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TRANSLATIONAL PHYSIOLOGY
1Departments of Electrical and Computer Systems Engineering and 2Physiology, Monash University, Melbourne, Australia
Submitted 3 March 2005; accepted in final form 16 March 2005
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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), stochastic resonance (SR) has since been proposed to occur in a variety of systems from a wide range of disciplines (for comprehensive reviews, see Gammaitoni et al. 1998; Wellens et al. 2004). The concept of SR was initially restricted to situations where "a dynamical system subject to both periodic forcing and random perturbation may show a resonance which is absent when either forcing or the perturbation is absent" (Benzi et al. 1981). Since then, the theory has been expanded, and in its most general form, the theory now describes any phenomenon whereby a nonlinear system can detect an otherwise undetectable stimulus with the addition of a random stimulus, i.e., noise, to the input.
SR phenomena can be broadly divided into two forms: "threshold SR" or "nondynamical SR," which simply requires a threshold, a subthreshold stimulus, and noise, and has been reported for a wide variety of biological systems (for review, see Moss et al. 2004), and "fully tuneable SR" or "dynamical SR" as defined by Gammaitoni (1995) and Gammaitoni et al. (1995), which additionally involves the matching of two time scales and has only been shown in a limited number of biological systems (see Fallon et al. 2004). Figure 1 shows the responses of two systems displaying both forms of SR and highlights the differences between the two (note that the response to a noise-alone stimulus is identical for both systems; bottom panels). The left panels of Fig. 1 show threshold SR, which is characterized by the optimal noise level (the amplitude of additional noise that results in the maximal output signal to noise ratio (SNR); arrows in figure) being equal to the system threshold. Additionally, for threshold SR, the optimal noise level is independent of the type of stimulus, which can even be aperiodic, and is shown by the alignment of the arrows in the middle and top panels with the noise-alone threshold (bottom panel). Conversely, fully tuneable SR (Fig. 1, right panel) is characterized by a dependence of the optimal noise on the frequency of the subthreshold periodic stimulus (note the shift in arrows between the middle and top panels). Specifically, the output SNR of the system is maximized when there is a matching of the frequency of noise induced response of the system (shown in the bottom panel) and the frequency of the subthreshold periodic stimulus. As the noise alone response of the system is monotonic, a subthreshold periodic stimulus of a low frequency will require less additional noise to maximize the output SNR, as a result of matching time scales, than a high-frequency subthreshold periodic stimulus (compare arrows indicating the optimal noise in the middle and top panels). Furthermore, the optimal noise value for any subthreshold stimulus frequency can be predicted from the response of the system to noise alone stimuli (compare the dotted lines in the bottom panel, with the arrows in the middle and top panels). Finally, it follows from the matching of time scales requirement, that the optimal noise level for any subthreshold periodic stimulus must be suprathreshold.
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), have already appeared. Whether such effects are due to threshold SR or fully tuneable SR is rarely discussed; in fact, it has been reported that "natural systems such as the animal and human brain, visual and auditory systems, and behavior lack the rigorous and quantitative theories needed to apply dynamical SR" (Moss et al. 2004). However, the type of SR being used will have a significant influence on the design of any noise-based devices, specifically the amplitudes of noise that should be used and the types of signals that can be enhanced. We therefore believe it is important to study which form(s) of SR may be operating, whether threshold SR or fully tuneable SR, in any particular biological system. We show conclusive evidence for fully tuneable SR in an in vivo preparation of slowly adapting type I (SA I) mechanoreceptors and in psychophysical experiments involving a tactile detection task. Preliminary results have previously appeared in abstract form (Fallon 2002; Fallon et al. 2001). |
METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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EXPERIMENTAL PREPARATION. Experiments were performed on seven cane toads (Bufo marinus) and had approval from the local Standing Committee for Ethics in Animal Experimentation. Animals were stunned and pithed, and a skin flap extending from the ventral midline to the ilium and from the sternum to the pelvic girdle was dissected free. Spinal nerves 4, 5, and 6 were freed along their length and cut at their point of entry into the vertebrae.
