Omnidirectional Sensory and Motor Volumes in Electric Fish
[['The motor volume is the swept volume of positions a body occupies for a given trajectory or set of trajectories .', 'It is a function of the way the body moves , as well as the geometric extent of the body .', 'To define it more clearly , we first present a definition of the time-limited reachable set .', 'We then use this definition to informally define the motor volume and the stopping motor volume ( the swept volume of the body over trajectories that bring the body to a halt ) , with the precise definitions presented in Text S1 .', 'For simplicity of description , we treat an organism as a rigid 3-D body .', 'We define the configuration space as the six-dimensional space representing the rigid-body degrees of freedom ( typically the ( x , y , z ) position of the center of mass , and θ , ϕ , and Ω , the yaw , pitch , and roll angles , respectively ) .', 'The state space X consists of the six configuration components and their time rates of change ( vx , vy , vz , vθ , vϕ , vΩ ) , resulting in a total of twelve dimensions .', 'The dynamics of the system are given by', 'where x is the instantaneous state and u is the instantaneous control input from the space', 'U℘ of all feasible instantaneous control inputs .', 'The time-limited reachable set R ( x0 , T ) is a construct from nonlinear control system theory [14–16] referring to all points in the state space X that can be reached by a system of the form given by Equation 1 from an initial state x0 given any feasible control history u of duration T . A feasible control history is a control input to the system as a function of time , such as muscle activations for a musculoskeletal system or steering wheel angles for a car .', 'The reachable set is defined as:', 'This is the set of states reachable in time exactly T . In our subsequent definitions , we will use the union of all reachable sets from t = 0 to t = T , denoted as:', 'To illustrate the concept of the time-limited reachable set , consider the simple one-dimensional case of a locomotive moving along a train track ( Figure 4A ) .', "The state space X is simply the locomotive's configuration space ( the x position of its center of mass ) , as well as the time rate of change of this one configuration component ( vx ) .", 'The initial condition x0= ( 0 m , 5 m/s ) .', 'The control inputs are limited to the set', 'U℘ = [–1 m/s2 , +1 m/s2] .', 'Figure 4A shows two dashed curves representing state space ( x , vx ) trajectories under constant acceleration ( +1 m/s2 ) and deceleration ( −1 m/s2 ) , starting from an initial velocity of 5 m/s and running for 5 s; all time-limited reachable sets of 5 s or less from this initial condition must be within these two curves .', 'For example , for t = 3 s , the time-limited reachable set R [ ( 0 m , 5 m/s ) , ≤ 3 s] is the union of sectors “1” through “3 . ”', 'Figure 4B shows time-limited reachable sets R [ ( 0 m , 0 m/s ) , ≤ 5 s] —identical conditions as for Figure 4A but with zero initial velocity .', 'For organisms with internal degrees of freedom ( multi–rigid-body systems with joints or flexible bodies ) , the same concepts apply but now the state space needs to include the additional degrees of freedom , and their time rates of change .', 'The motor volume ( Text S1 , Equation S1 ) is similar to the reachable set but with several important differences .', 'First , rather than a set of points in state space , the motor volume is the volume defined by the set of ( x , y , z ) coordinates of all positions occupied by the body over the time interval of interest .', 'Second , these points are in a coordinate frame that is aligned to the body at the onset of the behavior for which the motor volume is being constructed .', 'For the locomotive example , the thick green line between x = −2 and x = 38 in Figure 4A represents the motor volume ( more correctly , the motor “line” in this case ) MV ( 5 m/s , ≤ 5 s ) .', 'As defined in Text S1 , the motor volume is derived from the time-limited reachable set , where this set is computed from Equations 1 and 3 [17 , 18] .', 'However , in our case , we do not have access to either the equation of motion ( Equation 1 ) for the fish or to the set of feasible control histories; thus , we will estimate the motor volume empirically by examining motion capture data collected from over 100 prey capture sequences ( see Methods ) .', 'In this paper , we will consider the motor volume of the fish over all initial velocities that are typical within our prey capture sequence dataset .', 'Prey capture sequences were selected to begin about 0 . 