Vector Summation Model Research Articles

Interocular Phase Disparity Tuning of Binocular Contrast Summation Depends on Carrier Spatial Frequency and Orientation.

Binocular summation is a sensitive metric of binocular integration. As such, characterization of the mechanisms underlying binocular summation is a key step in translating and applying this knowledge to abnormal binocular systems afflicted with strabismus and amblyopia. Computational models of binocular summation have advocated the operation of mechanisms sensitive to the interocular phase disparity of first-order carrier gratings. This study investigated if such generalization depended on carrier spatial frequency and orientation. Monocular and binocular contrast detection thresholds were measured in nine observers with normal binocular vision. Stimuli comprised Gabor targets presented with one of three spatial frequencies (1, 3, and 9 cycles per degree [cpd]), two orientations (horizontal and vertical), and five interocular phase disparities (0, ± π /2, ± π radians). Horizontal and vertical fixation disparities were measured for each binocular threshold condition. Binocular summation ratios were computed by dividing the mean monocular detection threshold by the respective binocular detection threshold. Binocular summation ratio varied significantly with interocular phase disparity for the 1- and 3-cpd horizontal and vertical gratings. Phase dependency was reduced with the 9-cpd horizontal grating and absent for the 9-cpd vertical grating, even though binocular summation ratio exceeded predictions of probability summation. Computational modeling that incorporated the variability of fixation disparity into a vector summation model predicted a reduction in peak binocular summation ratio with increasing carrier spatial frequency but did not account for the reduction of phase sensitivity noted with the 9-cpd stimulus. Binocular summation magnitude is less dependent on interocular phase disparities for carrier spatial frequencies that exceed 3 cpd, especially with vertical gratings. Although vergence variability due to fixation disparities contributes to the overall reduction in binocular summation magnitude with increasing carrier spatial frequency, it does not provide a complete account for the lack of interocular phase disparity tuning noted with high grating spatial frequencies.

Summation of Temporal L-Cone- and M-Cone-Contrast in the Magno- and Parvocellular Retino-Geniculate Systems in Glaucoma.

PurposeThe purpose of this study was to characterize summation of temporal L- and M-cone contrasts in the parvo- (P-) and magnocellular (M-) pathways in glaucoma and the relationship between the respective temporal contrast sensitivities (tCS) and clinical parameters.MethodsPerifoveal tCS to isolated or combined L- and M-cone contrasts (with different contrast ratios, and therefore different luminance and chromatic components) were measured at different temporal frequencies (at 1 or 2 Hz and at 20 Hz) using triple silent substitution in 73 subjects (13 healthy, 25 with glaucoma, and 35 with perimetric glaucoma). A vector summation model was used to analyze whether perception was driven by the P-pathway, the M-pathway, or both. Using this model, L- and M-cone input strengths (AL, AM) and phase differences between L- and M-cone inputs were estimated.ResultsPerception was always mediated by the P-pathway at low frequencies, as indicated by a median phase angle of 179.84 degrees (cone opponency) and a median AL/AM ratio of 1.04 (balanced L- and M-cone input strengths). In contrast, perception was exclusively mediated by the M-pathway at higher frequencies (input strength not balanced: AL/AM = 2.94, median phase angles = 130.17 degrees). Differences in phase were not significant between diagnosis groups (Kruskal-Wallis = 0.092 for P- and 0.35 for M-pathway). We found differences between groups only for the M-pathway (L-cone tCS deviations at 20 Hz were significantly lower in the patients with glaucoma P = 0.014, with a strong tendency in M-cones P = 0.049). L-cone driven tCS deviations at 20 Hz were linearly correlated with perimetric mean defect (MD) and quadratically correlated with retinal nerve fiber layer (RNFL) thickness.ConclusionsUnaltered phase angles between L- and M-cone inputs in glaucoma indicated intact temporal processing. Only in the M-pathway, contrast sensitivity deviations were closely related to diagnosis group, MD, and RNFL thickness, indicating M-pathway involvement.

Frequency tuning and directional sensitivity of tympanal vibrations in the field cricket Gryllus bimaculatus.

