| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
1Department of Neuroscience, Baylor College of Medicine, Houston, Texas; 2Department of Zoology, Cambridge University, Cambridge, United Kingdom; 3Department of Bioengineering, Imperial College London, London, United Kingdom; and 4Computational and Applied Mathematics, Rice University, Houston, Texas
Submitted 23 June 2006; accepted in final form 26 September 2006
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
; Koch and Segev 2000; London and Häusser 2005; Migliore and Shepherd 2005; Stuart et al. 1999). Although most studies are conducted in vitro, several systems, notably insects, have emerged as viable in vivo preparations to study single-neuron computation (Borst and Haag 2002; Jacobs and Theunissen 2000; Ogawa et al. 2006; Pollack 2000). Among these, the locust lobula giant movement detector (LGMD) is particularly interesting because it has been proposed as a model for multiplicative computation (Gabbiani et al. 2002; for review, Gabbiani et al. 2004).
The LGMD is a large, identified neuron located in the third neuropile of the locust optic lobe (O'Shea and Williams 1974). Anatomical and electrophysiological evidence suggests that it receives excitatory retinotopic inputs from the locust eye sensitive to motion (Rowell et al. 1977; Strausfeld and Nässel 1981). The neuron is known to respond most strongly to objects approaching on a collision course with the animal (e.g., Guest and Gray 2006; Hatsopoulos et al. 1995; Rind andSimmons 1992; Schlotterer 1977), although it also exhibits less robust responses to other types of moving stimuli (Krapp and Gabbiani 2005; Rowell 1971; Simmons and Rind 1992). In the protocerebrum, the LGMD contacts the descending contralateral movement detector neuron (DCMD) neuron via a powerful synaptic connection (Killmann and Schürmann 1985; Killmann et al. 1999; O'Shea and Rowell 1975; Rind 1984). The DCMD fires in a 1:1 fashion with the LGMD and projects to thoracic motor centers, synapsing onto neurons involved in the generation of jumps and flight steering maneuvers (Burrows 1996; O'Shea et al. 1974; Simmons 1980). For this reason, the LGMD-DCMD circuit is believed to be involved in mediating collision avoidance and escape (Burrows and Rowell 1973; Gray et al. 2001; O'Shea et al. 1974; Rowell 1971; Santer et al. 2006; Schlotterer 1977). Evidence suggests that its role in collision avoidance involves detecting an angular size threshold as objects approach on a collision course (Gabbiani et al. 1999, 2002; Matheson et al. 2004). This computation is thought to be implemented by multiplying excitatory inputs sensitive to motion with inhibitory inputs sensitive to object size (Gabbiani et al. 2002).
Although previous work has treated the LGMD as a point neuron (Edwards 1982; Rind and Bramwell 1996), recent results suggest it may be electrotonically extensive (Gabbiani and Krapp 2006; Gabbiani et al. 2001; Krapp and Gabbiani 2005). Understanding the electrotonic structure of the LGMD is an important first step in elucidating the mechanisms of dendritic integration underlying its sensitivity to looming objects. For an electrotonically extensive neuron, the synaptic mapping of the neuron's inputs onto its dendritic compartments will be integral in shaping its response properties (Segev et al. 1995). This is especially interesting in light of recent work demonstrating the sampling of visual space by the locust eye to be highly anisotropic (Krapp and Gabbiani 2005) because the LGMD is believed to receive two inputs from each ommatidium (facet) on the eye in a retinotopically ordered manner (Rowell et al. 1977; Strausfeld and Nässel 1981). Based on this, it has been proposed that the interplay between electrotonic structure and synaptic connectivity of inputs from the eye may underlie the receptive field properties of the LGMD observed in response to local motion stimuli (Krapp and Gabbiani 2005). So far, much work has focused on describing the pattern of synaptic connectivity among neuron classes in the context of networks (Douglas and Martin 2004; Rolls and Treves 1998; Strausfeld and Nässel 1981; White 1989). Apart from a few systems (Borst and Egelhaaf 1992; Borst and Haag 2002; Jacobs and Theunissen 2000; Ogawa et al. 2006), surprisingly little is known about how the detailed mapping between inputs and dendritic compartments influences single-neuron responses to sensory stimuli. To study the interaction between electrotonic structure and synaptic mapping, our study employs a detailed description of the LGMD's excitatory dendritic field morphology. We first characterize quantitatively the morphology of the LGMD neuron and measure its inter-individual variability. Next, a compartmental model is fit to electrophysiological data to derive the passive electrotonic structure of the neuron. Finally, various connectivity schemes between the distribution of local visual inputs on the eye and the LGMD dendrites are tested in the model to determine how the different mappings shape the neuron's response properties to local motion stimuli.
|
METHODS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
LGMD STAININGS.
