Symmetry Breaking Analysis of Prism Adaptation’s Latent Aftereffect


should be sent to Till D. Frank, Department of Psychology, University of Connecticut, 406 Babbidge Road, Unit 1020, Storrs, CT 06269–1020. E-mail:


The effect of prism adaptation on movement is typically reduced when the movement at test (prisms off) differs on some dimension from the movement at training (prisms on). Some adaptation is latent, however, and only revealed through further testing in which the movement at training is fully reinstated. Applying a nonlinear attractor dynamic model (Frank, Blau, & Turvey, 2009) to available data (Blau, Stephen, Carello, & Turvey, 2009), we provide evidence for a causal link between the latent (or secondary) aftereffect and an additive force term that is known to account for symmetry breaking. The evidence is discussed in respect to the hypothesis that recalibration aftereffects reflect memory principles (encoding specificity, transfer-appropriate processing) oriented to time-translation invariance—when later testing conserves the conditions of earlier training. Forgetting or reduced adaptation effects follow from the loss of this invariance and are reversed by its reinstatement.

1. Introduction

As Redding and Wallace (1997, p. 162) remark in their extensive review and analysis of research on the phenomenon of adapting to prisms: “prism adaptation is not a single process.” Various and distinct adaptive processes contribute to the particulars of perception–action manifest during, and subsequent to, the wearing of prisms. In the present article, we employ tools of nonlinear dynamics to dissect and quantify one specific aspect of these adaptive processes: the induction of a latent recalibration aftereffect by a parametric change in the movement conditions from adaptation to re-adaptation.

In a recent investigation of prism adaptation, Blau et al. (2009) examined throwing underhand to a horizontal target with left shifting prisms and with a manipulation of the throwing arm’s moment of inertia through a weight attached above the elbow or at the wrist. Their investigation was directed, in part, at the frequent observation that the effect of prism adaptation on movement is typically reduced when movement in the test or re-adaptation phase (with prisms removed) is different from movement during the training or adaptation phase (with prisms; e.g., different posture, Baraduc & Wolpert, 2002; different speed, Kitazawa, Kimura, & Uka, 1997; different limb, Martin, Keating, Goodkin, Bastian, & Thach, 1996a, 1996b). In replication of the frequent observation, Blau et al. found that the aftereffect was weaker if arm moment of inertia at test was different from that during training. They also found, in agreement with the discovery by Fernández-Ruiz et al. (2000), that some adaptation is latent. It is manifest in further testing in which the movement at training is fully reinstated. Blau et al. observed that this latent or secondary aftereffect (AE2) (a) depended on the similarity of training and test moments of inertia, and (b) summed with the primary aftereffect (AE1) to yield a constant overall amount of aftereffect. More precisely, when due to experimental manipulations the primary aftereffect (AE1) is reduced by a certain amount x, then the secondary aftereffect (AE2) is increased by that amount x.

In framing and interpreting their experiment, Blau et al. (2009) gave emphasis to parallels between prism adaptation and implicit memory with verbal material. For the latter, memory performance is better when study and test are more similar (e.g., Franks, Bilbrey, Lien, & McNamara, 2000; Lukatela, Eaton, Moreno, & Turvey, 2007). To be more precise, memory performance for a given event or fact depends on the degree to which study and test engage very similar mental operations (Principle of Transfer-Appropriate Processing [TAP]; Morris, Bransford, & Franks, 1977) in very similar stimulus situations (Principle of Encoding Specificity [ES]; Tulving & Thomson, 1973). Accordingly, if adaptation of a behavior can be taken as a measure of implicit memory, then the magnitude of the primary aftereffect should depend on the degree that the task and/or stimulus situation at re-adaptation (the “test” phase with prisms off) matches the task and/or stimulus situation at adaptation (the “study” or “training” phase with prisms on).

The details of Blau et al.’s (2009) experiment that are of relevance to the present article are summarized in Figs. 1 and 2. Fig. 1 depicts the task. Fig. 2 depicts the conditions across the 90 trials for the five experimental groups, and the performance measure across trials for Groups 1, 3, and 5, the groups on which the modeling reported in the present article was focused. The test for the initial or primary aftereffect (AE1) was conducted in Trials 61–75 (Test 1). The test for the latent or secondary aftereffect (AE2) was conducted in Trials 76–90 (Test 2). The focus in Blau et al.’s experiment was on the effect of degree of similarity between the Test 2 conditions and the training conditions (Trials 31–60), and between the Test 2 conditions and the Test 1 conditions (see Fig. 2).

Figure 1.

 Prism glasses in the experimental study of Blau et al. (2009) shifted gaze in the x direction perpendicular to the throwing direction y. The attached mass (in Groups 2–5; see Fig. 2) biased the throw in the y direction toward the participant.

Figure 2.

 Left panel: The five experimental groups in Blau et al. (2009). Trials 0–30 were training trials, 31–60 were the prism adaptation trials, 61–75 without prisms comprised Test 1 (primary aftereffect, AE1), and 76–90 without prisms comprised Test 2 (secondary or latent aftereffect, AE2). Groups 1 (no added mass), 3 (mass added at biceps), and 5 (mass added at wrist) representing no change, moderate change, and larger change in arm moment of inertia were used to assess the latent aftereffect. The modeling in the present study was focused on these three groups. (Adapted from fig. 2 of Blau et al., 2009 with permission from Elsevier.) Right panel: The mean performance errors over Trials 30–90 of Groups 1, 3, and 5 as reported in Blau et al. (2009).