5 s before the onset of behavioral response and end at prey engulfment [11] , and thus contain both pre- and post-detection behavior .', 'The stopping motor volume ( Text S1 , Equation S2 ) designates the set of all ( x , y , z ) coordinates of all positions occupied by the body from a given initial velocity to the location at which the body can be halted in the shortest amount of time .', 'For the locomotive example , MVstop ( 5 m/s ) is equal to MV ( 5 m/s , ≤ 5 s ) with control inputs limited to the set', 'U℘ = [–1 m/s2] , and is shown by the red line in Figure 4A .', 'If the train had started at rest , then MVstop would collapse to a single point as illustrated by the red dot in Figure 4B .', 'In this paper , MVstop will be ascertained through analysis of motion capture data ( see Methods ) ; if a mechanical model exists , MVstop can be estimated by computing optimal stopping trajectories from each initial velocity .', 'The sensory volume ( SV ) for a given object is the volume defined by the set of ( x , y , z ) coordinates at which that object can be reliably detected , in body-fixed or sensory system–fixed coordinates .', 'The SV depends on a number of factors , such as target size , orientation , velocity , properties of the sensory system , properties of the detection algorithm , and so on .', 'In this paper , we will estimate SV ( prey ) for the active electrosensory detection of small water fleas ( D . magna ) .', 'With these definitions in hand , we can explore possible functional relationships between sensory and motor volumes .', 'Restricting our consideration to the stopping motor volume , consider Figure 5 , which shows three possible relationships between sensory and stopping volume geometries .', 'In Figure 5A ( “collision mode” ) , the sensory volume is smaller than the stopping volume .', 'We hypothesize that this situation should rarely occur in nature when the SV is for objects that the animal needs to avoid colliding with .', 'Anywhere that MVstop is not covered by the SV represents a region in which unintended collisions could occur .', 'Figure 5B ( “reactive mode” ) illustrates a situation in which the SV and MVstop are fully overlapping .', 'We hypothesize that this condition represents a functional lower bound on the size and shape of the SV , when the SV is computed for obstacles that the animal should avoid .', "Also , if the animal's prey capture strategy demands that it come to full stop to consume or “handle” the prey , then the reactive mode can be considered a lower bound on the size and shape of the SV for prey capture .", 'Finally , Figure 5C ( “deliberative mode” ) illustrates the case in which the SV is much larger than , and fully encloses MVstop .', 'This case is typical for visually guided predators in terrestrial environments hunting in full daylight , where visual range can far exceed stopping distance .', 'In the context of foraging behavior , the deliberative mode would generally lead to higher foraging efficiency .', 'However , for active-sensing systems , the sensing range could be limited by factors such as energetic costs , clutter , and quartic attenuation , among others , leading to a situation closer to the reactive mode ( Figure 5B ) than the deliberative mode ( Figure 5C ) .', 'If these constraining factors are particularly severe , then we predict that the SV should approximate MVstop , as in the reactive mode .', 'Finally , we hypothesize that if the SV and MVstop approximately match , then the animal may make behavioral adjustments to either the sensory volume ( e . g . , by changing the probe intensity ) or to the stopping volume ( e . g . , by changing their “cruising” velocity ) to maintain a matched relationship between the SV and the MVstop , particularly where the absence of such adjustments would bring the animal into collision mode .', 'Here we examine SV–MV relationships in the context of prey capture behavior of the black ghost knifefish .', 'From earlier behavioral studies , we know that the fish comes to a near halt before engulfing prey [11] .', 'Thus , we predict that SV ( prey ) should fully enclose MVstop over the velocities that characterize pre-detection swimming .', 'Our results will show that the relationship between SV ( prey ) and MVstop for the black ghost is best described by the reactive mode shown in Figure 5B .', 'For this mode , we predict that the animal might use behavioral adjustments to maintain a match between SV and MVstop .', 'We test this hypothesis by reanalyzing results of our prior work with prey capture behavior in water of different electrical conductivities , which influences the distance at which prey can be detected [11] .', 'The evidence suggests that as the SV ( prey ) shrinks in size due to increased water conductivity , the fish decreases its mean prey search velocity , which causes a corresponding decrease in the size of MVstop .'], ['Using synthetic prey capture trajectories , the sensory volume for active electrosensory prey detection in A . albifrons was estimated by computing a detection isosurface surrounding the fish , such that every point on the surface generates a threshold level of activation after summing and filtering the electrosensory afferent signals ( see Methods ) .', 'Every point contained within this bounding isosurface would yield a suprathreshold signal .', 'The estimated SV was found to be omnidirectional ( Figure 3 ) , extending in all directions from the body surface .', "On average , the sensory surface was 34 ± 5 mm ( N = 7 , 056 detection points , all numbers quoted as mean ± standard deviation