Summary. -- We consider the transmission of a periodic signal by noisy threshold devices. A general expression for the input-output characteristic is developed ...
IL NUOVO CIMENTO
VOL. 17 D, N. 7-8
Luglio-Agosto 1995
Stochastic Resonance in Threshold Devices (*). P, JUNG and G. MAYER-KRESS Departement of Physics, University of Illinois at Urbana-Champaign Urbana, IL 61801, USA (ricevuto il 24 Aprile 1995)
Summary. -- We consider the transmission of a periodic signal by noisy threshold devices. A general expression for the input-output characteristic is developed and applied to two particular threshold devices. It is shown that the amplitude of the signal output shows in the subthreshold regime a maximum as a function of the noise strength--the fingerprint of stochastic resonance. PACS 87.10 - General, theoretical, and mathematical biophysics (including logic of biosystems, quantum biology, and relevant aspects of thermodynamics, information theory, cybernetics, and bionics). PACS 01.30.Cc - Conference proceedings.
1. -
Introduction.
A lot of attention has been paid in the last years to the phenomenon of stochastic resonance. It can be understood as a synchronization of noise-induced activation with a periodic external forcing [1]. Considering a symmetric bistable system, the hopping between the two attractors becomes resonant, when the mean first passage time to leave one basin of attraction is approximately half the period of the periodic driving (for a more precise condition, see [2]). The main body of research in stochastic resonance has moved these days to applications such as signal processing, design of detectors, and neuronal processes. We are focusing in this paper on the application of stochastic resonance to neuronal-like processes. Neurons are by no means bistable systems and analogies with them are therefore questionable. Neurons are much better described as threshold devices, i.e. they fire whenever their input (membran voltage) is larger than a given threshold. In sect. 2, we derive a general expression for the input-output characteristic of threshold devices, driven by a periodic signal and Gaussian noise. In sect. 3, the general theory is applied to a particular threshold device which responds (*) Paper presented at the International Workshop ,Fluctuations in Physics and Biology: Stochastic Resonance, Signal Processing and Related Phenomena,, Elba, 5-10 June 1994. 827
828
P. JUNG
and
G. MAYER-KRESS
with a constant output. Spiking threshold devices, i.e. threshold devices which respond with a 6-spike, are discussed in sect. 4. Both threshold devices have in common that for weak amplitudes of the periodic signal, the amplitude of the transmitted signal shows a maximum as a function of the noise strength. This phenomenon is compared to stochastic resonance in bistable systems. 2. - N o i s y t h r e s h o l d devices: g e n e r a l theory.
A threshold device is characterized by its simple input-output relation, i.e. the output s(t) is zero when the input x(t) is smaller than a given threshold b and takes on a given function h(t) if the input crosses the threshold. Examples of threshold devices are relays [3], Schmitt triggers and (from a reductionist's view) neurons [4, 5]. We are discussing threshold devices with a constant output, i.e. (1)
s o ( x , ~, t) = SoO(X - b) = I 0, t So,
for x < b, for x > b ,
and with spiking output, i.e.
(2)
s~(x, 2, t) = goO(~) F~ 6 ( t -
t~),
n
where X(tn)- b = 0 and O(x) denotes the Heavyside step-function. The threshold device is driven by stationary Gaussian colored noise ~(t), defined in detail below. An important issue for threshold-crossing statistics is the degree of smoothness of the noise [7]. Smootheness is characterized by the high-frequency decay of the power spectrum S(w), obtained by a Fourier transform of the correlation function K ( t ) = (~(t)~(t')) (Wiener-Khintchine theorem), i.e. w
(3)
S ( w ) = f K(t) exp [ - i w t ] dr.
Inverting (3) one finds that the n-th moment of the power spectrum is directly related to the n-th derivative of the correlation function K(t) at t = 0, i.e.
