Biol. Cybern. 83, 471±480 (2000)

Prospects Stochastic resonance and signal detection in an energy detector ± implications for biological receptor systems Jakob Tougaard Centre for Sound Communication, Institute of Biology, Odense University, Campusvej 55, 5230 Odense M, Denmark Received: 22 December 1999 / Accepted in revised form: 14 April 2000

Abstract. Stochastic resonance is demonstrated in a simple energy detector model, as a non-monotonic relationship between signal-to-noise ratio and detection of a sinusoid signal in bandpass-limited Gaussian noise. The behaviour of the model detecting signals of various intensities and signal-to-noise ratios was investigated. Signi®cant improvement in detection was obtained by adding noise for mean signal intensities below the detection criterion of the detector. The range of usable noise levels, however, may be too small to be biologically meaningful. It is demonstrated that improving detection in the analysed model by adding noise to an otherwise undetectable signal is only at best as ecient as what can be obtained by adjusting the criterion to the signal-to-noise ratio. Improving detection by means of stochastic resonance is thus a sub-optimal strategy. It is speculated whether a demonstration of stochastic resonance in a biological system indicates any adaptive signi®cance. More than anything, it indicates the presence of a mismatch between receptor sensitivity and the signal-to-noise ratio of the experiment, not the cause of this mismatch.

1 Introduction Stochastic resonance ± the phenomenon that low levels of noise can improve detection of weak signals in certain non-linear detectors ± has received much attention in the last decade (see e.g. reviews by Wiesenfeld and Moss 1995; Bulsara and Gammaitoni 1996; Mitaim and Kosko 1998). The basic idea is that detection of subthreshold signals under certain circumstances can be facilitated by addition of noise. This may seem paradoxical, since noise is usually thought of as something

Correspondence to: J. Tougaard (Tel.: +45-65-502222, Fax: +45-65-930457, e-mail: [email protected])

detrimental that should be minimised whenever possible. This general view is supported by results from both information theory (Shannon and Weaver 1949) and the theory of signal detection (Green and Swets 1966). A key result from information theory is that the amount of information transferable through an information channel is monotonic with the signal-to-noise ratio (SNR). More noise leads to a smaller information transfer capacity. In line with this is the central result of signal detection theory that the detectability of a given signal is monotonic with the SNR. More noise leads to more errors in discriminating between signals and noise. Both statements, however, refer to optimal conditions and thus only set theoretical upper limits to the performance of a given system. The situation in sub-optimal systems may be quite di€erent and a monotonic relationship between the SNR and information transfer and detection is not automatically the case. Several authors have noted the possibility that biological sensory systems exploit stochastic resonance to enhance detection of very weak signals (Douglass et al. 1993; Collins et al. 1995; Wiesenfeld and Moss 1995; Heneghan et al. 1996; Moss et al. 1996; Bezrukov and Vodyanoy 1997b; Narins and Benedix 1997). Many sense organs, including sensitive mechano- and electroreceptors, routinely detect very weak signals in high background noise. A tempting thought is that this background noise in fact may enhance the detection of these weak signals, through the phenomenon of stochastic resonance. Stochastic resonance has been demonstrated in several electrophysiological preparations (Douglass et al. 1993; Braun et al. 1994; Gluckman et al. 1996; Levin and Miller 1996; Pei et al. 1996; Henry 1999) and in psychophysical experiments (Collins et al. 1997; Simonotto et al. 1997). Recently stochastic resonance has also been demonstrated in connection with natural food-searching behaviour (Russell et al. 1999). These experimental studies of stochastic resonance in biological systems have demonstrated enhanced detection of sub-threshold signals by an experimental increase in noise level (either the internal noise in the sense organ itself, or, in most cases, the external background noise).

472

Despite these experiments, an adaptive signi®cance of stochastic resonance in biological sensory systems has not yet been unequivocally demonstrated. Another, more common approach to stochastic resonance is theoretical, most often in the form of mathematical modelling (see Mitiam and Kosko 1998, for extensive references). The aim of the present study has been to investigate stochastic resonance with a theoretical approach, but nevertheless with a clear reference to biological systems. This is done by investigating stochastic resonance in a biologically relevant detector model and subsequently analysing the results from the point of view of the animal. 1.1 De®nition of stochastic resonance Stochastic resonance is traditionally de®ned as a nonmonotonic relationship between the SNR at the input of the detector (input SNR) and detection of some weak, ®xed intensity stimulus. The error in detection of a given signal normally increases with increasing noise level, but for detectors displaying stochastic resonance the opposite is found for certain ranges of noise levels. Thus, these detectors show a local decrease in errors in detection with increasing noise level. The most common stimulus used in stochastic resonance experiments is a continuous sinusoidal signal. In this case the performance of the detector is often quanti®ed by the so-called output SNR. The output SNR is equal to the di€erence between the peak in the power spectrum of the output, measured at the frequency of the sinusoid, and the spectrum noise level at frequencies immediately adjacent to the frequency of the sinusoid (see e.g. Douglass et al. 1993, for an example).1 Other authors have de®ned stochastic resonance in terms of information transfer (e.g. DeWeese and Bialek 1995; Heneghan et al. 1996; Levin and Miller 1996), seeing stochastic resonance as a local increase in information transfer through the detector at certain non-zero noise levels. This approach is in many ways comparable to the output SNR analysis, with the important exception that it opens the possibility of studying stochastic resonance phenomena related to random, non-periodic signals. This is referred to as aperiodic stochastic resonance (Collins et al. 1995; Heneghan et al. 1996), 1 The physical meaning of this measure, if any, is not at all obvious. Whereas the height of the peak in the spectrum at the frequency of the sinusoid expresses the power of the tone, the noise level in the spectrum is measured as a power density, expressing the power measured in frequency bands of a certain bandwidth (usually 1 Hz). This leaves the SNR with the unit Hz and not dimensionless as is usually the case. The power spectrum measure of SNR is not particularly well suited for studying stochastic resonance in biological systems. When inferring biological relevance of stochastic resonance from this measure, one must assume that the animal has some way of extracting the same information, that is, is able to perform an operation equivalent to making a power spectrum analysis over a considerable period of time. This is certainly not something that one can assume a priori, without a detailed knowledge of the function of the entire sensory system in question

