli. While regimes of chaotic oscil- lations are argued to result in poor tuning to sinusoidal driving, chaotically oscillating cells do provide a high sensitivity to low-frequency variations of external stimuli. 1 Introduction Perception of sensory stimuli in auditory and vestibular organs relies on active mech- anisms at work in the living organism. Manifestations of this active process are high sensitivity and frequency selectivity with respect to weak stimuli, nonlinear com- pression of stimuli with larger amplitudes, and spontaneous otoacoustic emissions [1]. From a nonlinear dynamics point of view, all these features are consistent with the operation of nonlinear oscillators within the inner ear [2, 3]. The biophysical im- plementations of these oscillators remain an important topic of hearing research [1, 4–6]. Several kinds of oscillatory behavior have experimentally been observed in hair cells, which constitute the essential element of the mechano-electrical Forced Oscillations and Resonance Forced Oscillations and Resonance Bởi: OpenStaxCollege You can cause the strings in a piano to vibrate simply by producing sound waves from your voice (credit: Matt Billings, Flickr) Sit in front of a piano sometime and sing a loud brief note at it with the dampers off its strings It will sing the same note back at you—the strings, having the same frequencies as your voice, are resonating in response to the forces from the sound waves that you sent to them Your voice and a piano’s strings is a good example of the fact that objects—in this case, piano strings—can be forced to oscillate but oscillate best at their natural frequency In this section, we shall briefly explore applying a periodic driving force acting on a simple harmonic oscillator The driving force puts energy into the system at a certain frequency, not necessarily the same as the natural frequency of the system The natural frequency is the frequency at which a system would oscillate if there were no driving and no damping force Most of us have played with toys involving an object supported on an elastic band, something like the paddle ball suspended from a finger in [link] Imagine the finger in the figure is your finger At first you hold your finger steady, and the ball bounces up and down with a small amount of damping If you move your finger up and down slowly, the ball will follow along without bouncing much on its own As you increase the frequency at which you move your finger up and down, the ball will respond by oscillating with increasing amplitude When you drive the ball at its natural frequency, the ball’s oscillations increase in amplitude with each oscillation for as long as you drive 1/5 Forced Oscillations and Resonance it The phenomenon of driving a system with a frequency equal to its natural frequency is called resonance A system being driven at its natural frequency is said to resonate As the driving frequency gets progressively higher than the resonant or natural frequency, the amplitude of the oscillations becomes smaller, until the oscillations nearly disappear and your finger simply moves up and down with little effect on the ball The paddle ball on its rubber band moves in response to the finger supporting it If the finger moves with the natural frequency f0 of the ball on the rubber band, then a resonance is achieved, and the amplitude of the ball’s oscillations increases dramatically At higher and lower driving frequencies, energy is transferred to the ball less efficiently, and it responds with loweramplitude oscillations [link] shows a graph of the amplitude of a damped harmonic oscillator as a function of the frequency of the periodic force driving it There are three curves on the graph, each representing a different amount of damping All three curves peak at the point where the frequency of the driving force equals the natural frequency of the harmonic oscillator The highest peak, or greatest response, is for the least amount of damping, because less energy is removed by the damping force 2/5 Forced Oscillations and Resonance Amplitude of a harmonic oscillator as a function of the frequency of the driving force The curves represent the same oscillator with the same natural frequency but with different amounts of damping Resonance occurs when the driving frequency equals the natural frequency, and the greatest response is for the least amount of damping The narrowest response is also for the least damping It is interesting that the widths of the resonance