(4)
d2nK(t) dt 2n
, =o
_ (-1) n 2----z
(V)(2n))-
(-1) n 2z
S(w)(o2~ dw.
f
The symmetry of the spectral density S ( w ) = S(-o~) causes the odd derivatives to vanish. For, e.g., Markovian processes, the spectral density decays proportional to -2, implying the non-existence of all the moments (~o2n / for n /> 1 and in turn that of the corresponding derivatives of the correlation function. An important statistical quantity for threshold crossing statistics is the threshold crossing rate rth given in case of a Gaussian stationary process by [3, 6]
(5)
rth = ~
~
exp
(w(~
"
Its existence requires the second moment of the spectral density to exist, i.e. a high-frequency decay of the spectral density of oJ -n ->4. We have chosen the following
STOCHASTIC RESONANCE IN THRESHOLD DEVICES
820
random process: (6)
~=~,
. 7 1 .1 ~ + 1 F 7172 (t)'v2
~ = . ( 1 +. 1 ) ~2't".
being a twice low-pass filtered white noise F(t) with the triter time scales v~ and 72. The Gaussian white noise F(t) has zero mean and its correlation function is given by
(r(t) F(t ')) = 2D 5 (t - t ').
(7)
Taking the Fourier transform of (6), we obtain for the spectral density of ~(t) (8)
2D
S(o~) =
(1 + w2v~)(1 +
w27~) '
decaying proportional to w-4 for large frequencies. The threshold crossing rate for this process is given by [7]
1
(9)
[b2]
r t h - 2z v~-~lv~ exp - ~ a
'
with the variance a = D / ( 7 1 + 72) of ~(t). In addition to the noise ~(t), we also consider a periodic signal at the input x(t) of our threshold device, i.e. x(t)=~(t)+Asin(t~t). Adding a periodic signal is equivalent to periodically changing the value of the threshold b. The transition probability density of the pair process (x(t), ~(t)) approaches for large times an asymptotic periodic distribution function which is given for the Gaussian process (6) by (10)
P(x, ~, t[x', ~', t') t-t'~>~ Pas(X , ~, t) =
1
exp [ (]x - A s i-n-x~(at 2 t )- ) ~ ] [ exp - (~-At2c~ 2ar
with the variances (11)
ax~ -
D
,
a~ =
T1 -I- 72
D 71"t2 (71 -~- 72)
The correlation function C~(t,t') of the output of the threshold device s(t), correspondingly, decays to the product of the mean values (s(x, ~, t))(s(x, ~, t')), where the mean values are calculated by using the asymptotic distribution
Pas (x, ~, t), i.e. ao
(12)
(s(x,~,t))= I d~ f dxs(x,~,t)Pa~(X,~,t). -co
-oo
"
P. JUNG and G. MAYER-KRESS
830
In view of the periodicity of the asympotic probability distribution, the mean value (s(x, ~, t)> is a periodic function in time and can be written as a Fourier series, i.e.
--
(13)
~
k~ exp[int~t].
Averaging the long-time approximation for the correlation function over the initial time t', we find T
C(r) = ~1 I Cs(t'+ r, t') = ~ -~~
(14)
Ik~ 12 exp[in~gT],
0
i.e. a periodic function in time. Correspondingly, the spectral density has 6-spikes at the multiples of the driving frequency, with weights, given by Ik~ 12, i.e. S((,) = 2~
(15)
~
]kn ]2
(~((t) -- n ~ ' ~ ) ,
n=
where T
1I
kn = -~
(16)
exp[-int)t](st(x, ~, t)>dt.
0 The 5-spikes are due to the external periodic signal and the weight gl--2~1 kl 12 indicates how much of the signal is actually transferred to the output of the threshold element. The complete spectral density consists also of a broad Lorentzian-like background, representing random operation of the threshold element. 3. - T h e
O-trigger.
In this section we present the results for signal transmission by a threshold-element with a constant output, i.e. for a O-trigger (see fig. 1). Evaluating the mean value (s~ (x, ~, t)} and the first Fourier coefficient for small signals A (for an expression, valid for arbitrary values of the signal strength A, see[7]), we obtain T
(17)
ik 112_ T
8~
a~/b2 exp
Ox~/b2 .
0
This leading-order approximation is analog to the linear response result for stochastic resonance in bistable systems[8]. The signal transmission Ikl 12 increases with increasing noise strength a ~ until it reaches a maximum. For further increasing noise, the signal transmission becomes smaller again. The maximum is located at (18)
ma~z b 2 .
(~ xx
Although the signal is smaller than the threshold, the output of the threshold element contains the signal. The mechanism for this subthreshold signal transmission is synchronized fluctuation-induced threshold crossing. Note also that the transmission Ikl 12 and its maximum does not depend on the driving frequency. This is in contrast
STOCHASTICRESONANCEIN THRESHOLDDEVICES
831
0.5
0.4
0.3 c~
0.2
0.1
o cJ 0.0
I
I
1.0
2.0
3:0
4:0
5.0
(Txx/b 2 Fig. 1. - The signal transmission of a O-trigger for weak signals (17) is plotted as a function of the normalized variance oxJb 2.
to conventional stochastic resonance in noisy bistable, periodically driven systems, where the maximum shifts with decreasing driving frequencies to smaller values of the noise.
