AMER. ZOOL., 41:1229–1240 (2001)
Vibration Sensitivity and a Computational Theory for Prey-Localizing Behavior in Sand Scorpions1 PHILIP H. BROWNELL2,*
AND
J. LEO
VAN
HEMMEN†
*Department of Zoology, Oregon State University, Corvallis, Oregon 97331 †Physik Department, TU Mu¨nchen, D-85747 Garching bei Mu¨nchen, Germany
SYNOPSIS. As burrowing, nocturnal predators of small arthropods, sand scorpions have evolved exquisite sensitivity to vibrational information that comes to them through the substrate they live on, dry sand. Over distances of a few decimeters, sand conducts low velocity (;50 m/sec) surface (Rayleigh) waves of sufficient amplitude and bandwidth (200, f ,500 Hz) to be biologically detectable. Eight acceleration-sensitive receptors (slit sensilla) at the tips of the scorpion’s circularly arranged legs detect surface waves generated by prey movements or ‘‘juddering’’ signals from other scorpions. From this input alone, direction of the disturbance source is calculated up to 20 cm distance. By ablating slit sensilla in various combinations on the eight legs, the contribution each makes in computing target location can be assessed. Other behavioral experiments show that differential timing of surface wave arrival at each sensor is most likely the cue that determines target location. Given the simplicity of this sensory system, a computational theory to account for wave source localization has been developed using a population of second-order neurons, each receiving excitatory input from one vibration receptor and inhibition from the triad of receptors opposite to it in the eight-element array. Input from a passing surface wave opens and closes a time widow, the width of which determines the firing probability of second-order neurons. Target direction is encoded as the relative excitation of these neurons, and stochastic optimization tunes the relative strengths of excitatory and inhibitory inputs for accuracy of response. The excellent agreement between predictions of the model and observed behavior of sand scorpions confirms a simple theory for computational mapping of surface vibration space.
VIBRATIONAL COMMUNICATION AND PREY LOCALIZING BEHAVIOR OF SAND SCORPIONS Beyond direct contact, vibrational communication implies the existence of a medium capable of conducting mechanical waves between a message sender and a message receiver. Judging from the steady growth of literature on vibration sensitivity in psammophilous animals (Salmon and Horsch, 1972; Brownell, 1977, 1984; Aicher and Tautz, 1990; Henschel, 1997; Brownell and Polis, 2001; Mason and Narins, 2001), sand appears be a particularly favorable substrate for conduction of mechanical information. In this paper we describe the prey localizing behavior of the Mojave 1 From the Symposium Vibration as a Communication Channel presented at the Annual Meeting of the Society for Integrative and Comparative Biology, 3–7 January 2001, at Chicago, Illinois. 2 E-mail:
[email protected]
Desert sand scorpion, Paruroctonus mesaensis, and some of the physical properties of loose sand that make it a good conductor of surface waves. The exquisite sensitivity of Paruroctonus to vibrational information is certainly not limited to signals emanating from prey alone, however, and likely mediates intraspecific vibrational communication in the form of ‘‘juddering’’ movements (sudden strong lurches of the body forward), tail thumping and tail wiping behaviors commonly observed in psammophilic scorpions (Tallarovic et al., 2000; Brownell and Polis, 2001). In this way an evolutionary connection may be drawn between passive detection of vibrational information for purposes of orientation to food and the more evolved process of active signal generation for purposes of communication between conspecifics. Sand scorpions make excellent models for analysis of surface vibrational infor-