Five LGMD neurons were impaled with sharp intracellular electrodes (60–100 M
) containing a solution of 2 M potassium acetate (KAc) and Lucifer yellow (2%; Molecular Probes, Carlsbad, CA). For these experiments, dissection and electrophysiology were identical to those described in Gabbiani et al. (2001; see also below). After iontophoretic injection (20–60 min), the brain was dissected out of the head capsule, fixed overnight in Millonig's buffer, dehydrated through an ascending alcohol series, and cleared (Strausfeld and Miller 1981; chapt. 9). This method has been shown to minimize tissue shrinkage (Bucher et al. 2000). The stained neurons were then viewed through a x40 objective on a confocal microscope (Zeiss; Göttingen, Germany), and multiple stacks of sections were acquired with 3.88 µm depth resolution. Typically, five to six stacks were required to cover the entire neuron. The number of images per stack varied according to the local complexity of the neuronal morphology and ranged between 5 and 55. All cells were taken from the right optic lobe.
ANATOMICAL EYE RECONSTRUCTIONS.
To assess the relationship between the geometry of the LGMD's excitatory dendritic field and that of the eye, we carefully removed the right eyes of five locusts. The eyes were painted with red nail polish to increase surface reflectance. They were imaged with a 488-nm laser line using a Nikon E350 confocal microscope and a x2.5 objective (Tokyo). Stacks were acquired with a 50 µm depth resolution and consisted of six images. Six to 10 points were selected at each depth along the perimeter of the eye using a MATLAB interface. In total, 
50 points per eye were used as input to the ellipsoid fitting algorithm described below.
VISUAL STIMULATION EXPERIMENTS.
Experiments were performed on adult female locusts (Schistocerca americana), 3–4 wk past the final molt. The legs, wings, and antennae were removed, and the animal was secured in a holder using vacuum grease. The head was bathed in cooled locust saline, opened, and cleared of fat and muscles. To improve recording stability, the gut was removed. The head was carefully separated from the thorax with the exception of the ventral nerve cords and four major trachea and rotated 90° about the medial-lateral axis. This procedure allowed for easier access to the optic lobe and minimized mechanical coupling between abdominal respiratory movements and the brain. The right eye was aligned so that the ventral-dorsal axis of the tilted head coincided with the anterior-posterior axis of the thorax and abdomen. The eye was then waxed firmly into position and care was taken to assure that the field of view was unobstructed. The optic lobe was desheathed with fine forceps, and the animal was placed in front of a video monitor. The brain was bathed in room-temperature locust saline. An extracellular hook electrode was placed around the contralateral connective to record from the DCMD. A holder was placed under the brain to minimize movement. Thin-walled borosilicate glass (1.2/0.9 mm OD/ID, World Precision Instruments, Sarasota, FL) was used to make electrodes for intracellular recordings on a horizontal puller (P-97, Sutter Instruments, Novato, CA). The electrodes were filled with 2 M KAc (resistances: 40–60 M). The LGMD was identified as the neuron the spikes of which corresponded one-to-one with those of the DCMD. After recording stabilization, the visual stimulation protocol described in the following text was applied. In some recordings, a small hyperpolarizing DC current was injected to stabilize the recording (no greater than –1 nA) that had no impact on the results. An Axoclamp-2B (Molecular Devices, Sunnyvale, CA) in bridge mode was employed for intracellular recording and current injection. The intracellular membrane potential was sampled at 20 kHz and stored via an A/D converter on a personal computer (PC, x86) running QNX6 (QNX Software Systems, Ottawa, Canada). Recording sessions lasted 1 h.
VISUAL STIMULATION PROTOCOLS.
Locusts were positioned so that their anterior-posterior axis was parallel to the front of the video monitor with the right eye 15.8 cm away from and facing the monitor. Visual stimuli were presented at a 200-Hz refresh rate, well above the temporal cut-off frequency of locust photoreceptors (80 Hz) (Howard et al. 1984). The stimulus consisted of a 7.6° diam black (0 cd/m2) disk rotating counter-clockwise on a white (90 cd/m2) background along a 10.4° diam path (Fig. 6A, inset). Rotational velocities of 1 and 4 cycle/s were employed. Each stimulus consisted of a single rotation and thus lasted 1 s and 250 ms, respectively. The stimuli in each trial were presented once in pseudo-random order at nine positions and two velocities for a total of 18 stimulus presentations. The stimuli were centered at each combination of elevations –30, 0, and 30° and azimuths of 60, 90, and 120°. An elevation of 0° corresponded to the eye equator and negative elevations to ventral locations. An azimuth of 0° corresponded to straight in front while an azimuth of 90° was lateral to the animal (e.g., Fig. 4A, bottom inset). In all cases, the stimulus angular size was computed without accounting for distortion as elevation and azimuth grew away from 0 and 90°, respectively (corresponding to the center of the eye and monitor), because the maximal angles employed here resulted at most in a 5% distortion. A delay of 5 s between stimulus presentations was employed and each nine-position trial was separated by a 15-s inter-trial interval. Each stimulus presentation at a given position on the eye was thus separated by 
1 min. Between 25 and 50 trials were performed in each of five locusts. The eye alignment procedure used in these experiments was not as precise as that employed in the extracellular recordings of Krapp and Gabbiani (2005) but was sufficient for our purposes (see RESULTS). Previous work showed that these visual stimuli are unlikely to activate feed-forward inhibition onto the LGMD (Gabbiani et al. 2005; Krapp and Gabbiani 2005).
|
|
|
|
|
|
LGMD RECONSTRUCTIONS AND COMPARTMENTAL MODELING.