curves shown in [link] depend on damping: the less the damping, the narrower the resonance The message is that if you want a driven oscillator to resonate at a very specific frequency, you need as little damping as possible Little damping is the case for piano strings and many other musical instruments Conversely, if you want small-amplitude oscillations, such as in a car’s suspension system, then you want heavy damping Heavy damping reduces the amplitude, but the tradeoff is that the system responds at more frequencies These features of driven harmonic oscillators apply to a huge variety of systems When you tune a radio, for example, you are adjusting its resonant frequency so that it only oscillates to the desired station’s broadcast (driving) frequency The more selective the radio is in discriminating between stations, the smaller its damping Magnetic resonance imaging (MRI) is a widely used medical diagnostic tool in which atomic nuclei (mostly hydrogen nuclei) are made to resonate by incoming radio waves (on the order of 100 MHz) A child on a swing is driven by a parent at the swing’s natural frequency to achieve maximum amplitude In all of these cases, the efficiency of energy transfer from the driving force into the oscillator is best at resonance Speed bumps and gravel roads prove that even a car’s suspension system is not immune to resonance In spite of finely engineered shock absorbers, which ordinarily convert mechanical ...Shalaby et al. Respiratory Research 2010, 11:82 http://respiratory-research.com/content/11/1/82 Open Access RESEARCH © 2010 Shalaby et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Research Combined forced oscillation and forced expiration measurements in mice for the assessment of airway hyperresponsiveness Karim H Shalaby 1 , Leslie G Gold 2 , Thomas F Schuessler 2 , James G Martin 1 and Annette Robichaud* 2 Abstract Background: Pulmonary function has been reported in mice using negative pressure-driven forced expiratory manoeuvres (NPFE) and the forced oscillation technique (FOT). However, both techniques have always been studied using separate cohorts of animals or systems. The objective of this study was to obtain NPFE and FOT measurements at baseline and following bronchoconstriction from a single cohort of mice using a combined system in order to assess both techniques through a refined approach. Methods: Groups of allergen- or sham-challenged ovalbumin-sensitized mice that were either vehicle (saline) or drug (dexamethasone 1 mg/kg ip)-treated were studied. Surgically prepared animals were connected to an extended flexiVent system (SCIREQ Inc., Montreal, Canada) permitting NPFE and FOT measurements. Lung function was assessed concomitantly by both techniques at baseline and following doubling concentrations of aerosolized methacholine (MCh; 31.25 - 250 mg/ml). The effect of the NPFE manoeuvre on respiratory mechanics was also studied. Results: The expected exaggerated MCh airway response of allergic mice and its inhibition by dexamethasone were detected by both techniques. We observed significant changes in FOT parameters at either the highest (Ers, H) or the two highest (Rrs, R N , G) MCh concentrations. The flow-volume (F-V) curves obtained following NPFE manoeuvres demonstrated similar MCh concentration-dependent changes. A dexamethasone-sensitive decrease in the area under the flow-volume curve at the highest MCh concentration was observed in the allergic mice. Two of the four NPFE parameters calculated from the F-V curves, FEV 0.1 and FEF50, also captured the expected changes but only at the highest MCh concentration. Normalization to baseline improved the sensitivity of NPFE parameters at detecting the exaggerated MCh airway response of allergic mice but had minimal impact on FOT responses. Finally, the combination with FOT allowed us to demonstrate that NPFE induced persistent airway closure that was reversible by deep lung inflation. Conclusions: We conclude that FOT and NPFE can be concurrently assessed in the same cohort of animals to determine airway mechanics and expiratory flow limitation during methacholine responses, and that the combination of the two techniques offers a refined control and an improved reproducibility of the NPFE. Background An excessive airway response to agonists such as metha- choline (MCh) or histamine is widely employed as a diag- nostic criterion for asthma [1]. Response is generally measured in human subjects through the spirometric assessment of maximal