The skin flap was secured in an experimental chamber to a solid plate by four pins that held the skin flap taut. The skin flap was perfused with amphibian Ringer solution (in mM: 111 NaCl, 2.5 KCl, 0.1 K2HPO4, 11 glucose, 2.4 NaHCO3 and 2.38 CaCl2) bubbled with carbogen (5% CO2 in O2), and maintained at room temperature (20–25°C). A small electro-magnetic actuator, tip diameter 1.2 mm, was used to mechanically stimulate the skin surface. The response of the electro-magnetic actuator was constant from DC to 50 Hz, above which the response was effectively two-pole low-pass filtered with near critical dampening. The spinal nerves were lifted into a paraffin oil-filled chamber used for recording. Functionally single afferents were obtained by removing the sheath of connective tissue from a spinal nerve and then subdividing the nerve into small filaments.
EXPERIMENTAL PROTOCOL. Afferent axons were identified as innervating SA I mechanoreceptors by their maintained discharge during the hold phase of a ramp-and-hold skin indentation and the lack of an OFF response. The singularity of an afferent was determined by the consistent nature of the recorded action potential and the absence of extremely short interspike intervals that are characteristic of multi-unit recordings. All test stimuli were superimposed on a subthreshold ramp-and-hold indentation (typically 75 µm, see Fig. 2) with a rest period of 60 s between stimuli to allow the receptor to recover (no change in stimulus thresholds were observed over the 60-min recording periods).
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The response of the receptor to subthreshold sinusoidal indentations with various amplitudes of additional noise was measured. Several frequencies of subthreshold sinusoidal signals were used, with the frequencies chosen to lie within the approximately linear region of the noise-alone response. This was expected to generate the greatest separation of the optimal noise amplitudes for each sinusoidal test frequency and resulted in the use of sinusoidal test frequencies between 1 and 20 Hz. The amplitude of each sinusoidal test signal was adjusted to be "near" but below its threshold.
DATA COLLECTION AND ANALYSIS. Action potentials were amplified using custom built amplifiers before being digitally recorded using a commercial data acquisition card (PCI-MIO-16E-4, National Instruments, Austin, TX) in a G3 Macintosh computer (Apple, Cupertino, CA.). All recording and analysis were done using custom software written within Igor Pro (Wavemetrics, Lake Oswego, OR).
The analysis of the responses of the mechanoreceptors to both noise-alone and combined subthreshold sinusoidal and suprathreshold noise stimuli has previously been described (Fallon et al. 2004). Briefly, the noise-alone response (the average discharge rate) was fitted with a curve based on Kramers' theorem, given by
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and 
are arbitrary constants. From the noise-alone response, it is possible to predict the amplitude of noise, DPRE, that should maximize the output SNR for a particular frequency of subthreshold periodic input. This is the amplitude of noise, when applied alone, that produces an average response rate equal to the particular subthreshold sinusoidal stimulus frequency (Fig. 1, bottom, dotted lines).
A measure based on the amount of modulation of the cycle histogram, SNRCYCLE, was calculated by fitting a sinusoid to the cycle histogram of the response to the combined noise and subthreshold periodic stimuli as shown in Fig. 2. SNRCYCLE was defined as the amplitude of the fitted sinusoid divided by the estimated error (SD) in the amplitude of the fitted sinusoid. SNRCYCLE was determined for the response to combined subthreshold sinusoidal and suprathreshold noise stimuli for a number of sinusoidal frequencies and noise amplitudes. The amplitude of noise that produced the maximum SNRCYCLE, DOPT, was calculated by fitting a logNormal curve of the form
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and are arbitrary constants. Evidence for fully tuneable stochastic resonance in SA I mechanoreceptors was assessed by correlating DOPT and DPRE, the measured and predicted optimal noise amplitudes, at different frequencies of subthreshold sinusoidal length change and comparing DOPT for each mechanoreceptor at the two different test stimulus frequencies. Psychophysical Experiments
SUBJECTS AND EXPERIMENTAL PREPARATION. Studies were conducted on six healthy adult volunteers of either sex, aged 20–31 yr. All subjects gave informed written consent to the experimental procedures, which where approved by the local Standing Committee on Ethics in Research on Humans.