It is important to underscore that the task of throwing as accurately as possible was held constant from training to tests. Only the stimulus situation, specifically, the throwing arm’s moment of inertia and its haptic perceptual consequences, changed. The inertia effect was in the y direction, whereas the prism effect was in the x direction (Fig. 1). Despite this contrast (and in concert with TAP and ES), when the haptic stimulus situation was unchanged from training to Test 1, Groups 1, 2, and 4 with different moments of inertia behaved similarly to each other. Conversely, when the haptic stimulus situation was changed from training to Test 1, Groups 3 and 5, instantiating a smaller and larger change, respectively, behaved differently from each other as shown in the right panel of Fig. 2 (and differently from Groups 1, 2, and 4).

2. Nonlinear attractor dynamics model and results

2.1. Attractors and detuning

Adaptation and re-adaptation data as shown in Fig. 2 (right panel) have frequently been analyzed by fitting the raw data to a monotonically decaying or increasing function (see, e.g., Blau et al., 2009; Brookes, Nicolson, & Fawcett, 2007; Fernandez-Ruiz & Diaz, 1999; Martin et al., 1996a, 1996b). In the present article, we evaluate adaptation and re-adaptation from a dynamical systems perspective. Within this perspective Schöner and Kelso (1988a, 1988b) suggested that learning any arbitrary bimanual rhythmic coordination (e.g., a phase relation of 45°) involves (a) an intrinsic attractor (e.g., 0° or in-phase) that is invariant over learning trials, and (b) a learning attractor that emerges over learning trials. Paralleling this suggestion, Frank et al. (2009) proposed a model of prism adaptation that involves an intrinsic attractor that is invariant over adaptation trials and an adaptation attractor that evolves over adaptation trials (“prism on” condition) and disappears during the re-adaptation trials (“prism off” condition; see left panel of Fig. 2). Their modeling includes a parameter δ that expresses the aforementioned training-to-test change in the throwing arm’s moment of inertia (and, thereby, its haptic stimulation) and allows the reconstruction of the latent aftereffect.

The use of δ to designate this parameter highlights the parameter’s relation to the notion of detuning in the coordination dynamics literature (Kelso, 1995; Park & Turvey, 2008). It does so in the following sense: The test conditions were either (a) “in resonance” with the training conditions, δ = 0 (e.g., Test 1 for Group 1), or (b) “out of resonance” with the training conditions, δ ≠ 0 (e.g., Test 1 for Group 5). In the present article, we investigate the extent to which a fit of the Blau et al.’s (2009) data to the model of Frank et al. (2009) provides evidence that the parameter δ is related to the experimentally manipulated difference between the movement conditions of training and test.

2.2. Nonlinear attractor dynamics model of prism adaptation

We consider a stochastic version of the nonlinear attractor dynamics (NAD) model of prism adaptation proposed by Frank et al. (2009). As originally formulated, the NAD model accounts for the stability and flexibility of the adaptation process. Before and after completion of the adaptation process, performance is stable. During the adaptation process, performance is (by definition) flexible.

Evolution of the performance error x is described by the original NAD model in a time-continuous framework. At any fixed time point, x is subjected to a so-called potential or attractor dynamics. That is, x is regarded as a coordinate of a potential energy function evolving in such a way that its potential energy value decays toward a minimum. This latter dynamical process is the stability property. The flexibility property, a key feature of Frank et al.’s (2009) NAD model, is that the potential energy function itself evolves as an integral aspect of the adaptation process.

The two features, stability and flexibility, are coupled in the NAD model. To this end, the total performance error potential is composed of a fixed (time-invariant) intrinsic potential and a potential that emerges during adaptation (adaptation potential). The latter potential reflects one part of the coupling between flexibility and stability. Another part of the coupling between flexibility and stability is given by the fact that the performance error acts as a driving force on the evolution process of the adaptation potential (see Eq. A1).

The evolution of the adaptation potential can be captured by means of the signed potential amplitude α. The sign of α describes in which direction the emerging adaptation potential shifts the performance error and in doing so makes it vanishing. The amount of α describes the overall scale of the potential. The original NAD model of Frank et al. (2009) includes two coupled evolution equations. One describes the evolution of the performance error (and as noted above accounts for performance stability). The other describes the emergence of the adaptation potential (and as noted above reflects performance flexibility). The second-order dynamical model can exhibit oscillatory and non-oscillatory solutions. In what follows, we restrict analyses of prism adaptation data to the non-oscillatory part of the Frank et al.’s NAD model. In this case, we can simplify using the principle of adiabatic elimination (see Haken, 2004). In doing so, we obtain a first-order dynamic model for the evolution of the signed amplitude α (see Appendix A). In the time-continuous framework, the adaptation dynamics for α reads


Equation (1) is a quantitative description of the phenomenon that performance is flexible under adaptation. Here, t is time defined on a continuous scale (see Frank et al., 2009). We will consider an event-based, time-discrete scale below. In Eq. (1), the parameter s denotes the effective prismatic shift, that is, the effect of shifting the gaze by prism glasses. The strength of the prismatic shift on a behavioral activity may vary across individuals (e.g., Warren & Platt, 1975). In Eq. (1), the parameter s denotes the experienced prismatic shift as measured in terms of the motor performance. That is, s corresponds to the initial motor performance error. This experienced shift need not necessarily correspond exactly to the degree that the prism glasses shift the target laterally relative to the participant’s sagittal plane.

The parameter  0 in Eq. (1) is the weight of the intrinsic error correction potential with respect to the emerging adaptation potential. The parameter δ reflects an additive force term of the error dynamics that tends to produce a positive error for δ > 0 and a negative error for δ < 0. (Eq. A1). To reiterate, it was shown in Frank et al. (2009) that a parameter such as δ, capturing the difference between training (adaptation trials) and test (re-adaptation trials), could be used to explain, at least qualitatively, the emergence of AE2, the latent aftereffect. The parameter κ > 0 characterizes how the strength of visual feedback about the performance error affects the adaptation process, that is, the emergence of the adaptation potential. (An alternative interpretation of the parameters κ is given below.) The most right-standing term in Eq. (1) is a fluctuating force composed of a normalized Langevin force Γ(t) and an amplitude Q > 0.

