User:Jonathan R. Williford/Mechanoreceptors and stochastic resonance

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    Stochastic Resonance (SR) is a counterintuitive phenomenon whereby noise under appropriate conditions can enhance the detection of weak signals rather than interfering with signal transmission. Originally described in nonlinear physical systems, SR is applicable as well for biological processes that are also frequently nonlinear. The first demonstration of SR in biology utilized the mechanoreceptive hairs of crayfish. Since mechanoreceptors are ubiquitous in the biological world, enabling animals and humans to sense their environment, we introduce the topic of mechanoreception as fundamental to a broad range of sensory modalities followed by SR and its application in the crayfish.


    Introduction to Mechanoreception

    The term mechanoreceptor applies to any sensory receptor that transduces mechanical energy into an electrical current, the universal signal currency of the nervous system. Although mechanoreception to the non-specialist is generally associated with skin sensations such as touch, vibration, and pressure, mechanoreceptors are also the primary receptors underlying many other senses such as, in vertebrates, audition, muscle and joint position, and the vestibular senses essential to balance and posture and contributing to the perception of self-motion and orientation relative to gravity. In lower (aquatic) vertebrates mechanoreceptors are employed for detecting water motion and pressure waves, e.g., the lateral line. In less familiar involuntary roles, mechanosensitivity contributes to such health related phenomena as regulation of blood pressure, cardiac overload and enlargement of the heart, osmotic homeostasis, and polycystic kidney disease. In addition to electrical signaling, mechanotransduction is now linked directly to biochemical signaling, e.g., where forces transmitted by or involving transport via the cytoskeleton affect changes including gene expression (Huang et al., 2004; Singh and Reuter, 2006).

    Sensory Mechanoreceptors

    For the skin senses, the primary cutaneous receptors include free nerve endings but more so nerve fibers variously encapsulated to afford specificity and/or amplification. The most common receptor, the Meissner corpuscle that lies just beneath the epidermis, is an elongated fiber encapsulated in connective tissue that responds to low-frequency vibrations but adapts rapidly with a decrease in firing rate following stimulus onset. Pacinian corpuscles, located in dermal layers of the skin as well as skeletal joints, are unmyelinated nerve terminals encapsulated in onion-like layers of cells that act as high-pass filters and contribute to a rapidly adapting, high frequency vibration sensitivity. At the other end of the spectrum, the slowly adapting Merkel’s disk is a branching pressure-sensitive nerve ending contacting the epidermis. Below the skin are mechanoreceptors of the musculoskeletal system, proprioceptors that provide information on body position. Embedded within and parallel to the somatic muscles are muscle spindle organs, specialized muscle fibers with a central non-contractile nuclear bag encircled by sensory neurons that monitor stretch. Sensitivity is maintained by efferent (motor) feedback to the muscle spindle to take up the slack during passive shortening as the main muscle contracts. Sensory nerve terminals embedded serially in tendons, the Golgi tendon organs, monitor muscle tension. The tendon organs are low-threshold, rapidly-adapting mechanoreceptors important for initiating spinal reflexes that also control muscle contraction.

    The primary sensory function in skin and proprioceptive receptors involves mechanoelectric transduction mechanisms resident in the dendritic terminals of afferent nerve fibers that project directly to the central nervous system. In contrast, mechanotransduction in the auditory, vestibular, and lateral line systems resides in epithelial receptor cells that relay signals synaptically to afferent fibers. The receptor cell common to these systems is the sensory hair cell, so named for the fine ciliary processes that arise from the apical surface of the cell membrane ( Figure 1, adapted from Holt et al., 1997). A stepwise series of parallel stereocilia, increasing in length until they meet a single, longer kinocilium, form an intricate, polarized transducing system. Displacement of these hairs toward the kinocilium results in an excitatory depolarization of the hair cell whereas motion away from the kinocilium results in hyperpolarization. These voltage changes modulate the release of excitatory neurotransmitter onto the postsynaptic afferent fiber. The structural and molecular mechanisms that underlie mechanotransduction have been studied extensively in hair cells beginning with the frog auditory and vestibular systems, providing the following gating-spring model (Hudspeth, 1985; see also Eatock et al., 2006). The tips of adjacent kinocilia are connected by a filamentous ‘tip link’ such that translation of the ciliary bundle toward the stereocilium imparts a mechanical force that activates cation channels near their tips ( Figure 1).