Subjects were blindfolded and seated comfortably with their right-forearm resting on a horizontal cushioned support. Localized indentations of the hairy skin on the dorsal surface of the hand (taking care to avoid hairs, superficial tendons, and blood vessels) were produced with the same electromagnetic actuator as used for the toad experiments.
EXPERIMENTAL PROTOCOL.
The threshold for detection of small sinusoidal stimuli was measured using test indentations consisting of 6 s of sinusoid, with an aural indication of the frequency of the sinusoidal indentations, with or without the addition of additional noise to the stimulus. Subjects were instructed to verbally report, within 30 s of the end of the test indentation, if they had felt a sinusoidal indentation that was of the same frequency as the aural cue. Subjects were asked to detect the presence of a specific frequency of sinusoidal indentation, rather than just any indentation for two reasons. First, a periodic stimulus is a critical part of fully tuneable SR, and second, the subject could readily perceive the noise-alone stimuli, which were not of interest. If unsure of the presence of the specific frequency sinusoidal indentation, the subjects were instructed to indicate that no sinusoidal indentation had occurred. A minimum inter-presentation interval of 30 s was used to allow for full creep recovery of the skin (Pubols 1982), which was confirmed by no change in the pure sinusoid (no-noise) detection threshold over the four hours of testing.
The threshold for detection was measured using a modified staircase technique (Cornsweet 1962) (Fig. 3). The technique begins with the application of a sinusoidal test indentation known to be readily detectable by the subject. In successive presentations the amplitude of the sinusoidal indentation was reduced until it was no longer detectable. The amplitude of the sinusoid was increased in successive presentations until it was again detectable. The procedure was repeated several times to determine a series of minimum amplitude detection points, indicated by the bold outline circles in Fig. 3. The detection threshold was calculated by averaging the minimum amplitude detection points.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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The responses of a SA I mechanoreceptor to a single presentation of both combined and individual subthreshold periodic and suprathreshold noise stimuli are shown in Fig. 2. By definition, the receptor did not respond to the subthreshold periodic stimulus alone (Fig. 2A). The suprathreshold noise-alone stimulus (Fig. 2B) produced a relatively sparse cycle histogram, due to the limited number of cycles (30) of sinusoidal stimulus. However, the action potentials were almost evenly distributed across all phases of the stimulus, giving a SNRCYCLE of 1.15. The combination of subthreshold sinusoidal and suprathreshold noise stimuli (Fig. 2C) resulted in a cycle histogram with a significant modulation of the action potential distribution across stimulus phase, giving a SNRCYCLE of 2.14.
A SA I mechanoreceptor exhibiting many features characteristic of fully tuneable SR is shown in Fig. 4. The average discharge rate during the 6 s of imposed noise-alone indentations superimposed on a subthreshold ramp-and-hold movement of 75 µm is shown in the bottom panel. The noise-alone threshold was 
50 µm, above which the average discharge rate increased with increasing noise amplitude toward a plateau of 30 i s–1. The noise-alone response was well fitted by a curve based on Kramers' rate, allowing for accurate predictions of DPRE from this curve. For the two test frequencies shown, 5 and 13 Hz, the predicted optimal noise amplitudes were 87 and 135 µm, respectively (Fig. 4, bottom, dotted lines). The combination of suprathreshold noise and subthreshold, periodic sinusoidal indentations produced SNRCYCLE data that were well fitted by a logNormal curve for each test frequency (Fig. 4, top). The resulting noise amplitudes that maximized SNRCYCLE were (DOPT ± SE) 95 ± 1 and 143 ± 1 µm for the 5- and 13-Hz subthreshold sinusoidal stimuli, respectively. The increase of DOPT with increased frequency of subthreshold periodic stimulus is a key feature of fully tuneable SR.
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0.001) to optimize the output SNR than did the lower frequency stimulus (circles).
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The results of a single staircase trial are shown in Fig. 3. The four smallest correct detections of the pure sinusoidal stimulus (circles) were 271, 431, 431, and 447 µm (bold outlines in Fig. 3) corresponding to a detection threshold of 395 ± 40 (SE) µm. The detection threshold for the combination of a sinusoid and 450-µm noise (squares) was 276 ± 5 µm. The normalized noise level and detection threshold for this trail were therefore 1.14 and 0.70, respectively.
