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Noise-enhanced performance in reading
Neurocomputing 50 (2003) 473 – 478
www.elsevier.com/locate/neucom
Letters
Noise-enhanced performance in reading Ralf Engberta; b; ∗ , Reinhold Kliegla a Department
b Center
of Psychology, University of Potsdam, P.O. Box 601553, 14415 Potsdam, Germany for Dynamics of Complex Systems, University of Potsdam, P.O. Box 601553, 14415 Potsdam, Germany
Received 1 August 2001; received in revised form 13 November 2001; accepted 15 November 2001
Abstract When we read a text, typical sequences of /xations on words form a rather complicated trajectory—almost like a random walk. This complex behaviour is the result of the interplay between lexical processing and programming of saccadic eye movements, the two processes that drive eye movements during reading. Stochastic 5uctuations are present in both subsystems. We use a minimal three-state model with residence time-dependent transition probabilities to study the role of stochasticity. In particular, we investigate the in5uence of noise on reading time by numerical simulations. Our model suggests that performance in reading is enhanced by an intermediate level of noise in lexical processing. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Reading; Eye movements; Saccade; Noise; Modelling
During reading, our eyes perform a complex sequence of /xations of words, separated by fast saccadic eye movements, which take our eyes across the words, sometimes with two or more /xations on a word and sometimes skipping a word or two. Eye movements are controlled by two di
∗
Corresponding author. Department of Psychology, University of Potsdam, P.O. Box 601553, 14415 Potsdam, Germany. E-mail addresses: [email protected] (R. Engbert), [email protected] (R. Kliegl). URLs: http://www.agnld.uni-potsdam.de/∼ralf, http://www.psych.uni-potsdam.de/people/kliegl/indexe.html c 2002 Elsevier Science B.V. All rights reserved. 0925-2312/03/$ - see front matter PII: S 0 9 2 5 - 2 3 1 2 ( 0 1 ) 0 0 7 1 1 - 1
474
R. Engbert, R. Kliegl / Neurocomputing 50 (2003) 473 – 478
or simply represent a “dumb” default (low-level) control system, which assumes certain mean saccade lengths and /xation durations. A crucial problem in models of eye movement control is that of the coupling between lexical processes and the saccadic motor system [10]. An analysis of this coupling mechanism is a challenging problem due to stochastic 5uctuations. Stochasticity plays an important role in lexical processing as well as in the saccadic motor systems [14]. Using numerical simulations of a recently proposed minimal model [4], we /nd that noise in lexical processing can enhance performance in reading. Analysis of eye movements in reading may be looked upon as a case study for the more general problem of scanning of visual scenes with higher structural complexity. Typical eye movements in reading form a complicated trajectory, both in space and time. In a /rst approximation, this random walk can be approximated by a series of /xations. The random walk over words is the consequence of saccades from word n to word k . For k = n, word n is re/xated which is typically observed for low-frequency words. If k ¿ n + 1, word n+1 is skipped. Word skipping is very likely for highly frequent words. Considering only forward saccades and re/xations, k ¿ n, is motivated by the hypothesis that this form of eye movements may represent a “default” model of eye movement control [14], with few internal states representing combinations of di
R. Engbert, R. Kliegl / Neurocomputing 50 (2003) 473 – 478
475
Fig. 1. The three-state model for the control of eye movements in reading. In state 1 word n is /xated and lexically processed within an average processing time ln . In states 2 and 3, we assume a parallel processing of lexical access, with average durations ln (foveal) and l? n (parafoveal) resp., and programming of saccades, with latencies sn and sn+1 . The transition from state 1 to state 3 is related to an autonomous initiation of a saccade, i.e. not driven by lexical access.