The stochastic evolution equation, Eq. (1), can be split into two parts. To this end, we consider the function


This function captures the deterministic impacts on the evolution of the adaptation amplitude α and is called the drift function. Equation (1) reads


The first-order dynamical model describes a potential dynamics involving the potential function V. That is, Eq. (3) can be written as



V(α) = −∫h(α)dα, and


for > 0, and


for = 0. Note that if |α| is large with respect to c, then Eq. (5) for the potential V reduces to Eq. (6) just as in the case = 0.

Figure 3, with the parameter α expressed in meters, illustrates the classical adaptation (A) and re-adaptation (R) process as expected for participants in Group 1 (no change in arm moment of inertia, see Fig. 2) and predicted by the NAD adaptation model. During the adaptation phase the potential V(α), which governs the emergence of the adaptation attractor, looks qualitatively as shown by A in Fig. 3A. The minimum is different from zero due to the impact of the prismatic shift. Fig. 3B (upper panel): the adaptation attractor amplitude increases from zero (open circle) to a stationary finite value (full circle) that corresponds to the potential minimum (minimum of A in panel A). During re-adaptation the potential V(α) has the qualitative form of R in Fig. 3A and exhibits a minimum at zero. Consequently, the amplitude dynamics converges to zero (transition indicated by open and full circle in Fig. 3B lower panel).

Figure 3.

 (A) Typical adaptation amplitude potentials V(α) during adaptation (labeled A) and re-adaptation (labeled R). (B) Details of the potentials shown in (A). Upper panel: During adaptations, the amplitude α makes a transition from 0 to a fixed-point value α ≠ 0. Lower panel: During re-adaptation, the adaptation amplitude makes a transition from α ≠ 0 to α = 0.

The prism adaptation dynamics predicted by the NAD model for experimental conditions involving a training-to-test change in movement details is more complex and illustrated in Fig. 4. The potential V(α) during the adaptation condition (Training) is illustrated by A in Fig. 4A. Under the first re-adaptation condition (Test 1) we are dealing with a potential V(α) that looks qualitatively similar to R1. The potential does not exhibit a minimum at zero, although the prismatic shift equals zero. The reason for this is that δ is different from zero and results in a shift of the minimum of V(α) out of the origin. Only during the second-adaptation condition (Test 2) shown by R2 does the potential V(α) exhibit a minimum at α = 0. The adaptation amplitude evolves qualitatively in three steps as illustrated in Fig. 4B (top panel to bottom panel).

Figure 4.

 (A) Typical adaptation amplitude potentials V(α) during adaptation (labeled A), primary (R1), and secondary (R2) re-adaptation. (B) Details of the potentials shown in (A). From top to bottom: adaptation, primary re-adaptation, and secondary re-adaptation. Transitions from initial (open circles) to final (full circles) adaptation amplitudes are indicated.

We can now consider the relation between α and the performance error x. Due to the aforementioned adiabatic elimination process, the performance error x can be computed from α by means of an invertible nonlinear mapping (see Appendix A)


such that Eq. (1) can be expressed alternatively in the non-closed form


From Eq. (8), it becomes clear that the parameter κ corresponds to one of the two constants that describe the coupling between stability (performance error dynamics) and flexibility (evolution of the adaptation attractor). The other coupling constant is given by the parameter c (see Eqs. A1 and A6). Moreover, the fact that Eq. (7) provides a mapping between the adaptation amplitude and the performance error implies that the stability related subprocesses can maximally change at a rate determined by the subprocesses accounting for performance flexibility. In other words, in the modeling effort presented above we focus on those cases in which the time-scale characterizing the performance flexibility under adaptation is the rate-limiting parameter for the overall dynamics, including both stability and flexibility. Mathematically speaking those cases involve a first-order dynamics and consequently are consistent with experimentally observed performance errors that decay with a non-oscillatory pattern under adaptation. In order to improve the link between model and experiment, and to estimate the parameters on the basis of experimental data, we consider in what follows a time-discrete, event-based counterpart of the time-continuous dynamical model, Eq. (1). First, we consider a sequence of performed actions = 1, 2, . . ., N. We denote with α(n) the adaptation amplitude when the nth action is performed. Likewise, we denote with x(n) the error observed in the nth performed action (nth trial). With these notations in mind, the first-order differential equation, Eq. (1), has as counterpart a first-order difference equation, which reads


In Eq. (9), we replaced the parameter κ by a parameter k. Both parameters have the same interpretation but different time units. The fluctuating force in the time-discrete case is described by the random variable ε. At any time step n, the variable ε is drawn from a normal distribution with variance equal to Q (see Frank, 2005; Risken, 1989). Equation (4) becomes