Female field crickets use phonotaxis to locate males by their calling song. Male song production and female behavioural sensitivity form a pair of matched frequency filters, which in Gryllus bimaculatus are tuned to a frequency of about 4.7 kHz. Directional sensitivity is supported by an elaborate system of acoustic tracheae, which make the ears function as pressure difference receivers. As a result, phase differences between left and right sound inputs are transformed into vibration amplitude differences. Here we critically tested the hypothesis that acoustic properties of internal transmissions play a major role in tuning directional sensitivity to the calling song frequency, by measuring tympanal vibrations as a function of sound direction and frequency. Rather than sharp frequency tuning of directional sensitivity corresponding to the calling song, we found broad frequency tuning, with optima shifted to higher frequencies. These findings agree with predictions from a vector summation model for combining external and internal sounds. We show that the model provides robust directional sensitivity that is, however, broadly tuned with an optimum well above the calling song frequency. We therefore advocate that additional filtering, e.g. at a higher (neuronal) level, significantly contributes to frequency tuning of directional sensitivity.

How to combine direction cues optimally

Many perceptual tasks require estimating a direction in space. Often several directional cues are available, such visual and gravitational cues to the subjective vertical, or visual and auditory cues to the direction of an object. In work on the subjective vertical, researchers have developed a heuristic vector summation model that has no deep theoretical motivation, but that accounts well for the direction and reliability of observers’ direction estimates when multiple cues are available, and that can accommodate directional cues ranging over all possible directions (Mittelstaedt, 1983). In work on combining visual and auditory cues to direction, researchers have used statistically motivated cue combination models that were originally developed for linear quantities such as depth, not circular or spherical quantities such as direction, and hence work only over a limited range of cue directions (Alais and Burr, 2004). Here we present a new model of directional cue combination that combines the advantages of both previous approaches. We develop a statistical theory of cue combination based on the von Mises distribution, the analog on the circle of the normal distribution on the line. We show that this theory differs in important ways from the theory of linear cue combination, e.g., a combined direction estimate can be less certain than any of the individual cues that were used to calculate it. We also show that the vector summation model developed empirically by previous investigators is an excellent approximation to our theory, meaning that it is a nearly optimal way of combining directional cues.

LINEAR VISUOMOTOR TRANSFORMATIONS IN MIDBRAIN SUPERIOR COLLICULUS CONTROL SACCADIC EYE-MOVEMENTS

It is well established that a localized population of neurons in the motor map of the midbrain superior colliculus (SC) drives a saccadic eye-head gaze shift. However, there is controversy as to how the brainstem saccade burst generators decode the SC activity. We focus on eye-movement generation by comparing two competing schemes from the recent literature that are both supported by neurophysiological evidence: the vector-averaging scheme versus the vector summation model. Whereas the former contains at least four nonlinearities to explain visuomotor planning and saccade execution, the latter relies predominantly on linear operations. We have demonstrated that the summation model accounts for the nonlinear main sequence of saccade kinematics, and predicted that this results from a spatial gradient in temporal burst profiles of SC cells: rostral cells have higher peak-firing rates and shorter burst durations than caudal cells. Yet, the number of spikes in their saccade-related bursts is identical. In contrast, the averaging model does not predict such activity profiles. We now also show that by incorporating the concept of predictive remapping in the spatial updating of saccade sequences, the phenomenon of target averaging in double-stimulation experiments, and the occurrence of goal-directed, but highly curved saccades in the double-step paradigm, can all be explained by the same linear summation mechanism. We argue that the linear model is more in line with neurophysiological data, while relying on fewer ad-hoc assumptions than the nonlinear vector-averaging scheme.

The human semicircular canal model of galvanic vestibular stimulation

A vector summation model of the action of galvanic stimuli on the semicircular canals has been shown to explain empirical balance and perceptual responses to binaural-bipolar stimuli. However, published data suggest binaural-monopolar stimuli evoke responses that are in the reverse direction of the model prediction. Here, we confirm this by measuring balance responses to binaural-monopolar stimulation as movements of the upper trunk. One explanation for the discrepancy is that the galvanic stimulus might evoke an oppositely directed balance response from the otolith organs that sums with and overrides the semicircular canal response. We tested this hypothesis by measuring sway responses across the full range of head pitch. The results showed some modulation of sway with pitch such that the maximal response occurred with the head in the primary position. However, the effect fell a long way short of that required to reverse the canal sway response. This indicates that the model is incomplete. Here, we examine alterations to the model that could explain both the bipolar and monopolar-evoked behavioural responses. An explanation was sought by remodelling the canal response with more recent data on the orientation of the individual canals. This improved matters but did not reverse the model prediction. However, the model response could be reversed by either rotating the entire labyrinth in the skull or by altering the gains of the individual canals. The most parsimonious solution was to use the more recent canal orientation data coupled with a small increase in posterior canal gain.

