June 2011
Volume 11, Issue 10
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Article  |   September 2011
A dual visual–local feedback model of the vergence eye movement system
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Journal of Vision September 2011, Vol.11, 21. doi:10.1167/11.10.21
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      Casper J. Erkelens; A dual visual–local feedback model of the vergence eye movement system. Journal of Vision 2011;11(10):21. doi: 10.1167/11.10.21.

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Abstract

Pure vergence movements are the eye movements that we make when we change our binocular fixation between targets differing in distance but not in direction relative to the head. Pure vergence is slow and controlled by visual feedback. Saccades are the rapid eye movements that we make between targets differing in direction. Saccades are extremely fast and controlled by a local, non-visual feedback loop. Usually, we change our fixation between targets that differ in both distance and direction. Then, vergence eye movements are combined with saccades. A number of models have been proposed to explain the dynamics of saccade-related vergence movements. The models have in common that visual input is ignored for the duration of the responses. This type of control is realistic for saccades but not for vergence. Here, I present computations performed to investigate if a model using dual visual and local feedback can replace the current models. Simulations and stability analysis lead to a model that computes an estimate of target vergence instead of retinal disparity and uses this signal as the main drive. Further analysis shows that the model describes the dynamics of pure vergence responses over the full physiological range, saccade-related vergence movements, and vergence adaptation. The structure of the model leads to new hypotheses about the control of vergence.