4.-
The J-trigger.
Using a well-known property of (~-distributions, the random output of the 5-trigger can be written as (19)
so(x, ~, t) =goO(~)
~ n
~
5 ( t - tn) =go~O(~)5(x(t)- b), -oo
where O(x) denotes the Heavyside step-function and t n are the zeros of x(t) - b. The mean value (so(x, ~, t)}, being up to the factor go the time-dependent threshold crossing rate, is then obtained by using (10) and reads in leading order of the signal strength A (20)
Ab
(s~ (x, ~, t)) = go rth 1 + - -
axx
sin (tgt) +
1 AY2 2
--
]
cos (~gt) .
According to the general theory, briefly sketched above, the strength of the signal-spike (i.e. the spike at the driving frequency) in the power spectrum is proportional to the square of the Fourier component kl of the mean value (s~ (x, ~, t))
832
P. JUNG and G. MAYER-KRESS 10 0
.J
\\\
cq
\\
\\
/'
~ 10-1
\ \
I \
\ \
10 ~2 i0 -I
,
,
,
,
,
,
,
,[
,
lO 0
,
\,
\\
....
\\
\
\ \,\
10 ~
~xx/b 2 Fig. 2. - The signal transmission of a b-trigger for weak signals (21) is plotted as a function of the normalized variance a,x/b 2 for e = 0 (solid line) and e = 0.5 (dashed line) (for a definition of e, see (22)). which is obtained from (20) as
(21)
Ikl 1 2 - - ' ~
1(A)2 [1
g2rt2h (Gxx/b2) 2 ~- (Gxx/b2)
-~-
=
~-
16z2vlv2
( a ~ / b 2)
( a ~ / b 2 ) 2 + ((7z~/b 2)
'
with (22)
e-
~r~2 T1 T2 jt~
2
As a function of the noise ~x~, Iki 12 shows, similar to the case of the O-trigger, a bell-shaped curve (see fig. 2). The decay for large fluctuations is also proportional to the inverse of the variance. There is an intermediate regime of the noise ~x~, where ]kl 12 decays proportional to the inverse square of the variance (see the first term in (21)). For large noise, however, the second term, which is due to the increase of the speed of the modulation (this term has been neglected in the adiabatic treatment of this problem in [8]) dominates. In adiabatic approaches [9] and [10] this term is not present and thus the signal strength ]kl t 2 decays proportional to o-2. Experimental results on stochastic resonance in the mechano-receptor system of cray-fishes, however, indicate--in agreement with (21)--a signal decay for large noise, being significantly weaker than o 2. It is also worth to mention that the position of the peak
STOCHASTIC
RESONANCE
IN THRESHOLD
833
DEVICES
is different from that in case of the O-trigger. The precise position is at /
m~x b2
(~x
(23)
1 2
_
1 e
+
1 ./ e ~
--
i +
1 e=" 4
.
In (23) as well as in (21), an important quantity is the ratio of the time scales of the filter 1/(V 1 V2 ) and the frequency of the external driving f2, described by e. For small e, i.e. in the adiabatic limit, the position of the peak is given by b2/2. For increasing driving frequency, the peak position gets shifted towards larger values, i.e. for small e we find (24)
1
a~=/b2 ~ _1 + - e 2
8
=
1
--
2
q- - -
~r~2VlV 2 .