All neurons were traced using code written in MATLAB (MathWorks, Natick, MA). Specifically, points were selected along the dendrite and radius was computed automatically by detecting edges based on rapid intensity changes between neighboring pixels. The stacks acquired for each cell were aligned using the overlapping segments of dendrite. The five reconstructions, labeled lgmda–lgmde are illustrated in Fig. 1. Compartmental modeling of the LGMD neuron was performed using the NEURON simulation package (version 5.4) (Hines and Carnevale 1997). Most simulations were carried out using the morphology of a single cell, lgmda (Fig. 1B), after converting its 1583 segments to aNEURON-compatible format. In all simulations, a time step of 5 µs was used, and the simulated membrane potential was stored at a sampling frequency of 5 kHz unless otherwise noted. A PC with a 2.4-GHz Intel (Santa Clara, CA) dual-processor running Red Hat Linux 9.0 (Red Hat, Raleigh, NC) was employed for all simulations.
PASSIVE PARAMETER FITS.
We used the data set described in Gabbiani and Krapp (2006) to constrain the passive membrane properties of the model. Subsequent simulations employed uniformly distributed passive parameters obtained from these fits. The resting membrane potential was set at –65 mV (Gabbiani and Krapp 2006). Only the membrane potential deflections to 500-ms-long hyperpolarizing current pulses of –1, –2, and –3 nA were employed because active conductances clearly affected the responses to more negative and positive currents pulses (see Gabbiani and Krapp 2006). The data obtained in response to each current pulse were averaged across six cells (1 penetration per cell, with 2 penetrations for 1 cell; 9–10 repetitions per penetration) and median filtered with a 1-ms time window, yielding an average trace.
Because each of the passive parameters (specific membrane resisitivity, Rm; specific membrane capacitance, Cm; axial resistivity Ra) dominates during a different portion of the current injection response (e.g., Ra is prominent during the first 2 ms) (Major and Evans 1994), we adopted a sequential procedure in which Rm was first fitted, followed by Cm and Ra. The membrane resistivity, Rm, was fit by minimizing the fit error between the simulated and experimental membrane potential deflection during the final 475 ms (i.e., at steady state) of the three current pulses (Fig. 2A). Because the first 25 ms included virtually all transients, the impact of Ra and Cm was negligible. First, the squared difference between the experimental and simulated trace was computed over the 475-ms interval. This quantity was then normalized by the time interval duration (475 ms) and the mean experimental membrane potential, yielding an error measure independent of current injection level. For a fixed Rm value, this normalized error was averaged across the three current values and all the Ra and Cm values considered, giving a fit error. We tested specific membrane resistivity (Rm) values of 2,000, 4,000, 4,500, 5,000, 5,500, 6,000, 8,000, 10,000, 12,000, 15,000, and 20,000 · cm2. The tested specific membrane capacitance (Cm) values varied from 0.5 to 1.75 µF/cm2 (in steps of 0.25). The axial resistivity (Ra) values ranged from 20 to 100 (in steps of 10), as well as 150 and 200 · cm.
Next we fitted the experimental membrane potential transients between 2 and 27 ms after current pulse onset with a single-exponential curve interpolating between the resting and steady-state membrane potential associated with Rm and the current injection level, yielding the membrane time constant, 
m. The first 2 ms of each current pulse was omitted because the membrane potential transient was affected by the equalization time constant of the cell in that window (see Fig. 2B) (see also Gabbiani and Krapp 2006). The optimal m was divided with the previously obtained Rm to obtain Cm (m = RmCm).
Ra was constrained using the experimental equalization time constant (eq) of the LGMD (Gabbiani and Krapp 2006). In a compartmental model, eq depends primarily on neuronal morphology and Ra (Holmes et al. 1992). We ran current injection simulations and stored the membrane potential at a high sampling rate (200 kHz) using the Rm and Cm values derived in the previous two steps, varying Ra across all 11 values given in the preceding text. Only current injections of –1 nA were employed, but identical results were obtained when –2 and –3nA current pulses were also considered. For each simulated membrane potential trace, we computed the equalization time constant by exponential peeling (see Fig. 2D). The logarithm of the membrane potential deflection minus an exponential curve based on m was plotted and the window between 0.3 to 0.8 ms after current injection onset was used to fit a straight line. The value of the simulated eq was obtained from this line's slope. The simulated eq was plotted as a function of Ra, and the value of Ra yielding eq in closest agreement with the experimental value was selected.
The simulated electrode was positioned at the junction of the excitatory dendritic field to the main process of the LGMD (see Fig. 1B). Electrodes positioned anywhere along the main segment of the excitatory field as well as in the dendritic segment running from the origin of the excitatory field to the spike initiation zone yielded similar results. Electrodes placed in the thinner dendrites of the excitatory dendritic field, near their tips, or in the inhibitory dendritic fields yielded different passive parameter values (data not shown). The thick dendrites close to the base of the excitatory dendritic field are the most likely penetration sites in the experiments of Gabbiani and Krapp (2006).