Figure 6 shows the response from a single subject. Although there are relatively few data points, the results could be fit by a logNormal curve for each test frequency. The resulting noise amplitudes that minimized the detection threshold (DOPT ± SE) were 0.9 ± 0.5 and 2.0 ± 0.1 for the 0.5- and 1-Hz sinusoidal stimuli, respectively. Pooled results from all six subjects are shown in Fig. 7 and show that the optimal noise level for the 1-Hz sinusoidal stimulus was significantly higher than the optimal noise level for the 0.5-Hz sinusoidal stimulus (paired t-test; P < 0.05). The increase of DOPT with increasing frequency of periodic stimulus is a key feature of fully tuneable stochastic resonance.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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A uniform distribution of noise amplitudes, rather than the more commonly used Gaussian distribution of amplitudes, was used for practical reasons in both our previous study of fully tuneable SR (Fallon et al. 2004) and this study. While a full theoretical analysis comparing the effects of the two types of noise has not been undertaken, modeling studies comparing the effects of uniform distributions of noise amplitude and Gaussian distributions of noise amplitude have indicated that there is no significant effect of the distribution used (Fallon 2001). Importantly, any clinical applications of SR using mechanical stimulation will be limited by practical considerations such as difficulties in producing a large instantaneous change in length, restrictions in absolute length changes possible, and the increased risk of injury with large instantaneous length changes. All of which will require the use of noise distributions that deviate from the idealized Gaussian distribution of noise amplitudes, for which there is a small, but finite, possibility of arbitrarily large amplitudes. Therefore that SR can occur with a variety of noise distributions as shown in the present study and in the SR literature (Nozaki et al. 1999), is an important finding.
Toad experiments
The results from the experiments with the SA I mechanoreceptors of the toad demonstrate the fully tuneable nature of ‘fully tuneable stochastic resonance’. In response to noise and a subthreshold periodic input, the maximum output SNR was achieved by tuning the amplitude of the noise while holding the frequency of the subthreshold periodic stimulus constant. That the responsible mechanism is fully tuneable stochastic resonance is further confirmed by the observation that the increase in output SNR is co-dependent on the amplitude of the noise and the frequency of the periodic input. That is, a higher frequency periodic stimulus required a larger noise amplitude to maximize the output SNR (Fig. 5). In addition, for a given frequency of subthreshold periodic stimulation, the empirically determined noise amplitude required to optimize the receptor's SNR was strongly correlated with that predicated from the receptor's response to noise alone. The SA I mechanoreceptor of the toad can therefore be added to the meager list of biological systems that have been shown to exhibit fully tuneable stochastic resonance (see Fallon et al. 2004).
Increases in output SNR with the addition of noise similar to those shown in Fig. 4 have previous been reported for rat SA I mechanoreceptors (Collins et al. 1996a). A major difference between the two studies is that Collins et al. used aperiodic stimuli, and therefore attributed the increase in output SNR to threshold SR, whereas this study used two different frequencies of subthreshold sinusoidal stimuli, and therefore can attribute the increase in output SNR to fully tuneable SR. These results emphasize that similar systems (rat and toad SA I) can exhibit either threshold SR or fully tuneable SR. In fact, it is likely that a single system could exhibit either, or both, threshold SR and fully tuneable SR if suitably stimulated; the implications of which are discussed in Function implications.
Psychophysical experiments
Simply because SA I mechanoreceptors of the toad exhibit fully tuneable SR under strictly controlled experimental conditions does not necessitate that a psychophysical experiment involving a tactile detection task (designed to use similar mechanoreceptors) will also exhibit fully tuneable SR. The computations involved in determining whether fully tuneable SR has occurred at the receptor level involve construction of cycle histograms, curve fitting, and time averaging, which has led some researchers to wonder whether "natural systems such as the animal and human brain" are capable of exhibiting fully tuneable SR (Moss et al. 2004). Conversely, there have been tantalizing reports involving visual perception tasks (Chialvo and Apkarian 1993) and tactile sensation in humans (Ivey et al. 1998) that have alluded to the possibility of fully tuneable SR. However, these reports failed to unequivocally show that the effect was fully tuneable SR and not threshold SR or some other effect.