of word n : ln = lb − lm log(Fn );
(1)
where lb and lm are constant parameters. The slope parameter lm determines the strength of the in5uence of word frequency on lexical processing time, whereas lb is the processing time of words, which occur rather seldom in normal language. When lexical access is done, the system switches to state 2; n is replaced by n + 1, when the transition is performed. This transition implies a shift of attention to word n+1 , while word n is still being /xated. Since parafoveal information is used for lexical access in state 2, ? ? we assume that lexical processing is in a preliminary stage, l? n = lb − lm log(Fn ) (with ? ? parameters lb and lm di
476
R. Engbert, R. Kliegl / Neurocomputing 50 (2003) 473 – 478
to word n+1 , i.e. the saccade program is not triggered by lexical access. The hypothesis underlying this type of transition is that the visual control system has some autonomy in programming a saccade which relates to a preferred rate of moving the eyes to the next word. 1 The variables ln , l? n , sn and an in Fig. 1 represent mean values for residence times in the corresponding states of the model. We now summarize the numerical implementation of our stochastic simulations [9]. An important concept for stochastic models of random processes is the transition probability rate, Wnm . To start with a general framework for stochastic transitions, we use the following transition rule [9]: If the system is in the state Sm at time t, having arrived there at time t − ( ¿ 0), the probability that it will step to some other state Sn in the next in/nitesimal time interval (t; t + dt) is Wnm () dt. 2 The total transition probability rate Wm () for a transition from Sn to a Sm=n is given by Wm () = n Wnm (). In the case of a Markov process, where Wm () is a constant, the pausing time is exponentially distributed. A typical experimentally observed distribution of /xation durations shows, however, a sharp maximum of relative frequency (probability) at 200 ms. Therefore, we now discuss the next more complicated case: a transition probability which increases linearly with . As an additional parameter, we introduce a refractory period 0 with vanishing transition probability rate. These two assumptions are described by 0 if ¡ 0 ; Wm () = (2) wm ( − 0 ) if ¿ 0 : In the following, the refractory time 0 is assumed to be proportional to the mean pausing time , i.e. 0 = with 0 6 ¡ 1 (a detailed discussion is given in [4]). For numerical simulations of our model, we use a corpus of sentences previously discussed in [4,14]. In the corresponding eye-tracking experiment participants read 48 sentences, each consisting of 8–14 words [16]. Parameters were estimated by a genetic algorithm (GA) approach (see [4] for details). Best-/t values for model parameters are ? lb = 263, lm = 11:2, l? b = 186, lm = 9:89, sn = 115, a = 325, l = 0:47, s = 0:64, and a = 0:23. As a result, /xation durations as well as skipping probabilities are in good agreement with experimental data. How does noise in5uence performance in terms of reading time? For optimal performance reading time has to be minimized. Noise in lexical processing as well as in the saccadic motor systems can potentially in5uence performance. In both cases, we vary the refractory time for the transitions (2) in order to study di
R. Engbert, R. Kliegl / Neurocomputing 50 (2003) 473 – 478
477
Fig. 2. Reading time per word Tf as a function of saccadic noise level (dashed line) from the motor systems and noise level in lexical access processes (solid line). There is only a weak in5uence from noise associated with the motor systems (saccades), while stochasticity in lexical access strongly e
probability parameter wm , but depends on relative refractory time , 1=2 4− = (1 − ) ≈ 0:52(1 − ):
(3)
Increasing the refractory time (by increasing ) results in a decrease of the ratio of standard deviation and mean value of the pausing time . As a measure of model performance in reading, we calculate mean /xation duration per word, Tf , from our simulations. Since the number of words read per unit time represents the 5ux of incoming information during reading, we may assume that Tf is a monotonic function of the signal-to-noise-ratio (SNR) of the reading process. The noise levels in the saccadic motor systems and in the lexical access process is varied by the proportions s and l , resp. Our central question is how the noise level in5uences reading time Tf . Our model simulations show that stochastic 5uctuations in lexical access in5uence reading time in a non-monotonic way, whereas noise in the saccadic motor system only marginally in5uences reading time (Fig. 2). Reading performance is optimal at an intermediate noise level, which is indicated by the pronounced minimum in Fig. 2. This phenomenon is a signature of stochastic resonance [6]. Noise-enhanced performance has been observed recently in many physiological systems [1,11,15]. Our theoretical study shows that control of eye movements may be an interesting system for further studies in biological stochastic resonance. These results may have strong implications on the role of lexical information processing for the control of eye movements. In current theoretical models [14], eye movements are mainly driven by lexical processing, implying an accurate control of /xation durations based on lexical diNculty. On the contrary, a low-level control strategy for moving the eyes during reading, predominantly based on simple text characteristics