The performance error x is related to the amplitude α in the manner


2.3. Model-based data analysis

In the following, we show how we estimated the parameters δ, c, k, and Q from the experimental data published in Blau et al. (2009) using Eqs. (9–11; for a comment on the growth curve analysis used in Blau et al. and the model-based analysis used in the current study, see Appendix C). Let xD(1), xD(2), . . ., xD(N) denote experimentally observed performance errors. We first selected a set of parameters k, δ, and c. Then we iterated the model and compared the model trajectory {x(1), x(2), . . ., x(N)} with the experimental data and calculated a model-fit-error. Subsequently, we varied the parameters k, δ, and c and computed the mismatch again. In total, we varied the parameters in a parameter space kmin, . . ., kmax, |δ| = 0, . . ., δmax, = 0, . . ., cmax. Note that according to the NAD adaptation model, non-vanishing parameters δ have the same sign as the experienced prismatic shift s. Since s was negative in the study by Blau et al. (2009; see Fig. 1A) we considered negative values for δ. We finally selected those parameters that minimize the model-fit-error. More precisely, we started each iteration with α(1) = 0 because at the beginning of the prism adaptation process the adaptation potential is not present (see also Frank et al., 2009). Furthermore, we assumed that the experienced prismatic shift s corresponds to the first performance error. Consequently, we put x(1) = xD(1). In sum, we computed the amplitudes α(n) and the performance errors x(n) using Eqs. (9) and (11) for ε = 0 with α(1) = 0 and x(1) = xD(1). For the 30 time steps of the adaptation phase (see Fig. 2), we used s ≠ 0 and δ = 0. For the subsequent 15 time steps (first part of the re-adaptation phase, “Test 1” condition), we had = 0 and |δ| ≥ 0. That is, we tested various δ parameters including δ = 0 as one possible parameter value. For the remaining 15 time steps (second part of the re-adaptation phase, “Test 2” condition), we had δ = 0. Between the adaptation and re-adaptation phase, we allowed the adaptation amplitude to decay as discussed in Frank et al. (2009; see also Fernández-Ruiz, Díaz, Aguilar, & Hall-Haro, 2004). The decay of the adaptation amplitude accounts for the fact that the aftereffect is usually smaller in magnitude than the prismatic shift. In order to mimic such a decay, we iterated the amplitude dynamics given by Eq. (9)m times, where m is a parameter that varied from = 0, . . ., mmax. We used kmin = 0.01, kmax = 1.0, dmax = 0.4 s2, cmax = 15, and mmax = 6. Note that in Eq. (7)δ occurs in combination with the term α2. The amplitude α in turn converges during the adaptation process to a value that is proportional to the prismatic shift s (Frank et al., 2009). Therefore, we scaled the interval for δ with the size of the squared prismatic shift. The model-fit-error was defined by


Having obtained the optimal parameters δ, c, k, and m that minimize the model error, we solved Eq. (9)—the time-discrete event-based form of the NAD prism adaptation model—for the optimal parameters. We then computed the strength Q of the fluctuating force. From the results of Friedrich and Peinke (1997) and Frank, Friedrich, and Beek (2006), Q can be computed from Eq. (9) as


The parameter Q needs to be estimated from the experimental data xD. To this end, we inverted Eq. (11). We computed pairs {αD(n), αD(+ 1)} from data pairs {xD(n), xD(+ 1)} and subsequently substituted the pairs {αD(n), αD(+ 1)} into Eq. (13). The five model parameters were estimated for each participant from the 30 data points in the training and the 30 data points in the Test 1 and Test 2 phases. That is, parameter estimates were calculated based on 60 data points.

3. Results

3.1. Theoretical and computed values of δ

For Test 1 in Fig. 2 (left panel), the theoretical δ values were δ = 0 for Group 1 and δ < 0 (in line with the negative prism) for Groups 3 and 5 (see Fig. 2, left panel), with the δ magnitude for Group 5 presumed to be larger than that for Group 3. As a check on the theoretical values, we varied δ from 0 to δmax and determined, for each participant in each of the three groups, the δ value that provided the best fit between model and data. The results are presented in Table 1 (minus a Group 3 participant who exhibited no prism adaptation; see Appendix B). The results indicate a close relation between the theoretical interpretation of δ (i.e., as 0 or negatively signed) and the optimal values of δ per group as determined by the NAD model.

Table 1. 
Model estimates of δ for Group 1 (no added mass), Group 3 (mass added at biceps), and Group 5 (mass added at wrist)
ParticipantGroup 1Group 3Group 5
1    0   0 −442
2−980−387 −480
3    0−289 −898
4    0−673 −720
5    0   0−1,037
6    0   0     0
7    0   0 −673
8    0   — −650

A one-way analysis of variance (anova) confirmed that δ differed across groups, F(2, 20) = 5.6, < .001, η2 = .36. Post hoc analyses (Tukey HSD) showed that Group 5 differed from both Groups 1 and 3 (both < .05; family-wise type I error 0.05), which did not differ from each other. As can be seen in Table 1, there were 12 participants with δ = 0 and 11 participants with δ < 0.

Dividing the data according to Table 1 yields the pattern of aftereffects shown in Fig. 5. Whereas observed AE1 was numerically larger, and almost significantly so, for the δ = 0 classification, t(21) = −1.98, = .06, observed AE2 was numerically and statistically larger for the δ finite (δ < 0) classification, = .01, t(21) = 2.82. The sums of AE1 and AE2 for the two classifications did not differ (> .05): 24.2 (SD = 6.9) for the 12 participants with δ = 0, 24.7 (SD = 9.7) for the 11 participants with δ < 0. The NAD model predicts the reciprocity of AE1 and AE2 implied by the foregoing identity of sums (Frank et al., 2009; see their fig. 8). The implication is evident in the present Fig. 5.

Figure 5.

 Magnitudes of the primary aftereffect (AE1) and secondary or latent aftereffect (AE2) as a function of the model estimates of δ given in Table 1. White bars: sample means (and SEs) of participants with δ = 0. Gray bars: sample means (and SEs) of participants with δ ≠ 0.

Figure 6 shows the original data and the computed deterministic (ε = 0) trajectories for Participant 5 from Group 1 (Fig. 6A) and Participant 4 from Group 5 (Fig. 6B) using Eqs. (9) and (13) with optimal parameters. All trajectories computed for δ = 0 looked qualitatively similar to the trajectory shown for Group 1 in Fig. 2 (i.e., the trajectories were smooth monotonically decaying functions for the re-adaptation phase composed of the trials = 61, . . ., 90). Likewise, trajectories computed for δ < 0 were qualitatively similar to the trajectory shown for Group 5 in Fig. 2 (i.e., the trajectories showed one peak in trials = 61, . . ., 75 and a second peak in trials = 76, . . ., 90).