Introduction
The striking feature of human oculomotor behavior is that the two eyes move as a “single double eye,” such that the two lines of sight intersect at the target (or at least very nearly so) when humans look at an object of interest. Natural targets usually differ in distance and direction relative to the head. It was initially believed that eye movements covering differences in distance and direction are generated by independent vergence and saccadic subsystems in human (Rashbass & Westheimer, 1961a; Westheimer & Mitchell, 1956) and in monkey (Judge & Cumming, 1986; Mays, Porter, Gamlin, & Tello, 1986). Other studies questioned this view and provided evidence for interaction between vergence and saccades (Enright, 1984; Erkelens, Steinman, & Collewijn, 1989; Ono, Nakamizo, & Steinbach, 1978; Sylvestre, Galiana, & Cullen, 2002; Zee, Fitzgibbon, & Optican, 1992). 
Currently, there are two leading theories explaining saccade–vergence interaction. One type of models is based on interaction between components of the conjugate saccadic system and vergence burst neurons (Busettini & Mays, 2005a, 2005b; Kumar, Han, Dell'Osso, Durand, & Leigh, 2005; Zhu, Hertle, & Yang, 2008). The other type of models proposes monocular saccadic burst neurons that are responsible for the intra-saccadic vergence enhancement (King & Zhou, 2002; Sylvestre et al., 2002; Van Horn & Cullen, 2008; Zhou & King, 1998). Zee et al. (1992) who were first in modeling saccadic vergence proposed both architectures as solutions for saccade–vergence interaction. 
In addition to the saccade–vergence models, a number of other models of the vergence subsystem have been proposed to explain the dynamics of pure vergence movements. Typically, the models differ by architecture and input. The goal of this study is to elucidate which input is effective in controlling vergence eye movements. I first discuss types of input related to disparity. Next, starting from the mechanics of the eyes, simulations of experimental data will indicate what neural signals should be present at the output level of the vergence subsystem. Evaluation of these signals will lead to a model that explains the dynamics of both pure and saccade-related vergence movements. 
Westheimer and Mitchell (1956) were the first to show that disparity is an effective stimulus for vergence in the absence of other cues. Later, disparity was shown to be the strongest stimulus in human (Semmlow & Wetzel, 1979) and in monkey (Cumming & Judge, 1986). Rashbass and Westheimer (1961b) concluded from responses to sinusoidal closed-loop and constant open-loop stimuli that vergence was controlled by continuous feedback of disparity. Since then, disparity is the stimulus that primarily has been used in models of vergence although Erkelens et al. (1989) found that vergence movements were faster and more accurate under conditions in which distance of targets was not just indicated by disparity. Still disparity was found to be the principle stimulus for vergence movements up to 34 degrees. Contributions from other sources were much weaker since movements were much smaller and irregular during monocular viewing and in the dark. It is known that vergence is also controlled by cross-coupling between accommodation and convergence (Schor, 1992). Contributions by accommodative vergence are not included in the current study. 
Absolute disparity is the difference between the vergence angle of the target (target vergence) and that of the eyes (ocular vergence). Relative disparity is the difference between the vergence angle of one target and that of another target. Absolute disparity is first represented by activity of individual neurons in V1 (Barlow, Blakemore, & Pettigrew, 1967; Pettigrew, Nikara, & Bishop, 1968). Relative disparity is first represented in V2. From there, many areas in the dorsal and ventral visual pathways make distinctive contributions to the processing of relative and absolute disparity to allow perception of stereoscopic depth and coordination of eye movements (Parker, 2007). Absolute disparity has been implemented in models of vergence in two ways. On the one hand, models that were developed to describe pure vergence used an estimate of absolute disparity as the driving signal (Krishnan & Stark, 1977; Maxwell, Tong, & Schor, 2010; Patel, Jiang, & Ogmen, 2001; Pobuda & Erkelens, 1993; Rashbass & Westheimer, 1961b; Schor & Kotulak, 1986). On the other hand, models developed to understand saccade-related vergence used “desired change in vergence angle” (Kumar et al., 2006; Zee et al., 1992) and “vergence command” (Busettini & Mays, 2005a) as the control signal. The terms indicate a signal that is equivalent to absolute disparity prior to ocular vergence responses, but, different from absolute disparity, the signal does not change during vergence responses. In other words, the driving signal is frozen for the duration of the complete vergence responses. Freezing the control signal is unrealistic because it is not consistent with the continuous visual feedback property of vergence. However, although relying on the unrealistic frozen input, the local feedback models produce realistic vergence responses. Therefore, it seems worthwhile to investigate models that combine visual feedback with a local feedback mechanism. The local feedback estimates the progress of the movement and will include the saccade-related vergence enhancements, if present. In the monocular models of the vergence–saccade interaction, the vergence local feedback will include the contribution to the vergence response by the asymmetry of the two monocular saccades. 
The computational study presented in this paper was undertaken to investigate if a single model can describe the dynamics of pure as well as saccade-related vergence movements in response to visual stimuli. The study is mainly devoted to modeling vergence movements made in response to step changes in target vergence because the various models in the literature were proposed to describe these responses. Simulations of responses to sinusoidal stimuli were included to study the generality of the model. 
Methods
Eye movements were simulated and analyzed using the Mathematica symbolic computing package. To simulate vergence eye movements as a function of time, the vergence control loop was combined with second-order linear filter approximations to the two ocular motor plants. In the Laplace domain, the transfer function H of each plant is given by 
H ( s ) = 1 ( τ 1 s + 1 ) ( τ 2 s + 1 ) ,
(1)
where the time constants of the low-pass filter are τ 1 = 8 ms and τ 2 = 150 ms, respectively. The approximation has been used in both saccadic (Van Gisbergen, Robinson, & Gielen, 1981) and vergence (Pobuda & Erkelens, 1993) models. The static gains of the plants are taken to be dimensionless implying that both input and output of the plants are expressed as angles (in degrees). The conversion from retinal disparity to muscle forces (the real input to the oculomotor plant) has been left out of the computations because it is largely unknown and irrelevant for studying the relationship between retinal disparity and ocular vergence. 
I followed the suggestion of Robinson (1966) to estimate the input signal to the plants that is required to produce realistic vergence eye movements in response to step changes in target vergence. Robinson suggested that the required motoneuron responses (supposed to be linearly related to the motoneuron's disparity-defined input signal) could be approximated by step changes in activity. The translation of the suggestion in stimulus terms is that the temporal profile of motoneuron activity is a delayed estimate of target vergence. Keller and Robinson (1972) and later Gamlin and Mays (1992) measured the profiles of motoneuron activity. Both studies found that instantaneous discharge rate showed step-like profiles. To test the hypothesis that plant input is a delayed estimate of target vergence, I computed vergence eye movements induced by stepwise changes in plant input, representing stepwise changes in the motoneuron activities of eye muscles. The simulated responses were compared to a set of voluntary vergence movements made between small, well-defined LED targets (Erkelens et al., 1989). The wide-range responses (amplitudes up to 34°), measured with high precision, pose an interesting challenge for any vergence model. The challenge for the models is to produce stable responses in combination with steady-state errors of less than 2% over a range of 34° (Erkelens et al., 1989) while having delays between stimuli and responses of about 160 ms (Rashbass & Westheimer, 1961b). Furthermore, adaptation experiments showed that peak velocity of vergence responses may even double without loss of stability (Munoz, Semmlow, Yuan, & Alvarez, 1999; Takagi et al., 2001). The authors of the adaptation studies have explained the adapted vergence responses by supposing that vergence is controlled by two components of which the transient open-loop component contains an adaptive gain and the sustained feedback component is not affected by adaptation. Based on simulation results of the present study, I shall propose a more likely explanation. 
Results
Figure 1A shows a schematic top view of the ocular plants and the basic components for the neural control of vergence (ϕ). Due to the fact that the eye muscles are arranged symmetrically for vergence, the controller can send the same plant input (PI) to both eyes. In the study of Erkelens et al. (1989), subjects made voluntary vergence movements from one LED target (A) to another (B, C, D, E) and back again (Figure 1). Shifts in target vergence (θ) are supposed to be induced by shifts in attention from one target to another. The supposition is realistic because responses to stabilized targets have shown that vergence is controlled by attended rather than fixated targets (Erkelens & Collewijn, 1991). Shifts in target vergence are assumed to occur stepwise (Figure 1C), and as a result, disparity changes stepwise at the same time too (Figure 1D). In response to the steps in target vergence and disparity vergence, movements were simulated by sending step-like plant input (Figure 1A), after a certain delay, to the vergence plant specified by Equation 1. Since the static gain of the plant is chosen to be unity and dimensionless, plant input (motoneuron activity) and output (vergence) have the same dimension. For that reason, plant input is expressed in terms of vergence. Figure 1C shows the simulated vergence movements. Disparity (Figure 1D) was computed as the difference between the target and ocular vergence traces of Figure 1C. Comparison of the computed profiles with the experimental vergence profiles in Figure 7 of Erkelens et al. shows that simulated and measured profiles are in fairly good agreement. It should be borne in mind that by using an approximation of the plant mechanics, simulations may lack fine details but should capture the main features of vergence movements. Vergence profiles show similar durations for the simulated and measured responses. Maximum velocity increases linearly with amplitude at a rate of about 5°/s per degree of vergence (Figure 1E), which is similar to the measured slopes shown in Figures 5 and 6 of Erkelens et al. The relationship between velocity and amplitude shows that the vergence system is linear in a range of 34°. From the agreement between simulation and data and given that the plant is modeled by a simple approximation, it seems justified to conclude that stepwise changes in plant input are a reasonable approximation of the actual motoneuron signals that generate vergence responses associated with changing fixation between targets differing in distance to the head. 
Figure 1
 
Simulations of vergence movements. (A) Schematic top view of the ocular plants and the basic components for the neural control of vergence. The controller responds to disparity (d) being the difference between target vergence (θ) and ocular vergence (ϕ). Plants receive input (PI) from the controller. (B) Stepwise changes in plant input associated with changing target vergence from one target (A) to another (B, C, D, E) and back (see Erkelens et al., 1989). (C) Profiles of target vergence and the computed vergence responses. (D) Disparity profiles. (E) Computed peak velocities as a function of vergence amplitude.
Figure 1
 