SYNAPTIC PARAMETERS.
The cholinergic, excitatory synapses impinging on the LGMD's excitatory dendritic field (Rind and Simmons 1998) were simulated using an alpha function
 |
syn is the time of peak (0.3 ms). The synaptic reversal potential, Erev, was set to 0 mV. These parameters were adapted from Bazhenov et al. (2001). The excitatory dendritic field was covered with 7,322 synapses. This number corresponds to the number of ommatidia on a locust compound eye (Krapp and Gabbiani 2005). The conductance, gmax, thus corresponds to the total contribution of a single ommatidium to the excitatory input on the excitatory dendritic field. It is important to note that because our model lacks active conductances and assumes each ommatidium contributes only one synapse, gmax represents an effective synaptic conductance (see DISCUSSION). It is thought that the LGMD actually receives synaptic contacts from at least two afferents per ommatidium (Rowell et al. 1977; Strausfeld and Nässel 1981), although implementing such a modification would not alter the observed results.
ELECTROTONIC STRUCTURE.
We studied the electrotonic structure in the model by activating single synapses at various dendritic positions and characterizing the response at the spike initiation zone (SIZ). Specifically, we computed two measures (Fig. 3) described in Zador et al. (1995).
The first measure is the log-attenuation (Lsyn-SIZ) of the synaptic potential at the spike initiation zone, defined as
 |
In this equation, Vsyn(t) and VSIZ(t) represent the membrane potential at the compartment where the synapse is located and the spike initiation zone compartment, respectively, and ln represents the natural logarithm. The integral was taken over the first 50 ms of the response. Lsyn-SIZ converges to the electrotonic distance X = x/
(where x is distance in micrometer and is the space constant, also in micrometer) in an idealized infinite cylinder as the frequency of the synaptic membrane potential transient approaches zero (i.e., as it becomes a DC step). The classical electrotonic distance applies only to cylinders, whereas Lsyn-SIZ can be applied to complex dendritic morphologies and to more physiological alpha-synapse stimuli.
The second measure is the centroid delay, Psyn-SIZ, a measure of propagation delay along the dendrites. The centroid of the membrane potential transient was computed at the synapse and at the SIZ using the following expression
 |
The centroid delay was simply
 |
In addition, we computed the ratio of peak membrane potential at the spike initiation zone to that at the synaptic site as well as the difference in arrival times for the peaks at these two locations (Fig. 3, C and D).
ELLIPSOID FIT OF EYES AND EXCITATORY DENDRITIC FIELDS.
Ellipsoid fitting was performed using the same algorithm for both the locust eyes and the LGMD excitatory dendrites (Fig. 4A). The excitatory dendritic field was first converted into a cloud of points, with a point selected for every 5 µm of dendritic length. This resulted in 1,500 points. Eye data points were obtained as described in Eye reconstructions. The ellipsoid best fitting the data points was specified by nine parameters: three center coordinates, three axes lengths, and three rotation angles between the coordinate and ellipsoid axes. These nine parameters were first selected by an initial guess and then optimized by an iterative algorithm. The initial guess was obtained by computing the centroid of the point cloud and its two dominant axes of symmetry. The centroid was used for the coordinates of the ellipsoid center. The axes of symmetry were used to generate initial lengths of the ellipsoid axes (the 2 shorter axes were set to equal lengths) and the three angles of rotation between the coordinate axes and those of the ellipsoid. For the dendritic fields, this initial procedure was conducted using a 25 µm "coarse" point clouding.
The nine ellipsoid parameters were optimized by minimizing the error function defined as the summed square distance between each data point and the closest point on the ellipsoid surface. The MATLAB least-square fitting function, lsqnonlin, was employed to minimize the fit error, based on the Levenberg-Marquardt method with line search (Moré 1977). During an iteration of the algorithm, the three center coordinates and three axis lengths were first fit simultaneously by a call to the least-squares fitting function. The three rotation angles were then fit by a subsequent call. In both cases, the fitting function typically converged after 50 steps. Five iterations of this procedure were repeated to obtain the final parameters.
OPTICAL AXIS AND DENDRITIC LENGTH DENSITIES.
The locust eye consists of ommatidia, each of which samples light from a particular direction—its optical axis. We computed both the density of optical axes on the eye ellipsoid and the dendritic length density on the excitatory dendritic field ellipsoid (Fig. 4, B–E). Dendritic length density represents the amount of dendritic length in a given region of the ellipsoid fitted to the excitatory dendritic field. A comparison of these densities (RESULTS) allowed us to characterize the general properties of the first of the three synaptic mappings between visual space and the LGMD excitatory dendritic field described in the following text. The density of optical axes on the eye was calculated using the density distribution of optical axes in visual space (Krapp and Gabbiani 2005) and the average of the five eye-fitted ellipsoids. For each optical axis, the number of axes falling within a 5° radius cap centered around it was computed (see Krapp and Gabbiani 2005 for details) and divided by the area of the cap on the average eye ellipsoid. The lgmda excitatory dendritic field was used to compute dendritic length density. First, the lgmda excitatory dendritic field was broken up into 5-µm-long segments. The angular coordinates of each segment's endpoints were obtained. A 5° cap centered at a particular endpoint was constructed and the number of dendritic segment endpoints bound by the cap was counted. The density was obtained by dividing this number by the corresponding cap surface area on the lgmda-fitted ellipsoid. The surface areas of 5° ellipsoidal caps on the eye or on the excitatory dendritic ellipsoid were computed by numerical integration, since no analytic formula exists (Tee 2000).