Figure 6.

 (A) Data from Participant 5 of Group 1 and best-fit line computed from the NAD model. (B) Data from Participant 4 of Group 5 and best-fit line.

3.2. Observed and computed values of AE1 and AE2

Figure 7 summarizes the observed and computed magnitudes of AE1 (primary aftereffect) and AE2 (latent aftereffect) for the δ values identified in Table 1. Inspection reveals that the NAD model (represented by gray bars) successfully generated the observed values (represented by white bars).

Figure 7.

 Observed and NAD modeling of the primary and secondary aftereffects (with SE bars).

3.3. Dependence of parameters c, k, m, and Q on group

As identified in Table 1, Groups 1, 3, and 5 were distinguished by the mean numerical value of δ, the parameter that represents the difference between the movement conditions in the adaptation trials, Trials 31–60, and the movement conditions in re-adaptation trials 61–75. We asked whether Groups 1, 3, and 5 were distinguished by any parameters other than the symmetry breaking parameter δ. That is, did they differ according to (a) parameter c, the relative strength of the adaptation error potential to the intrinsic error potential; (b) parameter k, the influence of visual feedback on error correction during adaptation (Trials 31–60); (c) parameter m, the decay rate of adaptation prior to re-adaptation trials; and (d) the parameter Q, the noise amplitude? Table 2 summarizes the group values for the four parameters. Table 2 suggests no systematic dependence on group for any of the four parameters and none was revealed by anova (all Fs < 1).

Table 2. 
Model parameters for Group 1 (no added mass), Group 3 (mass added at biceps), and Group 5 (mass added at wrist)
11.4 (3.9)0.07 (0.02)3.7 (3.1)140 (60)
20.3 (0.8)0.08 (0.04)2.6 (3.2)100 (40)
33.2 (5.5)0.06 (0.02)2.6 (2.5)160 (80)

4. Discussion

We have highlighted the observation that research on implicit memory with verbal material suggests that memory performance is best when study and test are most similar, that is, when for a given event or fact, processing mode and stimulus conditions at study are reproduced at test. In more general terms, memory performance is maximal when study and test are fully symmetric. Or, conversely, “forgetting is consequent to breaking symmetry, specifically, violating time-translation invariance.” In the present article, these notions have been generalized to prism adaptation. For the NAD model, the degree of symmetry breaking associated with an aspect of prism adaptation—namely, the adapted behavior—is defined by the parameter δ. In the model development and model-based data analyses comprising the present article we have sought to determine the linkage (a) between δ and the group manipulations made by Blau et al. (2009) and depicted in Fig. 2, and (b) between δ and the observed experimental consequences of the group manipulations, also depicted in Fig. 2.

With respect to training and test conditions, Groups 1, 3, and 5 were distinguished by no change, moderate change, and larger change, respectively, in the throwing arm’s inertia. Table 1 reveals that the NAD model’s determination of δ corresponded to the foregoing contrast: δ was closer to a mean of 0 for Group 1 and furthest from a mean of 0 for Group 5. The lack of perfect correspondence (all Group 1 δ values at 0 and all Group 3 and Group 5 δ values < 0, with the negativity larger for Group 5) might indicate (a) imperfections in the modeling, or (b) contributors to symmetry breaking in addition to, or other than, the attachment and location of a weight on the throwing arm, or both (a) and (b). A promise of the model-based determination of δ is that it can be used to detect differences in degree of symmetry breaking when experimental manipulations are seemingly identical.

The major test of δ is in regard to the primary and latent aftereffects. We found that the values of δ (Table 1) computed from the NAD model paralleled the observed differences between AE1 and AE2 (Fig. 5) and generated the observed magnitudes of AE1 and AE2 both locally (Fig. 6) and in the mean (Fig. 7). In the model development, we had tied the definition of δ to the relation between the throwing arm’s inertia at training (Trials 31–60) and its inertia at Test 1 (Trials 61–75). The sufficiency of this definition is apparent from the results: The latent aftereffect was modeled just as well as the primary aftereffect (see Figs. 6 and 7).

In comparison to δ we detected no systematic relations between performance error and the other parameters. We might have found an influence of the throwing arm’s inertia on k, that is, participants in Groups 3 and 5 could have relied on visual feedback more than participants in Group 1. Likewise, we might have found an influence of the throwing arm’s inertia on Q on the assumption that muscle variability would be least for Group 1 and most for Group 5. It is important to underscore that each of the four parameters in Table 2 could be manipulated experimentally. The expectation would be that such manipulations would have specific consequences on observed and predicted error performance. The delay parameter m, for example, has been shown to affect AE1 (Fernández-Ruiz et al., 2004) and could be expected to affect AE2. The larger theoretical significance of these parameter manipulations is conveyed by the Boltzman factor (developed below). In this context, it is important to note that the NAD model was developed to explain the basic findings in Blau et al. (2009). By contrast, the experiment by Blau et al. (2009) was not designed to test predictions of the NAD model. Therefore, it does not come as a surprise that we did not find statistically significant effects for all other model parameters listed in Table 2.