Simulations of vergence movements. (A) Schematic top view of the ocular plants and the basic components for the neural control of vergence. The controller responds to disparity (d) being the difference between target vergence (θ) and ocular vergence (ϕ). Plants receive input (PI) from the controller. (B) Stepwise changes in plant input associated with changing target vergence from one target (A) to another (B, C, D, E) and back (see Erkelens et al., 1989). (C) Profiles of target vergence and the computed vergence responses. (D) Disparity profiles. (E) Computed peak velocities as a function of vergence amplitude.
By comparing plant input signals (Figure 1B) with disparity signals (Figure 1D), it follows that the retinal disparity signal by itself cannot serve as the driving signal for vergence responses. After a certain interval following the stepwise changes in target vergence, disparity decreases monotonically to close to zero whereas plant input remains constant. As a consequence, the gain element transferring disparity into plant input would have to be strongly disparity-amplitude-dependent. Furthermore, the gain between disparity and target vergence would eventually have to become so high (on the order of 50) that the vergence would become unstable. In conclusion, the disparity signal alone cannot possibly maintain the required tonic level of motoneuron activity. However, a well-accepted idea in the literature of eye movements is that commands are given in the form of velocity signals that are subsequently integrated over time by neural integrators. Inspired by this idea, I designed a simple feedback model in which vergence responses are generated by retinal disparity and its time integral as parallel input signals (Figure 2A). After being delayed by 160 ms (Δt), the retinal disparity signal is processed by three elements. The element VC, short for velocity command, converts the disparity signal into a vergence velocity signal. According to the linear relationship between vergence amplitude (which is similar to initial disparity) and vergence velocity (Figure 1E), the gain of VC should be a constant close to 5°/s per degree of vergence (s−1). The elements NI, the perfect neural integrators, convert the vergence velocity signal into vergence position signals. The pulse command PC must have the dimension of time in order to generate a signal that can be added to the integrators NI. After being divided by a factor of two, the NIs produce the sustained plant input (PI) for the left and right eyes that is required to make ocular vergence ϕ equal to target vergence θ in static conditions. In dynamical conditions, PI follows from the output of both NI and PC: 
P I ( t ) = P I ( 0 ) + 1 2 [ V C P C d ( t ) + N I ( t ) ] ,
(2)
where 
N I ( t ) = V C 0 t d ( t i ) d t i ,
(3)
and 
d ( t ) = d ( t Δ t ) = θ ( t Δ t ) φ ( t Δ t ) .
(4)
Unprimed symbols indicate physical quantities and primed symbols indicate neural signals in all equations and models of this study. Single- and double-primed symbols may just differ by a delay or result from computations of different sets of signals. The left graph of Figure 2B shows simulations of the model for three values of VC smaller than or equal to the optimal value of 5 s−1. Higher values of VC cannot be chosen because they lead to unstable responses. If the value of VC is sufficiently low, the vergence response (red in Figure 2B) is stable but too slow to be realistic. Increasing the vergence speed by increasing VC renders the response underdamped (blue and green traces). These examples show that another input signal is essential for the generation of fast and damped vergence responses. The third element PC, pulse command, produces a signal that generates transient motoneuron activity. The right graph of Figure 2B shows three simulations for different PC gains. Combinations of VC and PC that are required to generate vergence responses of sufficient speed produce underdamped vergence movements. The observed underdamped behavior means that the adaptation results of Munoz et al. (1999) and Takagi et al. (2001), showing that vergence responses remain damped while their speed increases are not simulated by the model. Other evidence for the model to be incorrect is that any saccade-related enhancement would be detected only after a visual latency. As shown in Busettini and Mays (2005b), the post-saccadic vergence takes into account the saccadic-related enhancement much earlier than after a visual latency. Therefore, I consider it unlikely that a model exclusively based on visual feedback represents the control of vergence movements. 
Figure 2
 
Visual feedback model. (A) In the model, disparity is converted by the velocity command VC into a velocity signal. The velocity signal is converted into plant inputs (PIs) via two parallel paths. The velocity signal is converted into position signals by the pulse command PC and by temporal integration (NI). (B) Vergence responses ϕ simulated with three different values for VC and PC. (C) Vergence responses for the same values for VC and PC but without any delay.
Figure 2
 