SYNAPTIC MAPPINGS.
Three synaptic mappings were employed to assign inputs from specific regions of visual space to synapses on specific segments of the LGMD's excitatory dendritic field. The first mapping is based on a neighborhood-preserving transformation between visual space, as sampled by the ommatidia, and the excitatory dendritic field. The remaining mappings were used to investigate the impact of the assumptions made in the first mapping. In all cases, the excitatory dendritic field was broken into 7,322 segments of equal length (0.95 µm), corresponding to the number of ommatidia (optical axes) on the locust eye (Krapp and Gabbiani 2005). An alternative would have been to distribute synapses onto dendritic segments of equal surface area. However, dendritic surface area turned out to be much less well suited for this purpose than dendritic length (RESULTS). Moreover, in anatomical studies, the number of synapses is also commonly characterized per unit length (e.g., White 1989), facilitating comparison with our study (DISCUSSION).
Topographic mapping (Fig. 5C, top).
This mapping assumed that the 7,322 ommatidia sampling visual space (represented as optical axes in Fig. 5A) are mapped in a uniform, neighborhood-preserving manner onto the corresponding 7,322 dendritic segments of the LGMD excitatory dendritic field. To the best of our knowledge, no general method to generate such a mapping has been proposed. We thus developed a method based on two Kohonen self-organizing maps (SOMs; Fig. 5B). This algorithm was selected because it was originally designed to yield a neighborhood-preserving mapping from one space onto another. A detailed description of the SOM algorithm can be found in Kohonen (2001; esp. chapt. 3). Two passes of the SOM algorithm were employed, followed by a corrective step to guarantee a one-to-one correspondence between source and target spaces. The first, coarse mapping pass involved 50,000 iterations while the second, refining pass involved 100,000 iterations. In each pass, the two parameters of the SOM, 
(learning-rate; unitless) and 
(neighborhood radius; in fraction of target grid side length), determined the mapping. Both parameters were assigned an initial value and decreased linearly to a final value for a given pass.
The first SOM was used to map the ommatidia onto an 86 x 86 equally spaced square grid with 74 points of the last row omitted (total: 7,322 grid points; Fig. 5B, top). In the coarse and refining mapping passes, decreased from 0.95 to 0.4 and from 0.5 to 0.2, respectively, whereas decreased from 0.6 to 0.1 and from 0.05 to 0.005, respectively. The SOM algorithm does not guarantee that two ommatidia will be mapped onto two different grid points (i.e., it is not necessarily 1 to 1) and for 20% of the ommatidia, several were mapped onto the same target grid point. For these, we applied the following correction algorithm, resulting in a final one-to-one and neighborhood-preserving map. First, a list of ommatidia to be remapped was obtained by going over the sets of ommatidia mapped onto the same target grid point and selecting all but the one with the minimal angle between its optical axis and the optical axes of its four nearest neighbors in the target grid. After this step, there were as many unfilled target points in the target map as unmapped ommatidia in the list. Ommatidia in the list were randomly selected one after the other, and the desired location in the target grid where they would minimize their nearest-neighbor inter-optical axis angle was computed. A line was drawn between the desired location and the closest unoccupied target point in the target grid. The ommatidia assigned to points along that line were successively shifted by one position toward the unoccupied target point, resulting in the desired location being unoccupied. The ommatidium was then assigned to the desired location.
The second SOM was used to map dendritic points onto the same grid (Fig. 5B, bottom). In the coarse and refining mapping passes, decreased from 0.8 to 0.4 and from 0.5 to 0.2, respectively, whereas decreased from 0.6 to 0.05 and from 0.05 to 0.005, respectively. Euclidean distance in dendritic space was computed and the same correction procedure as for the ommatidia was employed. The final map between ommatidia and their corresponding dendritic segments was obtained by combining the two SOMs with the boundaries of both grids aligned so that corresponding locations in visual and dendritic space were in register (see RESULTS and Fig. 4A). Because this map preserves the distribution of ommatidial optical axes in visual space, it can be described as topographic (Fig. 5D). Several runs of the procedure were performed to verify the similarity of the resulting mappings.
Uniform, neighborhood-preserving mapping (Fig. 5C, middle).
The uniform mapping mimicked the synaptic arrangement expected if ommatidia on the locust eye were to sample visual space uniformly which is in sharp contrast to their actual arrangement. Thus this mapping allowed us to test the role that the eye's sampling of visual space plays in shaping the LGMD's receptive field to local motion stimuli. For this mapping, only the SOM from dendritic space into the 86 x 86 point grid was employed. The coordinates of points in this grid were then converted into visual coordinates by assigning one grid axis to elevation, ranging from –90 to 90° in equal steps and the other to azimuth (0–180°). Because the local neighbor relations of the dendritic field were preserved in the target grid, it was possible to align it realistically with visual space (see RESULTS and Fig. 4A). That is, the grid axis corresponding to the anterior-posterior dendritic axis was assigned to azimuth, and the axis corresponding to the dorso-ventral dendritic axis was assigned to elevation (Fig. 5E).