478
R. Engbert, R. Kliegl / Neurocomputing 50 (2003) 473 – 478
like word lengths, would lead to increased noise in lexical processing time. Our results on numerical simulations of a simpli/ed model [4] suggest that such a strategy could improve performance. We thank Eric D. Reichle, Carnegie Mellon University, for providing us with the corpus of sentences for our simulations and AndrQe Longtin, University of Ottawa, for valuable comments on the manuscript. This work was supported by Deutsche Forschungsgemeinschaft (DFG grants KL 955=3-1 and KL 955=3-2). References [1] J.J. Collins, Biophysics: /shing for function in noise, Nature 402 (1999) 241–242. [2] M.B. Elowitz, S. Leibler, A synthetic oscillatory network of transcriptional regulators, Nature 403 (2000) 335–338. [3] R. Engbert, F.R. Drepper, Chance and chaos in population biology—models of recurrent epidemics and food chain dynamics, Chaos Solitons Fractals 4 (1994) 1147–1169. [4] R. Engbert, R. Kliegl, Mathematical models of eye movements in reading: a possible role for autonomous saccades, Biol. Cybernet. 85 (2001) 77–87. [5] R. Fricke, J. Schnakenberg, Monte-Carlo simulations of an inhomogeneous reaction-di
www.elsevier.com/locate/neucom
Letters
Noise-enhanced performance in reading Ralf Engberta; b; ∗ , Reinhold Kliegla a Department
b Center
of Psychology, University of Potsdam, P.O. Box 601553, 14415 Potsdam, Germany for Dynamics of Complex Systems, University of Potsdam, P.O. Box 601553, 14415 Potsdam, Germany
Received 1 August 2001; received in revised form 13 November 2001; accepted 15 November 2001
Abstract When we read a text, typical sequences of /xations on words form a rather complicated trajectory—almost like a random walk. This complex behaviour is the result of the interplay between lexical processing and programming of saccadic eye movements, the two processes that drive eye movements during reading. Stochastic 5uctuations are present in both subsystems. We use a minimal three-state model with residence time-dependent transition probabilities to study the role of stochasticity. In particular, we investigate the in5uence of noise on reading time by numerical simulations. Our model suggests that performance in reading is enhanced by an intermediate level of noise in lexical processing. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Reading; Eye movements; Saccade; Noise; Modelling
During reading, our eyes perform a complex sequence of /xations of words, separated by fast saccadic eye movements, which take our eyes across the words, sometimes with two or more /xations on a word and sometimes skipping a word or two. Eye movements are controlled by two di
∗
Corresponding author. Department of Psychology, University of Potsdam, P.O. Box 601553, 14415 Potsdam, Germany. E-mail addresses: [email protected] (R. Engbert), [email protected] (R. Kliegl). URLs: http://www.agnld.uni-potsdam.de/∼ralf, http://www.psych.uni-potsdam.de/people/kliegl/indexe.html c 2002 Elsevier Science B.V. All rights reserved. 0925-2312/03/$ - see front matter PII: S 0 9 2 5 - 2 3 1 2 ( 0 1 ) 0 0 7 1 1 - 1
474
R. Engbert, R. Kliegl / Neurocomputing 50 (2003) 473 – 478
or simply represent a “dumb” default (low-level) control system, which assumes certain mean saccade lengths and /xation durations. A crucial problem in models of eye movement control is that of the coupling between lexical processes and the saccadic motor system [10]. An analysis of this coupling mechanism is a challenging problem due to stochastic 5uctuations. Stochasticity plays an important role in lexical processing as well as in the saccadic motor systems [14]. Using numerical simulations of a recently proposed minimal model [4], we /nd that noise in lexical processing can enhance performance in reading. Analysis of eye movements in reading may be looked upon as a case study for the more general problem of scanning of visual scenes with higher structural complexity. Typical eye movements in reading form a complicated trajectory, both in space and time. In a /rst approximation, this random walk can be approximated by a series of /xations. The random walk over words is the consequence of saccades from word n to word k . For k = n, word n is re/xated which is typically observed for low-frequency words. If k ¿ n + 1, word n+1 is skipped. Word skipping is very likely for highly frequent words. Considering only forward saccades and re/xations, k ¿ n, is motivated by the hypothesis that this form of eye movements may represent a “default” model of eye movement control [14], with few internal states representing combinations of di
R. Engbert, R. Kliegl / Neurocomputing 50 (2003) 473 – 478
475
Fig. 1. The three-state model for the control of eye movements in reading. In state 1 word n is /xated and lexically processed within an average processing time ln . In states 2 and 3, we assume a parallel processing of lexical access, with average durations ln (foveal) and l? n (parafoveal) resp., and programming of saccades, with latencies sn and sn+1 . The transition from state 1 to state 3 is related to an autonomous initiation of a saccade, i.e. not driven by lexical access.