    Cilia model.jpg
    Figure 1:
    Since the apical ciliary surface of both auditory and vestibular hair cells is bathed in a high-potassium concentration endolymph, inward potassium current depolarizes the hair cell and in turn activates voltage-gated calcium channels in the cell soma. The resulting influx of calcium ions facilitates the release of neurotransmitter. This mechanism imparts great sensitivity, with threshold movements of ~ 0.3 nm, and response times on the order of tens of microseconds for receptor potentials. Such rapid responses are characteristic of mechanoelectric transduction in contrast to the slower response of photo- and chemotransduction whose receptors rely heavily on indirect G-protein activated second messenger systems.

    The mechanism of hair cell transduction as the basis of multiple sensory organs is therefore a function of mechanical coupling to environmental forces. In the cochlea of the inner ear, hair cells are supported by the basilar membrane with their ciliary bundles bathed in a viscous endolymphatic fluid or in contact with the tectorial membrane. The differential movement of the cochlear membranes, or fluid motions, mediated by the middle ear bones results in shearing forces that bend the cilia. Similarly, within the vestibular labyrinth (see Highstein et al., 2004), hair cells of the utricle and saccule macula (the sensory epithelium) are overlain by an otolithic membrane embedded with calcium carbonate crystals. The greater density of these otoliths imparts a similar shearing force onto the ciliary bundles whenever the head alters its position with respect to gravity or experiences acceleration. In the semicircular canals the hair bundles of the sensory epithelium extend into a compliant gelatinous cupula that occludes the canal. Thus, movements of the endolymphatic fluids deflect the cupula in response to angular acceleration of the head. An equivalent system is found in the lateral line of fish and some amphibians, a fluid-filled skin canal with numerous pores opening to the outside. Here, neuromasts lining the canals of the trunk and head feature ciliary tufts embedded in a gelatinous cupula that extends into the canal. Fluid movements inside the canals deflect the cupula as a result of pressure differences alongside the fish. Additional, superficial neuromasts are scattered over the skin surface of fish and amphibians, as illustrated in a superficial neuromast of the bristlenose catfish, Ancistrus ( Figure 2, M.H. Hofmann, unpublished; k, kinocilia; s, stereocilia).

    Figure 2:

    Hair cell morphology highlights a conservative evolutionary feature of cell biology as it relates to sensory systems. Although the mechanically-gated ion channels are located in the kinocilia, somewhat of a misnomer since they lack the complete axoneme of 9 + 2 microtubules characteristic of cilia and flagella, the presence of a kinocilium with its ciliary axoneme and polarizing role in directional sensitivity invites comparison with other sensory receptors that feature non-motile cilia. For example, vertebrate rods and cones arise during development as extensions of a ciliary membrane whose invaginations and vesicles form the outer segments containing photopigments. Olfactory cells also feature cilia, the membranes of which contain the odorant binding receptors and G-proteins that initiate sensory transduction. Likewise, modified ciliary structures are characteristic of many primary mechanosensory neurons where dendrites contain the 9 pairs of ciliary fibrils extending from a ciliary base. Indeed, the cilium arose early on in eukaryotic organisms as a cellular compartment for sensing the environment, as in ciliate protozoans where it functions simultaneously as a sensory antenna and locomotory organelle. For example, a mechanical stimulus applied to the anterior cilia of Paramecium triggers a spike-like membrane potential and calcium current that reverses the effective stroke of ciliary beating resulting in reversed swimming.

    Mechanoreception in Invertebrates

    Cilia aside, mechanoreception is poorly understood in invertebrate animals, with the notable exception of arthropods. Many of these organisms (cnidarians, various phyla of worms, mollusks, echinoderms, and protochordates) feature a soft flexible integument (~skin) without specialized mechanosensory organs. Rather, branching free nerve endings in the epithelial tissues, about which little is known other than their morphological description, appear to mediate tactile responses. Various invertebrates feature a sensory epithelium that extends into bristles or papillae that function as sensory hairs. Statocysts, invaginations lined with sensory hairs deflected by a statolith, are recognizable in animals as diverse as jellyfish, mollusks, and a few worms and echinoderms, but corresponding physiological analyses are largely unavailable. Exceptions include the body wall stretch receptors of annelid worms, the tactile sense organs richly endowed in cephalopod suckers, and arthropods. Nevertheless, fruit fly (Drosophila) and soil nematode (C. elegans) invertebrates now serve as important models for the genetic analysis of mechanoreceptor ion channels, including ciliary proteins homologous with those of vertebrates.