unless otherwise noted ) from the fish's body surface .", 'As described in more detail below , this is also consistent with the empirically determined Daphnia detection distance of 28 ± 8 mm ( N = 38 ) reported in an earlier behavioral study [19] once the sensorimotor delay time is factored in .', 'In addition to using synthetically generated linear prey capture trajectories , we also tested detection performance using empirically measured fish and prey trajectories from a previous study , which have more complex spatiotemporal profiles [11] .', 'The computationally estimated detection distance was 33 ± 7 mm ( N = 38 trials × 10 repeats , or 380 , see Methods ) , compared with the measured detection distance of 28 ± 8 mm ( N = 38 trials ) .', 'However , the latter empirical value is actually the “reactive distance” at which a motor response is first observed and does not include the sensorimotor delay between detection and movement .', 'The estimated sensorimotor delay for behavioral reactions in the knifefish is 115 ms ( see Methods ) .', 'Incorporating the sensorimotor delay , we obtain an estimate for the “neural” detection distance in the empirical data of 35 ± 9 mm ( N = 38 ) , which is in good agreement with the simulation result of 33 ± 7 mm for these trajectories .', 'We examined the motor volume as defined above for the entire body , MVbody as well as the motor volume for the mouth alone , MVmouth , at fourteen discrete times , ranging from 117 to 700 ms . Motor volumes at three of these time points are illustrated in Figure 6 .', 'The motor volume that maximally overlaps the SV ( t = 432 ms ) is shown in interactive 3-D Figure S4 along with the SV .', 'Because of the unusually high maneuverability of the fish , including its backward-swimming capability , the shape of the body motor volume is omnidirectional and approximately cylindrical on short time scales , extending both in front of the head and behind the tail of the fish .', 'The size and shape MVmouth , while also omnidirectional , is more compact in the rostrocaudal direction .', 'Figure 7 shows the relationship between the SV and the stopping motor volume .', 'The SV fully encloses MVstop except for a rostral protrusion .'], ['In the present study , we find that the time-limited motor volumes for A . albifrons are omnidirectional and extend equally in front and behind the fish ( Figure 6; interactive 3-D version Figures S2 , S3 , and S5 ) .', 'This confirms the previously reported propensity of weakly electric knifefish to spend a significant fraction of time swimming backward at velocities comparable to their forward-swimming velocities [13] .', 'The motor volume is found to be cylindrical , indicating that the fish has considerable lateral and dorsoventral mobility [11 , 12] .', "The time-limited motor volume introduced in this study provides a quantitative measure of the fish's motor capabilities as a function of time interval .", 'The body motor volume was closer in shape to the sensory volume and exhibited greater overlap than the mouth motor volume ( Figure 6 ) .', 'Functionally , the mouth motor volume is more closely associated with prey capture behavior , whereas the body volume may be more relevant to obstacle avoidance and general navigation in complex environments .', 'The sensory volume seems better matched to the body motor volume , which suggests that general navigational capabilities , habitat complexity , and obstacle avoidance should all be considered when examining relationships between sensory and motor volumes in the context of prey capture behavior .', 'We find that the active sensory volume for prey detection is also omnidirectional and approximately cylindrical ( Figure 3 ) .', 'The omnidirectional shape of the SV is similar to the shape of the isopotential surfaces of the self-generated electric field surrounding the fish ( Figure 1A ) , but exhibits less of a bulge in the caudal tail region due to anisotropies in the sensor density ( Figure 1B ) , field intensity ( Figure 1A ) , reduced sensory surface area due to the tapering body morphology ( Figure 1B ) , and the sensitivity of the primary electrosensory afferents to prey-induced perturbations [20 , 21] .', 'The relative importance of these factors in determining the precise shape of the sensory volume was not examined .', 'It will be particularly interesting in future studies to explore the extent to which the higher electroreceptor density in the head region of the fish influences prey-detection distance versus spatial resolution of prey position .', 'The simulations of prey detection in this study combined quantitative models of each of these factors in order to arrive at an estimated sensory volume for electrosensory-mediated prey detection .', 'As shown in Figure 2 , there was good agreement between the computationally estimated sensory volume and the empirical distribution of prey detections found in an earlier study [11] .', 'The empirical study had relatively low N ( 38 prey capture events for the water conductivity that yielded maximum detection range ) , and the detection points were biased