Visual feedback model. (A) In the model, disparity is converted by the velocity command VC into a velocity signal. The velocity signal is converted into plant inputs (PIs) via two parallel paths. The velocity signal is converted into position signals by the pulse command PC and by temporal integration (NI). (B) Vergence responses ϕ simulated with three different values for VC and PC. (C) Vergence responses for the same values for VC and PC but without any delay.
The cause of the model's instability is the relatively long visual delay. This is illustrated by simulations of the model in which the delay is set to 0 ms (Figure 2C). Now, the vergence responses become fast and damped. The problem created by the delay is that disparity in combination with its time integral cannot produce the required stepwise changes of plant input. A solution for this problem is to use target vergence instead of disparity as the driving stimulus because target vergence of static targets is constant by nature. Since target vergence is not signaled by any individual visual or motor signal, it can only be represented by a combination of signals. Target vergence θ may be represented by a signal θ′ that is obtained by combining an estimate of disparity d′ and a neural signal indicating ocular vergence ϕ′. At the location in the model where θ′ is computed, θ′ is given by the following equation: 
θ ( t ) = d ( t ) + φ ( t ) .
(5)
Corollary discharge, also called efference copy, of the vergence state is assumed to supply the signal ϕ′ in the model (Figure 3A). The delay of the corollary signal is indicated by Δt c. Since signal d′(t) represents disparity of the stimulus d(t − Δt v) and signal ϕ′(t) represents ocular vergence ϕ(t − Δt c − Δt e), the signal θ′(t) can only represent target vergence of the stimulus θ(t − Δt v) accurately if Δt v = Δt c + Δt e. To study stability effects due to variability in the delays, I have divided the total time Δt of the feedback loop into visual (Δt v) and motor delays (Δt m). Visual delays have been measured by Schmolesky et al. (1998). Relative to stimulus onset, the authors measured delays of neural activity in V1 (44 ms), V2 (59 ms), V3 and MT (72 ms), and FEF (80 ms) in monkey. Assuming that target vergence is first represented at the level of the inferior temporal cortex where signals arrive slightly later than in MT and assuming that visual delays are similar in human and monkey, I have chosen Δt v = 75 ms. In accordance with Zee et al. (1992), I assume that Δt e = 3 ms and thus much shorter than the delay of visual feedback. As a result, Δt c = 72 ms. In order to keep the duration of the total delay of the loop at 160 ms, the motor delay has to be chosen as Δt m = 85 ms. The term motor delay as it is used here is slightly deceptive because, among other processing times, the delay includes time needed for the decision-making process in area FEF. The model presented in Figure 3A resembles the local feedback model for the generation of pure saccades, originally proposed by Robinson et al. (Robinson, 1975; Zee, Optican, Cook, Robinson, & Engel, 1976). The saccadic model was later extended by a similarly structured vergence model to describe pure vergence movements and to explain saccade–vergence interaction by shared omnipause neuron gating (Zee et al., 1992). A major difference between the Zee model and the current model is the neural representation of target vergence. The current model enables vergence to respond continuously to visual feedback. In order to extract a disparity signal from the target vergence signal, I propose that the signal representing target vergence, θ′, is compared with a local neural estimate of the vergence state, ϕ′. Ideally, the resulting signal d″ is a short-latency estimate of the retinal disparity signal d. Similarly as d′ in the previous model (Figure 2A), d″ drives a set of premotor neurons to produce the velocity command VC. VC's output is passed to the pulse command PC and both neural integrators NI. PC generates the initiation of vergence responses, whereas the NIs produce the eye position commands that are needed to keep vergence at the desired steady state. The vergence position command is fed back after a short time delay of Δt e to provide the local efference copy of ϕ necessary for the computation of d′. The plant input PI is described by the following equations: 
P I ( t ) = P I ( 0 ) = 1 2 [ V C P C d ( t ) + N I ( t ) ] ,
(6)
where 
N I ( t ) = V C 0 t d ( t i ) d t i ,
(7)
and 
d ( t ) = θ ( t ) φ ( t ) ,
(8)
where 
φ ( t ) = N I ( t Δ t e ) ,
(9)
and 
θ ( t ) = θ ( t Δ t m ) ,
(10)
where 
θ ( t Δ t m ) = d ( t Δ t m ) + φ ( t Δ t m ) ,
(11)
where 
d ( t Δ t m ) = θ ( t Δ t m Δ t v ) φ ( t Δ t m Δ t v ) ,
(12)
and 
φ ( t Δ t m ) = φ ( t Δ t m Δ t c ) .
(13)
Figure 3B shows simulations by the model for three VC and PC values. Vergence responses are damped but slow (red trace) for small VC values. Higher VC values result in underdamped responses (blue and green traces). The model's behavior is inappropriate because VC alone should produce slow and damped responses that by addition of the PC pulse component should become sufficiently fast. As mentioned before, input signal d″ should be a short-latency estimate of the disparity stimulus d. In other words, it is not the visual feedback signal d′, with its long delay, driving the response but the short-delay estimate d″. By doing so, the stability of the system should be greatly increased. The left graph of Figure 3C shows that this is not the case due to the fact that θ′ does not represent target vergence θ well during the vergence response. The reason for the mismatch is that the chosen corollary discharge signal (i.e., the output of the vergence–velocity integrator NI) represents ocular vergence in static but not in dynamic conditions. The solution to this problem is to reconstruct an estimate of the actual eye rotations from the neural commands using a dynamic model of the plant (DP in Figure 4A). The right graph of Figure 3 shows that the signal Le′ − Re′ of this model is better than ϕ′ in representing ϕ. Now, θ′ correctly signals the stepwise change in target vergence. The signal NI is not suited to serve as input for DP because DP requires the same input as does the plant, and therefore, its input should also include the PC signal. An appropriate input signal for DP is a signal that equals signal PI. Consequently, in the set of Equations 6 to 13, which describes the plant input PI of the model, Equation 11 is replaced by 
θ ( t Δ t m ) = d ( t Δ t m ) + L e ( t Δ t m ) Re ( t Δ t m ) .
(14)
Now, target vergence is accurately represented in both static and dynamic conditions (see signal θ″ in Figure 4B). Figure 4C shows simulations of the model for various PC values. Both VC and PC contribute to the speed of vergence responses. The PC signal increases speed and decreases duration of the responses. High PC values generate overshoot of the vergence response, however, without causing further oscillations. After overshooting target vergence, responses decay gradually to their steady state (Figure 4C, blue and green traces). The combination of overshoot and decay resembles vergence responses that have been observed in vergence adaptation experiments (Munoz et al., 1999; Takagi et al., 2001). The model cannot generate overshoot of the vergence responses if PC is absent, irrespective how high the value of VC. The reason is that without the contribution of PC, PI cannot surpass or even reach the stepwise change that is required for the production of realistic vergence responses (Figure 4C, red trace). Plant input changes about stepwise for specific combinations of VC and PC values (Figure 4D, red trace). This example shows that contributions of pulse-like (PC) and integrated (NI) input signals in parallel are needed to produce step changes of motoneuron activity. Higher PC values cause pulse-step-like plant input signals and some overshoot of the vergence responses (Figure 4C, blue and green traces). Since delays related to different processes play a role in the model, it is necessary to simulate vergence responses for several combinations of the delays Δt c, Δt v, and Δt m. The model is only realistic if it allows for some natural variability in the delays. The combination Δt c and Δt v is critical because it determines how closely the neural representation of target vergence matches the stimulus. Figure 4E shows simulations in which Δt c and Δt v differ by 10 ms from the ideal condition where Δt c and Δt v are identical. The mismatches between the delays cause only moderate deviations in the later part of the vergence responses. A striking property of the model is that neither motor delay Δt m nor visual delay Δt v affects the dynamics of the vergence responses, provided that the visual delay Δt v is about equal to the corollary delay Δt c. Visual, efferent, and motor delays only affect the latency of vergence responses. Local feedback determines the dynamics. 
Figure 3
 
Visual–local feedback model. (A) In the model, the disparity signal d′ is converted into the target vergence signal θ′ by adding d′ to a corollary of the NI signal representing ocular vergence in static conditions. The target vergence signal θ′ is delayed by Δt m before it (θ″) reaches the local feedback controller. The local feedback signal is delayed by Δt e before it is subtracted from the θ″ signal to produce the short-latency copy of the disparity stimulus (d″). Disparity is delayed by Δt v and corollary discharge by Δt c + Δt e. (B) Vergence responses ϕ simulated with three different values for VC and PC. (C) Signals at the location of the white star in the model for VC = 2 s−1 and PC = 0 s.
Figure 3
 