although the mechanism of stochastic resonance is exactly the same as for continuous, periodic signals. A third way of analysing stochastic resonance is through signal detection theory (Green and Swets 1966), which is the approach used in the following. Here the performance of the detector is measured directly as its ability to correctly discriminate signals in noise from background noise (Collins et al. 1997; Inchiosa and Bulsara 1996). This measure seems most relevant for assessing stochastic resonance in biological sensory systems, because one measures something of immediate importance to the animal, namely the detection of (individual) signals in a noisy environment. An energy detector model was chosen as the basis for the analysis performed below. Energy detector models of various types are routinely used to model physiological receptor systems, especially in audition (e.g. Plomp and Bouman 1959; Green 1985; Au and Moore 1988; Tougaard 1996, 1998), but also in, for example, mechanoreception (Checkosky and Bolanowski 1992, 1994). The present model contains a binary decision criterion, which is a strong non-linear component, and the model is thus a priori expected to display stochastic resonance e€ects. As real animals have to deal with signals of many di€erent intensities in an often highly variable background noise, it seems important to investigate the performance of the model under a similar range of di€erent conditions. The intention of the analysis is thus to gain fundamental insight into stochastic resonance phenomena in a biologically relevant model and at the same time investigate its possible biological signi®cance, by analysing the behaviour of the model with methods one would normally use for describing the behaviour of a real receptor or sensory system. 2 The model The detector model used is an energy detector, consisting of a squaring device followed by an integration over the time interval T (the integration time). The integral is approximated by a discrete summation over 2WT samples of the signal, where W is the bandwidth of the noise (see below). This approximation is very good, except for very low values of 2WT (Urkowitz 1967). The model is essentially identical to the energy detector described by Green and Swets (1966, p. 174). Detection is based on the quantity I, termed the decision variable and expressing the average intensity (power) of the signal. I is given as 1 Iˆ T

ZT 0

s2 …t†dt 

2WT 1 X x2 2WT mˆ1 m

…1†

where s…t† is the input signal and xm is the amplitude value of the m-th sample of the input signal. The detector is presented with two di€erent inputs ± either Gaussian noise alone or Gaussian noise plus a sine wave (the sine wave is in the following referred to as the signal, the noise alone as the N condition, and the signal

473

plus noise as the SN condition). The goal of the detector is to correctly identify the sine wave as the signal and reject the noise, when it is presented alone.2 2.1 Decision variable under N conditions The noise is assumed to be white Gaussian noise with bandwidth W and root-mean-squared (RMS) amplitude AN . The output IN is then given as  2WT 2WT  1 X A2 X x Nm 2 IN ˆ x2Nm ˆ N …2† 2WT mˆ1 2WT mˆ1 AN The term xN denotes a Gaussian-distributed variable with zero mean and variance A2N . Because no more than 2WT samples are used it is reasonable to assume that the individual xN 's are independent and by normalisation we thus obtain a sum, which is v2 -distributed with 2WT degrees of freedom. The mean and the standard deviation of the output IN is then given as lN ˆ

A2N

2WT

2WT ˆ A2N ;

rN ˆ

p 4WT ˆ p 2WT WT A2N

A2N

…3†

2.2 Decision variable under SN conditions The sine wave signal is assumed to be independent of the noise, with duration T and RMS amplitude AS . Output ISN is given as ISN

2WT 1 X ˆ …xNm ‡ xSm †2 2WT mˆ1

ˆ

2WT 2WT 2WT 1 X 1 X 1 X x2Nm ‡ x2Sm ‡ 2xNm xSn …4† 2WT mˆ1 2WT mˆ1 2WT mˆ1

As the signal and noise are uncorrelated, and both xN and xS have means of zero, the last term in (4) converges towards zero for increasing values of WT . ISN thus equals the sum of the output under the N condition (IN ) plus the output given the signal alone as input (IS ). Thus, q lSN ˆ lN ‡ lS ; rSN ˆ r2N ‡ r2S where lS deviation, the signal equals A2S

and rS are the mean and the standard respectively, of IS . Given that the period of exactly equals an integer multiple of T , lS (by de®nition) and rS equals zero.3 This