toward the dorsal surface of the fish , because prey were introduced into the tank from above .', "The computational approach used here allowed us to obtain a more complete and less biased estimate of the fish's sensory detection volume .", 'Our prediction that the fish avoids collision mode is supported by Figure 7 ( interactive 3-D version Figure S6 ) , which shows nearly full enclosure of the stopping volume by the sensory volume .', 'The stopping volume was taken from just before detection ( which always occurred during forward movement ) to the point of zero forward velocity as the fish reverses to capture the prey .', 'Thus , unlike the time-limited motor volume , which sampled all initial velocities including negative velocities , the stopping volume is more strongly biased in the forward direction .', 'However , the extent of the sensing range is restricted to nearly the lower limit of the reactive mode ( Figure 5B ) where the SV and MV are matched .', 'We expect this is due to constraints that include the metabolic cost for emitting energy into the environment and the interference caused by interaction between the emitted signal and nearby clutter .', 'Energetic costs scale steeply with sensing range , approximately as a quartic power of the range due to geometric spreading effects on the outbound and return paths of active probe signal [1 , 3] .', 'For quartic scaling , doubling the active-sensing range would require a 16-fold increase in emitted energy .', 'To appreciate the effect of this scaling , consider that an active 15-g A . albifrons requires approximately 300 J/day [22] .', 'If we assume that ≈1% is used for the field , as estimated for another weakly electric fish [23] , then 3 J/day is needed for the field .', 'To double the detection distance for D . magna from the measured ≈30 mm to ≈60 mm would therefore require 48 J/day for the field; to double this again to ≈120 mm ( still less than one body length for a 15-g fish ) would then require 768 J/day , more than double the entire energy budget of the fish .', 'Thus , the high energetic costs associated with extending the active-sensing range is likely to place strong selective pressure on the shape and extent of the active sensory volume .', 'In comparison , the high acuity passive visual system of a typical raptor allows it to spot prey from over a kilometer away , or about 10 , 000 body lengths .', 'Although it is more difficult to provide a quantitative metric for the interference effects from clutter , the great reduction in emission power observed in dolphins and bats when surrounded by clutter or when nearing a target ( see Introduction ) suggests that this may also be an important factor in limiting the desirable range of an emitted signal .', 'Returning to the automobile scenario , driving at night in a dense fog provides a practical example of where backscatter is a limiting constraint .', 'Increasing headlight intensity under these conditions ( e . g . , switching from low beams to high beams ) can actually degrade detection performance because the “noise” from backscattered light makes it more difficult to detect the “signal” that is reflected back from a target of interest .', 'Given that the black ghost is in reactive mode , we predict that the fish may make behavioral adjustments to either the sensory volume or to the stopping volume to maintain a matched relationship between the SV and the MVstop , particularly where the absence of such adjustments would bring the animal into collision mode .', 'We are able to qualitatively evaluate this hypothesis by examining how search ( predetection ) swimming velocity varies with water conductivity .', 'We have shown that water conductivity changes the range at which prey are detected [11] .', 'In the previous study [11] , we found that the mean detection distance decreased from 28 mm at a water conductivity of 35 μS/cm to 12 mm at 600 μS/cm .', "Over this conductivity range , the fish's predetection swimming velocity decreased 30% from 99 mm/s to 71 mm/s .", 'At the shorter detection distances associated with higher conductivity water ( 600 μS/cm ) , we have previously estimated that multiple sensory modalities , including passive electrosense and lateral line mechanosense , are playing a role [24] .', 'Thus , quantitative evaluation of the matching hypothesis would require modeling the SV for these other sensory modalities , which is outside the scope of this study .', 'The size of the estimated sensory volume , and to a lesser degree the shape of the volume , are influenced by the properties of the neural detection algorithm .', 'A more sensitive algorithm will result in a larger detection range .', "The detection algorithm used here is not intended to model the fish's actual detection performance in detail .", 'Doing so would require a more extensive analysis of additional factors , such as sensory reafference associated with tail bending , environmental background noise , contributions of other sensory modalities , neuroanatomical constraints on sensory convergence , etc .', 'Rather , the detection volume reconstructed