forced expiratory manoeuvres fol- lowing the administration of progressively increasing concentrations of the constrictive agonist [1]. Forced expiratory manoeuvres have been favoured because of their relative technical simplicity and the widespread availability of inexpensive equipment. However, forced expirations are dependent on patient cooperation, which is not possible to obtain in very young patients [2], and techniques such as forced oscillatory mechanics [3] and the squeeze technique for forced expirations have been applied in these circumstances [4-6]. * Correspondence: annette.robichaud@scireq.com 2 SCIREQ Scientific Respiratory Equipment Vietnam Journal of Mechanics, NCNST of Vietnam T. XX, 1998, No 3 (16 - 23) IN T E R A C T I O N B E T W E E N N O N L IN E A R P A R A M E T R I C A N D FO R C E D O SCILLATIO N S N g u y e n V a n D a o , N g u y e n V a n D i n h , T r a n K im C hi The interaction of nonlinear oscillations is an important and interesting prob lem, which has attracted the attention of many researchers. Minorsky N. [5] has stated “Perhaps the whole theory of nonlinear oscillations could be formed on the basis of interaction”. The interaction between the forced and “linear” parametric oscillations when the coefficient of the harmonic function of time is linear relative to the position has been studied in [l, 4], In this paper this kind of interaction is considered for “nonlinear” parametric oscillation with cubic nonlinearity of the modulation depth. The asymptotic method of nonlinear mechanics [ 1 ] is used. Our attention is focused on the stationary oscillations and their stability. Different resonance curves are obtained. 1. E quation of m otion and approxim ate solution Let us consider a nonlinear system governed by the differential equation where e > 0 is the small parameter; h > 0 is the damping coefficient; 7 > 0 , p > 0 , r > 0, u; > 0 are the constant parameters; eA = u 2 — 1 is the detuning parameter, where the natural frequency is equal to unity; and 6 > 0 is the phase shift between two excitations. The frequency of the forced excitation is nearly equal to the own frequency u, and the frequency of the nonlinear parametric excitation is nearly twice ELS large. So, both excitations are in fundamental resonance. They will interact one to another. Introducing new variables a and rp instead of X and X as follows, X + U 2X = E A x — h i — 7 1 3 + 2 p i 3 c o s2 w i + rcos[uit — (5) , ( 1 . 1) X = a cos 6. X = —acưsinớ, 9 = ut -H ĩp, ( 1.2 ) we have a system of two equations which is fully equivalent to (1.1) da e „ dxb 6 — = — — F sin Ớ, a - j- = — — F cos Ớ, at u at u (1.3 ) 16 ere F = A x — h i — 7 1 3 + 2px3 COS 2ut + r cos(wi — (5). ie equations (1.3) belong to the standard form, for which the asymptotic method applied [1]. Thus, in the first approximation we can replace the right hand sides (1.3) by their averaged values in time. We have the following averaged equations: le r e da e dtp £ ~dt= ~ 4 Z adt^~ 2 Ũ 90' /0 = 2h.ua + pa3 sin 20 + 2r s in ( 0 + 6), go = a E + p a 3 COS 2iỊ) + r cos(t/> + (5), 3 , E = A — - 7 a . 4 (1.4) (1.5) The stationary solution (ao^xpo) of the equations (1.4) are determined by da dĩp uations — = 0, ■— = 0 or dt dt / 0 = 2h u a 0 + pa.Q sin 2 0 0 + 2r sin (0 o + £) = 0, gQ = a0E 0 + p al C O S 2xị)0 + r cos(rjj0 + Ổ) = 0, H 6) 3 2 E 0 — A — -'ya.Q, 4 equivalently / l = fo COSTCO - <7o sin ĩpo 3 r = ~ rs in <5 + - sin(2i/>0 + <5) + 2h u a 0 COS 0 0 + (paổ — .EcOao sinV>0 = 0 , 1 2_ (1.7) = f Q s m i p o + (Jq co s xpQ = — ~ r COS 6 — - c o s (2t/>0 + <5) + 2hua0 sin 0 0 + (p °0 + Eo)ao COS 00 = 0. 2 2 elow , for sim p licity, w e con sid er only the case (5 = 0 . To elim ina te COS 2 t/)0 an d n2ĩịjo from (1.6) and (1.7), we use the combinations / = ~fo - Pao /i = rhuao - ịpa^ipaị - E0) - — 2phua,Q COS ipo = 0, sin xJjQ T T 3 9 = 2^0 + Paổỡi = 2 flo^o + 2 rpa° + 2p/iwao sin 00 ^2 + p a £(p a0 + £ o ) (1.8) r L~2 cos 00 = 0 17 The condition for equivalence of (1.6) and (1.8) is r2 Ỷ 4p2flo* As usual, equations (1.8) are considered as two linear algebraic equations relative to two unknowns u an(i V : u = sint/>o; V = COS 0 0 . T h e elim in a tio n o f th e p hase 00 can be d on e by using the relationship u2 + V 2 = 1. Two cases must be identified: 1. The “ordinary” case when the determinant D of the coefficients of u and V in (1.8) is different from zero, where D 2phujaị p aị{páị - Eo) - r 2 D — 4p2h 2u 2cLo + [r2 — pa^(pa 2. The “critical* case when D = 0 r2 2 + Pao(Pao + E o) 2phua,Q - E o )} [ V— + pa40{pal + E 0 Journal of Technical Physics. 