2 It is worth noting that an energy detector is not the optimal solution for detecting a sinusoid signal in Gaussian noise. For optimal detection (i.e. minimal number of errors) a crosscorrelation or matched ®lter type of detector should be employed (Green and Swets 1966). It is, however, not the aim of the current article to enter a discussion on optimal detection across di€erent types of detectors and focus will thus remain solely on the energy detector model 3 If the frequency of the sine wave signal is suciently high (fS  1=T †, rS and lS also approach zero and A2S , respectively

means that all variance in ISN originates from the noise alone. This assumption shall be relaxed later in the discussion. We are now left with the following expressions for the mean and standard deviation of the distribution of ISN : lSN ˆ A2N ‡ A2S ;

A2N rSN ˆ rN ˆ p WT

…5†

If 2WT is not too small, the distributions of IN and ISN can, according to the central limit theorem, be well approximated by two normal distributions with identical standard deviation rN and a di€erence in means of A2S . This corresponds to the well-known Gaussian, equal variance situation of signal detection theory, where the di€erence between the two distributions can be characterised by an index of detectability (Green and Swets 1966). This index (d 0 ) is the di€erence in means between the two distributions, expressed in units of standard deviations: d0 ˆ

A2 p lSN lN lS ˆ ˆ 2S WT rN rN AN

…6†

The detectability of a sine wave signal in Gaussian noise for this energy detector is thus directly proportional to the SNR of the signal, calculated from the signal and noise intensities. 2.3 Discrimination Detection of the signal is based on a decision criterion a (expressed in the same units as I). Whenever I exceeds a, the detector reports signal present; otherwise signal absent is reported. Four possible combinations of stimulus and response of the detector thus exist, following the conventional terminology of signal detection theory: hit, miss, false alarm, and correct rejection. The performance of the detector can be evaluated from hit rates and false alarm rates associated with a given value of the decision criterion a. These rates are calculated as     a lSN lSN a ˆU Hit rate ˆ 1 U rSN rSN     a lN lN a False alarm rate ˆ 1 U ˆU …7† rN rN where R t U…t† is the cumulated Gaussian distribution, equal to 1 /…s†ds, where /…s† is the Gaussian (normal) probability density function with zero mean and unit variance (Green and Swets 1966). Given also the probabilities of signal-plus-noise and noise-only presentations, the overall proportion of presentations correctly classi®ed (pCorr ) can be calculated as pCorr ˆ Hit rate
p…SN† ‡ …1

False alarm rate†
p…N† …8†

We shall assume p…N† ˆ p…SN† ˆ 0:5 in all the following, in which case pCorr equals the mean of the hit rate and the correct rejection rate.

474

2.4 Optimal criterion and noise level For given intensities of signal and noise, a criterion exists, which maximises pCorr . Given that the two distributions are both normal and with equal standard deviations, this optimal criterion is given as (Green and Swets 1966): p A2S WT A2N A2S d0 2 2 p  …9† ‡ ˆ A aopt ˆ lN ‡ r ˆ AN ‡ N 2 2 2A2N WT The optimal criterion is thus independent of the timebandwidth product WT and thus also of the standard deviation of IN . The value of the maximised pCorr (using the optimal criterion aopt ) does, however, depend on WT [as calculated from (7) and (8)]. Instead of optimising the decision criterion an alternative approach is to optimise the noise level for given signal intensity and criterion value. The proportion correct as a function of noise intensity is given as       1 a lN a lSN ‡ 1 U pCorr …A2N † ˆ U 2 rN rSN p   2 1 …a AN † WT ˆ U 2 A2N p   …a …A2S ‡ A2N †† WT ‡ 1 U A2N   p  1 a WT p ˆ U WT 2 A2N p  p …a A2S † WT ‡U WT A2N The optimal noise level is the level maximising pCorr …A2N †. This noise level …A2Nopt ) is found from the ®rst derivative of pCorr …A2N †: pCorr0 …A2Nopt †

ˆ0

m " p p! p …a A2S † WT 1 …a A2S † WT
/ WT 2 A2Nopt …A2Nopt †2 !# p p a WT a WT p
/ WT ˆ0 ; A2Nopt …A2N †2 opt

where / is the Gaussian probability density function m …a

 A2S †
e

1 2

p WT

2

p …a A2 † WT S 2 A Nopt

 ˆa
e

1 2



p a WT A2 Nopt

p WT

2

This equation has no solutions for A2S  a, indicating that the error rate increases monotonically with noise level in this situation. In other words, stochastic resonance cannot be present for signal intensities above a. For situations where A2S < a we get from the above:

" p " p#2 1 p …a A2S † WT 1 a WT WT 2 2 A2Nopt A2Nopt   a A2S ˆ ln a

#2 p WT

m ……A2S †2

2aA2S †WT

2…A2Nopt †2

‡

A2S WT ˆ ln b; A2Nopt

where b ˆ

a a

A2S

m 2 ln b
…A2Nopt †2

2WT
A2S
A2Nopt

…A2S

2a†WT
A2S ˆ 0

This second-order equation of A2N has the solutions: A2Nopt ˆ

WT


A2S

q  …WT
A2S †2 ‡ 4WT
A2S
ln b
…A2S 2a† 2 ln b …10†

Both solutions are real, one always negative, the other always positive.4 Ignoring the negative solution as having no physical meaning, this means, that for all values of WT , a, and A2S , where A2S < a, a unique, non-zero noise level exists, which minimises the total number of errors. The parameter b, de®ned above as …a A2S †=a, is a convenient way of expressing the di€erence between criterion and signal intensity. This parameter, termed the criterion/signal mismatch, is central in the description of stochastic resonance phenomena, as will be discussed below. 3 Behaviour of the model In the following, a distinction will be made between the decision criterion (a) and the threshold of the detector. Whereas the decision criterion is an inherent parameter of the detector, the threshold is here de®ned as that particular signal intensity, at a given noise level, that results in 75% correct classi®cations of signals and noise. The value of 75% correct is arbitrary, but in correspondence with common practice in psychophysical experiments, such as the two-alternatives forced choice paradigm (Green and Swets 1966). The threshold thus re¯ects the actual sensitivity of the detector, at a given noise level, and is what can be observed from an outside inspection of the system.5 The detection of signal intensities smaller than a in the presence of di€erent levels of noise is illustrated in Figs. 1 and 2, calculated from (7) and (8). All parame4

This can be realised by observing that both WT
A2S and ln b
…A2S 2a† are always positive (recalling that A2S < a) plus the p fact p that the numerator of (10) has the form a  a2 ‡ b and that  2 a < a ‡ b for all positive values of a and b 5 Although the observed threshold of a detector thus di€ers from the decision criterion a, it approaches a asymptotically as the noise decreases towards zero

475

Fig. 1A±E. Distributions of the output of the receiver under noise alone (IN , solid line) and signal plus noise conditions (ISN , dotted line), shown for ®ve di€erent levels of input noise. The vertical dotted line indicates the decision criterion (a) used in Fig. 2. WT ˆ 25 and b ˆ 2=3. The index of detectability, d 0 , expressing the separation of the two distributions in units of standard deviations, is indicated together with the signal-to-noise ratio (SNR)

ters, except the noise level, were kept constant. At low noise levels (high SNR), as in Fig. 1A, the detectability of the signal is high (distributions of IN and ISN well separated). However, because virtually no part of the distributions are above the criterion level a, nothing is detected by the detector. This corresponds to hit and false alarm rates of virtually zero (Fig. 2A). As the noise intensity (A2N ) is increased, both distributions shift to the right, causing ®rst an increase in detection of the signal (hits), and at higher noise levels also an increase in the false alarms (Fig. 2A). At the same time the standard deviation of the distributions increases as well, causing a decrease in detectability (Fig. 2B). The proportion of correct discriminations (the performance) as a function of noise level is shown in Fig. 2C, both with a ®xed criterion (solid line) and a criterion optimally adjusted to the SNR (dotted line). There is a pronounced peak in the performance at a non-zero noise level for the ®xed criterion ± a clear indication of a stochastic resonance e€ect. In contrast to this is the monotonically decreasing performance with noise level for the optimally adjusted criterion. Two parameters determine the magnitude and extent of the stochastic resonance e€ect, as seen in the proportion correct curve (Fig. 2C, solid line). These are the time-bandwidth product, WT , and the criterion/signal mismatch, b. Figure 3 shows the in¯uence of changing the timebandwidth product of the detector. The best performance obtainable increases signi®cantly as WT does, approaching 1 (perfect discrimination) asymptotically (Fig. 3A). This can also be seen directly from (3) and

Fig. 2A±C. Detection of the signal at di€erent SNRs. A Hit and false alarm rates associated with a ®xed decision criterion. B Index of detectability (d 0 ). C The overall proportion of correct classi®cations (mean of hit rate and correct rejection rate), both with a ®xed decision criterion (solid line) and with a criterion optimally adjusted to the SNR, as described in the text (dotted line). Note that the SNR is plotted on a reversed axis, in agreement with common practice of stochastic resonance literature, where output of the detector is plotted as a function of increasing input noise levels. WT ˆ 25 and b ˆ 2=3

(5) ± an increase in WT leads to a decrease in r, causing a better separation of the two distributions. The increase in detection, however, is at the expense of a narrowing of the range of noise levels for which the performance is increased above chance level (Fig. 3B). The e€ect of changing the criterion/signal mismatch parameter b is shown in Fig. 4A, for situations where A2S < a. The greater the mismatch between signal and criterion (larger b), the more noise needed to improve