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mation, as much for the physical simplicity of the substrate they inhabit as for the elegant simplicity of the predatory behavior they display. In many respects the scorpion’s orientation to prey disturbing the flat, featureless surface of a sand dune is the same sensory problem that orb spiders face in orienting to prey caught up in a web, or that water striders encounter in locating prey struggling at the surface of a pond. With these similarities in mind, we expect the study of predatory behavior in sand scorpions to transcend vibrational media and address the more general question of how animals make sense of mechanical information propagating through space. For this we present a computational model conceived for the specific circumstances and behaviors of sand scorpions but simple enough to apply as a general theory for perception of vibrational information conducted along free surfaces. Vibration-source localizing behavior quantified Paruroctonus mesaensis is an ambush predator of insects and other scorpions, always hunting at night from a motionless rest position outside its burrow on the sand surface. When prospective prey approach within 20 cm, the sand scorpion first assumes an alert posture with its body slightly elevated from the substrate and the pedipalps (organs of prey capture) extended and open forward. Subsequent disturbances of the sand by the prey evoke a sequence of quick rotations with forward movements toward the target until the pedipalps can grasp and immobilize the prey for stinging. Scorpions are naturally fluorescent under portable blacklights (Fig. 1) so that images of this predatory orientation response (POR) can be captured on film at night and subsequently measured for accuracy in the laboratory. The two components of the POR— the angle of turning and the distance moved toward the prey—were both shown to be highly accurate within 10 cm distance (Brownell and Farley, 1979a), confirming the impressions from field observations that prey within this radius are nearly always captured in a single orientation response. A second orientation behavior of Paru-
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FIG. 1. The Mojave sand scorpion, Paruroctonus mesaensis, in its hunting stance with tarsi resting on sand in a circular array. Photo taken under ultraviolet illumination shows natural fluorescence of the cuticle.
roctonus, the defensive orientation response (DOR), has more utility for experimental analysis of vibration sensing behavior because it is easily and repeatably evoked in the laboratory when animals are agitated into defensive posture. The DOR is characterized by the same accurate rotation of the body to align the pedipalps toward a vibrational disturbance but there is no forward movement. Figure 2 shows an example of the accuracy of DOR for stimuli presented at 8–10 cm distance. With this angular component of vibration source localization assayable under controlled conditions, we have a means of demonstrating that the signals used by the scorpion to determine stimulus location are mechanical waves propagating along the surface of sand (Brownell, 1977, 1984). Wave propagation in sand When considering the prospects for vibrational communication between animals, our attention is drawn immediately to the solid substrates through which the signal must pass. From a physical perspective, most natural substrates are massive, rigid and highly inelastic to conduction of mechanical waves when compared to air and water. Unlike air and water which, by com-
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FIG. 2. Accuracy of orientation to vibrational stimuli presented to Paruroctonus. A. The angle of the turning response (ordinate) to target stimuli presented at various angles (abscissa) 8–10 cm away is accurate (●) compared to perfect response (dashed line). B. When vibration receptors on the left legs (BCSS L1–L4, darts in inset) were inactivated by puncture with a pin, turning behavior was biased toward 908 right, indicating the ablated animal’s perception of stimulus direction was calculated from the remaining (R1–R4) receptors. D, O; responses of two animals, both with L1–L4 sensors ablated.
