Ординатура / Офтальмология / Английские материалы / Eye Movements A Window on Mind and Brain_Van Gompel_2007

initialization in step 0 of the algorithm), we assume that all fixations are well-located (cf., Nuthmann et al., 2005). Based on this zero-order approximation, we obtain our zero-order estimate of the distribution parameters 0 L and 0 L for the probability distribution of well-located fixations. Using the procedure discussed above, we compute the first estimate of the probability distribution of mislocated fixations, ML1 x (steps 1 and 2 of the algorithm). The fact that welland mislocated fixations sum up to

this idea many times, we end up with an iterative estimation scheme. A self-consistency check of this approach is the asymptotic convergence of the probability distributions for welland mislocated fixations to values WL x and ML x, respectively (step 4).

2.3. Numerical results

After the theoretical derivation of the iterative algorithm, we carried out numerical simulations based on experimentally observed distributions of landing positions (see Nuthmann et al., 2005, for materials and methods). We computed all distributions for word lengths ranging from 3 to 8 letters. First, it is important to note that for iteration step 1 (wordbased probabilities) we excluded the first and last fixations of each sentence. Second, the calculation of averages over word lengths were limited to a maximum value of a word length of 8 letters for reasons of statistical power (i.e., number of cases). We chose a value of = 10−2 as a criterion of convergence (step 4), which resulted in a termination of the algorithm after only 4 iterations.3

A glance at the numerical results illustrated in Figure 1 indicates that our algorithm reduces the standard deviation of the landing position distributions considerably. Because mislocated fixations occur most likely near word boundaries, the removal of the contributions from mislocated fixations mainly affects the tails of the normal distributions. This effect can be derived from McConkie et al. (1988)’s argument. Overall, the probabilities of a word receiving a mislocated fixation depends strongly on word length. Probabilities for word lengths from 3 to 8 ranged from 25.3 to 5.9%. In the next section, we check the algorithm using numerical simulations of the SWIFT model.

3. Mislocated fixations: Model simulations

3.1. The SWIFT model

The SWIFT model (Engbert et al., 2005; see also Engbert, Longtin, & Kliegl, 2002; Engbert, Kliegl, & Longtin, 2004; Laubrock, Kliegl, & Engbert, 2006; Richter, Engbert, &

3 A second check with = 10−6 took 8 iterations.

Figure 1. Iterative algorithm to estimate proportions of mislocated fixations from empirical data. Left panels show landing position distributions for well-located fixations as a function of word length. Letter 0 corresponds to the space to the left of the word. Also presented is the best-fitting normal curve for each distribution. Right panels display the proportion of mislocated fixations as a function of word length and landing position. First row of panels: zeroth iteration step; second row: first iteration step; third row: fourth iteration step where the algorithm converged (for = 0.01).

Figure 2. Testing the algorithm with SWIFT simulations. Panel (a) shows landing position distributions. Also presented is the best-fitting normal curve for each distribution. Panel (b) displays the proportion of mislocated fixations. The estimated curves are calculated from extrapolations of the distributions in (a). Exact results are directly computed from model simulations.

mislocated fixations (and well-located fixations). Second, since SWIFT also mimics the experimentally observed landing position distributions (compare Figure 1a with Figure 2a or see the discussion of within-word landing positions in Engbert et al., 2005), our results lend support to the conclusion that mislocated fixations occur frequently (up to more than 20% for short words) and play a major role in normal reading.

3.3. Exploring the magnitude of different cases of mislocated fixations

Using numerical simulations of the SWIFT model, we now investigate the prevalence of different types of mislocated fixations (Figure 3). After removing the first and last fixation in a sentence it turned out that 23.2% of all fixations generated by the model (200 realizations) were mislocated. Table 2 provides a complete synopsis of all mislocated fixation cases.

We distinguish between mislocated fixations due to an undershoot vs overshoot of the intended target word. The undershoot cases comprise failed skipping, unintended forward refixation, and undershot regression (Figure 3, cases I to III). For example, a forward refixation is unintended (case II) if the eyes actually planned to leave the launch word (n) and move to the next word (n + 1 but instead remained on the launch word. Undershoot cases cover 69% of all mislocated fixations. Note that failed skipping (case I, 41.8%) is by far the most frequent case of mislocated fixations. Of all fixations, 3.3% are unintended forward refixations (case II). Somewhat surprisingly, the SWIFT model also produces a

Figure 3. Most important cases of mislocated fixations, if the realized saccade target shows a maximum deviation of one word from the intended saccade target. According to the nomenclature used in the figure, word n is the realized target word. The dashed lines indicate the intended saccades, which are misguided due to saccadic error (solid lines represent executed saccades).

certain amount of undershot regressions (case III, 3.0% of all fixations). This effect is related to the fact that a saccade range error applies also to regressions in SWIFT. We expect, however, that this behavior of the model can easily be optimized by changing the oculomotor parameters for regressions.

In summary, the SWIFT model predicts a rather specific pattern of mislocated fixations. The overshoot cases consist of unintended skipping, failed forward refixation, overshot regression, and failed backward refixation (Figure 3, cases IV to VII). Overshot interword regressions (case VI) as well as failed backward refixations (case VII) hardly ever occur. However, 4.3% of all fixations are failed forward refixations (case V) while 2.3% of all fixations are unintended skippings (case IV).