    Mechanoreceptors in Arthropods

    The study of arthropod mechanoreceptors rivals that of vertebrates and the diversity of internal and external receptors is enormous. Like vertebrates, arthropods feature a rigid (exo) skeleton, which also serves as the external integument. With a relatively non compliant exterior surface devoid of epithelium and undifferentiated free nerve endings, crustaceans, insects, spiders and their allies feature a wide variety of receptors specialized to detect both the outside world as well as body structures internally. Many are analogous to those of the skeletonized vertebrates, e.g., proprioceptors. Here, stretch sensitive sensory cells are embedded within strands of muscle or elastic elements that span the joints of body and appendage segments. The muscle receptor organ (MRO) of crayfish and lobster is a clear analog of the vertebrate muscle spindle organ. Here, the dendrites of two sensory neurons, one phasic and one tonic, are embedded within muscle strands stretched as the abdomen flexes. Sensitivity is modulated by efferent feedback to the muscle fibers that adjusts length and tension in the muscle over a range of postures.

    As in vertebrates, arthropod mechanoreceptors underlie the function of multiple sensory modalities, including the tactile senses, audition, and sensitivity to gravity. The scolopidial organ is a common element of many of these systems and the site of sensory transduction where one or more bipolar receptor cells with a ciliary distal process are ensheathed and capped by various cuticular end organs. For example, the chordotonal organs are scolopidia embedded in connective tissue attached to and stretched by joint movements. Campaniform organs, oval patches of flexible cuticle, cap receptors that monitor cuticular stress. Often located in the leg bases, these receptors detect loading relative to orientation and gravity. Thin membranous cuticle forms the insect tympanum, the end organ that caps the ciliary scolopidium of auditory organs located in the thorax or forelegs.

    Most visible are the external, cuticular setal hairs flexibly articulated with the exoskeleton that are sensitive to touch or fluid motion in the environment. These hairs exhibit a wide variety of shapes and articulations consistent with their sensitivity and function. For example, the thin filiform hairs on the cercal appendages of insects (crickets, cockroaches) easily detect subtle wind currents and mediate effective escape responses (Levin and Miller, 1996). These hairs exhibit a range of sensitivities to wind velocity and direction and are perhaps the most sensitive of all mechanoreceptors, responding theoretically at the level of thermal noise (Shimozawa et al., 2003). Chelicerates (spiders, etc.) also exhibit a range of low-threshold wind sensitive setae. In crustaceans, hairs range from short, stiff tactile setae to long feathered seta deflected by the weakest of water currents. Stretch sensitive receptor cell dendrites extend to various extents into the hair shaft, or chords relay hair movements to the more proximal transducing scolopidia. In a specialized function, setal hairs also form gravitational sensory organs, the statocysts. These cuticular invaginations, prominent within crustaceans, contain a statolith supported by a dense population of fine hairs that are deflected according to the position of the animal.

    Moss f23.jpg
    Figure 3:
    Moss f11.gif
    Figure 4:

    The crayfish mechanoreceptors, chosen for the study of random noise effects (Section 2.3), are within a graded population of well-studied setal hairs on the surface of the tailfan appendages ( Figure 3). The afferent fibers of short stiff hairs, phasic and rapidly adapting, activate the system of giant fibers that mediate tailflip escape responses. Our experiments sampled long feathered setal hairs articulated flexibly and responding tonically to gentle water motion ( Figure 3b). Each hair is innervated by the distal, ciliated dendrites of two bipolar sensory cells (Mellon, 1963). These hairs are directionally selective, bending in a planar motion defined by the geometry of their articulating socket. Directionality is further encoded in the response properties of the innervating receptors, each receptor cell firing a burst of spikes when the hair is deflected in the direction opposite to the other (Douglass et al., 1998). Figure 4 is a polar plot in the x-y plane of the sensitivity of a selected hair (Douglass et al., 1998). Arrows indicate the mean vectors of the paired anterior-sensitive (solid line) and posterior-sensitive (dashed line) sensory neurons.

    Moss f12.jpg
    Figure 5:

    The response function of a generalized sensory receptor as a function of stimulus intensity is shown in Figure 5, where the symbols represent experimental data. In the next section we discuss “threshold” stochastic resonance. For calculations of the characteristics of stochastic resonance we use an idealized rectangular function for the threshold as shown by the blue lines and the right hand scale in Figure 5 (Adapted from Mellon, 1968).