parison, are excellent conductors of sound, light and volatile chemical substances, solid media are difficult to move and absorbent to mechanical waves. But dry sand, like that found in desert dunes is a surprisingly good conductor of the wave type that carries most of the energy away from surface disturbances—the Rayleigh wave. The physical characteristics of sand waves are summarized in Figure 3. ‘‘Substrate vibration’’ is an imprecise term used to describe mechanical movements of a solid substrate. Beyond a short distance from the source of disturbance, mechanical motions in solids propagate as one of four elastic waves (White, 1965; Graff, 1975). Two of these are body waves—the compressional and shear waves, and two conduct along the surface—the Rayleigh and Love waves (Fig. 3A). The properties of these waves and the proportion of source energy transported by each depends on physical properties of the substrate and the disturbance. These values for dry, loose sand have been
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measured (Brownell, 1977; Aicher and Tautz, 1990) as summarized in Figure 3a. Over distances of a few decimeters, loose sand is a surprisingly good conductor of low velocity (;50 m/sec) surface (mostly Rayleigh) waves, the wave type that conducts more than 70% of the mechanical energy away from the source. Furthermore, regardless of the disturbance that creates vibrational energy at the source, most of the surface wave component propagating beyond a few centimeters is contained within a bandwidth of 300–400 Hz (Fig. 3B). At these frequencies frictional loss of energy, measured as the coefficient of absorption, a10, and expressed in units of decibels attenuation per centimeter distance, is sufficiently low that nearly all of the signal loss with distance is attributable to geometric spreading of the wave front (Fig. 3c). These are effectively the same factors of attenuation existing at the surface of water. Vibration sensing In sand scorpions, vibrational stimuli are sensed by slit sensilla embedded in the cuticle (Brownell and Farley, 1979b). As in other arachnids, slit sensilla are typically located near joints in the exoskeleton where they monitor changes in tension during locomotion (Barth, 1985). In sand scorpions the 6 to 7 slits just proximal to the basitarsal–tarsal joint of each leg (Fig. 4) are particularly large and responsive to small amplitude accelerations of the tarsi resting on the substrate (e.g., vibrational stimuli). Together, the eight sense organs act as an array of accelerometers, arranged around the circle of contact points the scorpion’s legs (tarsi) make with the substrate (Fig. 1). Behavioral experiments show this simple field of receptors generates the input from which target direction is accurately calculated (Brownell and Farley, 1979c). Morphological and physiological analysis of the basitarsal slit sensilla (BCSS) show them to be of typical arachnid structure, with two neurons innervating each slit. One BCSS with six slits contains 12 afferent neurons, but only one or a few of these respond reliably to threshold levels of stimulation. This can be shown electrophysiologically by accelerating the tarsus at small
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FIG. 3. Wave propagation in dry sand. A. Diagram and table summarizing physical properties of surface and body waves propagating away from a source disturbance in dry sand. B. Left panel: Surface wave recordings from two accelerometers placed at 21 cm (upper trace) and 81.5 cm (lower trace) from an episodic surface disturbance. Right panel: Fast Fourier transform spectra for the recordings to the left, showing most energy of the wave is contained in a narrow band centered at about 350 Hz (from Aicher and Tautz, 1990, with permission). C. A comparison of factors contributing to attenuation of the surface wave (i.e., geometric spreading, and absorbance at a10 5 0.23 db/cm) shows most of the loss is due to simple spreading of the wave front. Threshold acceleration for BCSS stimulation is below 0.01 m/sec2 (ref. Fig. 4).
amplitudes and variable frequency (Fig. 4). At low frequencies (;100 Hz) the receptor cells in one slit respond with phase-locked regularity relative to periodic motions of the stimulus. At higher frequencies (.500 Hz), unit responses (‘‘spikes’’) from the BCSS appear to remain phase-locked even though failing to respond to every cycle of oscillation. This apparent phase-locking of the sensory response to peaks in the acceleration stimulus is the first indication that highly accurate timing of the stimulus arrival at each leg may be important to sensory processing in this system. By varying the frequency of the vibrating stimulus and measuring the threshold amplitude necessary to evoke a sensory response, it is possible to show that acceleration sensitivity over the bandwidth of 200 to 500 Hz, the same bandwidth containing most of the vibrational energy of sand (Fig. 3b), is below 1 cm/sec2. Such sensitivity accounts for the animal’s ability to detect strong signals
from prey (i.e., sand burrowing cockroaches) beyond 30 cm distance (cf. Fig. 3c). Mechanisms of source localization If the eight BCSS are the only source of sensory input required for accurate determination of vibration source location, then it should be possible to ‘‘blind’’ sand scorpions to vibrational stimuli by ablating just these receptors. Furthermore, since there are only eight of them, selective elimination of part of the sensory field should preserve some level of sensitivity to vibrational stimuli while compromising accuracy of the turning response in a predictable pattern. As predicted, animals with all eight of their BCSS ablated are unresponsive to local disturbance of the substrate, and those with two or more sensilla intact continue to respond to stimulation but with systematic patterns of inaccuracy. An example of the latter is shown in Figure 2, for animals missing the left half of their sensory field
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FIG. 4. Vibration receptors and sensory response. A. Diagram of the tarsus (T) in P. mesaensis showing location (SS) of the BCSS just above a terminal leg joint. (M, C, MC, LC, PS; hairs and claws aiding support of the tarsus on sand). B. Photomicrograph of slit sensilla (ss) with their mechanosensory neurons (m) labeled by neuronal tracer dye. C. Electrophysiological response, Ve, recorded from a single BCSS slit. Sensory units respond at fixed phase intervals relative to vibrational accelerations, Vd, applied in realistic bursts to the tarsus. Calibrations: upper traces, 1 mV; lower traces, 1 cm/sec2.