16, 2, 213— 225, 1975. Polish Academy o f Sciences. Institute o f Fundamental Technological Research, warszawa INTERACTION BETWEEN PARAMETRIC AND FORCED OSCILLATIONS IN MULTIDIMENSIONAL SYSTEMS NGUYEN VAN DAO (HANOI) This paper is devoted to the investigation of the interaction between parametric and forced non-linear oscillations in multidimensional system described by two non-linear differential equations of the second order. The two modes (.X, y) of the system considered are excited by sinusoidal forces. The two modes are coupled non-linearly by means of the product of their coordinates. Under certain conditions the oscillation of the first mode (x) excites parametrically the oscillation of the second one and so the two oscillations of the second mode (parametric and forced oscillations) may coexist and there exists some kind of interaction between them. We shall now consider the stationary oscillations of the modes and their stability. 1. Equations of Motion. Stationary Oscillations Let us consider the oscillations of the system with two degrees of freedom described by a set of two differentia] equations of the type x+ / 2 ^ + £/2(/i0ir + ccc3 + cv2 x) = Osinyf, (1 1) ' ỷ + (ư2y + eco2(hỷ + fỉy3 + bx2y) = eơ)2p cos(ví-f <5), where h0 > 0, h > 0, a, Cy b, Q,p > 0, /?, Ổ, are constants and £ is a small parameter. We assume the following relations between the frequencies cư,r, and y: (1.2) cư2 = Ơ2V2 + eơ)2A , y = ev, nX # m y , where Ơ, e are rational numbers, A is detuming of frequencies and m, n are integers. First, we transform the Eqs. (1.1) by means of the formulae X = <7siny/-f ax COSỠ!, X = yqcosyt— Aai sinfli, 0 -3 ) y — ỠCOSỚ, ỳ = -ơrasine, q = J ĩ ị p , where al , ỚJ, Ơ, 6 are the new variables which will be determined later. 214 Nguyen Van Dao Substituting (1.3) into (1.1) and transforming it, we obtain the following equations in the standard form: da ị dĩ dxp l = EẰ(h0x + ccx3 + cy2x)sinOi , (1.4) ai —J— = eXỌìqX + QLX3 -+■ cy^x )CO SỚỊ, at = eơv<Ị>(x,y,ỷ, 0 sinớ + 0 0 2), at ady ~dT = eơv&(x, y, ỳ, t)cosd + 0(e2), where where (1.5) &(x9 y, ỳ, 0 = Ay + hỳ + bx2y + Py3 — pcos(vt+ Ỗ), y)ị = ỚJ — At, rp = d — ơvt Averaging the right-hand parts of (1.4) over the time, we receive the equations of the first approximation for the unknowns a, al9y>: 1 ) for ơ # e, ơ # 1 . £//i0 - «1. (1.6) n 2 eơ~vz a = -y-ha+ b , 3 ỡ . 2~ + 4 ^ 8 a1 = — — h^ax (1.7) £V à = — ~Y [vha +/?sin(y>— ($)]+ axp = 3) for e = Ơ # 1 £T zla + ợ2ứ + — /fa3 -p cos(y — Ô) ỏi = ” 2 ^°ứl’ (1.8) à = — eơr ơvha + - 7- a2a sin 2 w 4 £ơ = -y r + ; Interaction between parametric and forced oscillations 215 4) for e = Ơ = 1 ■ EẢ u a\ — 2~ 0ứl» (1.9) a — — vha+ -^-q2asin2y)Jrpsin(y)—ỗ)\+ = b 3 b Aa+ ~Y q2a+ ậa3— -Ỵ q2acos2y)—pcos(y — Ô) + where the nonwritten terms disappear when flj = 0. By analogy with N e s s [1], the first case is called the non-resonant case, the second case — the harmonic resonant case, the third (1.8) — parametric resonant case and the last (1.9) — harmonic and parametric resonant case. Obviously, the most interesting is the last resonant case. We shall inves tigate it in more detail. The stationary solution of the system (1.9) is the one which is determined from equa tions = ả = ỹ) = 0 or flj = 0, (1.10) r / j f l - f <72 f l s i n 2 y j + / ? s i n ( Y > — Ô) = 0 , Aa-\- — q2a+ pa3— Ỵ q2acos2ĩp—pcos(y-ô) = 0. Eliminating \p in the two last equations of (1.10), we obtain the following relation for the amplitude a = a0 = const of the stationary oscillation of the coordinate y: (1.11) A/ = 0, where M = (w2 + u2 — V 2) 2 — /? 2 [ (w — acos2 <5)2 + (w*f Z>sin2<5)2], (1.12) u = vha0, v = ~ q 2a0, IV = - a0 ỊA + q2 + ậaị The relation (1.11) is expressed in the parameters of the initial system as al (1.13) -F I - * 2 b 2 3 l\ 2,2 b1 e + 2 q + 4 H + ^ 2 - T 6 ^ 1 —k2 b 3 o 2 b \ b 2 . . - —— h q2 + + -j-ợ2cos2<5 + I co/2 H ~ .. .Forced Oscillations and Resonance it The phenomenon of driving a system with a frequency equal to its natural frequency is called resonance A system being driven... amplitude of the oscillations becomes smaller, until the oscillations nearly disappear and your finger simply moves up and down with little effect on the ball The paddle ball on its rubber band moves... frequency f0 of the ball on the rubber band, then a resonance is achieved, and the amplitude of the ball’s oscillations increases dramatically At higher and lower driving frequencies, energy is