Visual–local feedback model. (A) In the model, the disparity signal d′ is converted into the target vergence signal θ′ by adding d′ to a corollary of the NI signal representing ocular vergence in static conditions. The target vergence signal θ′ is delayed by Δt m before it (θ″) reaches the local feedback controller. The local feedback signal is delayed by Δt e before it is subtracted from the θ″ signal to produce the short-latency copy of the disparity stimulus (d″). Disparity is delayed by Δt v and corollary discharge by Δt c + Δt e. (B) Vergence responses ϕ simulated with three different values for VC and PC. (C) Signals at the location of the white star in the model for VC = 2 s−1 and PC = 0 s.
Figure 4
 
Improved model. (A) In the model, the disparity signal d′ is converted into target vergence θ′ by adding d′ to the neural estimate of ocular vergence (Le′ − Re′). Disparity is delayed by Δt v and the efference copy signals (DP) by Δt c. The target vergence signal θ′ is delayed by Δt m before it reaches the local feedback controller. The local feedback signal is delayed by Δt e before it is subtracted from the delayed θ′ signal to produce the short-latency copy of the disparity signal (d″). The red arrow indicates where the output of saccadic burst neurons affects the vergence control loop to produce intra-saccadic vergence enhancement. (B) Signals at the location of the red star in the model for VC = 5 s−1 and PC = 0 s. (C) Vergence responses simulated with one value for VC and three different values for PC. (D) Plant input signals PI for the vergence responses shown in (C). (E) Vergence responses simulated with three combinations of visual and efference copy delays.
Figure 4
 
Improved model. (A) In the model, the disparity signal d′ is converted into target vergence θ′ by adding d′ to the neural estimate of ocular vergence (Le′ − Re′). Disparity is delayed by Δt v and the efference copy signals (DP) by Δt c. The target vergence signal θ′ is delayed by Δt m before it reaches the local feedback controller. The local feedback signal is delayed by Δt e before it is subtracted from the delayed θ′ signal to produce the short-latency copy of the disparity signal (d″). The red arrow indicates where the output of saccadic burst neurons affects the vergence control loop to produce intra-saccadic vergence enhancement. (B) Signals at the location of the red star in the model for VC = 5 s−1 and PC = 0 s. (C) Vergence responses simulated with one value for VC and three different values for PC. (D) Plant input signals PI for the vergence responses shown in (C). (E) Vergence responses simulated with three combinations of visual and efference copy delays.
Different from existing models for saccades and vergence, the dual visual–local feedback model is equipped with non-resettable local feedback integrators. Figure 5 shows examples of vergence responses illustrating that using resettable integrators would require a complex timing mechanism. Figure 5A shows simulated vergence movements in response to two successive steps in target vergence of unequal size. Input signal θ″ is indeed a delayed copy of target vergence, and signal ϕ″, the local feedback signal, approaches target vergence in static vergence conditions. To allow d″ to be the short-delay estimate of d, the feedback integrator should not be reset to zero before the completion of the total response. In the example of Figure 5A, this means that the integrator can first be reset after 1.5 to 2 s. In the example of Figure 5B, there are no appropriate times for resetting the integrators as long as the stimulus keeps oscillating. These examples show that equipping the model with non-resettable integrators simplifies the model greatly. Calculated gain–phase relationships between oscillating target vergence and ocular vergence are shown in Figure 5C. Comparison of the model results with measured gain–phase relationships (Erkelens & Collewijn, 1985, 1991; Pobuda & Erkelens, 1993; Rashbass & Westheimer, 1961b) shows that phase lags produced by the model are somewhat longer than those of real vergence responses. Phase increases about linearly with frequency corresponding with a delay of about 200 ms in the feedback loop. Simulated gains are slightly higher than measured gains reported in the literature, which may indicate that changing disparities are detected and processed less optimally by the oculomotor system than disparities that remain constant for about 200 ms. 
Figure 5
 
Behavior of the improved model. (A) Target vergence (dotted line) and ocular vergence (continuous line) movements (left panel) and signals θ″, ϕ′, and d″ (right panel) simulated by the model in response to two successive steps in target vergence. The first step in target vergence occurred at t = 0 s and the second one at t = 0.4 s. (B) Vergence movements induced by sinusoidal oscillations of target vergence (dotted lines) of frequencies of 0.125 Hz and 0.5 Hz, respectively. (C) Gain–phase relationship between simulated ocular vergence and target vergence.
Figure 5
 