(i.e., all receptors on the left legs ablated). The manipulated animals still respond vigorously to stimuli presented at all angles, but the turning response is, itself, unilateral (toward the intact right side). Moreover, even for targets presented toward the intact side of the animal, the accuracy of turning is biased toward the mean direction of the remaining legs, that is, about 908 right. Various combinations of sensory ablation similar to this one (Brownell and Farley, 1979c; cf. Fig. 7) give clear indication that input from all sensors is integrated during the assessment of target location, not just those few sensors closest to the source. Thus, the BCSS are both necessary and sufficient input for directing the sand scorpion’s turning response and some form of central integration of their combined input will specify direction. Another way to access the mechanism of source localization is to alter the pattern of sensory stimulation by placing fully intact animals on manipulated substrates. One approach is to introduce a thin (;1 mm) gap of air into the substrate beneath a scorpion so that its left and right legs rest on separate and vibrationally uncoupled half-spaces (Brownell and Farley, 1979a, c). Under
these circumstances the animal behaves as though only half its legs are receiving input, the left legs or the right legs, depending on which side of the gap the target is presented. A variation of this experimental manipulation uses piezoelectric crystals to vibrate the two half-spaces independently, thus gaining control over the amplitude and timing of substrate movements beneath the right and left legs (Fig. 5). These investigations show that either relative amplitude or timing of stimulus presentation can be used independently as a cue for source localization (Brownell and Farley, 1979c). However, when amplitude and timing cues are presented in opposition to each other (e.g., a dichotomous stimulus where amplitude predicts turns in one direction and timing in the opposite direction), scorpions demonstrate preference for the timing cue by turning in the direction of the platform that moves first rather than at greatest amplitude. This experiment also shows that sand scorpions can detect time intervals between stimulation of the legs as small as 0.2 msec, well below the transit time required for a Rayleigh wave to cross the expanse of the sensory field (diameter/VR ø 1 msec).
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computational analysis. The success or failure of such a model, as judged by how well it predicts observed behavior, will direct our attention to what needs further clarification through neurobiological analysis.
FIG. 5. Experiment demonstrating the importance of stimulus timing as the cue for source localization. When right and left legs of a scorpion are placed on separate platforms separated by a narrow gap, the two halves of the sensory field can be stimulated independently at different times and amplitudes relative to each other using pulse trains from two piezoelectric transducers (TR, TL). Plotted points show mean angle of turn (6SE) for variable time intervals (Dt) between movements of the left and right platforms. The inset shows time-shifted (Dt) pulse trains (125 Hz, 1 sec duration) applied to left (TL) and right (TR) platforms. (After Brownell and Farley, 1979c).