Can vector summation describe the orientation system of juvenile ospreys and honey buzzards? – An analysis of ring recoveries and satellite tracking

Juvenile bird migrants are generally believed to use a clock‐and‐compass migratory orientation strategy. According to such a strategy migrants accomplish their migration by flying a number of successive flight steps with direction and number of steps controlled by an endogenous programme. One powerful way of testing this is by comparing predictions from a model of such a strategy with observed patterns. We used data from ringing and satellite‐based radio telemetry to investigate the orientation system of juvenile ospreys (Pandion haliaetus) and honey buzzards (Pernis apivorus) migrating from Sweden to tropical west Africa. The ring recoveries showed a much larger scatter in the orientation of ospreys than of honey buzzards, but there was only a slight such difference in the satellite tracks. These tracks of individuals of both species were rather straight with a high directional concentration per step. The honey buzzard data showed a close fit to a simple vector summation model, which is expected if birds follow a clock‐and‐compass strategy. However, the osprey data did not fit such a simple model, as ring recoveries showed a significantly greater deviation at short distances than predicted on the basis of long distance data. Satellite tracking also indicated less concentrated orientation on short distances. The pattern observed for the osprey can generally be explained by an extended vector summation model, including an important element of pre‐migration dispersal. The existence of extensive dispersal in the osprey stands in contrast to the apparent absence of such dispersal in the honey buzzard. The explanation for this difference between the species is unclear. The model of orientation by vector summation is very sensitive to the existence of differences in mean direction between individuals. Assuming such differences, as tentatively indicated by the satellite tracking data, makes simple compass orientation by vector summation inconsistent with the distribution of ring recoveries at long distances, with a high proportion of misoriented birds falling outside the normal winter range.

ON-pathway dysfunction and timing properties of the flicker ERG in carriers of X-linked retinitis pigmentosa.

Carriers of X-linked retinitis pigmentosa (XLRP) frequently show prolonged implicit times of the flicker electroretinogram (ERG). This study tested the hypothesis that a preferential response attenuation within the cone depolarizing (ON) bipolar cell (DBC) pathway is a major contributing factor. Light-adapted, full-field ERGs were recorded from 10 XLRP carriers and 12 visually normal control subjects. Fundamental amplitudes and phases of ERG responses to sinusoidally flickering stimuli at temporal frequencies ranging from 8 to 96 Hz were analyzed within the framework of a recent vector summation model of the cone system ERG to test for evidence of a response attenuation within the DBC pathway. In addition, ERG responses to sawtooth flicker were examined for a reduced b- to d-wave amplitude ratio, indicative of ON pathway dysfunction. The carriers' fundamental response phases at 32 Hz correlated significantly with their log ratios of response amplitudes at 32 versus 12 Hz (r = 0.89, P < 0.001) and with their log b- to d-wave amplitude ratios (r = 0.71, P < 0.05), both of which were used as indices of response attenuation within the DBC pathway. A control experiment demonstrated that a reduced sensitivity of cone phototransduction made at most only a minimal contribution to the timing changes in the carriers' flicker ERG responses. The overall pattern of results indicates that a preferential response attenuation within the DBC pathway is the primary source of timing changes in the flicker ERGs of these carriers of XLRP. These findings illustrate the value of analyzing ERG responses to flickering stimuli at multiple temporal frequencies to evaluate mechanisms of disease action in photoreceptor degenerations.

Magnetic Position Determination by Homing Pigeons?

How pigeons return home from unfamiliar release sites is a long-standing puzzle in animal behaviour. Walker (1998, 1999) has described a “vector summation model” which “identifies a novel coordinate that pigeons could use with magnetic total intensity to determine position”. The model is not applicable in a magnetic field generated simply by a geocentric dipole, but requires a field perturbed by higher-order sources. Tests are devised to simulate the addition of both regional and local magnetic anomalies to a geocentric dipole field, and to calculate the directions of the home loft from a number of release sites. The results indicate that a pigeon would be unlikely to derive useful information from the model.