In the stationary case, the distribution function P of α is a Boltzmann distribution of the form


In particular, for δ = 0, the distribution P corresponds to the Maxwell–Boltzmann distribution


The parameter β is the Boltzmann factor. In our case (see formal development up to Eq. 8 above), we have


The feedback parameter κ and the noise amplitude Q can be regarded as counterparts to the damping constant and noise amplitude of systems in thermodynamic equilibrium. The fluctuation–dissipation theorem of equilibrium system states that the ratio of κ and Q is inversely related to a macroscopic measurable variable, the temperature T. In other words, the Boltzmann factor β = κ/Q is a macroscopic quantity inversely related to the temperature T. This implies that all equilibrium systems with the same factor β exhibit the same ratio of dissipation and noise. Adaptation processes in perception–action tasks occur in systems whose functioning is maintained by energy flows. Consequently, we are dealing with systems operating far from thermodynamic equilibrium rather than equilibrium systems. However, it has been argued that for systems that operate far from thermodynamic equilibrium a similar fluctuation–dissipation theorem holds. Accordingly, the ratio κ/Q of a non-equilibrium system reflects a characteristic system property that varies in a lawful way with experimental manipulations (Frank, Patanarapeelert, & Beek, 2008; Frank et al., 2006). If prism adaptation is a self-organizing process at some remove from equilibrium, as we assume, then we should expect future investigations that manipulate feedback (κ or k) and noise (Q) to reveal that β correlates with major observables of the prism adaptation process. For example, in the present data (without explicit manipulations of k and Q) there were indications that β and AE1 were positively correlated (= .47, = .02).

Returning to δ, prism adaptation experiments have engaged many different kinds of manipulations that violate time-translation invariance. We identified several in the introduction, all having to do with changing an aspect of the behavior from adaptation trials to re-adaptation trials. Such a change has been the focus of the present analysis. Other kinds of changes, however, are more aptly identified as changes in the environment in which a behavior is performed (e.g., Norris, Greger, Martin, & Thach, 2001; Rossi, 1972), or as changes in the behavior itself (e.g., Martin et al., 1996a, 1996b). These changes can also be expected to function as δ. That is, they should shift the minimum of V(α) out of the origin in the manner shown in Fig. 4 (via function R1 in panel A and the middle plot in panel B).

These justifiable expectations raise the question of what happens when there is more than one violation of time-translation invariance? For the well-studied case of detuned (or imperfect) dynamics of interlimb coordination, the answer is that the multiple instances of δ combine despite their categorical differences, as if sharing a common currency (Park & Turvey, 2008). Future prism adaptation experiments that involve more than one violation of time-translation invariance will prove valuable to the development of the NAD model and the understanding of adaptive processes.

As a final comment on the modeling, we should note that data from the baseline condition (Trials 1–30; see Fig. 2) were not incorporated. Future studies may exploit these data to improve the parameter estimating procedure. Moreover, in line with the theoretical work that produced the NAD model of prism adaptation (Frank et al., 2009), it was assumed in the present study that parameters are constant during the phases of the experiment (baseline, training, tests). Future efforts might be devoted to estimating model parameters separately for each phase.

4.1. Postscript I: Implications for the prism paradigm

Wearing prism glasses induces two kinds of shifts, one in respect to where environmental things are seen to be, and one in respect to where limbs are felt to be. Adjusting a visible target to straight ahead without body movement gives a measure of the visual shift (VS), and pointing straight ahead with an unseen limb gives a measure of the proprioceptive shift (PS; Redding & Wallace, 1997). A well-established finding is that in combination VS and PS produce the observed total shift (TS) measured by target pointing with the unseen limb (Redding & Wallace, 1997, pp. 60–61). The implication is of nested reference frames, with that of the haptic perceptual system apparently subordinate to that of the visual perceptual system. This implication is confirmed by direct manipulation of both reference frames.

The reference frame for the unseen limb may not be the geometric or anatomical axes of the limb but, rather, the axes of the mass distribution of the limb. A cross-like splint attached to the limb and weighted asymmetrically breaks the coincidence of the arm’s anatomical and physical reference frames (Pagano & Turvey, 1995). The resultant shift in the direction of the arm’s center of mass (van de Langenberg, Kingma, & Beek, 2008) changes the limb’s felt position. When prism glasses break the coincidence of the pointing arm’s visual and actual positions and induce PS along with VS, an attached asymmetrically weighted splint induces an additional amount of PS, with the resultant TS now equal to VS plus PS (prisms) plus PS (splint; Riley & Turvey, 2001). PS (splint) is invariant over PS (prisms).

In the present context, there is no PS (arm load) to be added to VS and PS (prisms). Differences in arm loading did not affect throwing performance in the x axis (Fig. 1) in either the pre-prism trials or in the adaptation trials (Blau et al., 2009; Fernández-Ruiz et al., 2000). “Arm load” in the experiment modeled in the present article is an example of variables that lie outside the standard classification scheme of the prism adaptation paradigm. The determination of variables like “arm load” is via new kinds of assays, namely, Test 1’s dissimilarity to Training and Test 2’s dissimilarity to Test 1, as identified in Fig. 2. One way of expressing these assays is that the former evaluates “forgetting” (lost prism adaptation); the latter evaluates the “temporariness of forgetting” (recovered prism adaptation). The parsing of TS into VS and PS obtained through the measures described above is too coarse to reveal the governing dynamics. Context changes of varying degrees of subtlety (captured by δ) are integral to those dynamics and, ideally, detectable by extensions of the aforementioned methods.

4.2. Postscript II: Potential generality of forgetting as symmetry breaking

To reiterate, our departure point for pursuing the symmetry parameter δ was a parallel drawn between re-adaptation of a prisms-adapted, previously “studied,” throwing skill, and remembering verbal material previously “studied” implicitly. We noted that both phenomena seemingly depended on the degree to which mental (TAP) and environmental (ES) contexts were unchanged from “study” to “test.”

An early and influential expositor of this invariance perspective was McGeoch (1932, 1942). He rejected the interpretation of forgetting as permanent loss from a store of past experiences in favor of temporary retrieval failure. His hypothesized causes of failure were competing memories at recall and the functional equivalents of variant TAP (the notion of inappropriate set or mental reference frame) and variant ES (the notion of altered environmental conditions of stimulation). The contemporary human memory literature, for infants as well as adults, provides substantial evidence and theory consistent with McGeoch’s invariance perspective (see Capaldi & Neath, 1995; Hayne & Rovee-Collier, 1995; Neath & Surprenant, 2003).