476

Fig. 3A,B. In¯uence of the time-bandwidth product (WT ) on the performance of the detector. A Performance of the detector at optimal noise level as a function of WT . B The proportion of correct classi®cations at di€erent noise levels, plotted for three di€erent values of WT . The criterion/signal mismatch b ˆ 5=6 for all curves

detection and the smaller the gain from adding the noise. If the criterion is lower than the signal (A2S > a, equal to b < 0), the stochastic resonance e€ect disappears and a monotonic decrease in performance with noise level is seen (Fig. 4B). In Figs. 1±4, the detection of a ®xed signal intensity was plotted against increasing noise levels. This is the customary way of illustrating stochastic resonance effects. In Fig. 5, the situation is reversed and detection performance is shown as a function of increasing signal intensity, for ®ve di€erent noise levels (WT and a constant). This type of plot re¯ects the standard way of displaying sensitivity of a receptor, by measuring detection at increasing signal intensities for a constant noise level. This type of plot informs us more directly about the performance of the detector, seen from a biological point of view, than do traditional stochastic resonance ®gures (as Figs. 2C, 3B, 4A). The bene®cial e€ect of adding limited amounts of noise to the input of the detector is evident from Fig. 5, showing detection of signals of variable intensity with a ®xed criterion and at di€erent noise intensities. In the absence of noise, the function of the detector is strictly binary (Fig. 5, solid line). Either it performs at chance level (signal below criterion level) or with 100% ®delity (signal above criterion level).

Fig. 4A,B. In¯uence of criterion/signal mismatch (b) on the performance of the detector. A Detection of a signal with intensity below a, measured as the proportion of correct classi®cations at di€erent noise levels and for three di€erent values of b. B Detection of a signal with an intensity greater than a, at three di€erent values of b. WT ˆ 25 for all curves

If noise is added, three things happen: (1) the detector threshold drops, (2) the response curve becomes increasingly less step-like, referred to as linearisationby-noise (Chialvo et al. 1997), and (3) the asymptotic

Fig. 5. Detection of signals of increasing intensity, measured as the proportion correct classi®cations of signals and noise, at ®ve di€erent levels of noise. WT ˆ 200 for all curves

477

performance of the detector at high signal intensities drops to less than 100% (due to an increased number of false alarms with increasing noise level). At the lowest noise level shown ( 1:5 dB re. a), the threshold of the detector drops by about 6 dB, without any loss in ®delity for high signal levels. Adding more noise ( 0:5 dB re a) yields a further 3.5 dB in the threshold, but now at the expense of an increased false alarm rate of about 6%, resulting in a less than perfect performance at high stimulus levels. At higher noise levels, the performance of the detector quickly deteriorates and approaches chance level for noise intensities above the criterion level (the detector saturates). 4 Discussion Stochastic resonance is clearly demonstrated in the present energy detector model, seen by the non-monotonic relation between detection and input noise level (Figs. 2C, 3A, 4A). This is not surprising, since the present detector, with a ®xed criterion level, a, as the non-linear element, is in essence a level-crossing detector, and stochastic resonance phenomena are well known for this class of detectors (Gingl et al. 1995; Jung 1995; Loerincz et al. 1996). The phenomenon of stochastic resonance can be understood as a simple consequence of a mismatch between the decision criterion of the detector and the signal intensity (DeWeese and Bialek 1995). Adding noise can facilitate detection of a weak signal, because the noise adds to it and helps in ``lifting'' it above the criterion (Fig. 1). Bulsara and Gammaitoni (1996) and DeWeese and Bialek (1995) both point out that stochastic resonance in this respect can be considered equivalent to the addition of random noise to improve the function of analog-to-digital converters at very low signal levels (dithering). It is evident from Fig. 1 that stochastic resonance only works within a con®ned range of noise intensities, since the detector completely saturates at higher levels (Fig. 1E). From Fig. 2C it is seen that adding noise to improve detection of an otherwise sub-threshold signal is only at best equivalent to what can be obtained by optimally adjusting the criterion of the detector. Adding noise can thus be seen as a way to compensate for the criterion/signal mismatch (DeWeese and Bialek 1995). If the criterion is constantly adjusted optimally to the SNR, detection performance becomes monotonic with detectability and the stochastic resonance e€ect disappears (Fig. 2C, dotted line). Stochastic resonance is a phenomenon intimately linked to suboptimal criteria and is thus likely to be found in any system with a ®xed decision criterion. The stochastic resonance e€ect as such is thus independent of the exact nature of signals and noise and is not an inherent property of the noisy signal, as stated by Jung (1995) and Loerincz et al. (1996). Stochastic resonance is intimately linked to the decision criterion and will work even if the only di€erence between signal and noise is intensity, as demonstrated by Heneghan et al. (1996)