Behavior of the improved model. (A) Target vergence (dotted line) and ocular vergence (continuous line) movements (left panel) and signals θ″, ϕ′, and d″ (right panel) simulated by the model in response to two successive steps in target vergence. The first step in target vergence occurred at t = 0 s and the second one at t = 0.4 s. (B) Vergence movements induced by sinusoidal oscillations of target vergence (dotted lines) of frequencies of 0.125 Hz and 0.5 Hz, respectively. (C) Gain–phase relationship between simulated ocular vergence and target vergence.
The Zee model has been developed to explain saccade–vergence interaction (Zee et al., 1992). To investigate whether the dual visual–local feedback model can also describe enhancement of vergence responses during saccades, a saccadic model is added to the current vergence model. The combined model can simulate vergence responses between targets differing in distance and direction. The saccadic system is modeled by the positional local feedback system as it was implemented by Zee et al. (1992). The desired change in conjugate direction serves as its input, which is compared with an efference copy of the change in conjugate direction to produce the instantaneous motor error. This signal drives the saccadic burst neurons to produce a velocity command as long as the burst neurons are disinhibited by a pause of the omnipause neurons. The saccadic system and vergence system operate in parallel and their output is added at the level of the motoneurons to produce the plant input of the left and right eyes. The omnipause neurons do not connect to the vergence system so that vergence can respond to changes in target vergence at all times. The only difference of the vergence component in the extended model with the one shown in Figure 5A is that the vergence velocity command VC is made susceptible to output of saccadic burst neurons (SB): 
V C n e w = V C ( 1 + G S B / S B max ) ,
(15)
where G is a gain factor and SBmax is the maximum output of the saccadic burst neurons so that SB/SBmax represents normalized saccadic burst output that runs between 0 and 1. The chosen enhancement of vergence during saccades resembles one of the possible interactions that were proposed by Busettini and Mays (2005b). A key difference is that the dual visual–local feedback model has a multiplicative interaction with the vergence velocity command, whereas the model of Busettini and Mays (2005b) has a multiplicative interaction with an estimate of the vergence error, which is then converted to the actual motor command. Figure 6 shows simulations of eye movements in response to targets differing in distance and direction. To show the effect of vergence enhancement by saccades clearly, eye movements without (Figure 6A) and with (Figure 6B) saccadic–vergence interaction are plotted next to each other. Since the model is linear, absolute amplitudes of saccades and vergence are irrelevant. The size of enhancement is related to the amplitudes of vergence and saccades relative to each other. Figure 6 shows simulations in which the ratio between vergence and saccadic amplitude was 1 to 4. This ratio allowed comparison with large enhancement effects reported by Erkelens et al. (1989) who showed that vergence responses were completed for about 80% to 90% during the saccades. The model could achieve similar enhancements by choosing values for G up to 2, so that the vergence velocity command VC was maximally tripled during saccades. The simulated saccadic and vergence velocity and position profiles are very similar to the measured ones shown by Erkelens et al. and qualitatively similar to those of Zee et al. (1992) and Busettini and Mays (2005a) who used larger vergence–saccade ratios. Comparison of the profiles without (Figure 6A) and with (Figure 6B) saccadic–vergence interaction shows that interaction enhances intra-saccadic vergence from about 50% to 90% of the total vergence response and decreases post-saccadic smooth vergence from 50% to 10%. Comparison of input signals in Figure 6 shows that absence or presence of intra-saccadic vergence enhancement does not have an effect on the representation of target vergence (θ″) in the feedback loop. Enhanced vergence changes the neural representations of ocular vergence (ϕ″) and retinal disparity (d″) oppositely, so that their sum, θ″, remains unchanged. As a result, the differences in time delays of the long and local feedback loops do not cause any instability or interruption of the enhanced vergence response. The vergence velocity command VC is the only element of the model where the multiplicative interaction can occur without changing the relationship between visual and local feedback. VC is the only element that is shared by the three feedback loops of the model (visual, efferent, and local), and therefore, the multiplicative interaction affects the signals in all loops in a similar way. 
Figure 6
 
Simulated saccadic–vergence interaction. (A) Velocity and position profiles of saccades and vergence and signals θ″, ϕ″, and d″ simulated by the combined saccade and improved vergence model without any interaction between the saccadic and vergence components. (B) Similar profiles and signals as in (A), but now vergence is enhanced during the saccade. The dotted lines mark the end of the conjugate saccades.
Figure 6
 