The potential breadth of the time-translation invariance perspective on forgetting is highlighted further by inquiry into extinction of Pavlovian learning—the gradual loss of a conditioned response (CR) following repeated presentations of the conditioned stimulus (CS) in the absence of the unconditioned stimulus (US) that was paired originally with the CS. Contemporary research on the phenomena of renewal, restoration, reinstatement, and US devaluation, along with traditional and continuing observations of spontaneous recovery, reveals that to extinguish the CR is not to terminate the relation between the CS and US (Bouton & Moody, 2004; Capaldi & Neath, 1995; Domjan, 2006). In the phenomenon of renewal, the CR level achieved during acquisition is recovered following a change in the context in which the CR was extinguished. In the phenomenon of restoration, the CR level achieved during extinction is re-established by renewing the context in which the CR was extinguished.

A mature invariance perspective is beholden to the determination of context, in both its TAP and ES forms. For the memory experimentalist, the setting of an experiment divides into variables distributed over organism and environment that are seemingly obvious in their relation to the experimental task and those that seemingly bear no causal relation to the experimental task whatsoever (e.g., Wickens, 1987). Typically, only the rationally transparent variables are identified as context. Experiments by Thomas and colleagues (Thomas, 1985; Thomas, Stengel, Sherman, & Woodford, 1987) highlight the explanatory problems that the latter strategy gives rise to. Pigeons that learned a color-discrimination problem when standing on a flat (inclined) surface failed the problem when the surface was inclined (flat). In this experiment, most of the changes from “study” to “test” were in the class of rationally opaque: shadows and angular relations among hips, knees, ankles, and digits. What could they possibly have to do with the acquired color discrimination?

The explanatory significance of non-obvious rationally opaque variables to the operative context is a defining characteristic of Gottlieb’s probabilistic epigenetic approach to development (Gottlieb, 1997; Johnston, 1997; Miller, 1997). For example, the freezing response exhibited by ducklings to the maternal alarm call is linked developmentally, and non-obviously, to their own perinatal (in the shell, prior to hatching) vocalizations. Absence of the latter (experimentally induced by temporary devocalization) can eliminate the adaptive freezing behavior normally seen in hatched ducklings (for a summary, see Miller, 1997). As Johnston remarks, from a focus on identifying precursors rationally related to outcomes, the finding by Miller is prima facie implausible: “There is no coherent rational argument that might lead one to suppose that the duckling has to hear its own embryonic call before showing an adaptive response to the alarm call” (Johnston, 1997, p. 511).

A separate line of inquiry into freezing behavior reveals outcomes that accord with the invariance perspective, extending the perspective’s range from issues of memory to issues of development. In the laboratory, ducklings can be reared “socially” or “non-socially” during the period from hatching to testing for alarm call behavior (usually around 24 h after hatching). Social rearing means that 8–12 ducklings (typical mallard clutch size) interact within the nest. Non-social rearing means that each duckling in a brood is reared in its own individual box, with the individual boxes so constructed and arranged so as to allow auditory interaction but not visual or tactile interaction. The testing for alarm call responsiveness can also be social or non-social, that is, ducklings can be tested as a group, that is, as the brood (as occurs in the wild), or they can be tested individually. The surprising outcome is that responsiveness depends on the rearing–testing relation: The incidence of freezing to the maternal alarm call was equal for social–social and non-social–non-social and approximately 50% higher than non-social–social and social–non-social (Blaich, Miller, & Hicinbothom, 1989; Miller, 1997).

In conclusion, to observations of forgetting and reduced adaptation as (a) following from a loss of “study”–”test” invariance and (b) reversible by the reinstatement of “study”–”test” invariance can be added an observation of incomplete development. At some level of discourse these observations in their generality might bear on the counters to archival interpretations of prior events advanced by Bergson (1896/1912; memory without storage), Elsasser (1986; information transfer without spatial–temporal contiguity), and Robbins (2002, 2008); direct memory).


Appendix A: Derivation of the first-order NAD model from the second-order NAD model proposed in Frank et al. (2009)

In Frank et al. (2009), a deterministic model was proposed that describes the evolution of the performance error x(t) and the dynamics of the signed amplitude α(t) of the adaptation potential. The model is described in terms of two coupled first-order differential equations.


Here, D describes the amount to which the conditions under which adaptation occurs differ from the conditions under which re-adaptation occurs. This parameter becomes crucial for our understanding of the latent aftereffect. The parameter D represents an additive force that tends to produce a positive error for > 0 and a negative error for < 0. The parameter  0 describes the strength of the intrinsic error correction potential. Similarly, the parameter > 0 describes a time-invariant measure for the strength of the potential that emerges during adaptation. The total strength of the potential is given by the product B|α|. The parameter s corresponds to the experienced prismatic shift. The function G describes a sigmoid force that results in an increase or decrease of the adaptation amplitude if the performance error x is different from zero. The two, coupled, first-order differential equations can be written as a single second-order differential equation




This equation can be regarded as the evolution equation of a particle with generalized coordinate α and mass equal to unity.