here is intended as an estimate of “best-case” detection performance based solely on changes in active electrosensory afferent spike activity .', 'Echolocating bats emit ultrasonic energy into the environment to detect prey [25] .', "While the precise size and shape of the bat's sensory volume will vary with many factors ( species , call intensity , duration , etc . ) , the sensory volume for echolocation is generally a cone that extends in front of the head of the bat with an angular range of approximately ±30° in azimuth and elevation [26] .", 'The angular coverage may extend as much as ±75° relative to the body axis when head and pinnae movements are included [25 , 27 , 28] .', 'For the detection of flying insects by pipistrelle bats , the sensory volume extends at least 100–200 cm in front of the animal [25 , 29] based on the reactive distance to prey .', "The detailed shape of the bat's motor volume has not been reported .", 'The stopping distance can be estimated by combining information on the initial velocity of the bat , maximal deceleration , and sensorimotor time delay .', 'Taking a representative bat cruising velocity of 5 m/s [25] , a maximal deceleration of 15 m/s2 ( estimated from a sample trajectory in [25] ) , and an estimated sensorimotor delay of 100–200 ms [30] yields an estimated stopping distance in the range of 130–180 cm .', 'Although there is a great deal of uncertainty in these estimates , the stopping distance of the bat seems comparable to the sensory range for prey detection .', 'This suggests that bats , like electric fish , have an active sensory volume for prey detection that may be comparable to their stopping volume .', 'Quantitative comparisons of sensory and motor volumes for a single bat species would help clarify these relationships .', 'Odontocetes ( toothed whales , dolphins , and porpoises ) also use ultrasonic energy for prey detection .', 'Dolphins can detect prey-sized objects at distances on the order of 100 m [6 , 31] .', 'The 100-m sensing range of dolphins is certain to be significantly beyond their stopping volume , although there is little published data with which to make quantitative comparisons .', 'This suggests that energetic and clutter-related constraints on active sensing may not be as significant for dolphins as they are for bats and electric fish .', 'Both the active electrosensory volume for prey detection and the time-limited motor volume of A . albifrons are omnidirectional and approximately cylindrical .', 'This is in striking contrast to most other animals , which tend to exhibit a strong forward bias in both sensory and motor volumes .', 'This forward bias is observed for passive-sensing systems such as visually guided fish [32–37] .', 'Figure 8 ( interactive 3-D version Figure S7 ) compares the omnidirectional prey sensory volume in the black ghost knifefish ( A . albifrons ) with a more typical forward-biased passive sensory volume .', 'The latter is illustrated by the volume for visually mediated prey detection in the stone moroko ( Pseudorasbora parva ) , a fish of comparable size that also feeds on Daphnia [36] .', 'The angular coverage of the visual volume is approximately 100° in azimuth and 60° in elevation , with a range that varies from about 60–120 mm , depending on environmental conditions [36]; the 120-mm range is shown in Figure 8 .', 'The active electrosensory volume of A . albifrons and the passive visual volume of P . parva for prey detection are similar in size ( approximately 1 , 000 cm3 each ) but quite different in shape .', 'We propose that the ratio of the sensory volume to stopping volume ( SV:MVstop ) is a useful metric for both active and passive sensory systems when considering whether sensorimotor control systems are in collision , reactive , or deliberative mode .', 'Collision mode occurs when the ratio is below unity .', 'Reactive mode occurs when the ratio approximates unity , as appears to be the case for knifefish and bats .', 'In this mode , sensorimotor control algorithms are likely to be reactive , with relatively fast , direct coupling between sensation and action .', 'Movement options are largely conditioned by mechanical considerations such as inertia and minimal turning radius .', 'For example , some sensor-based motion planning algorithms in robotics are based on estimating the stopping volume for the nearest obstacle; as the robot becomes more massive , the range of any active-sensing system for obstacle detection must be extended accordingly [38] .', 'Deliberative mode occurs when the ratio is large , as for dolphin echolocation and for many passive visual and auditory systems .', 'In this mode , an animal can acquire sensory data from targets that are far outside its stopping volume .', 'This allows the animal a greater range of movement options , because there is adequate time for complex motion planning before reaching the target [39] .', 'For example , in the context of prey capture behavior , a dolphin with a high SV:MVstop ratio is able to engage in long-range tracking of distant prey , whereas a weakly electric fish with a ratio near