of word n : ln = lb − lm log(Fn );
(1)
where lb and lm are constant parameters. The slope parameter lm determines the strength of the in5uence of word frequency on lexical processing time, whereas lb is the processing time of words, which occur rather seldom in normal language. When lexical access is done, the system switches to state 2; n is replaced by n + 1, when the transition is performed. This transition implies a shift of attention to word n+1 , while word n is still being /xated. Since parafoveal information is used for lexical access in state 2, ? ? we assume that lexical processing is in a preliminary stage, l? n = lb − lm log(Fn ) (with ? ? parameters lb and lm di
476
R. Engbert, R. Kliegl / Neurocomputing 50 (2003) 473 – 478
to word n+1 , i.e. the saccade program is not triggered by lexical access. The hypothesis underlying this type of transition is that the visual control system has some autonomy in programming a saccade which relates to a preferred rate of moving the eyes to the next word. 1 The variables ln , l? n , sn and an in Fig. 1 represent mean values for residence times in the corresponding states of the model. We now summarize the numerical implementation of our stochastic simulations [9]. An important concept for stochastic models of random processes is the transition probability rate, Wnm . To start with a general framework for stochastic transitions, we use the following transition rule [9]: If the system is in the state Sm at time t, having arrived there at time t − ( ¿ 0), the probability that it will step to some other state Sn in the next in/nitesimal time interval (t; t + dt) is Wnm () dt. 2 The total transition probability rate Wm () for a transition from Sn to a Sm=n is given by Wm () = n Wnm (). In the case of a Markov process, where Wm () is a constant, the pausing time is exponentially distributed. A typical experimentally observed distribution of /xation durations shows, however, a sharp maximum of relative frequency (probability) at 200 ms. Therefore, we now discuss the next more complicated case: a transition probability which increases linearly with . As an additional parameter, we introduce a refractory period 0 with vanishing transition probability rate. These two assumptions are described by 0 if ¡ 0 ; Wm () = (2) wm ( − 0 ) if ¿ 0 : In the following, the refractory time 0 is assumed to be proportional to the mean pausing time , i.e. 0 = with 0 6 ¡ 1 (a detailed discussion is given in [4]). For numerical simulations of our model, we use a corpus of sentences previously discussed in [4,14]. In the corresponding eye-tracking experiment participants read 48 sentences, each consisting of 8–14 words [16]. Parameters were estimated by a genetic algorithm (GA) approach (see [4] for details). Best-/t values for model parameters are ? lb = 263, lm = 11:2, l? b = 186, lm = 9:89, sn = 115, a = 325, l = 0:47, s = 0:64, and a = 0:23. As a result, /xation durations as well as skipping probabilities are in good agreement with experimental data. How does noise in5uence performance in terms of reading time? For optimal performance reading time has to be minimized. Noise in lexical processing as well as in the saccadic motor systems can potentially in5uence performance. In both cases, we vary the refractory time for the transitions (2) in order to study di
R. Engbert, R. Kliegl / Neurocomputing 50 (2003) 473 – 478
477
Fig. 2. Reading time per word Tf as a function of saccadic noise level (dashed line) from the motor systems and noise level in lexical access processes (solid line). There is only a weak in5uence from noise associated with the motor systems (saccades), while stochasticity in lexical access strongly e
probability parameter wm , but depends on relative refractory time , 1=2 4− = (1 − ) ≈ 0:52(1 − ):
(3)
Increasing the refractory time (by increasing ) results in a decrease of the ratio of standard deviation and mean value of the pausing time . As a measure of model performance in reading, we calculate mean /xation duration per word, Tf , from our simulations. Since the number of words read per unit time represents the 5ux of incoming information during reading, we may assume that Tf is a monotonic function of the signal-to-noise-ratio (SNR) of the reading process. The noise levels in the saccadic motor systems and in the lexical access process is varied by the proportions s and l , resp. Our central question is how the noise level in5uences reading time Tf . Our model simulations show that stochastic 5uctuations in lexical access in5uence reading time in a non-monotonic way, whereas noise in the saccadic motor system only marginally in5uences reading time (Fig. 2). Reading performance is optimal at an intermediate noise level, which is indicated by the pronounced minimum in Fig. 2. This phenomenon is a signature of stochastic resonance [6]. Noise-enhanced performance has been observed recently in many physiological systems [1,11,15]. Our theoretical study shows that control of eye movements may be an interesting system for further studies in biological stochastic resonance. These results may have strong implications on the role of lexical information processing for the control of eye movements. In current theoretical models [14], eye movements are mainly driven by lexical processing, implying an accurate control of /xation durations based on lexical diNculty. On the contrary, a low-level control strategy for moving the eyes during reading, predominantly based on simple text characteristics
478
R. Engbert, R. Kliegl / Neurocomputing 50 (2003) 473 – 478
like word lengths, would lead to increased noise in lexical processing time. Our results on numerical simulations of a simpli/ed model [4] suggest that such a strategy could improve performance. We thank Eric D. Reichle, Carnegie Mellon University, for providing us with the corpus of sentences for our simulations and AndrQe Longtin, University of Ottawa, for valuable comments on the manuscript. This work was supported by Deutsche Forschungsgemeinschaft (DFG grants KL 955=3-1 and KL 955=3-2). References [1] J.J. Collins, Biophysics: /shing for function in noise, Nature 402 (1999) 241–242. [2] M.B. Elowitz, S. Leibler, A synthetic oscillatory network of transcriptional regulators, Nature 403 (2000) 335–338. [3] R. Engbert, F.R. Drepper, Chance and chaos in population biology—models of recurrent epidemics and food chain dynamics, Chaos Solitons Fractals 4 (1994) 1147–1169. [4] R. Engbert, R. Kliegl, Mathematical models of eye movements in reading: a possible role for autonomous saccades, Biol. Cybernet. 85 (2001) 77–87. [5] R. Fricke, J. Schnakenberg, Monte-Carlo simulations of an inhomogeneous reaction-di