Simulated saccadic–vergence interaction. (A) Velocity and position profiles of saccades and vergence and signals θ″, ϕ″, and d″ simulated by the combined saccade and improved vergence model without any interaction between the saccadic and vergence components. (B) Similar profiles and signals as in (A), but now vergence is enhanced during the saccade. The dotted lines mark the end of the conjugate saccades.
Discussion
Zee et al. (1992) proposed the model in which vergence is driven by desired change in vergence position. The model was developed to explain saccade–vergence interaction and was inspired by an adapted version (Scudder, 1988) of the local feedback model for saccades (Robinson, 1975). The concept of local feedback for the control of vergence has been adopted and further developed in other studies (Busettini & Mays, 2005a; Kumar et al., 2005; Maxwell et al., 2010; Mays & Gamlin, 1995; Zhu et al., 2008). The present computational analysis supports this type of model by showing that alternative models, in which retinal disparity is the prime driver of vergence, do not work. The current model suggests a binocular neural mechanism that is able to drive vergence movements. However, it may well be that vergence is driven by a neural mechanism having monocular components as has been proposed by a number of studies (King & Zhou, 2002; Sylvestre et al., 2002; Van Horn & Cullen, 2008; Zhou & King, 1998). Until now, the two types of models are functionally equivalent. 
Zee et al. (1992) assumed that, analogous to the saccade model, the input signal “desired change in vergence” remains fixed during the latency and execution of responses. Given the delay and duration of vergence responses, the fixed input implies that the model does not respond to changes in target vergence for periods of up to about 1 s. This is not realistic. Records of vergence responses to step (Erkelens, 1987; Rashbass & Westheimer, 1961b), ramp (Hung, Semmlow, & Ciuffreda, 1986; Semmlow, Hung, & Ciuffreda, 1986), sinusoidal (Erkelens & Collewijn, 1985; Pobuda & Erkelens, 1993; Rashbass & Westheimer, 1961b), and double-step (Erkelens, 1987; Pobuda & Erkelens, 1993) changes in target vergence showed that vergence responds continuously to changes in target vergence. The present study presents a realistic modification of the Zee model. The model describes how vergence can respond to varying, long-latency, visual stimuli and can show stable dynamics due to short-latency, local feedback. The requirement for uninterrupted feedback implies that target vergence has to be derived from instantaneous signals. Target vergence cannot be derived from retinal information alone because a retinal signal does not dissociate target motion from ocular motion. The combination of absolute disparity and ocular vergence signals, both signals known to be present in the brain, provides a good representation of target vergence. In principle, ocular vergence may be indicated by two different neural signals, namely, by afferent signals originating from proprioceptors in eye muscles and by a corollary or copy of the vergence command to the plants. I have modeled ocular vergence as a corollary signal because of the existing evidence favoring the use of corollary signals in the control of saccades (Sommer & Wurtz, 2008). Simulations showed that in order to be useful the corollary signal should be sophisticated in that it should closely capture the dynamics of the plants. A consequence of dual visual and local feedback is that such a control is incompatible with modeling local feedback as a resettable integrator. The concept was proposed for the local feedback loop of saccades (Jürgens, Becker, & Kornhuber, 1981; Scudder, 1988) to explain the absence of signals in the superior colliculus that encode target position relative to the head. Electrical stimulation experiments provided evidence for reset of the integrator (Nichols & Sparks, 1995). However, the electrical stimulation results were not reproduced during the natural generation of saccades (Goossens & van Opstal, 1997). To date, the existence of a resettable integrator is still debated (Corneil, Hing, Bautista, & Munoz, 1999; Kardamakis & Moschovakis, 2006) and still lacks direct neurophysiological proof. 
There is a longstanding debate about the presence of prediction or anticipation of stimulus motion in the control of vergence (Rashbass & Westheimer, 1961b). Prediction of target motion would explain why phase lags in response to oscillating stimuli are shorter than allowed by the delay of visual feedback. Until now, prediction of target motion has not been included in models of vergence control. There is a favored location where prediction can be included in the dual visual–local feedback model, namely, at the location where the target vergence signal θ′ is computed (Figure 4A). Prediction of the target vergence signal effectively means shortening of Δt m. As has been argued earlier, Δt m does not affect the dynamics of vergence responses. Shortening of Δt m causes that ocular vergence will follow target vergence with less phase lag, keeping disparity small. The effects of small disparities are that the target remains projected close to the foveae and retinal speeds stay low. 
Based on the existing literature on vergence control and the present study, I hypothesize that vergence is driven by two different disparity-related mechanisms. One mechanism is driven by target vergence of an attended target. This system is modeled and analyzed in the present study. A second reflexive mechanism is driven by retinal disparity of the total visual field. There are several arguments that suggest that the two mechanisms are distinct and in competition with each other. During binocular vision, the visual field may contain a range of retinal disparities depending on objects' distances relative to the head. Binocular foveation of a particular object establishes that this object's retinal disparity is minimized at the expense of disparities of the other objects in the visual field. Changing fixation requires selection of another target by visual attention and usually involves both saccadic and vergence movements of the eyes (Erkelens et al., 1989). Zee et al. (1992) proposed a saccade-like mechanism for the generation of vergence movements made between static targets. The dual visual–local feedback model contains a saccade-like mechanism that can drive all vergence responses. Selection and attention play key roles in the generation of saccades. Selection and attention must also be vital mechanisms in the control of pure vergence responses. Without these mechanisms, it would not be possible to change vergence from one binocularly foveated object to another. It is well documented that the frontal eye fields (FEFs) are involved in visual attention and the generation of saccades. Recent findings suggest that the two roles of the FEF work together to select both the features of a target and the appropriate movement to foveate it (Schafer & Moore, 2007). Gamlin and Yoon (2000) were the first to report vergence-related cells in and near FEF. Recently, Akao, Kurkin, Fukushima, and Fukushima (2005) found that the FEF also carries visual signals appropriate to be converted into motor commands for vergence. These results suggest that voluntary saccades and vergence are generated by similar mechanisms. 
The attention-related vergence system is probably not the only disparity-based controller of vergence. Evidence for inattentive control is provided by vergence eye movements that are driven by disparity in the absence of binocularly attended targets. Eye movements, recorded during the viewing of large oscillating random-dot stereograms (Julesz, 1971), showed ongoing vergence responses even if the subjects perceived the stereograms in rivalry (Erkelens & Collewijn, 1985). Transient vergence responses lasting up to 10 s were measured if the stereograms were stabilized such that subjects could not fuse them to a single stereoscopic percept (Erkelens, 1987). While perceiving binocular rivalry, the subjects attended monocular targets, and thus, the vergence responses must have been induced by the inattentive system alone. Torsional vergence (Howard & Zacher, 1991), vertical vergence (Howard, Allison, & Zacher, 1997), and horizontal vergence (Regan, Erkelens, & Collewijn, 1986) that are induced by large fused stereograms may also reflect activity of the inattentive system. However, since the stereograms usually contain binocularly attended targets, vergence may be driven by both the attentive and inattentive systems. How the two systems interact is largely unknown. There is some evidence that the inattentive system is not fully suppressed by the attentive system. Stevenson, Lott, and Yang (1997) showed that the instruction to fixate a stationary target reduced but did not null vergence responses induced by the oscillating disparity of the surrounding stereogram. There is also evidence that the two systems do not act fully independent of each other. Busettini, Fitzgibbon, and Miles (2001) showed that vergence responses to large stereograms are prominent in the wake of saccades but are smaller later on. Busettini et al. interpreted the results in terms of post-saccadic enhancement of vergence responses. An alternative interpretation that fits in the concept of having two interacting systems is that vergence responses induced by the inattentive system are suppressed to a certain extent during binocular fixation in between saccades. According to this interpretation, the post-saccadic responses reveal contributions of the inattentive system in conditions that contributions of the attentive system are not yet fully deployed. A remarkable finding of Busettini et al. was that latencies of post-saccadic vergence responses were just about 80 ms and, thus, much shorter than those of attentive vergence responses. The difference in latency is in line with my hypothesis that the two systems are driven by different stimuli, the attentive system by target vergence of the attended object, and the inattentive system by absolute disparities of all objects in the visual field. 
Acknowledgments
I thank an anonymous reviewer for detailed and helpful comments. 
Commercial relationships: none. 
Corresponding author: Casper J. Erkelens. 
Email: c.j.erkelens@uu.nl. 
Address: Department of Physics, Faculty of Science, Helmholtz Institute, Utrecht University, PrincetonPlein 5, 3584 CC Utrecht, The Netherlands. 
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Figure 1
 
Simulations of vergence movements. (A) Schematic top view of the ocular plants and the basic components for the neural control of vergence. The controller responds to disparity (d) being the difference between target vergence (θ) and ocular vergence (ϕ). Plants receive input (PI) from the controller. (B) Stepwise changes in plant input associated with changing target vergence from one target (A) to another (B, C, D, E) and back (see Erkelens et al., 1989). (C) Profiles of target vergence and the computed vergence responses. (D) Disparity profiles. (E) Computed peak velocities as a function of vergence amplitude.
Figure 1
 