unity must use a reactive strategy for chance encounters with nearby prey .', 'Quantitative analyses of sensory and motor volumes for both active and passive-sensing systems can highlight important functional relationships between sensing , movement , and behavioral control in animals .'], ['In a previously published study [11] , adult weakly electric fish ( A . albifrons ) were videotaped in a light-tight enclosure under infrared illumination .', 'Individual water fleas ( D . magna , 2–3 mm in length ) were introduced near the water surface and drifted downward; prey capture behavior was recorded using a pair of video cameras oriented along orthogonal axes .', "Relative to the fish's velocity ( ∼100 mm/s ) the prey were relatively stationary ( prey velocity < 20 mm/s ) .", 'Prey capture events ( from shortly before detection to capture ) were subsequently digitized , and 3-D motion trajectories of the fish surface and prey were obtained using a model-based tracking system with a spatial resolution of 0 . 5 mm and a temporal resolution of 1/60 s [40] .', 'The time of prey detection was defined by the onset of an abrupt longitudinal deceleration as the fish reversed swimming direction to capture the prey [11] .', 'These reversals are characteristic of most prey capture encounters .', 'This is related to the fact that the fish tends to swim forward with its head pitched downward , such that the dorsum forms the leading edge as the fish moves through the water .', 'Initial prey encounters thus tend to be uniformly distributed along the entire length of the body , so a reversal of swimming direction is typically required to intercept the prey .', 'The volume of space supporting prey detection by the active electric sense was estimated computationally using measurements and empirically constrained models of the prey , electric field , fish body and sensor distribution , electrosensory images , afferent firing dynamics , and behavior .', 'Model parameter values and their sources are summarized in Table 1 .', 'Electric field .', 'The electric field vector Efish ( mV/cm ) at a 3-D point in space x was computed using an empirically tested model of the electric field [41] .', 'This model sums the individual contributions to the field from each of a series of charged poles used to model the electric organ of the fish:', 'where x is a point in space ( cm ) ,', 'is the location of pole i of n total poles , q is a normalization constant ( mV cm ) that scales the overall magnitude of the field , σmes is the conductivity of the water that the field measurements were performed in ( which establishes the q value ) , and σmod is the conductivity of the water for the simulated field .', 'The quantity q is analogous to electric charge in an electrostatic model and is distributed such that the first m poles have a “charge” of q/m and the remaining poles have a charge of –q/ ( n –', 'm ) , resulting in a total net charge of zero .', 'For our simulations , n = 267 , m = 266 ( all but one pole at the tail was positive ) , q = 10 , and the pole locations xp ran from the nose to the tail of the fish along the central axis of the fish body with equidistant spacing .', 'These values resulted in field values within 10% of measurements of the electric field vector Efish of A . albifrons obtained by other researchers ( B . Rasnow , C . Assad , P . Stoddard , unpublished data ) in water of conductivity σmes = 210 μS/cm using a multiaxis electrode array [20 , 42] ( Figure 1A ) .', 'The', 'term scales the field strength to the water conductivity used in simulation σmod = 35 μS/cm .', 'This scaling is based on empirical measurements of the knifefish field at different water conductivities [43] , which suggest the electric organ can be idealized as constant current source .', "We selected 35 μS/cm because our earlier study [11] found that the detection range was highest for trials at this conductivity , and this conductivity is most similar to rivers of the fish's native habitat .", 'Body model with electroreceptor distribution .', 'We used a prior survey [44] of the density of probability type ( P-type ) tuberous electroreceptor organs ( hereafter electroreceptors ) on the surface of A . albifrons .', 'These are the dominant electroreceptor type for A . albifrons [45 , 46] .', 'Each electroreceptor is connected uniquely to a primary afferent , which generates action potentials with a probability that varies with stimulus intensity .', 'The receptor locations were mapped onto a high-resolution polygonal model of the fish derived from a 3-D scan of a body cast [19 , 24] in accordance with the measured sensor densities [44] ( Figure 1B ) .', 'This resulted in a total of 13 , 857 mapped electroreceptors , in close agreement with neuroanatomically derived counts from A . albifrons [44] .', 'Prey model .', 'Based on prior measurements of live prey ( D . magna ) , it was modeled as a 1 . 