Physica A: Statistical Mechanics and its Applications
Strong anticipation: Sensitivity to long-range correlations in synchronization behavior
Introduction
Explanations of anticipation in cognitive science and neuroscience typically attribute internal predictive models to organisms. Dubois [1] labels anticipation of a model-dependent kind as weak and contrasts it with strong anticipation that arises from lawful regularities embedded in the ordinary mode of functioning of a system and its environment. A prominent example of model-independent strong anticipation is anticipating synchronization [2], [3], a temporal relation that emerges from time delays in appropriately coupled “master” and “slave” dynamical systems. The slave system anticipates the master system by synchronizing with a state that lies in the future of the master system. Anticipating synchronization may be very general. It can be demonstrated empirically in both physical, e.g., Ref. [4], and biological, e.g., Ref. [5], systems and has been shown through modeling to accommodate defining characteristics of proactive regulation in physiological and neural systems [6], [7]. Importantly, the fact of anticipating synchronization renders the notion of strong anticipation as neither impossible nor mysterious.
Most broadly construed, strong anticipation is an assertion of non-local dependence of an organism’s current behavior on the global temporal structure within its environment. An organism and its environment together make up a single system with temporal structure that extends far beyond the short range. On the one hand, current states of the organism-environment system depend on initial conditions. On the other hand, they depend on the possibilities for future events. In anticipation, the organism acts at once on its past history and on the potential outcomes afforded by its fit with the environment. (For a discussion of perceiving possibilities for action, and ontological hypotheses for prospective control of behavior, see Ref. [8].) The broad construal of strong anticipation is elaborated in Section 3. It is prefaced by a synopsis of an anticipatory phenomenon in a widely used and simple experimental procedure. The phenomenon and procedure are points of departure for the experiment reported in the present article.
Section snippets
Negative mean asynchrony
The notion of anticipation is prominent in studies of person-to-metronome synchronization. Radil, Mates, Ilmberger, and Pöppel [9], for example, reported a mean asynchrony, an “anticipation tendency” [10, p. 973], of −30 ms between taps and onsets of metronome beats. However, there are questions about the usefulness of negative mean asynchrony (NMA) as a marker of anticipation in general [10]. Participants in synchronization experiments exhibit strong individual differences in NMA [11], [12].
A role for long-range correlations in strong anticipation
The case of synchrony with a chaotic signal brings synchronization research beyond its familiar frontiers. It is a research problem not unlike beat perception in music psychology [19]. Music is not limited to metronomic regularity. The baseline for simplicity would be a single tone repeated at regular intervals. Music can vary both in ordinal complexity and temporal complexity [20]. Ordinal complexity involves a variety of notes along the octave. Temporal complexity involves an irregularity of
Chaos and anticipation
There is an extensive literature in cognitive science on the human ability to anticipate chaotic signals. In a vigilance task [30], a target moved along a horizontal line in a trajectory defined by the chaotic regime of the logistic map. Participants were instructed, at each time point, to anticipate the next location of a target on the screen. Error decreased as participants continued, and their responses converged on a parabolic lag 1 autocorrelation, characteristic of the logistic map. Later
Method
The chaotic signals were produced by a C program using Eq. (1) and parameters and [44]. For each Ikeda [36], [37] metronome signal, the map began at randomly selected initial conditions and continued with one of three settings of delay: , 5, or 10. The values of the resulting Ikeda map were normalized to range from 1 to 1.5, and these values became the inter-onset intervals (in seconds) of the chaotic signal. Each chaotic signal consisted of 1000 onsets. With each new onset, the
Results
A key premise of this experiment is that the chaotic signal presented to participants is unpredictable by linear standards. The linear redundancy of the Ikeda signals may be used to evaluate the signals’ linear predictability. Linear redundancy is an information-theoretic statistic indexing the linear dependence across time, varying inversely with entropy [45], [46], [47], [48]. For any observable , the linear redundancy is computed over a set of lagged copies , . Over time
Conclusion
In this paper, we have presented a finding consonant with the hypothesis, advanced by Anderson and Schooler [25], [26], that learning and memory reflect near-optimal statistical inference about the patterning of environmental events. In their view, adaptation of behavior to the environment’s statistical structure requires neither explicit statistical inference nor explicit prediction. Using Dubois’s [1] terminology, the adaptation is a necessary consequence of a strongly anticipatory system.
As
Acknowledgement
Partial support was provided by NSF grant BCS-0643271.
References (51)
- et al.
Physica A
(2003) Brain Cognition
(2002)- et al.
Physica D
(1987) - et al.
Hum. Movement. Sci.
(2004) Organ. Behav. Hum. Decis. Process.
(1997)- et al.
Physica A
(2004) Physica D
(1995)- et al.
Physica A
(2002) Phys. Rev. Lett.
(2001)