Simulations of vergence movements. (A) Schematic top view of the ocular plants and the basic components for the neural control of vergence. The controller responds to disparity (d) being the difference between target vergence (θ) and ocular vergence (ϕ). Plants receive input (PI) from the controller. (B) Stepwise changes in plant input associated with changing target vergence from one target (A) to another (B, C, D, E) and back (see Erkelens et al., 1989). (C) Profiles of target vergence and the computed vergence responses. (D) Disparity profiles. (E) Computed peak velocities as a function of vergence amplitude.
Figure 2
 
Visual feedback model. (A) In the model, disparity is converted by the velocity command VC into a velocity signal. The velocity signal is converted into plant inputs (PIs) via two parallel paths. The velocity signal is converted into position signals by the pulse command PC and by temporal integration (NI). (B) Vergence responses ϕ simulated with three different values for VC and PC. (C) Vergence responses for the same values for VC and PC but without any delay.
Figure 2
 
Visual feedback model. (A) In the model, disparity is converted by the velocity command VC into a velocity signal. The velocity signal is converted into plant inputs (PIs) via two parallel paths. The velocity signal is converted into position signals by the pulse command PC and by temporal integration (NI). (B) Vergence responses ϕ simulated with three different values for VC and PC. (C) Vergence responses for the same values for VC and PC but without any delay.
Figure 3
 
Visual–local feedback model. (A) In the model, the disparity signal d′ is converted into the target vergence signal θ′ by adding d′ to a corollary of the NI signal representing ocular vergence in static conditions. The target vergence signal θ′ is delayed by Δt m before it (θ″) reaches the local feedback controller. The local feedback signal is delayed by Δt e before it is subtracted from the θ″ signal to produce the short-latency copy of the disparity stimulus (d″). Disparity is delayed by Δt v and corollary discharge by Δt c + Δt e. (B) Vergence responses ϕ simulated with three different values for VC and PC. (C) Signals at the location of the white star in the model for VC = 2 s−1 and PC = 0 s.
Figure 3
 
Visual–local feedback model. (A) In the model, the disparity signal d′ is converted into the target vergence signal θ′ by adding d′ to a corollary of the NI signal representing ocular vergence in static conditions. The target vergence signal θ′ is delayed by Δt m before it (θ″) reaches the local feedback controller. The local feedback signal is delayed by Δt e before it is subtracted from the θ″ signal to produce the short-latency copy of the disparity stimulus (d″). Disparity is delayed by Δt v and corollary discharge by Δt c + Δt e. (B) Vergence responses ϕ simulated with three different values for VC and PC. (C) Signals at the location of the white star in the model for VC = 2 s−1 and PC = 0 s.
Figure 4
 
Improved model. (A) In the model, the disparity signal d′ is converted into target vergence θ′ by adding d′ to the neural estimate of ocular vergence (Le′ − Re′). Disparity is delayed by Δt v and the efference copy signals (DP) by Δt c. The target vergence signal θ′ is delayed by Δt m before it reaches the local feedback controller. The local feedback signal is delayed by Δt e before it is subtracted from the delayed θ′ signal to produce the short-latency copy of the disparity signal (d″). The red arrow indicates where the output of saccadic burst neurons affects the vergence control loop to produce intra-saccadic vergence enhancement. (B) Signals at the location of the red star in the model for VC = 5 s−1 and PC = 0 s. (C) Vergence responses simulated with one value for VC and three different values for PC. (D) Plant input signals PI for the vergence responses shown in (C). (E) Vergence responses simulated with three combinations of visual and efference copy delays.
Figure 4
 
Improved model. (A) In the model, the disparity signal d′ is converted into target vergence θ′ by adding d′ to the neural estimate of ocular vergence (Le′ − Re′). Disparity is delayed by Δt v and the efference copy signals (DP) by Δt c. The target vergence signal θ′ is delayed by Δt m before it reaches the local feedback controller. The local feedback signal is delayed by Δt e before it is subtracted from the delayed θ′ signal to produce the short-latency copy of the disparity signal (d″). The red arrow indicates where the output of saccadic burst neurons affects the vergence control loop to produce intra-saccadic vergence enhancement. (B) Signals at the location of the red star in the model for VC = 5 s−1 and PC = 0 s. (C) Vergence responses simulated with one value for VC and three different values for PC. (D) Plant input signals PI for the vergence responses shown in (C). (E) Vergence responses simulated with three combinations of visual and efference copy delays.
Figure 5
 
Behavior of the improved model. (A) Target vergence (dotted line) and ocular vergence (continuous line) movements (left panel) and signals θ″, ϕ′, and d″ (right panel) simulated by the model in response to two successive steps in target vergence. The first step in target vergence occurred at t = 0 s and the second one at t = 0.4 s. (B) Vergence movements induced by sinusoidal oscillations of target vergence (dotted lines) of frequencies of 0.125 Hz and 0.5 Hz, respectively. (C) Gain–phase relationship between simulated ocular vergence and target vergence.
Figure 5
 
Behavior of the improved model. (A) Target vergence (dotted line) and ocular vergence (continuous line) movements (left panel) and signals θ″, ϕ′, and d″ (right panel) simulated by the model in response to two successive steps in target vergence. The first step in target vergence occurred at t = 0 s and the second one at t = 0.4 s. (B) Vergence movements induced by sinusoidal oscillations of target vergence (dotted lines) of frequencies of 0.125 Hz and 0.5 Hz, respectively. (C) Gain–phase relationship between simulated ocular vergence and target vergence.
Figure 6
 
Simulated saccadic–vergence interaction. (A) Velocity and position profiles of saccades and vergence and signals θ″, ϕ″, and d″ simulated by the combined saccade and improved vergence model without any interaction between the saccadic and vergence components. (B) Similar profiles and signals as in (A), but now vergence is enhanced during the saccade. The dotted lines mark the end of the conjugate saccades.
Figure 6
 
Simulated saccadic–vergence interaction. (A) Velocity and position profiles of saccades and vergence and signals θ″, ϕ″, and d″ simulated by the combined saccade and improved vergence model without any interaction between the saccadic and vergence components. (B) Similar profiles and signals as in (A), but now vergence is enhanced during the saccade. The dotted lines mark the end of the conjugate saccades.
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