16
The existence of a maximum in ]kl 12 as a function of the fluctuations means that the sensitivity of the threshold device is optimized at a finite noise level. A very similar behavior has been observed in periodically driven noisy symmetric bistable systems. The noise-induced hopping between the stable states is synchronized with the external driving which periodically raises and lowers the height of the barrier. This synchronization works best, when the dwell-time tdw in one of the stable states is approximately half the period T of the periodic signal. In other words, the statistical time scale tow has to agree approximately with a deterministic time scale T/2. In our threshold crossing dynamics, something very similar happens: The effective threshold is periodically raised and lowered with the external driving. One might therefore expect that the condition for stochastic resonance is given when the inverse of the threshold crossing rate agrees with the period of the periodic signal. But this is not the case! Instead, we fred a condition for the peak which depends only very weakly on the period of the external signal, merely a static-resonance condition. How can this be understood? To answer this question, we have to revisit periodically driven noisy bistable systems--however with asymmetric stable states. Such a system can be approximately described in terms of an asymmetric 2-state system [11]. The dynamics of such a system can be visualized as mainly random motion in the globally stable state with some occasional excursions to the metastable state. The resemblence to threshold crossing dynamics is evident. In [11], the response of an asymmetric noisy two-state system on periodic forcing has been analyzed in detail. It has been found that the asymmetry exponentially suppresses the response of the system on periodic forcing (see also [14][13]), yielding a response amplitude which depends only very weakly on the frequency of the external signal. The approximative resonance condition, i.e. the matching of the external frequency with the inverse mean first passage time, is therefore only valid for symmetric potentials (see also [12]). The behavior of the threshold devices is thus actually not very different from asymmetric bistable systems.
We wish to thank F. Moss, A~ Hertz and A. Jackson for stimulating discussions. P J wishes to thank the Deutsche Forschungsgemeinschaft for financial support within the Heisenberg Programm.
834
e. JUNG and G. MAYER-KRESS
REFERENCES [1] BENZI R., SUTERAA. and VULPIANIA., J. Phys. A, 14 (1981) L453; NICHOLISC., Tellus, 34 (1982) 1; MCNAMARAB. and WIESENFELD K., Phys. Rev. A, 39 (1989) 4854; GAMMAITONIL., MARCHESONIF., MANICHELLA-SAETTAE. and SANTUCCIS., Phys. Rev. Lett., 62 (1989) 349; JUNG P. and H.~i.NGGIP., Europhys. Lett., 8 (1989) 505; Phys. Rev. A, 41 (1990) 2977; GANG HU, NICOLIS G. and NICOLIS C., Phys. Rev. A, 42 (1990) 2030; Moss F., Ber. Bunsenges. Phys. Chem., 95 (1991) 303; JUNG P. and H~NGGIP., Phys. Rev. A, 44 (1991) 8032; Moss F., BULSARAA. and SHLESINGERM. (Editors), J. Stat. Phys., 70 (1993) 1; JUNG P., Phys. Rep., 234 (1993) 175; J. K. DOUGLAS,WILKENSL., PENTAZELOUE. and Moss F. Nature, 365 (1993) 337; MOSS F., in Frontiers in Applied Mathematics, edited by G. WEISS (SIAM, Philadelphia) 1992; WIESENFELD K., PIERSON D, PENTAZELOUE., DAMES C. and Moss. F., Phys. Rev. Lett., 72 (1994) 2125. [2] Fox R. F. and YAN-NAN LU, Phys. Rev. E, 48 (1993) 3390. [3] STRATONOVICHR. L., Theory of Random Noise, Vol. II (Gordon and Breach, New York, London, Paris) 1967. [4] MCCULLOCH W. S. and PITTS W., Bull. Math. Biophys., 5 (1943) 115. [5] LONGTINA., BULSARAA. R. and Moss F., Phys. Rev. Lett., 67 (1991) 656. [6] RICE S. 0., in Noise and Stochastic Processes, edited by N. WAX (Dover Publications) 1954, p. 133. [7] JUNG P., Phys. Rev. E, 50 (1994) 2513. [8] GINGL Z., KIss L. and Moss F., Europhys. Lett., 29 (1995) 191. [9] JUNG P. and BAaT[ZSSEK R., in Fluctuations and Order: The new Synthesis, edited by M. MILLONAS, INLS-Series (Springer) 1994. [10] BARTUSSEK R., HANGGI P. and JUNG P., Phys. Rev. E, 49 (1994) 3930. [11] DYKMANM. I., MANNELLAR., McCLINTOCKP. V. E. and STOCKS N. G., Phys. Rev. Lett., 65 (1990) 2606. [12] WIESENFELD K., PIERSON D., PENTAZELOUE., DAMES C. and Moss F., Phys. Rev. Left., 72 (1994) 2125. [13] DYKMANM. I., LUCHINSKYD. G., MCCLINTOCKP. V. E., STEIN N. D. and STOCKS N. G., Phys. Rev. A, 46 (1992) R1713. [14] HANGGI P. and THOMAS H., Phys. Rep., 88 (1982) 207.