with a Gaussian noise pulse as signal in a Gaussian noise background. Two parameters are central for determining the magnitude of the stochastic resonance e€ect in the present energy detector. This is the time-bandwidth product of the detector and the criterion/signal mismatch, b. These parameters are likely to be central in determining the stochastic resonance e€ect in other types of detectors as well. 4.1 Importance of the time-bandwidth product The time-bandwidth product of the detector, which is the product of the integration time of the detector and the bandwidth of the noise signal, determines the variance of both IN and ISN [(3) and (5)]. The smaller WT , the greater the variance of the distributions. Increased variance ± for a constant RMS signal amplitude ± results in a greater overlap between the distributions of IN and ISN and thus leads to an increased error rate, even with an optimal criterion level. This is also seen from the detectability, which is monotonic with WT [(6)]. A simple way to increase WT is to increase the integration time of the detector (average over a longer time interval). This, however, may not be an option for biological signals, which are often of short duration or varying in time, or both. Several authors have pointed out that it is instead possible to enhance the stochastic resonance e€ect by addition of a number of identical receptors in parallel (e.g. Collins et al. 1995; Chialvo et al. 1997; Gailey et al. 1997). Given that certain important assumptions can be ful®lled, adding receptors in parallel is computationally equivalent to an increased WT , since detecting a signal with a system of n detectors, each with a time constant T , is equivalent to detection with one single receptor with a time constant of nT . The important advantage of using receptors in parallel is that the duration of the signal can be T , while the e€ective integration time is n times larger. The most important assumption for actually achieving this, and the assumption most likely not to be ful®lled in biological systems, is that the noise presented to each receptor must be independent of the noise at all other receptors. Averaging in parallel on a signal with identical noise presented to all receptors will not improve detection. Only additional noise added by the receptors themselves will be removed in the averaging and thus will result in an increasingly perfect representation of the noisy signal presented to the detectors. Alternatively, WT can be increased by increasing the bandwidth of the noise, either directly, if the noise is internal to the detector, or, if the noise is external, by increasing the bandwidth of the receptor (i.e. a widening of the ®lter characteristics of the receiving system). The latter, however, also directly a€ects the detection, through an increase in noise intensity received by the system and thus leads to a direct increase in lN and lSN .

478

4.2 Importance of the criterion

4.4 Detection of signals of di€erent intensity

The most straightforward factor determining the extent and magnitude of the stochastic resonance e€ect is the criterion of the detector, or more precisely, the criterion/ signal mismatch (Fig. 4). The signal intensity itself, A2S must be below a (b > 0) for a stochastic resonance e€ect to be present [(10)]. The less di€erence between criterion and signal (the smaller positive b), the more pronounced the stochastic resonance e€ect will be and the wider the range of SNRs where the detector performs above chance level (Fig. 4A). This seems intuitively clear, because with a signal just below criterion level, little noise needs to be added to lift the signal over the criterion. At the same time, since the needed noise level is low, the variance of IN (and hence also ISN ) is low. This means that the noise level can be increased substantially, without severely a€ecting performance. At some point, however, the noise variance becomes large enough for the noise itself to exceed the criterion, leading to an increased false alarm rate and hence a deteriorated overall performance of the detector.

So far, most studies of stochastic resonance, both theoretical and experimental, have dealt with detection of ®xed intensity signals in a variable noise background. This, however, is not a realistic situation compared to the function of biological receptors. These are constantly exposed to considerable variation in the intensity of both signals and noise. For a receptor to have practical value to an animal, it has to be functional under both a range of di€erent SNRs and absolute noise levels. From Fig. 5 it is evident that large drops in threshold can be achieved by increasing WT of the detector. The downside to lowering the threshold by increasing WT is that the performance of the detector becomes increasingly more dependent on the exact level of noise, especially if very low thresholds are desired. With a high time-bandwidth product, minute changes in input noise (less than 1 dB) may totally destroy the bene®cial e€ect of the noise and either remove the stochastic resonance improvement on sensitivity (at lower than optimal noise levels) or saturate the detector completely (at higher than optimal levels). Two di€erent strategies thus exist to improve detection of weak, sub-threshold signals. Either the criterion of the detector can be adjusted (lowered) or the noise in the system can be increased (exploitation of the stochastic resonance e€ect). Both ways it will be possible to remove the mismatch between criterion and signal. In the ®rst case the criterion is matched to the SNR, in the latter the SNR; is matched to the criterion. Adjusting the criterion, however, rather than adding noise, will always be at least as ecient as adding noise and in most cases substantially better. From this follows the important understanding that stochastic resonance cannot improve detection in any way that could not have been achieved to the same degree (or better), by adjusting the criterion level optimally. The mechanism of stochastic resonance is not, as is sometimes misunderstood, able to make otherwise non-detectable signals detectable (pointed out very precisely by Dykman and McClintock 1998). One can, however, imagine real life situations where it may be impossible to adjust the criterion of the detector and where stochastic resonance thus may be the only practically available solution for improving detection.