Thus, the temporal responsiveness of the sensory system appears to be well within sufficiency for resolving which legs are closest to the wave source. With these essential facts in hand, we can begin to construct a neural model for perception of vibrational space around the sand scorpion. A COMPUTATIONAL MODEL OF VIBRATION SOURCE LOCALIZATION If timing of vibrational stimulation to a field of eight vibration sensors is sufficient information for the sand scorpion to determine source direction, how might neuronal circuitry in the animal’s brain transform information in the temporal domain (time of stimulus arrival at each receptor) into information in a spatial domain (the angle of the turn to be made)? The mapping of time into space is a characteristic feature of auditory processing networks for which there is a considerable literature (Carr and Konishi, 1990; Buonomano and Merzenich, 1995; Grothe, 2000), but little has been done to illuminate the mechanisms for perception of mechanical waves propagating along free surfaces. It is instructive to begin such analysis by constructing a theory of how localization might work, one simple enough to be expressed mathematically for
Formulation of the model We begin formulation of the model (Stu¨rzl et al., 2000) by representing the spatial structure of the scorpion’s vibrational sensory field in mathematical terms, then calculating the temporal information available to the animal during the passage of a simulated natural signal, the Rayleigh wave. The tarsi of adult Paruroctonus and their vibration sensors are arranged in a circle (Fig. 1) of radius R ø 2.5 cm in adults. Furthermore, in their hunting stance, each tarsus is held at relatively fixed positions given by angles gk 5 6188, 6548, 6908, 61408, where 08 is directly anterior (see inset Fig. 2). We label these inputs individually as 1 # k # 8 moving clockwise from the right front leg. The Rayleigh wave generated by a stimulus at angle wS and distance r from the leg-circle center approximates a plane wave when r $ 8 cm. For a given stimulus angle, wS, the time difference Dt (gk, glzwS) between the arrival of a wave at two BCSS on tarsi at angles gk and gl is then Dt(gk , gl z wS ) 5 {R/vR}[cos(wS 2 gl ) 2 cos(wS 2 gk )] (1) so that Dt ∈ [2Dt0, Dt0] with Dt0 5 2R/vR. A quick inspection of Eq. 1 shows that for opposed legs (gk 5 gl 1 1808) the cosine term is 2cos(wS 2 gl, giving 2R (circumference of the leg circle) divided by propagation velocity of the Raleigh wave (vR) as the maximal time interval the scorpion can experience. That is, Dt ;1 msec. The next step in model formulation is to make an informed guess at the neuronal mechanism that could transform time information into patterned neuronal activity in the scorpion’s brain. While we know comparatively little about the neuroanatomy of scorpion central nervous systems, we have fairly detailed studies of spiders showing ring-shaped integrative neurons in the sub-
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FIG. 6. Inhibitory triad model to account for vibration source localization in sand scorpions. The circular array of BCSS receptors (outer circles), at fixed angles g relative to the sensory field center, innervate eight command neurons (inner circles). Two of the eight associated inhibitory neurons (grey) are shown (for clarity, only those corresponding to R3 and L2). Inhibitory and excitatory synapses are represented as open or darkened circles, respectively. This arrangement of neurons and interactions is hypothetical but consistent with other time-measuring circuits in vertebrates (Carr and Konishi, 1990). (After Stu¨rzl et al., 2000).
oesophageal ganglion (SOG) where fiber tracts from the eight legs converge (Babu and Barth, 1984; Babu, 1985; Gronenberg, 1989, 1990; Anton and Barth, 1993). Similarly, we know from behavioral observations of sand scorpions that vibration source localization requires integrative input from multiple receptors within the ring of eight, with receptors nearest and farthest away from the target having the greatest impact on accuracy (Brownell and Farley, 1979c). Figure 6 shows a diagrammatic representation of interneuronal circuitry that meets these requirements using conventional synaptic connectivity between a small set of interneurons. Central to its construction are eight excitatory command neurons and inhibitory partners which will evaluate the sensory input and calculate an appropriate motor output for rotation toward the target. Each BCSS, representing a direction gk,