Random map (Fig. 5C, bottom).
This mapping assumed that the 7,322 optical axes sampling visual space are mapped randomly onto segments of the LGMD's dendritic tree. Thus this mapping allowed us to test the role that retinotopy plays in shaping the LGMD's receptive field to local motion stimuli. Ten map instances were generated by a pseudo-random number generator and used in simulations.
RELATION BETWEEN PEAK SPIKE FREQUENCY AND MEMBRANE DEPOLARIZATION DURING VISUAL STIMULAION.
We derived a peak spike frequency to membrane potential transform to compare experimental and simulated receptive fields. The intracellular LGMD membrane potential recorded in the visual stimulation experiments (see Experimental methods) was first processed to detect spikes. The shortest inter-spike interval (ISI) within the first 250 ms of stimulation was used to compute the peak instantaneous spike frequency. The intracellular membrane potential was median filtered using a 1-ms window to suppress spikes. The median-filtered trace was used to obtain the peak membrane potential in the first 250 ms of visual stimulation. Peak spike frequency, 
, and peak membrane potential, 
m, were averaged for a given animal across all stimulus presentations at a given location (9 locations total; 20–50 presentations per location). A linear regression was then fit to these nine points for each animal, = m + 
. The mean slope and intercept of the resulting regressions (n = 5 animals) were used to derive an equivalent peak spike frequency to peak membrane potential transform
 | (1) |
= 1/ was found to have a value of 0.05 mV/Hz and 
= –/ was equal to 5.94 mV (see RESULTS and Fig. 6). The values of used in Eq. 1 did not fall below 7 Hz.
STIMULATED VISUAL STIMULATION.
To conduct simulated visual stimulation in the LGMD model and thereby obtain artificial receptive fields (RFs) for local motion stimuli, the visual hemifield was divided into a uniform grid (150° in elevation by 180° in azimuth). The model's RF was obtained for each of the three synaptic mappings by "stimulating" it at each location. The stimulus consisted of simultaneous activation of a number of synapses equal to that covered by a disk of 7.6° diameter at that location according to the map under consideration. The surface area of the disk thus matched that used in visual stimulation experiments. To account for local variations in optical axes density, the activated synapses were selected as follows. First, the number of optical axes inside a 20 x 20° rectangle centered at the grid location was counted. The number of activated synapses, n, was obtained by scaling this number by the ratio of disk to rectangle surface area {[
· (7.6/2)2]/202 
1/9}. Then, n synapses were randomly selected within the set receiving input from the 20 x 20° rectangle, and 10 such simulations were run per stimulus location. The resulting peak membrane potential at the point where the excitatory dendritic field (field A, Fig. 1B) contacts the main process of the LGMD, origin-A, was recorded and averaged (Fig. 7, B–E). This site was selected because it is presumed to be close to the electrode location in the visual stimulation experiments. For each of the maps illustrated in Fig. 7, the variability of the RF profiles was low within the 10 runs (see in particular Fig. 7E). Thus subsequent simulations (Fig. 8) were conducted using a single set of input synapses.
|
|
) will occur over the area swept by the edge of the disc during that time period. Assuming a photoreceptor acceptance angle of 2.5° (Wilson 1975), only those ommatidia with an optical axis center within 1.25° (i.e., half of 2.5°) of the disk edge will be stimulated. This corresponds to a stimulated area of 60°2, which did not yield substantially different results from simulations using the area of the disk (45°2). In fact, the sensitivity profile of the RFs plotted in Fig. 7 was only weakly dependent on the typical number of synapses stimulated until the number of synapses reached a much larger value. EFFECT OF INTER-SYNAPTIC DISTANCE. We investigated the consequences of placing the stimulated synapses at various distances from each other along dendritic segments of the excitatory dendritic field while preserving the overall mapping of visual space defined by the topographic map. For a given stimulus location in visual space, n synapses were selected, as described in the preceding text, based on the topographic map. Their center of mass (COM) was computed in dendritic space and the point closest to the COM on the excitatory dendritic field was determined. The n synapses were then placed along the dendritic branch containing the point closest to the COM. The first synapse was placed at this point and subsequent ones were alternatively placed on each side of the first synapse at fixed distances along the dendrite. The distance values tested were 0.2, 1, and 5 µm.
EFFECT OF SUBLINEAR SUMMATION ON RF SHAPE.