On a Wing and a Vector: a Model for Magnetic Navigation by Homing Pigeons

The ability of pigeons to home directly from distant, unfamiliar release sites has defied explanation because it has been impossible to identify the mechanism by which the birds determine their current position relative to their home loft. A variety of magnetic effects on homing orientation have implicated magnetic total intensity in position determination but no testable models for magnetic navigation by homing pigeons have resulted. Here, a vector summation model is proposed which identifies a novel coordinate that pigeons could use with magnetic total intensity to determine position. The model makes predictions about the accuracy of homing and patterns of homing orientation over local and regional scales. The model requires no unusual computational or cognitive abilities. It is, however, consistent with a significant volume of pigeon homing data and can be tested in a variety of ways.

On the effects of a subharmonic masker on the loudness of a pure tone

Loudness matching functions between a 1-kHz tone in quiet and in the presence of a 70 dB(SPL) 0.5-kHz subharmonic masker were studied to determine the alterations in loudness due to intensity shifts at the stimulus frequency arising from the generation of the second harmonic of the masker. Such effects are understood to be due to a quadratic nonlinearity occurring in the inner ear transduction process. In the experiment, masked thresholds were measured at 30° phase angle intervals of the stimulus. Matching functions were obtained for phase conditions corresponding to minimum and maximum thresholds. Both the shape of the matching functions and their relative displacement from each other were consistent with the predictions of the vector summation model [Clack et al., J. Acoust. Soc. Am. 52, 536–541 (1972); Schubert, J. Acoust. Soc. Am. 45, 790–791 (1969)]. Assuming that the threshold shifts were due to constructive and destructive interference between the 1-kHz tone and the second harmonic of the masker, a single effective matching function was constructed by means of the vector summation model. This calculation demonstrated the ability of loudness matching to probe nonlinear cochlear mechanisms. Matching functions generated from several proposed loudness functions were compared with the data.

Integration across directions in dynamic random dot displays: vector summation or winner take all?

Recent studies have clearly demonstrated that the activity of directionally selective neuronal populations in the middle temporal (MT) and medial superior temporal (MST) cortical areas plays a direct role in the judgment of the direction of visual motion. However, the way in which the information is derived from a population of neurons remains unknown. Two principal models have been suggested in the past: the vector summation model suggests that the responses of neurons encoding all directions of motion are weighted and pooled to obtained an accurate estimate of the mean direction of motion; the winner-take-all model is based on a competition between different direction-specific channels, so that decisions are cast in favor of the channel generating the strongest directional signal. To discriminate between these two models we generated random dot stimuli that contained an asymmetric distribution of directions of motion. Human subjects were asked to adjust the global direction of motion to the upward vertical direction. When the directional signals were of similar strength, subjects tended to perceive global motion in the mean direction of motion (corresponding to vector summation), but as one directional signal became more prominent, most subjects' settings diverged from the mean towards the modal direction of motion. Some subjects could either match the mean or the modal direction of motion in the display, depending on the task instructions. These results suggest that the perceptual judgment of direction of motion is not based on any rigid algorithm generating a single valued output. Rather, human observers are able to judge different aspects of the distribution of activity in a cortical area depending on the task requirements.

Perceptual integration in heterogeneous taste percepts.

When mixtures of dissimilar tasting compounds are perceived, specific taste sensations are integrated into a percept. Four models are evaluated that were designed to predict the total taste intensity of a mixture percept from the total taste intensities of the unmixed components or from the specific taste intensities of the components within the mixture percept. The predictions of the two dominant component models are inferior to those made by the two other models. Predictions made by the vector summation model are accurate, but the model contains theoretical weaknesses that make it an invalid psychological integration model. The sum of sensations model predicts total intensity with an accuracy similar to the vector summation model, but it is more parsimonious. Furthermore, it can account for all of the empirical phenomena observed.