4.3 Relaxing the assumptions of the model Some important assumptions underlie the model. These assumptions can, however, be relaxed considerably, without changing the general conclusions. The central part of the requirements for stochastic resonance to be present is the criterion a and that the decision variable I is monotonic with stimulus energy. In this way, addition of noise to any signal will increase the total energy received by the detector, thus increasing the probability that the signal will be detected. The exact shape and nature of the distributions of IN and ISN is thus not important for the stochastic resonance e€ect itself, only for the actual performance observed. The decision criterion, a, in the current model is ®xed. Stimuli resulting in values of I above a are always detected, stimuli below, never. In most biological systems, the criterion is not discrete and ®xed, but speci®ed statistically. In other words, for all levels of stimulation, there is a non-zero probability that the detector will report ``signal present'', although with an increasing probability of response, the higher the stimulus intensity, and virtually zero for low intensities. A statistical decision criterion will smooth the transition between detection and non-detection, in some sense comparable to the addition of an extra internal noise source. This internal noise source will limit the performance of the detector at low levels of external noise, whereas the external noise dominates for higher levels.6 6 Bezrukov and Vodyanoy (1995, 1997a) have shown that a threshold is not required at all to generate a stochastic resonance e€ect. A parallel system of many independent receptors (e.g. voltage-dependent membrane channels), each with an input-dependent probability of response (i.e. opening in case of the membrane channels), can also display stochastic resonance behaviour

4.5 Stochastic resonance in biological receptors It is tempting to look for stochastic resonance phenomena in biological receptor systems, as many of these routinely detect very weak signals embedded in high levels of background noise. However, as discussed above, the phenomenon of stochastic resonance can be considered no more than an epi-phenomenon, related to a non-optimal criterion level. It may thus be that a demonstration of stochastic resonance in a biological system is not in itself signi®cant, since it can be considered a simple consequence of a ®xed criterion of the receptor, not matched to the signal. Because stochastic resonance is an inherent property of the

479

non-linear mechanisms involved in signal transduction and detection, it will always be present, whether bene®cial or not, as long as a sub-optimal criterion is present. We can then turn things completely upside down and realise that if stochastic resonance can be demonstrated in a sensory system, it is above all only an indication that the criterion of the receptor is suboptimal, seen in relation to the given signal and at the given SNR. Stochastic resonance phenomena as such may thus not be the central issue from a biological point of view. Rather, they should be looked at as important indicators of a sub-optimal match of the receptor to the particular SNR of a given experiment. A central question to ask is then why the given animal has a sub-optimal detector (seen from a purely signal detection point of view). Part of the answer may lie in the background noise level of the experiment being too low, compared to the natural setting of the animal. The detector may thus be well matched to the natural SNR and only display stochastic resonance behaviour under laboratory conditions. A second important factor to consider is the statistics of detection faced by most animals. In most situations signals meaningful to the animal are rare events, occurring only now and then. To keep the false alarm rate at an acceptable level, the threshold of the receptors thus needs to be well above the background noise level. The sensitivity of a given receptor is then likely to be an evolutionary compromise between the values of detecting weak signals and the cost of too many false alarms, weighted by the statistics of the signals. Weak signals may thus occur too rarely and be of too little value for the animal to bother with them, although it might be perfectly possible to increase detection by an increase in receptor sensitivity. It might be that the factor governing the receptor sensitivity in this case is the maximum noise level the animal is likely to experience on a regular basis. The important factor determining the sensitivity may thus be the false alarm rate rather than the detection of signals as such. 5 Conclusion Demonstration of stochastic resonance in a biological receptor system, as stated above, more than anything informs us that the threshold of the receptor in question is not matched to the SNR of the experiment. This has some rather paradoxical consequences. If stochastic resonance is in fact exploited by a given receptor system to compensate for this mismatch, we will be unable to demonstrate the mismatch, since it is no longer there for us to see. In other words, systems where stochastic resonance can be demonstrated experimentally are unlikely to exploit this e€ect. Secondly, if the threshold of a particular system is found to be optimal for the normal SNR experienced by the receptor, we cannot immediately know whether this optimal adjustment is caused by a change in sensitivity of the detector or through an exploitation of the stochastic resonance e€ect by an increase in the noise level

of the detector, since we only observe the endpoint of the optimisation process. The answer may be buried in the evolutionary history of the particular receptor and is thus unlikely to be answered by electrophysiological or behavioural means alone. Acknowledgements. This manuscript bene®ted substantially from discussions with several people: Simon Boel Pedersen, Annemarie Surlykke, Morten Kannewor€, Marianne E. Jensen, and Lee A. Miller. All are thanked for their quali®ed inputs. This study was made possible by ®nancial support from the Danish National Research Foundation, Centre for Sound Communication.