    Stochastic Resonance

    Stochastic resonance (SR) is a counterintuitive phenomenon occurring in some nonlinear systems, whereby the addition of random noise to a weak signal causes it to become more detectable or enhances the transmission of the information in the signal through the system. The origins of SR, its early demonstrations in dynamical systems and its important applications to global climate modeling have been amply reviewed elsewhere in Scholarpedia: (Nicolis and Rouvas-Nicolis, Stochastic Resonance). We will therefore concentrate in this section on applications of SR in biology, and we will only briefly mention its migration into chemistry and medical science.

    Moss f21.png
    Figure 6:

    Dynamical SR

    For many years after its discovery SR was thought to exist only in stochastic dynamical systems (Gammaitoni, et al, 1998). A paradigmatic and often studied system is the overdamped Brownian motion of particles in a one-dimensional double well potential,

    \[U(x)=-x^2/2+x^4/4\ .\]

    The dynamics of a single particle in one or the other well subject to random noise, \(\xi(t)\ ,\) plus a weak, information carrying signal, \(\varepsilon\ ,\) is governed by a stochastic Langevin equation, \[\dot{x}=x-x^3+\xi(t)+\varepsilon \cos \omega t\ ,\] where \(\varepsilon\) is small compared with the height of the barrier in the potential well. An example is shown in Figure 6, where (a) shows the potential, (b) shows a time series of the “particle” randomly switching between the left and right wells (the noise on top of the waveform has been removed so that only the barrier crossings are shown), and (c) is the power spectrum \(P(f)\) versus the frequency (3 harmonics are visible), where the signal strength S is the sharp peak (fundamental) on top of the background noise, N. The signal-to-noise ratio, SNR = S/N. Now consider what happens as the noise intensity is systematically increased. The SNR, in this example, or some other measure of order, information, detectability, or more recently success in natural selection (Garcia et al., 2007) passes through a maximum at an optimal value of the noise intensity, \(D=\sqrt{<\xi^2>}/2\ .\) This characteristic noise optimization is the defining signature of SR, as discussed further below.

    Moss f22.gif
    Figure 7:

    Threshold or Non Dynamical SR

    In 1995 a quite different view of SR emerged (Gingl et al., 1995, Pierson et al., 1995). The new picture was not based on dynamics at all. Instead, the description became purely statistical, so that stochastic differential equations as descriptors were unnecessary. This perception came to be called threshold or non dynamical SR. The process can be described quite simply as depicted in Figure 7. First one must focus on (b). The threshold is shown by the dashed black line. Note the sub threshold signal, indicated by the sinusoidal black line. The mean of this signal (blue line) lies a distance \(\Delta_0\) below the threshold. The noise (red trace) is added to the signal and causes the time sequence of positive-going threshold crossing events (spikes) shown in (a). The power spectrum of the spike train, shown in (c) is a bit different from that of the dynamical example discussed above. The spectrum of the noise, N, is flat as shown by the “noisy” line at about – 29 db in this case. The signal features are the two sharp peaks at \(f_0\) and its 2nd harmonic at \(2f_0\ .\) As before, we calculate the SNR as the ratio of the amplitude of the signal feature above the noise to the noise level. The SNR (in decibels) versus the noise intensity, \(D\ ,\) is shown in (d). We note that the SNR enhancement is maximal at an optimal noise intensity, \(D_{\rm o} \approx 0.53\ .\) A simple, approximate theory gives for the SNR (in dB) (Pierson et al., 1995):