We examined the relative impact of sublinear synaptic summation on the simulated RFs of the topographic and uniform synaptic mappings. For this purpose, both models were stimulated by the 7.6° disk at two positions in the visual field: elevation 0°, azimuth 30°, corresponding to the high acuity region of the topographic map, and elevation 0°, azimuth 135°, corresponding to the region of maximal response in the topographic map. The activated synapses were then split into two random subgroups with an equal number of synapses, A and B, and sublinearity was assessed by computing the ratio of origin-A when both subgroups were activated simultaneously and separately
 |
This measure characterizes the fraction of linear summation, and is equal to 1 if the responses to the two subgroup (A, B) stimulations sum linearly to generate the combined response (A & B); if the combined response is less than the sum of the individual subgroups (i.e., summation is sublinear), 
will be <1. In addition, we split the synapses into subgroups based on the elevation of their corresponding ommatidial optical axes ("high" and "low" elevation subgroups). Simulations were conducted with gmax values of 50 and 100 nS.
|
RESULTS |
|---|
|
TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
|---|
) (see also Fig. 6A). In contrast, the density of ommatidia sampling visual space is highest in the front, close to the eye equator and gradually decreases along the equator toward the back of the eye, as well as away from the equator (Krapp and Gabbiani 2005) (see also Fig. 5A). To investigate the relative impact of electrotonic structure and synaptic input location on the LGMD sensitivity distribution, we built a compartmental model of the neuron. We used this model to explore the functional consequences of applying different ways of mapping local synaptic inputs onto the excitatory dendritic field. Morphological properties of the LGMD
Figure 1A illustrates the morphology of five LGMD neurons reconstructed from confocal stacks after staining with a fluorescent dye (METHODS). Qualitatively, the morphologies of the five neurons were very similar. Figure 1B shows a magnified view of the lgmda neuron. As first described by O'Shea and Williams (1974), the LGMD possesses three distinct dendritic fields, labeled A–C in Fig. 1B. Anatomical and electrophysiological evidence suggests that dendritic field A receives local excitatory motion sensitive inputs and that dendritic fields B and C receive ON and OFF inhibitory, GABAergic inputs, respectively (see Gabbiani et al. 2004 for review). The LGMD possesses two sites of action potential initiation (O'Shea 1975), one of which is believed to initiate action potentials in response to visual inputs. This site will be referred to as the SIZ (labeled in Fig. 1B). Its location was identified from the reconstructions as a narrowing along the primary process of the LGMD to a minimal diameter of 1.7 µm in lgmda (mean: 1.8 µm, SE: 0.3, n = 5). On average, it was located 259 ± 25 (SE) µm away from the origin of dendritic field A (n = 5). The second site of action potential initiation is located in the protocerebrum and is implicated in the generation of spikes in response to auditory inputs (O'Shea 1975). The protocerebrum is also the location of the LGMD axon terminals that can be seen as much less extensive arborizations compared with the dendritic fields at the bottom of Fig. 1B. Some of these axon terminals contact the DCMD neuron (Killmann et al. 1999; O'Shea and Williams 1974). The two inhibitory dendritic fields (B and C in Fig. 1B) were connected to the primary process between the origin of dendritic field A and the SIZ with field C's site of connection always closer to the SIZ than that of field B (except in 1 case, where both fields shared a common connection site: lgmde). The mean distance of the connection sites of fields B and C from the origin of dendritic field A was 102 ± 9 (SE) µm (n = 7) and 131 ± 10 (SE) µm (n = 5), respectively. In two of the five neurons, field B was connected to the primary process at two locations; for these cases, both positions were included in the preceding distance calculation. The soma was always connected to the primary process between the origin of dendritic field A and the connection site of field B [distance to origin of field A: 49.1 ±14.8 (SE) µm, n = 5]. Table 1 reports the total length of dendrites, the number of inter-branch point segments, the average segment radius, and the total dendritic surface areas of the individual dendritic fields. Inter-branch point segments were defined as the sections of dendrite between individual branch points or dendritic tips and branch points. The variability in the number of segments was low for both fields A and C and higher for field B. Field A always contained the most segments, whereas field C always had the least, except for lgmdc, where field C had the same number of segments as field B (Table 1). The variability in total dendritic length between cells was smaller for fields A and C, relative to field B. For any individual neuron, the order of dendritic field lengths, from highest to lowest, was always A, B, and C (Table 1). The variability in mean radius was substantially greater: a two- to threefold difference in mean radius between the cell with highest mean radii in all fields (lgmda) and the lowest ones (lgmde) was observed. Mean radius was highest for field A in all cells and lowest for field C. These length and radius relationships were preserved when surface area was computed, though the overall variability was higher in this measure, up to a factor of 3.4. Overall, the morphology of the LGMD was qualitatively conserved across animals, but showed quantitative inter-animal variability.
|
We used responses to hyperpolarizing current pulses obtained in vivo (experimental data from Gabbiani and Krapp 2006) to constrain the passive electrotonic parameters of the LGMD. The final passive model's responses to current pulses of –1, –2, and –3 nA is compared with the experimental membrane potential traces in Fig. 2A. The fit is best at –1 nA, as inward rectification becomes increasingly apparent for more negative current pulses (Gabbiani and Krapp 2006). The passive model was constructed by finding values for the specific membrane resistivity, Rm, the specific capacitance, Cm, and the axial resistivity, Ra, in that order (see METHODS). Figure 2B shows the fit error as a function of Rm. An optimal fit was obtained for Rm = 4,500 cm2 (arrow). This value is comparable to those obtained in other insect visual interneurons in vivo (Borst and Haag 1996). To determine Cm, we used the relation m = Rm · Cm after fitting the membrane potential relaxation to steady state by a single exponential (Fig. 2C). The fit ignored the first 2 ms of the pulse because in that early phase the dynamics of the membrane potential was also affected by the equalization time constant, as explained in the following text. This procedure yielded a membrane time constant (m) of 6.6 ms, close to the average value of 7.3 ms determined from individual LGMD neurons by Gabbiani and Krapp (2006). The equivalent Cm value amounted to 1.5 µF/cm2.