In order to account for performance variability, we supplement the two evolution equations for x and α with fluctuating forces. For sake of simplicity, we assume that these fluctuating forces can be expressed in terms of so-called Langevin forces (Risken, 1989). The stochastic evolution equations thus obtained read


The parameter > 0 is the strength of the fluctuating force that acts error dynamics. The total fluctuating force is given by √Kξ(t). The function ξ(t) is a Langevin fluctuating force normalized to 2 (Risken, 1989). Likewise, the expression √QГ(t) represents a fluctuating force that acts on the signed amplitude of the adaptation attractor. Here, > 0 is the strength of the force and Г is a Langevin fluctuating force normalized to 2 (Risken, 1989). For our purposes, it is sufficient to consider the linearized version of G at = 0. In this case, we have


where κ > 0 measures the strength of the linearized force. In the deterministic case (= 0), the two coupled first-order dynamical model can be written in terms of a single second-order dynamical system (see above) which depending on the parameters exhibits oscillatory or non-oscillatory solutions (Frank et al., 2009). The experimental data presented in Blau et al. (2009) show adaptation processes that initially (i.e., during the first 10 throws) exhibit a clearly monotonic (i.e., non-oscillatory) behavior. Our goal in the present study is to focus on this monotonic behavior. Monotonic behavior, in general, is consistent with first-order dynamical models. Consequently, we apply the second-order dynamical model (A1) in a parameter domain in which the model effectively behaves like a first-order dynamical system. The parameter domain in this context is given by parameters A, B, D, that are large with respect to κ. In this case, the concept of adiabatic elimination can be applied and the second-order dynamical system reduces to a first-order dynamic system (Haken, 2004). According to the principle of adiabatic elimination, the variable x(t) is a fast evolving variable, whereas the variable α(t) is a slowly evolving variable. For any given value of α, the fast evolving variable x converges quickly to its fixed point. Therefore, the details of the dynamics of the fast evolving variable can be neglected and the dynamics of the whole system is governed solely by the dynamics of the slowly evolving variable. Mathematically speaking, adiabatic elimination implies that we can put dx/dt = 0 (Haken, 2004). As a by-product, it can also be shown the fluctuating force that occurs in the evolution equations of the fast evolving variable x(t) can be neglected (Frank, Daffertshofer, Beek, & Haken, 1999; Haken, 2004). Consequently, from the evolution equation of x(t) in Eq. (A1) it follows that


Dividing the whole equation by B, we obtain


We introduce next the parameter δ as


which reflects the impact of the context-change D relative to the impact of the adaptation process B. Likewise, we introduce the relative parameter


which reflects the impact A of the intrinsic potential relative to the impact B of the adaptation process. Having defined these two parameters, Eq. (A4) becomes


This equation can be solved for x:


Substituting Eq. (A8) into Eq. (A2), we obtain


Equations (A8) and (A9) correspond to Eqs. (7) and (1), respectively.

Appendix B: Special cases

The data and the best-fit model trajectory for Participant 2 of Group 1 are shown in Fig. 8A. The model fit yields a δ < 0. Consequently, the fitted trajectory shows a second peak in the re-adaptation phase.

Figure 8.

 (A) Data and the best-fitting model trajectory (solid line) for Participant 2 of Group 1. (B) Data of Participant 8 of Group 3. Dashed lines indicate model predictions obtained for κ = 0 (see Appendix B for details).

The data for Participant 8 of Group 3 is shown in Fig. 8B. The participant exhibited neither adaptation nor re-adaptation. In addition, during the test phase (Trials 60–90) in Trial 77 there is a negative data point that differs considerably from all other data points of the test phase.

Failure of prism adaptation and absence of re-adaptation have been reported by Martin et al. (1996a, 1996b) for humans suffering from disorders of the cerebellum in a throwing task like that of the present research. Several data panels in their Figs. 2 and 4 match Fig. 8B. A study with a patient suffering from bilateral damage to regions of the posterior parietal cortex also reveals failure of prism adaptation (Newport, Brown, Husain, Mort, & Jackson, 2006; see also Pratik, Sainburg, & Haaland, 2011). As far as we know, Participant 8 of Group 3 had neither of the aforementioned disorders.

A more conservative approach to our anomalous participant is provided by the finding of Warren and Platt (1975) that degree of prism adaptation is positively correlated with indices of visual abilities and negatively correlated with indices of motor abilities. Warren and Platt suggested that whereas participants with better visual abilities tended to solve the problem of missing the visual target through greater reliance on visual information, participants with better motor abilities tended to solve the problem of missing the visual target through greater reliance on haptic information. We might conjecture therefore that Participant 8 in Group 3 was a rather dramatic case of the latter kind. In our model (see Eqs. 1 and 8) the parameter κ > 0 characterizes how the strength of visual feedback about the performance error affects the adaptation process, that is, the emergence of the adaptation potential. Our conjecture, therefore, is that for the anomalous participant, κ was less than some minimal value required for the emergence of the adaptation potential.

For κ = 0 our model predicts that s in the adaptation phase and = 0 (assuming δ = 0) in the re-adaptation phase. For Participant 8 of Group 3, we may determine s by computing the average of the throws 31–60. The dashed lines in Fig. 8B correspond to the model predictions thus obtained. Note that in the case of κ = 0 other model parameters such as δ and c are not well defined and cannot be estimated.

Appendix C: Growth curve analyses and dynamical systems theory

In fig. 2 of the study by Blau et al. (2009), the prism adaptation and re-adaptation data of Blau et al. are captured by means of growth curve analysis—a descriptive analysis of time series data with the aim to detect whether or not experimental manipulations affect the temporal structure of a time series. By contrast, the aim of the dynamic systems approach expressed in the present article through Eqs. (1–13) is to explore and develop a dynamic law for prism adaptation and re-adaptation (akin to Newton’s law that governs the motion of physical bodies). Such a dynamic law is a promising tool for studying the link between experimental manipulations and causes that (a) are conceptualized in terms of dynamic forces and (b) result in the effects observed on the behavioral level during prism adaptation. Both approaches (growth curve analysis and dynamic systems theory) are complementary and can benefit from each other. For example, in Blau et al. growth curve analysis allowed the identification of the second re-adaptation phase (“Test 2,” i.e., Trials 75–90) as a function of no mass added, mass added at biceps, or mass added at wrist (see Fig. 2 of the present study). The dynamic model developed in Frank et al. (2009) and in the present article explains to a certain extent why this is the case, that is, what causes the observations that Blau et al. were able to report by using growth curve analysis.