    \[ SNR = 10\log_{10}\left[\frac{2\sigma_0\Delta_0^2B^2}{\sqrt{3}D^4}\right]\exp\left[-\frac{\Delta_0^2}{D^2}\right] \] where \(B\) is the peak amplitude of the subthreshold signal, and \(\sigma_0\) is the bandwidth of the noise. [The noise in the dynamical case is usually assumed to be of infinite bandwidth, since that makes certain calculations exact. Here the noise must be of limited bandwidth, but that is no restriction, since all noise in the natural world as well as in designed instruments is in fact band limited.] The maximum of SNR in the formula above is accounted for by the factors \(D^{-4}\) in the prefactor and \(D^{-2}\) in the exponent. Figure 7 reveals an interesting fact. Only three ingredients are necessary for threshold SR: a threshold, a subthreshold signal and noise. Moreover, since the process is only statistical, theoretical calculations for specific examples are much simpler than is the case for dynamical SR. Since the three ingredients are ubiquitous in nature, and because the threshold view is more intuitive, SR has migrated into many diverse fields, so that by now there is an enormous literature on the subject in virtually all areas of science and engineering: (909,000 hits on Google as of January 2007). As examples of this migration we very briefly describe and cite work in only two fields outside of physics and biology: chemistry and medical science. In chemistry, the Belousov-Zhabotinsky reaction (BZ) has long been used to demonstrate nonlinear dynamical phenomena, particularly the nucleation and propagation of spiral waves. A light-sensitive, two-dimensional realization of the BZ reaction has been used to demonstrate spatiotemporal SR (Kádár et al., 1998). Noise, applied over the surface in independently fluctuating cells, has been shown to enhance the propagation of BZ traveling waves, and the enhancement is maximal at optimal noise intensity. In medical science, two demonstrations (of many) can be cited here. First, noise enhanced propagation of electromyographic (EMG) pulses was observed in the human median nerve (Chiou-Tan et al., 1996). In this experiment, the noise was generated internally by the incoherent firings of many compound motor neurons. The firing rate (noise intensity) was controlled externally by contractions by the subject of the abductor pollicis muscle against a calibrated external force gauge. Standard EMG stimuli were applied transcutaneously above the median nerve on the right upper arm. In the absence of muscle contractions the stimulus amplitudes were adjusted to be just below or near the threshold of detection by electrodes on the middle finger of the right hand. As subjects contracted their brevis muscles against the force gauge, the subthreshold EMG signal emerged from the noise with increased amplitude. A different version of this same experiment using human muscle spindles was also carried out (Cordo et al., 1996) as well as the ground-breaking use of SR to enhance human balance in healthy and pathological subjects (Priplata et al., 2003).

    Stochastic Resonance in Biology

    The introduction of SR into experimental biology, based on theoretical predictions by Longtin et al. (1991), was its first demonstration in any field other than physics (Douglass et al., 1993). Above we learned that mechanoreceptors are widespread among sensory modalities in biology. A particularly accessible and simple mechanoreceptor exists on the tail fan of the crayfish in the form of an array of hydrodynamically sensitive hairs innervated by sensory neurons. There are hairs of both long and short length, but here we focus on the long (about 250 \(\mu\)m) hairs that sense water disturbances caused by the approach of a predator (among other things). Figure 3 shows this arrangement (a) with an inset (b) that shows a close-up view of several hairs. Glass pipette suction electrodes were attached to a sensory neuron in the root (a) and connected to the amplifier on the left. The entire tailfan was moved sinusoidally through fluid at about 10 Hz. Thus the hairs were exposed to a periodic, hydrodynamic stimulus. The amplitude of the stimulus was adjusted to be near subthreshold, and spike trains were recorded.

    Moss f24.gif
    Figure 8:
    Moss f25.gif
    Figure 9:

    Figure 8 shows three power spectra of spike trains recorded at near optimal noise intensity using a neuron model (Fitzhugh-Nagumo). Stimulated at 55 Hz, the signal features (sharp peaks at about 55 Hz) are clearly visible. It is obvious that their amplitude changes with noise intensity with the maximum amplitude obtained at the optimal value of noise. The lower panel also shows clearly that noise larger than optimum raises the noise floor and reduces the relative amplitude of the signal feature. The SNRs measured from these (and additional) power spectra versus the noise intensity are shown by the diamonds in Figure 9. The diamonds are from the simulation using the Fitzhugh-Nagumo neuron model to represent the crayfish mechanoreceptor and its associated sensory neuron. The actual crayfish mechanoreceptor data, stimulated at 55 Hz, are shown by the triangles. These data in Figure 9 show that the SNR is maximized at an optimal value of the noise intensity, clearly demonstrating SR in a biological experiment for the first time.

    The threshold view of SR also stimulated experiments in animal (Russell et al., 1999, Freund et al., 2001) and human (Ward 2002a, Ward et al., 2002b) behavior based on perceptive thresholds that are well known in both. SR applied to perception in behavioral experiments is called Behavioral SR.


    Chiou-Tan, F.Y., Magee, K., Robinson, L., Nelson, M., Tuel, S., Krouskop, T. and Moss, F. Enhancement of subthreshold sensory nerve action potentials during muscle tension mediated noise. Intern. J. Bifurc. and Chaos 6, 1389-1396 (1996).