Finally, we fit the intracellular resistivity (Ra). This parameter proved more difficult to fit, because it contributes the least to membrane potential responses (see, e.g., Johnston and Wu 1995). We constrained Ra by obtaining estimates for eq as a function of Ra in the model while holding Cm and Rm fixed to the values obtained in the first two fitting steps. The model value of eq was obtained from membrane potential responses to simulated current injection pulses using the standard peeling procedure (METHODS) as depicted in Fig. 2D. The resulting eq versus Ra plot is shown in Fig. 2E. Because the mean experimental eq was found to be 0.34 ms (Gabbiani and Krapp 2006) (dotted line in Fig. 2E), this implied that Ra is approximately equal to 60 cm in our model (using the median eq, 0.26 ms, yielded the same result). We estimated the dependence of passive parameters on morphology by performing the same fits using the lgmde and lgmdb morphologies, the neurons with the largest and smallest surface area, respectively. Overall, the value of Rm ranged from 2,000 to 4,500 cm2 with the smallest Rm corresponding to the smallest surface area. The values of Ra ranged from 40 to 60 cm and were only weakly dependent on the particular morphology employed. A Cm value of 1.5 µF/cm2 predicted a membrane time constant about twice as fast for the neuron with the smallest surface area. This value lies on the lower end of experimentally observed values (Gabbiani and Krapp 2006).
LGMD is not electrotonically compact
Several lines of evidence suggest that the LGMD is not electrotonically compact in vivo (Gabbiani and Krapp 2006; Gabbiani et al. 2001; Krapp and Gabbiani 2005). We used the passive model derived in the previous section to test this hypothesis quantitatively. Synapses were placed at various locations across the dendritic field and measures of attenuation and delay for dendritic signal propagation were computed. Figure 3A shows the log-attenuation of the membrane potential (Lsyn-SIZ, see METHODS) (Zador et al. 1995) between the synapse compartment and the SIZ. The large attenuation of even relatively proximal inputs (red arrowhead and inset in Fig. 3A) demonstrates a significant filtering prior to the SIZ of the LGMD. The log-attenuation converges to the electrotonic distance in an infinite cylinder as stimulus frequency approaches zero (Zador et al. 1995). The dimensionless electrotonic distance, X, is defined as X = x/ where x is distance (in µm) and is the space constant of the cylinder (Johnston and Wu 1995). A neuron is considered electrotonically compact if its dendritic arbors spans 
0.1 (i.e., X 0.1). Functionally, compact neurons exhibit minimal attenuation for inputs coming from even the most distal synapses. The LGMD can be considered electrotonically large as the distal tips typically attain an Lsyn-SIZ value of around 3. Figure 3B demonstrates that excitatory postsynaptic potentials propagating along the dendrites experience significant centroid delays: up to 7 ms from the most distal dendritic tips of the excitatory field to the SIZ. Figure 3C depicts Lsyn-SIZ and the ratio of peak membrane potential at the SIZ to that at the synaptic location as a function of its distance from dendritic field A's origin. For synapses beyond 200 µm from the origin of field A, only 10% of the peak synaptic membrane potential deflection is attained at the SIZ. Figure 3D shows the centroid delay and delay of EPSP peak between the SIZ and the synaptic location as a function of its distance from the origin of field A. EPSP peak delay attains maximal values of 2.5 ms, about a third that of centroid delay. As explained in the following text, the dendritic tips of field A are expected to sample the frontal visual field and the thicker dendrites the caudal visual field. Thus our results are consistent with experimental findings showing weaker responses to frontal visual stimuli relative to caudal and lateral stimuli, as well as a greater latency of response to frontal stimuli (Krapp and Gabbiani 2005). Lsyn-SIZ and centroid delays reached similar maximum values in the inhibitory fields, which are thus also electrotonically extensive (not shown). The noncompactness of dendritic field A may seem surprising when compared with vertebrate neurons with similar dendritic anatomies, such as Purkinje cells, which have been shown to have dendritic fields with average electrotonic sizes of 0.1 space constants (Roth and Häusser 2001). However, the Rm values reported here and for many other insect visual interneurons are typically an order of magnitude lower than those observed in vertebrate neurons, resulting in a smaller space constant and hence larger electrotonic size (Johnston and Wu 1995). Using the Rm and Ra values derived in the preceding text and a dendritic diameter d of 10 µm to derive analytically ( =
) yields an estimated value of 1,369 µm for the space constant and an electrotonic distance of <0.5 (dimensionless) for the most distal dendritic tips. This value is still larger than the typical 0.1 value
t · V(t)dt/