Reductions in detection thresholds with the addition of noise, such as those in Fig. 6, have previous been reported (Collins et al. 1996b, 1997); however, the mechanism involved was proposed to be threshold SR. The dependence of the optimal noise level on the frequency of the sinusoidal test signal shown in Fig. 7 is at odds with such an interpretation, but is in agreement with the predictions of fully tuneable SR. The results from the tactile detection task reported here is therefore the first unequivocal demonstration of fully tuneable SR in humans.
Functional implications
Information transfer is optimized for systems exhibiting threshold (or nondynamical) SR when the additional input noise is approximately equal to the threshold. Under such circumstances, the system's response will be optimal to input stimuli of all frequencies. Conversely, for systems exhibiting fully tuneable (or dynamical) SR, the level of additional input noise must be tuned to an appropriate suprathreshold level, dependent on the frequency of input stimulus. The system's response will then be optimized for a narrow range of stimulus frequencies.
The application of noise to the sole of the foot (or the knee) has been shown to reduce sway (for review, see Collins et al. 2003). Sway can be reduced in both the elderly and the young by the application of "subsensory" (90% of perceptual threshold) mechanical noise to the sole of the foot (Priplata et al. 2002). The proposed mechanism for the reduction in sway is an increase in the sensitivity of the detection of sway via SR. Because the frequency of anterior-posterior sway during quiet stance with eyes open is in the order of 0.1 Hz and does not significantly change over time unless the subject is perturbed (Loughlin and Redfern 2001), it is possible that either threshold SR, or fully tuneable SR may be operating. Reports suggesting the clinical use of SR to ameliorate age-related impairments in balance (Priplata et al. 2003) highlight the need to establish which form(s) of SR may be operating.
When applying additional noise to the sole of the foot in an attempt decrease sway, the precise level of noise chosen will be influenced by many factors. If the reduction of sway is occurring via fully tuneable SR, as the reduction in detection thresholds observed in this report were, the level of noise should be adjusted to the suprathreshold level of noise that will optimize the detection of particularly frequencies of sway, at the expense of detection of other mechanical stimuli. Conversely, if the reduction of sway is occurring via threshold SR, the level of noise should be adjusted to the threshold level of noise. In both situations, the pertinent threshold of noise may not in fact be the perceptual threshold, but the mechanoreceptor's threshold.
A further consideration is that the SR phenomenon may not be even be occurring in mechanoreceptors, but may in fact occur at some processing station within the CNS. While the reports of fully tuneable SR to date have be limited to slowly adapting type receptors, there is no theoretical reason why fully tuneable SR should be limited to slowly adapting systems and fully tuneable SR could in fact occur in other types of receptors or within neural networks. It is therefore conceivable that during the integrating of signals from a variety of receptors, fully tuneable, or threshold, SR could occur.
Finally, for any type of clinical application, the patient's comfort must be a consideration. For the fully tuneable SR showed in this report, the optimal level of noise was sometimes an order of magnitude greater than perceptual threshold, which clearly would not be satisfactory in a clinical setting.
In summary, fully tuneable SR can occur in SA I mechanoreceptors in the toad and in psychophysical experiments involving tactile detection tasks. Given the different properties of fully tuneable (or dynamical) SR and threshold (or nondynamical) SR, consideration of which form(s) of SR are occurring must be given thought, particularly in the development of any applications that are designed to take advantage of the extra information available via SR.
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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ACKNOWLEDGMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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FOOTNOTES |
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Address for reprint requests and other correspondence: J. B. Fallon, Bionic Ear Inst., Dept. of Otolaryngology, 2nd Floor, 32 Gisborne St., East Melbourne, 3002 Victoria, Australia (E-mail: James.Fallon{at}ieee.org)
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Chialvo DR and Apkarian AV. Modulated noisy biological dynamics: three examples. J Stat Phys 70: 375–391, 1993.[CrossRef]
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Collins JJ, Priplata AA, Gravelle DC, Niemi J, Harry J, and Lipsitz LA. Noise-enhanced human sensorimotor function. IEEE Eng Med Biol Mag 22: 76–83, 2003.
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