with 1 # k # 8, excites its corresponding command neuron and the associated inhibitory interneuron k. The latter also receives excitation from BCSS gk21 and gk11 adjacent to gk, thus forming an inhibitory triad that will act with synaptic delay (DtI 5 0.7 msec) to block the action of the command neuron coding for direction gk´ opposite gk in the sensory field. Thus, for each command neuron encoding for turns in one direction (gk), the inhibitory triad antagonistic to it is centered and represented by k¯ 5 [(k 1 3), mod 8] 1 1. The circuitry of Figure 6 and Eq. 1 provides a foundation for computing differences in rate of firing for the command neurons from the differences in timing of sensory input they receive from the BCSS. For this computation the concept of a time window is introduced as the physical parameter that will control excitability of the command
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neurons in our model. For convenience of discussion, let us assume that R3 is closest to a wave source and that M sensory neurons associated with this BCSS can, but need not3, fire first as the Rayleigh wave passes beneath. Excitatory input from R3 opens a time window of increased excitability for its corresponding command neuron, and simultaneously activates the inhibitory triad that will suppress (after synaptic delay DtI 5 0.7 msec) activity of the command neuron for L2, opposite R3 in the array. As the Rayleigh wave passes through the sensory field, other BCSS are excited probabilistically as well, and these, in turn, excite their corresponding command neurons and associated inhibitory partners. Thus, in the general terms of our model, as the stimulus angle wS varies, the width of each command neuron’s time window is the order of 2Dt 1 DI where Dt is given by Eq. 14, DI 5 a7ms, and a negative width indicates that inhibition comes before excitation. For command neuron k opposite direction wS of the stimulus, inhibition will arrive before the excitation from its corresponding BCSS and there is low probability of that cell firing. On the other hand, if wS ø gk, excitatory input may trigger a spike before its opposing inhibitory triad can suppress it. Eventually, all time windows for increased rate of firing in the interneurons are reliably closed by action of the inhibitory triads. Representing time as a population vector code Since the currency of information transfer to motorneurons is a rate of spike discharge, the neural circuitry controlling the angle of turn must produce a rate code specifying target location. Conversion of time information to a rate code in our model is performed by means of a population vector, a representation commonly used in other motor system modeling applications 3 Two neurons behind each slit respond probabilistically to the passing Rayleigh wave. Thus, angles of turn computed by the model will be expressed as a probability density (see Fig. 7). 4 We take conduction delays from the slit sensilla to the SOG to be equal for all legs.
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(Georgopoulos et al., 1986; Salinas and Abbott, 1994; Lewis, 1999). The N 5 8 command neurons constitute the population that decides the direction of turn by taking the sum of the eight vectors corresponding to the directions of the eight legs, weighted by their relative rates of firing. For each command neuron k there is a spike rate nk roughly determined by Dt, the time window each passing Rayleigh wave opens and closes as it passes beneath BCSS k and k’s inhibitory triad, respectively. Using the complex number exp(igk) to represent directions on a two-dimensional surface, we define the population vector n exp(if) by
One N
ne if 5
k51
k
igk
⇒ f 5 arg
One N
k51
k
igk
(2)
with its argument f being the direction the animal perceives as the desired angle of turn. Since the nk are stochastic variables so too will be the population vector’s argument f 5 f(wS). Given the stimulus angle wS, its probability density P(f) is to be computed (Lamperti, 1998). Testing the model In constructing a realistic stimulus to present to our model, we are fortunate that the physical characteristics of Rayleigh waves in sand are known (Fig. 3; Aicher and Tautz 1990) and relatively simple to reconstruct in mathematical terms. Natural vibrational stimuli in sand vary in duration but even those of short duration, say TS 5 100 msec, are capable of generating accurate turning responses. We can synthesize a reasonable approximation of the natural signal by assuming a discrete Gaussian distribution of cosine waves with a mean of 300 Hz and a standard deviation of 50 Hz. Thus we have used y(t)/y0 5 100 [SkD(fk)cos(2pfkt 1 xk)]/ [SkD(fk)], fk 5 300 1 (k 2 150) Hz, where D(.) is a Gaussian with ^ f & 5 300 Hz, the standard deviation is 50 Hz, and the xk, 0 # k # 300, are independent equidistributed random variables. The sensory response of the BCSS is computed by applying the stimulus y(t) to each organ with the assumption that each