References Au WWL, Moore PWB (1988) The perception of complex echoes by an echolocating dolphin. In: Nachtigall PE, Moore PWB (eds) Animal sonar. Processes & performance. Plenum, New York, pp 295±299 Bezrukov SM, Vodyanoy I (1995) Noise-induced enhancement of signal transduction across voltage-dependent ion channels. Nature 378: 362±365 Bezrukov SM, Vodyanoy I (1997a) Stochastic resonance in nondynamical systems without response thresholds. Nature 385: 319±321 Bezrukov SM, Vodyanoy I (1997b) Signal transduction across alamethicin ion channels in the presence of noise. Biophys J 73: 2456±2464 Braun HA, Wissing H, SchaÈfer K, Hirsch MC (1994) Oscillation and noise determine signal transduction in shark multimodal sensory cells. Nature 267: 270±271 Bulsara AR, Gammaitoni L (1996) Tuning in to noise. Phys Today, March: 39±45 Checkosky CM, Bolanowski SJ (1992) E€ect of stimulus duration on the response properties of Pacinian corpuscles: implications for the neural code. J Acoust Soc Am 91: 3372±3380 Checkosky CM, Bolanowski SJ (1994) The e€ect of stimulus duration on frequency-response functions in the pacinian (P) channel. Somatosens Mot Res 11: 47±56 Chialvo DR, Longtin A, MuÈller-Gerking J (1997) Stochastic resonance in models of neuronal ensembles. Phys Rev E 55: 1798± 1808 Collins JJ, Chow CC, Imho€ TT (1995) Stochastic resonance without tuning. Nature 376: 236±238 Collins JJ, Imho€ TT, Grigg P (1997) Noise-mediated enhancements and decrements in human tactile sensation. Phys Rev E 56: 923±926 DeWeese M, Bialek W (1995) Information ¯ow in sensory neurons. Nuovo Cimento 17D: 733±741 Douglass JK, Wilkens L, Pantazelou E, Moss F (1993) Noise enhancement of information transfer in cray®sh mechanoreceptors by stochastic resonance. Nature 365: 337±340 Dykman MI, McClintock PVE (1998) What can stochastic resonance do? Nature 391: 344 Gailey PC, Neiman A, Collins JJ, Moss F (1997) Stochastic resonance in ensembles of nondynamical elements: the role of internal noise. Phys Rev Lett 79: 4701±4704 Gingl Z, Kiss LB, Moss F (1995) Non-dynamical stochastic resonance: theory and experiments with white and arbitrarily coloured noise. Europhys Lett 29: 191±196 Gluckman BJ, Neto€ TI, Neel EJ, Ditto WL, Spano ML, Schi€ SJ (1996) Stochastic resonance in a neuronal network from mammalian brain. Phys Rev Lett 77: 4098±4101 Green DM (1985) Temporal factors in psychoacoustics. In: Michelsen A (ed) Time resolution in auditory systems. Springer, Berlin Heidelberg New York, pp 122±140 Green DM, Swets JA (1966) Signal detection theory and psychophysics. Wiley, New York

480 Heneghan C, Chow CC, Collins JJ, Imho€ TT, Lowen SB, Teich MC (1996) Information measures quantifying aperiodic stochastic resonace. Phys Rev E 54: 2228±2231 Henry KR (1999) Noise improves transfer of near-threshold, phase-locked activity of the cochlear nerve: evidence for stochastic resonance? J Comp Physiol [A] 184: 577±584 Inchiosa ME, Bulsara AR (1996) Signal detection statistics of stochastic resonators. Phys Rev E 53: R2021±R2024 Jung P (1995) Stochastic resonance and optimal design of threshold detectors. Phys Lett A 207: 93±104 Levin JE, Miller JP (1996) Broadband neural encoding in the cricket cercal sensory system enhanced by stochastic resonance. Nature 380: 165±168 Loerincz K, Gingl Z, Kiss LB (1996) A stochastic resonator is able to greatly improve signal-to-noise ratio. Phys Lett A 224: 63±67 Mitaim S, Kosko B (1998) Adaptive stochastic resonance. Proc IEEE 86: 2152±2183 Moss F, Chioutan F, Klinke R (1996) Will there be noise in their ears? Nat Med 2: 860±862 Narins PM, Benedix JJH (1997) Does stochastic resonance play a role in hearing? In: Lewis ER (ed) Diversity in auditory mechanisms. World Scienti®c, Singapore, pp 83±90

Pei X, Wilkens LA, Moss F (1996) Light enhances hydrodynamic signalling in the multimodal caudal photoreceptor interneurones of the cray®sh. J Neurophysiol 76: 3002±3011 Plomp R, Bouman A (1959) Relation between hearing threshold and duration for tone pulses. J Acoust Soc Am 31: 749±758 Russell DF, Wilkens LA, Moss F (1999) Use of behavioural stochastic resonance by paddle ®sh for feeding. Nature 402: 291±294 Shannon CE, Weaver W (1949) The mathematical theory of communication. University of Illinois Press, Urbana Simonotto E, Riani M, Roberts M, Twitty J, Moss F (1997) Visual perception of stochastic resonance. Phys Rev Lett 78: 1186±1189 Tougaard J (1996) Energy detection and temporal integration in the noctuid A1 auditory receptor. J Comp Physiol [A] 178: 669±677 Tougaard J (1998) Detection of short pure tone stimuli in the noctuid ear: what are temporal integration and integration time all about? J Comp Physiol [A] 183: 563±572 Urkowitz H (1967) Energy detection of unknown deterministic signals. Proc IEEE 55: 523±531 Wiesenfeld K, Moss F (1995) Stochastic resonance and the bene®ts of noise: from ice ages to cray®sh and SQUIDs. Nature 373: 33±36