    • Cordo, P.; Inglis, J. T.; Verschueren, S.; Collins, J. J.; Merfeld, D. M.; Rosenblum, S.; Buckley, S. and Moss, F. (1996). Noise in human muscle spindles. Nature 383 (6603): 769-770. 
    • Douglass, J. K. and Wilkens, L. A. (1998). Directional selectivities of near-field filiform hair mechanoreceptors on the crayfish tailfan (Crustacea: Decapoda). Journal of Comparative Physiology A: Sensory, Neural, and Behavioral Physiology 183 (1): 23-34. 
    • Douglass, J. K.; Wilkens, L.; Pantazelou, E. and Moss, F. (1993). Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance. Nature 365 (6444): 337-340. 
    • Freund, J. A.; Kienert, J.; Schimansky-Geier, L.; Beisner, B.; Neiman, A.; Russell, D. F.; Yakusheva, T. and Moss, F. (2001). Behavioral stochastic resonance: How a noisy army betrays its outpost. Physical Review E 63 (3). 
    • Garcia, R.; Moss, F.; Nihongi, A.; Strickler, J. R.; Göller, S.; Erdmann, U.; Schimansky-Geier, L. and Sokolov, I. M. (2007). Optimal foraging by zooplankton within patches: The case of Daphnia. Mathematical Biosciences 207 (2): 165-188. 
    • Gingl, Z.; Kiss, L. B. and Moss, F. (1995). Non-Dynamical Stochastic Resonance: Theory and Experiments with White and Arbitrarily Coloured Noise. Europhys. Lett. 29 (3): 191-196. 

    Holt, J.R., Corey, D.P., and Eatock, R.A. Mechanoelectrical transduction and adaptation in hair cells of the mouse utricle, a low-frequency vestibular organ. J. Neurosci. 17, 8739-8748 (1997).

    • Huang, H. (2004). Cell mechanics and mechanotransduction: pathways, probes, and physiology. AJP: Cell Physiology 287 (1): C1-C11. 
    • Hudspeth, A. (1985). The cellular basis of hearing: the biophysics of hair cells. Science 230 (4727): 745-752. 
    • Showalter, K.; Kádár, S. and Wang, J. (1998). None. Nature 391 (6669): 770-772. 
    • Levin, J. E. and Miller, J. P. (1996). Broadband neural encoding in the cricket cereal sensory system enhanced by stochastic resonance. Nature 380 (6570): 165-168. 
    • Longtin, A.; Bulsara, A. and Moss, F. (1991). Time-interval sequences in bistable systems and the noise-induced transmission of information by sensory neurons. Phys. Rev. Lett. 67 (5): 656-659. 

    Mellon, DeF. Electrical responses from dually innervated tactile receptors on the thorax of the crayfish. J. Exp. Biol. 40, 127-148 (1963).

    Nicolis, G. and Rouvas-Nicolis, C. in Encyclopedia of Dynamical Systems, Random dynamical systems, stochastic resonance, Scholarpedia.

    Pierson, D., O’Gorman, D. and Moss, F., Stochastic Resonance: Tutorial and Update, Intern. J. Bifurcation & Chaos 4, 1-15 (1995).

    • Priplata, A. A.; Niemi, J. B.; Harry, J. D.; Lipsitz, L. A. and Collins, J. J. (2003). Vibrating insoles and balance control in elderly people. The Lancet 362 (9390): 1123-1124. 

    Russell, D., Wilkens, L., and Moss, F. Use of behavioral stochastic resonance by paddlefish for feeding. Nature, 402, 219-223 (1999).

    • Singla, V. (2006). The Primary Cilium as the Cell's Antenna: Signaling at a Sensory Organelle. Science 313 (5787): 629-633. 

    Ward, L.M., Dynamical Cognitive Science (Cambridge, MA, MIT Press 2002a).

    • Ward, L. M.; Neiman, A. and Moss, F. (2002). Stochastic resonance in psychophysics and in animal behavior. Biological Cybernetics 87 (2): 91-101. 

    Internal references

    • Zhabotinsky, A. (2007). Belousov-Zhabotinsky reaction. Scholarpedia 2 (9): 1435. 
    • Izhikevich, E. (2006). Bursting. Scholarpedia 1 (3): 1300. 
    • Meiss, J. (2007). Dynamical systems. Scholarpedia 2 (2): 1629. 
    • Johnson, D. (2006). Signal-to-noise ratio. Scholarpedia 1 (12): 2088. 
    • Rouvas-Nicolis, C. and Nicolis, G. (2007). Stochastic resonance. Scholarpedia 2 (11): 1474. 
    • Cullen, K. and Sadeghi, S. (2008). Vestibular system. Scholarpedia 3 (1): 3013. 

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    See also

    1/f Noise, Excitability, Resonance, Signal-to-Noise Ratio, Stochastic Resonance

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