VIBRATION SENSORY SYSTEM slit responds independently5. Spike generation is then governed by an inhomogeneous Poisson process (Kempter et al., 1998) with density pF (t) 5
5
0 A ln[1 1 y(t)/y0 ]
for y(t) , 0 for y(t) $ 0
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model of Hodgkin and Huxley6 (Koch, 1999). Depending on neuronal potential V at time t, the resulting excitatory and inhibitory postsynaptic currents (E/IPSCs) give rise to the total input current I(V, t) 5
O [g a(t 2 t ; t )(V 2 V) E
t E
E
E
f
(3) where A 5 250 s interjects phase locking around 300 Hz. The logarithmic function follows from Weber’s law governing the generic relationship between stimulus intensity and sensory excitation (Dusenbery, 1992). An ‘‘inhomogeneous Poisson Process’’ with density pF means that (i) the probability of an event in an interval of duration Dt at time t equals pF(t)Dt, the inhomogeneity stemming from an explicit dependence of pF(t) upon the time t, (ii) the probability of two or more events during Dt is o(Dt), thus negligible as Dt becomes small, and (iii) events in disjoint intervals are independent. Finally, the input from the entire field of receptors, as stimulated by our ‘‘natural’’ signal, is applied to the model of interneuron circuitry proposed in Figure 6. In quantitative terms, the spikes generated by BCSS gk are ‘‘fed’’ to its command neuron as those of the triad centered at k¯ opposite gk inhibit the command neuron with synaptic delay DI (50.7 msec). We thus generate two independent Poisson processes of spikes labeled by f arriving at times tfE from gk and tfI from gk¯ with densities pF,E(t) 5 pF(t) and pF,I(t) 5u pF(t 1 Dt 2 DI). As before, Dt 5 Dt (gk, g¯lzwS) is given by (1) and ¯l is taken to be the member of the inhibitory triad of gk that fires first. These independent excitatory and inhibitory inputs are applied to command neurons using the well-known excitability 21
5 The sensory response is a train of stimulus-locked action potentials (cf. Fig. 4). BCSS spikes triggered by different amplitude maxima in the Rayleigh waves train are assumed to be phase-locked to the stimulus. They are taken to be independent and non-interacting events because of the sharp peak near 300 Hz for the vibrational power spectrum of sand (Brownell, 1979; Aicher and Tautz, 1990); cf. Fig. 3B).
1 gI a(t 2 t If ; tI )(VI 2 V)]
(4)
where VE 5 40 mV, VI 5 25 mV, while gE and gI are synaptic conductances. The postsynaptic potential a(t; tsyn) :5 (t/tsyn) exp[1 2 t/tsyn] vanishes for t , 0. Using time constants characteristic of arthropod neurons, we have tsyn 5 tE 5 tI 5 1 ms, sufficiently less than the oscillation period of ;3 ms for Rayleigh waves at 300–350 Hz. These computed currents are applied to the command neurons controlling angle of turn and the sum of their activities (Eq. 2) yields the computed angle of turn f(wS). Figure 7 presents a graphical display of the probability density P(f) computed from our theoretical model superimposed on experimental data obtained from scorpions. In addition to predicting the turning behavior of normal animals with intact sensory field, the model accounts rather well for the altered behavior of animals with partially ablated sensory fields (Brownell and Farley, 1979a; Brownell, 1984). The dark shadings of Figure 7 also encompass the intrinsic scatter observed in behavioral data. Moreover, the importance of incorporating a triad of receptors for the antagonistic inhibitory input of our model, rather than a singlet, can be seen in the inadequacy of the latter to predict the observed behavior (see, e.g., Fig. 7e, f). Optimizing model performance Throughout development of this model, care has been taken to incorporate the probabilistic nature of ‘‘real world’’ sensory processing (e.g., stimulus waveform variance, probabilistic nature of sensory spike generation) and to minimize complexity of 6 The parameters are the canonical values, with T 5 188C. Given the E/IPSCs chosen, it was found that a command neuron did not fire as long as 20.517 msec , tE 2 tI , 1.107 msec where tE and tI are the arrival times of the ‘‘excitatory’’ and ‘‘inhibitory’’ spike.