Functional properties of parietal visual neurons: radial organization of directionalities within the visual field

Parietal visual neurons (PVNs) were studied in waking monkeys as they executed a simple fixation-detection task. Test visual stimuli of varied direction, speed, and extent were presented during the fixation period; these stimuli did not control behavior. Most PVNs subtend large, bilateral receptive fields and are exquisitely sensitive to stimulus motion and direction but insensitive to stimulus speed. The directional preferences of PVNs along meridians are opponently organized, with the preferred directions pointing either inward toward or outward away from the fixation point. Evidence presented in the preceding paper (Motter et al., 1987) indicates that opponent directionality along a single meridian is produced by a feed-forward inhibition of 20 degrees-30 degrees spatial extent. The observations fit a double-Gaussian model of superimposed but unequal excitatory and inhibitory receptive fields: When the former is larger, inward directionality results; when smaller, outward directionality results. We examine here the distribution of the meridional directional preferences in the visual field. Tests showed that opponent organization is not produced by differences in local directional properties in different parts of the receptive field. The distribution of response intensities from one meridian to another is adequately described by a sine wave function. These data indicate a best radial direction for each neuron with a broad distribution of response intensities over successive meridians. Thus, any single PVN, with rare exceptions, cannot signal radial stimulus direction precisely. We then determined how accurately the population response predicted radial stimulus direction by the application of a linear vector summation model. The resulting population vector varied from stimulus direction by an average of 9 degrees. Whether or not the perception of the direction of motion depends upon a population vector remains uncertain. PVNs are especially sensitive to object movement in the visual surround, particularly in the periphery of the visual field. This, combined with their large receptive fields and their wide but flat sensitivity to stimulus speed, makes them especially sensitive to optic flow. This is discussed in relation to the role of the parietal visual system in the visual guidance of projected movements of the arm and hand, in the guidance of locomotion, and in evoking the illusion of vection.

Integration of spectral and temporal cues separated in time and frequency.

Subjects discriminated a "standard" pair of tone bursts (T1, T2) from a "comparison" pair (T1 + delta t, T2 + delta f), containing increments in the duration delta t of the first burst and/or the frequency delta f of the second burst. The threshold (d' = 2.0) for delta t was measured as a function of delta f, and the threshold for delta f as a function of delta t. The integration of increments in duration and frequency was studied as a function of the spectral and temporal separation between T1 and T2. A trade-off between the values of delta t and delta f required for d' = 2.0 performance was observed. This integration takes place when delta t, delta f occur simultaneously in the same spectral region, and when they occur separated by up to 120 ms, or by up to a full octave. The efficiency of integration was similar for all conditions of temporal and spectral separation studied, because the discriminability of delta t and of delta f is also nearly uniform across experimental conditions. The results from all experimental conditions are adequately described by a vector summation model derived from TSD. In a subsidiary experiment, subjects categorized pure tones varying in duration and frequency as "high" or "low" in pitch and "long" or "short" in duration. It was found that combined variations in duration and frequency result in essentially independent perceptual processes, although pitch has a small effect upon the perceived duration. It is concluded that spectral-temporal integration is a general ability operating in a variety of stimulus conditions.(ABSTRACT TRUNCATED AT 250 WORDS)

Loudness (or annoyance) summation of combined community noises

The purpose of the experiment was to study how the loudness (or annoyance) of the constituents in pairs of community noises combines into total loudness (or total annoyance). Three community noises (pile driving, jack hammering, street traffic) were combined pairwise at different sound levels. The observers judged the total loudness of the combined noises as well as the loudness of each component noise when heard alone. Three models of loudness (or annoyance) summation for noise were tested: a vector summation model, a model assuming that the loudnesses of the masked constituent noises add arithmetically, and a simple model stating that the total loudness equals the loudest of the component noises when heard alone. All three models fitted the data satisfactorily from a statistical point of view. The ’’loudest component’’ model was favored because it produced a prediction error of only 14% in the ’’worst case’’ and can, therefore, serve as a ’’rule of thumb’’ for many practical purposes.

Lateralization and the MLD: The Vector Summation Model

At the Houston meeting, the first author reported the results of an experiment requiring discrimination between various combinations of interaural time and intensity cues. Performance could be described quite well by a simple vector summation of performances with each cue alone. The angle, η, between the vectors provided an estimate of the correlation of the two cue dimensions in the psychological discrimination space. Similar data reported recently by Jeffress and McFadden [J. Acoust. Soc. Amer. 52, 1197–1206 (1972); 50, 1484–1493 (1971)] provided the opportunity to extend the vector summation model into an MLD paradigm. As in our earlier work, one angle, η, along with an efficiency ratio, reflecting the relative sensitivity of each observer to interaural time and intensity differences, predicted performance well for all values of alpha; that is, performance for cue conditions in which time and intensity differences are consonant or dissonant can be described by the same vector summation process. An analysis of changes in efficiency ratio and η across observers and frequency, along with a comparison of these parameters in the detection and lateralization tasks will be presented.