Noise-enhanced propagation in a dissipative chain of triggers

Noise-enhanced propagation in a dissipative chain of triggers S. Morfu.C. Comte.M. Bilbault and P. Marquié Université de Bourgogne, LE2I, (F.R.E.) C.N.R.S. 2309 BP 40-210 DION cedex, FRANCE email: smorfu@u-bourgogne.fr February 24, 2002 Abstract: We study the influence of spatiotemporal noise on the propagation of square waves in an electrical dissipative chain of triggers. By numerical simulation, we show that noise plays an active role to improve signal transmission. Using the Signal to Noise Ratio at each cell, we estimate the propagation length. It appears that there s an optimum amount of noise that maximizes this length. This specific case of stochastic resonance shows that noise enhances propagation. 1 Introduction In linear systems, noise induces negative effects since the output Signal to Noise Ratio (S.N.R.) is a monotonous decreasing function of noise intensity, whereas the behavior is completely different with nonlinear systems. Indeed, in such systems, it has been shown that an appropriate external noise added to a weak signal of information, called subthreshold signal, can enhance the S.N.R. and then the signal detection [Gammaitoni et al., 199]. This effect known as Stochastic Resonance (SR), since its introduction to explain the periodic recurrence of ice ages in climate dynamics [Benzi et al., 191; Benzi et al., 193; Nicolis, 192], has been investigated in many fields [Gammaitoni et al., 199] like visual perception [Simonotto et al., 199; Ditzinger et al., 2000] or biology [Longtin, 1993; Gebeshuber, 2000; Zeng et al., 2000]. This phenomenon has been studied first in unicellular nonlinear systems, like the Schmitt trigger [Fauve & Heslot, 193; Melnikov, 1993; Godivier & Chapeau-Blondeau, 199; Gammaitoni et al., 1999], and more recently, considering lattices of coupled excitable cells [Lindner et al., 1995; Zhang et al., 199; Locher et al., 199; Lindner et al., 199; Chapeau-Blondeau, 1999; Chapeau- Blondeau & Rojas-Varela, 2000]. In particular, Chapeau-Blondeau [Chapeau-Blondeau, 1999] has shown that the propagation of a low amplitude or subthreshold wave in a nonlinear line of two states threshold elements could be assisted by the addition of noise. Considering overthreshold signals in a dissipative lattice, one might wonder if a spatiotemporal noise could also improve their propagation. In this paper, we study the response of a discrete dissipative electrical line to an initial periodic rectangular pulse train in presence of noise. For an appropriate value of the noise amplitude, we show that the propagation length is enhanced, compared to the case without noise. First, we present the discrete electrical transmission line and study the propagation of square pulses without noise. Adding noise and by a measurement of the S.N.R., a stochastic resonant behavior will appear, leading to the enhancement of the propagation length. 1

4 + 4 + 2 Presentation of the electrical lattice The electrical lattice consists in the succession of elementary cells. Each cell contains an RC low pass filter and a simple comparator with saturation and threshold voltages V OH and U th (figure 1). Figure 1: Schematic representation of the electrical lattice. R and C are linear components, while the comparators present an infinite input impedance and a threshold voltage U th. Applying Kirchoff laws leads to the differential equation between voltages U n 1 and U n, respectively at capacitors n and n 1: τ du n dt + U n = f(u n 1 ), (1) where τ = RC is the time constant of the low pass filters. In eq. (1), the nonlinear function f is the comparator characteristic expressed by: f(u n ) = V n = V OH 2 ( 1+tanh [k(u n U th )] ), (2) where k is the slope of the transfert characteristic. Let us consider, in this section, the response of the system without noise. The signal V 0 (t) launched at the input of the line is a periodic rectangular pulse train of amplitude V OH, period T s = 9τ and width t 0. The low state duration between two successive pulses is sufficient to consider separately each pulse (see figure 2), that is, the capacitors have time enough to be completely discharged (U n (T s ) 0). Let us consider now the case k in the nonlinearity. Then, f tends to an Heaviside function H, and therefore: V n = V OH H(U n U th ). (3) Expressing the charge and discharge of the first capacitor leads straightforwardly to the width t 1 of a pulse after the first comparator: [ ( VOH U th t 1 = t 0 + τ ln 1 exp ( t )] 0 τ ). U th In figure 2 are represented two pulses of the initial voltage signal V 0 (t), and the input and output voltages of the first comparator, respectively U 1 and V 1. Simple calculations allow to determine the recursion expression for the width t n of the resulting rectangular pulse V n (t) at the output of comparator n, that is: [ ( VOH U th t n = t n 1 +τ ln U th τ (4) 0 D 0 1 exp ( t n 1 Figure 2: Propagation of the initial train of pulses V 0 (t) throught the first cell. Voltages U 1 (t) and V 1 (t) represent respectively the voltages at the input and output of the first comparator. )] ). 2