5-mm-radius conductive sphere with an electrical conductance of σobj = 300 μS/cm [19 , 24] .', 'Idealizing the prey as a sphere allows us to use an analytical model for the stimulus caused by the prey , described below .', 'Electrosensory image formation .', 'The voltage perturbation Δϕ ( mV ) at an electroreceptor on the fish surface , arising from a small spherical target object , was computed using an empirically tested model [47]:', 'where Efish ( mV/cm ) is the electric field vector at the prey , r ( cm ) is the vector from the center of the spherical object to the electroreceptor on the fish surface , a is the radius of the sphere ( cm ) , σobj is the conductivity of the sphere , and σw is the conductivity of the water ( μS/cm ) .', 'Simulations were run with water conductivity σw set to 35 μS/cm ( see Methods , Electric field ) .', 'Primary afferent spiking activity .', 'Computed voltage perturbations at each sensory receptor on the fish body were transformed into primary afferent spiking activity using a previously published adaptive threshold model of P-type ( probability coding ) primary electrosensory afferent response dynamics [48] .', 'This model gives rise to negative correlations in the interspike interval ( ISI ) sequence , which lead to long-term spike train regularization .', 'This correlation structure has been shown to increase information transfer and improve detection performance for weak signals [48–50] .', 'The electric organ discharge ( EOD ) frequency was taken as 1 kHz , which is typical of A . albifrons [51] .', 'P-type afferents fire at most one spike per EOD cycle .', 'Thus , afferent activity was modeled as a binary spike train with a sampling period equal to the EOD period ( 1 ms ) .', 'On each EOD cycle', '( n ) , the following update rules are evaluated in order:', 'Equation 6 implements low-pass filtering of the voltage perturbation Δϕ[n] with gain β and time constant τm .', 'The state variable u[n] can be conceptualized as a membrane potential; it is initialized to 0 , corresponding to the steady-state value with no stimulus present ( Δϕ = 0 ) .', 'Equation 7 adds random noise to u[n] to creat
['Introduction', 'Results', 'Discussion', 'Materials and Methods', 'Supporting Information']
['Active sensing organisms , such as bats , dolphins , and weakly electric fish , generate a 3-D space for active sensation by emitting self-generated energy into the environment .', 'For a weakly electric fish , we demonstrate that the electrosensory space for prey detection has an unusual , omnidirectional shape .', "We compare this sensory volume with the animal's motor volume—the volume swept out by the body over selected time intervals and over the time it takes to come to a stop from typical hunting velocities .", "We find that the motor volume has a similar omnidirectional shape , which can be attributed to the fish's backward-swimming capabilities and body dynamics .", 'We assessed the electrosensory space for prey detection by analyzing simulated changes in spiking activity of primary electrosensory afferents during empirically measured and synthetic prey capture trials .', "The animal's motor volume was reconstructed from video recordings of body motion during prey capture behavior .", 'Our results suggest that in weakly electric fish , there is a close connection between the shape of the sensory and motor volumes .', 'We consider three general spatial relationships between 3-D sensory and motor volumes in active and passive-sensing animals , and we examine hypotheses about these relationships in the context of the volumes we quantify for weakly electric fish .', 'We propose that the ratio of the sensory volume to the motor volume provides insight into behavioral control strategies across all animals .']
['Most animals , including humans , have sensory and motor capabilities that are biased in the forward direction .', 'The black ghost knifefish , a nocturnal , weakly electric fish from the Amazon , is an interesting exception to this general rule .', 'We demonstrate that these fish have sensing and motor capabilities that are omnidirectional .', "By combining video analysis of prey capture trajectories with computational modeling of the fish's electrosensory capabilities , we were able to quantify and compare the 3-D volumes for sensation and movement .", 'We found that the volume in which prey are detected is similar in size to the volume needed by the fish to stop .', 'We suggest that this coupling may arise from constraints that the animal faces when using self-generated energy to probe its environment .', "This is similar to the way in which the angular coverage and range of an automobile's headlights are designed to match certain motion characteristics of the vehicle , such as its typical cruising speed , turning angle , and stopping distance .", 'We suggest that the degree of overlap between sensory and movement volumes can provide insight into the types of control strategies that are best suited for guiding behavior .']
['neuroscience', 'computational biology']
Most animals have sensory and motor capabilities that are biased in the forward direction. The black ghost, a nocturnal weakly electric fish, is an exception to this general rule. We demonstrate that these fish have omnidirectional sensing and motor capabilities.