3.4. Exploring responses to mislocated fixations

The exploration of mislocated fixations raises the interesting question how the eyemovement control system responds to a mislocated fixation. This is obviously another problem that can be studied with the help of model simulations. In the SWIFT model,

Figure 4. Simulations with the SWIFT model. Every word-length category is represented with two panels. Upper panels display landing position distributions; lower panels show the proportion of different saccade types after well-located initial fixations on a word. Shaded areas indicate u-shaped refixation curves.

predominantly land at the end of the current word (in case of an intended one-word skip) or on any word to the left of the intended target word (in case of an intended multiple- word skip).4 Indeed, landing position distributions for failed skipping saccades are clearly shifted to the right, having their maximum at the last letter of the word (Figure 5, upper panels). Thus, these saccades predominantly land at the end of words.

Two types of saccades apparently play a major role when failed skipping occurs (Figure 5, lower panels): The SWIFT model primarily responds with a skipping of the next word(s) or simply a forward saccade to the next word. Refixations and regressions seem to play a minor role.5 For illustration, let us consider the case of a failed oneword skipping. If the misguided saccade is followed by a saccade to the next word (one-word forward saccade), the error of initially not hitting this word is corrected.

4 Please note that this prediction does not imply that most fixations landing at the end of a word are mislocated in the sense that they undershot the intended target word. Rather, fixations at the end of words also comprise saccades that did land on the selected target word but overshot the center of this word.

5 Note that results for responses to landing positions at word beginnings are not very stable since these landing positions are infrequent.

Figure 5. Simulations with the SWIFT model. Landing positions (upper panels) and proportions of subsequent saccade types after failed skipping (i.e., the most frequently occurring case of mislocated fixations; lower panels).

However, it is possible that this particular word was fully processed from the parafovea during the mislocated fixation. In this case, a corrective saccade to this word is no longer necessary. Instead, the eyes proceed with skipping the word.

4. Exploring mislocated fixations and the IOVP effect in the SWIFT model

The observation that mislocated fixations most often happen near word boundaries suggests a theoretical link to the IOVP effect: If we assume that an error-correction saccade-program is immediately started as a response to a mislocated fixation, this might reduce the average fixation durations near word boundaries (Nuthmann et al., 2005). This conjecture is valid as long as initiations of new saccade programs occur on average with a finite interval after a fixation begins. Next, we investigate this issue in the SWIFT model.

4.1. Error correction of mislocated fixations in SWIFT

In SWIFT (Engbert et al., 2005), the assumption of an immediate start of an errorcorrection saccade-program is implemented as Principle VI: If there is currently no labile

saccade program active, a new saccade program is immediately started. The target of this saccade will be determined at the end of the labile saccade stage according to the general rule (Principle IV).

In eye-movement research in reading, fixation durations are generally interpreted in terms of latencies for subsequent saccades6 (cf., Radach & Heller, 2000). Thus, it is assumed that the next saccade program is started close to the beginning of the current fixation. In SWIFT, however, saccade programs are generated autonomously and in parallel (see Fig. 19.2 in Kliegl & Engbert, 2003, for an illustration). This has several implications. First, different from the E-Z Reader model (Reichle et al., 2003), saccades are not triggered by a cognitive event. Second, fixation durations are basically realizations of a random variable; saccade latency is randomly sampled from a gamma distribution. However, lexical activation can delay the initiation of a saccade program via the inhibition by foveal lexical activity (Principle III). Third, saccade latency is not equal to fixation duration. In general, there are two possibilities for the initiation of a saccade program terminating the current fixation. Either the program is initiated within the current fixation or it was initiated even before the start of the current fixation. In addition, we now introduced a corrective saccade program in response to a mislocated fixation (see Engbert et al., 2005, for a discussion of the neural plausibility of an immediate triggering process for error-correcting saccades). If there is currently no labile saccade program active, a new saccade program is initiated at the beginning of the (mislocated) fixation. Thus, in case of a mislocated fixation, SWIFT’s autonomous saccade timer is overruled. As a consequence, the program for the saccade terminating a mislocated fixations is initiated earlier (on average) than in the case of a well-located fixation, which leads to the reduced fixation duration for mislocated fixations. This reduction, however, is only achieved if there is a substantial proportion of saccade programs initiated after the start of the current fixation.

In Figure 6, we consider saccade programs that were executed and thus not canceled later in time. Computed is the time difference (in ms) between the start of such a saccade program and the start of the current fixation. When considering well-located fixations, we obtain a relatively broad distribution with most saccade programs initiated during the current fixation. However, the picture drastically changes for mislocated fixations. First, there are saccade programs that were initiated before the start of these mislocated fixations (ca. 15% of all cases, Figure 6b), reflecting the situation that there was a labile saccade program active in the moment the eyes landed on an unintended word. However, most of the time this was not the case leading to the immediate start of a new saccade program (Principle VI). Therefore, the curve presented in Figure 6a has a very pronounced peak at the start of the current fixation. As simulations suggest that mislocated fixations comprise about 23% of all fixations, this peak, though attenuated, reappears when all fixations are considered.

It is important to note that the correction mechanism cannot increase fixation duration since the corrective saccade program is only initiated if there is currently no labile saccade program active. In addition, as long as the probability for an active saccade program at

6 Saccade latency is defined as the time needed to plan and execute a saccade.