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FIG. 7. Comparison of model predictions with observed turning behavior of Paruroctonus. In each panel, the response angle f of turn (ordinate) is plotted as a function of the stimulus angle wS (abscissa). (A) the response of an animal with fully intact sensory field shows fairly accurate orientation (6158) at all angles of target presentation. (B)–(F) Animals with BCSS ablated on some legs (dots) show systematic errors in turning accuracy. Both the probability density P(f) (dark shadings) and experimental points (dots or triangles, taken from Brownell and Farley, 1979c) are indicated. When the inhibitory triad is replaced by a single inhibitory neuron in the model, the mean response (dashed line) gives less satisfying fit to the observed behavior. (From Stu¨rzl et al., 2000).
the circuitry required to account for the observed behavior. We now assess the impact of some of our assumptions on model performance. The influence that the number N of command neurons has on precision of the computed angle of turn can be seen in the standard deviation of response, sN, expressed as
sf2 ø
8sk2 N(^n k &max 2 ^nk &min ) 2
(5)
Here 1nk,max and 1nk,min are the maximum and minimum number of spikes computed for command neurons as a function of Dt, and s2k 5 Var (nk) (Stu¨rzl et al., 2000). Figure 8 shows that for M 5 2 sensory neu-
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FIG. 8. Standard deviation sf of the turning angle f as a function of the conductance gE appearing in Eq. 4 for fixed ratios x 5 gI/gE. For 1 # x # 3 the mean error sf shows a broad minimum with center near the dashed line. Stochastic optimization is realized through tuning the synaptic efficacies gI and gE. The values of sF near the minima agree with the scatter of the behavioral data shown in Figure 7. (From Stu¨rzl et al., 2000).
rons/command neurons and N 5 8 a precision of turning within 13–15 degrees error is easily obtained. This is within range of the observed natural behavior (Fig. 2) indicating that the number of sensory and interneuronal elements required to execute the behavior can be minimized. Likewise, noise is inherent in the integrated response of command neurons assessing target direction since BCSS sensory neurons are excited probabilistically by a variable waveform, and the firing times, though phase-locked to the Rayleigh wave, are not precisely predictable. Under these circumstances it is natural to ask if plasticity of synaptic processing within the model can be tuned to minimize the effects of noise on accuracy, i.e., by ‘‘Stochastic Optimization,’’ a reinterpretation (Stu¨rzl et al., 2000) of stochastic resonance (Wiesenfeld and Moss, 1995). To answer this question we have varied the synaptic conductance gE in (5) for various fixed ratios of inhibitory vs. excitatory synaptic effectiveness (gI/gE). The broad minima seen in Figure 8 for the standard deviation of targeting accuracy (sN) at intermediate values of gI/gE strongly suggest that the system is optimized for precision, presumably through evolutionary adaptation (Grothe, 2000; Jaramillo and Wiesenfeld, 1998).
CONCLUSIONS The results summarized here from behavioral, sensory, and computational analysis of vibration source localizing behavior of sand scorpions all point to an underlying simplicity in the mechanisms animals may use for perception of vibrational space. Perhaps these mechanisms were particularly evident in sand scorpions because of the economy of sensory structure and function involved, the clean nature of the vibrational signals detected, and the accessibility of orientational behaviors for experimental analysis. Consistent with this biology, the computational theory presented here uses a few-neuron model and conventional synaptic interactions to generate one of the simplest neuronal maps yet described for processing of spatio-temporal information from the outside world (van Hemmen, 2001). Moreover, the excellent agreement between predictions of theory and behavioral observations suggest that the neuronal mechanisms for perception of vibrational space will to some extent be similar to those used for localization in auditory space. Finally, we have shown that stochastic optimization can be a driving force for evolution of ever more accurate perception of source location, simply by tuning the relative effectiveness of excitatory and inhibi-
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