& & ' ' 2 5 5 ' & % = C A D + A I = N I Figure 3: Propagation of the square wave in the lattice. The solid lines represent, from left to right, one pulse at the output of comparators 1, 12 and 19, the dashed lines the voltages U 1, U 12, U 19 at their input (for a sake of clarity, only one period of each voltage U n (t) and V n (t) is represented). Results have been obtained by direct simulation of equations (1) and (2). The value of the width at the input of the line is t 0 = 3τ, while the other parameters are U th = 0.5 V, V OH = 1 V, k = 100, τ = 1 s. Figure 4: Propagation length n max versus the width of the pulses t 0 at the input: the solid line is the theoretical law obtained by the series (4); crosses are simulation results from equations (1) and (2). Parameters are U th = 0.5 V, V OH = 1 V, k = 100, τ = 1 s. This expression shows that t n decreases versus n, revealing the dissipative effect. As a consequence, there exists a maximal propagation length n max (last cell number for which t n > 0) over which propagation stops (see figure 3 for the special case n max =19). Indeed, the charge of capacitor at cell n = n max + 1 is not sufficient to reach the threshold value U th. The propagation length has been obtained by direct simulation of eqs. (1) and (2) with a fourth order Runge-Kutta algorithm. Figure 4 shows a good agreement between theoretical and simulation results and provides a measurement of the propagation length n max without noise. 3 Noise effects As we have defined the propagation properties in the lattice without noise, we will present, in this section, the results obtained by considering spatiotemporal noise effects. So, spatiotemporal zero mean white noise of amplitude A is added to % & ' &. H A G K A? O 0 Figure 5: Power Spectrum of the cell number 22. Parameters are U th = 0.5 V, V OH = 1 V, k = 100, τ = 1 s, T S = 9τ, t 0 = 3 τ, A = 0.15 V. each voltage U n (t). For a given zero mean uniform noise over [ A, A], one hundred simulations are performed with the same initial condition (periodic square pulse train with period T S = 9τ and width 3τ). We calculate then, for each cell, the Power Spectrum (P.S.) of signal V n (t). Each P.S. consists of an amount of spectral peaks at multiple integers of 1/T s emerging from a 1/f background noise (figure 5). 3

& & Using the P.S., the S.N.R. can be defined, at frequency 1/T S, as the ratio between the coherent signal and the background noise powers, respectively ( ) S S.N.R. = 10 log 10. (5) N = N Here, the signal power S is obtained by subtracting the noise background N, estimated around 1/T s, from the total power at the frequency 1/T s [Lindner et al., 199]. On figure 6 is represented the S.N.R. as a function of noise amplitude A for several cell numbers. Two behaviors appear then: (i) Cells 10, 15 and 1 for which n n max = 19, display a monotonous decreasing S.N.R. curve versus noise amplitude. Indeed, the coherent signal can reach these cells in absence of noise, so noise has a negative influence on propagation. (ii) The S.N.R. curves of more distant cells (n > n max ) show that there exists an amount of noise amplitude that maximizes the S.N.R., revealing the S.R. effect. In coupled noisy systems, a standard propagation length n max definition, for a given noise amplitude, 5 4 @ * ) Figure : Propagation length n max versus noise amplitude A. Parameters are U th = 0.5 V, V OH = 1 V, k = 100, T S = 9τ, t 0 = 3 τ, τ = 1 s. corresponds to the last cell whose S.N.R. is greater than a reference level, 0 db for instance [Chapeau- Blondeau, 1999]. Under these conditions, figure shows this propagation length with respect to noise amplitude A. There is a range of noise amplitudes, namely 0 < A < 0.35 V, that give a propagation length greater than n max. An optimum appears then for A = 0.225 allowing the information signal to reach the 31 th cell (instead of the 19 th without noise). In that sense, this increasing of propagation performance shows that noise enhances propagation. Note that, in the case of temporal noise (the same noise is added to each signal U n (t)), a more spectacular enhancement of propagation is obtained with an optimum reached for A = 0.2, giving a propagation length three times greater than without noise. 4 Conclusion ) Figure 6: S.N.R. versus noise amplitude A. From top to bottom cell numbers 10, 15, 1, 22, 23, and 2. Parameters are U th = 0.5 V, V OH = 1 V, k = 100, T S = 9τ, t 0 = 3 τ, τ = 1 s. In this letter, we have considered the propagation of an overthreshold signal through a nonlinear dissipative chain of triggers. We have analytically proved in this exemple that, without noise, the dissipation leads to a maximal distance of propagation n max. Then, adding an appropriate amount of noise enhances propagation, since it allows the information signal to propagate farther in the lat- 4

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