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Progress in Biophysics and Molecular Biology 113 (2013) 92e96

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Progress in Biophysics and Molecular Biology

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Review

Measuring the temporal extension of the Now

Susie VrobelThe Institute for Fractal Research, Ernst-Ludwig-Ring 2, 61231 Bad Nauheim, Germany

a r t i c l e i n f o

Article history:Available online 27 March 2013

Keywords:Anticipatory systemFractal timeEndo-observer-participantBoundary complexity

E-mail address: Susanne.Vrobel@t-online.de.

0079-6107/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.pbiomolbio.2013.03.009

a b s t r a c t

Anticipatory systems require a model of time which takes account of both successive and simultaneousrhythms. Such a model should also incorporate the fact that both past and future determine the presentstate of anticipatory systems across multiple scales, from physical to biological and social ones. MyTheory of Fractal Time meets these requirements and enables us to compare the Now’s temporalcomplexity of endo-observer-participants in terms of their boundary complexity.

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Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922. Nested Nows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923. Nested rhythms: time bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934. Fractal time: the extended Now . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935. Temporal interfaces: endo- vs. exo-perspectives and assignment conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

1. Introduction

A linear concept of time is unsuitable for describing antici-patory systems. This is mainly a result of the fact that a linearmodel explains the present state of a system as a reaction to thepast. For anticipatory systems, however, the present is alsodetermined by future events. This influence of the future occursacross multiple scales: Physical, biological and social rhythmswith short cycles influence longer ones they are embedded inand vice-versa.

To describe such temporally overlapping, nested systems, amodel of time is needed which is able to relate successive andsimultaneous rhythms. A model based on my Theory of FractalTime meets this requirement by differentiating between Dtlength,Dtdepth and Dtdensity. It allows us to compare the temporal

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complexity of anticipatory systems in terms of the density of theirtime bubbles, which correlates to their range of potential internalresponses.

The complexity of anticipatory systems differs with the vantagepoint. From an endo-perspective, interfaces may become trans-parent e that is to say, invisible to the observer-participant. From apseudo exo-perspective, i.e. from the point of viewof another endo-observer, the interfaces of other observer-participants may well bevisible. The difference between these two perspectives is quantifiedas the boundary complexity, which describes the number of spatio-temporal interfaces transparent to the endo-observer-participant.Boundary complexity can be measured in spatial extensions andtemporal delays.

2. Nested Nows

Anticipatory systems act within a temporal extension e theirNow ewhich is shaped by both their memory of the past and their

S. Vrobel / Progress in Biophysics and Molecular Biology 113 (2013) 92e96 93

expectation of the future. The first description of such an extendedNow, which contains the structures of present, past and future, wasgiven by the German phenomenologist Edmund Husserl (1980). Heasked a simple question, namely: How come we are able to hear atune and not just a series of uncorrelated successive notes? Husserlclaimed that while we hear a note being played, the preceding notestill lingers on in our memory and the note we anticipate to followalso becomes part of our consciousness of the present. His answerhad far-reaching consequences: The Now was not a point-like cutbetween the future and the past, but had extension, as it containedpast, present and future structures.

But this is not the end of the story. An extended Now in Hus-serl’s sense was not simply extended in one dimension, as onemay assume if past, present and future events were lined up onone level of description, like beads on a string. His ingenious stepwas to add a second temporal dimension (although he did notrefer to it as such) which evolved from the overlapping of thememory of the past and the anticipation of the future in theconsciousness of the present. And this extended present continuedto grow, as the preceding notes would sink down further intomemory each time a new note was played. Analogously, each newnote would trigger the expectation that this was not the end andthat there was another note to follow the present one. And as eachnew note entered the consciousness of the present, the structureof this present became more and more complex. A cascade ofnested memories and expectations stretched the present furtherand further.

Husserl’s extended, nested Now also contains the seed for theneed to distinguish between two temporal dimensions: successionand simultaneity. Whereas musical notes which are played one byone create the dimension of succession, the notes which wesimultaneously remember, hear and anticipate generate thedimension of simultaneity. Together, they generate a time bubblewhose extension is both context-dependent and varies with eachanticipatory system’s internal make-up, such as its perceptualapparatus and cognitive complexity.

The notions of succession and simultaneity are both necessaryto describe the temporal perspectives of anticipatory systems. Buthow exactly canwemeasure a temporal perspectivee that is to say,an extended Now?

3. Nested rhythms: time bubbles

Measuring time means comparing rhythms. The ticking of amechanical clock may serve as a yardstick for comparing the lengthof slower clocks, such as the rotation of planets around a sun. Whatwe do in such cases is to compare successive units of time at onelevel of description with those at a different level.

When it comes to measuring biological rhythms such as brainwaves, however, measuring the mere number of successive units oftime at one level of temporal resolution, say at 14 Hz, neglects avital quality of anticipatory systems, namely, their embeddednessin other rhythms.

Within our bodies, shorter rhythms are nested into slower ones:Neural oscillations, for instance, are embedded in slower metabolicrhythms (Buzsáki, 2006). And our entire body is, again, nested bothin a physical and social context, which provides further temporalframeworks into which we are embedded. Examples are tidalrhythms, which are, themselves, embedded in astronomical ones.

All these rhythms are nested: Short rhythms are embedded inlonger ones, which are, in turn, embedded in even longer ones andso on. The resulting nesting cascade contains both the anticipatorysystem and its context. The question as to where the boundary ofthe anticipatory system is to be found is, as we shall see below, amatter of negotiation.

As the concept of a nesting cascade presupposes that of a hier-archy, a differentiation is necessary at this point. Stanley Salthedifferentiates between the notions of the compositional as opposedto the subsumptive hierarchy (Salthe, 1997, 2012a, 2012b). Theformer is reductionist in essence andmanifests as containment. It isa hierarchy in which the sum of the parts are not more than thewhole. The latter manifests as change and involvement, which al-lows for emergence and evolution.

Nestedness, if the term strictly means containment, can bedescribed by the logic of the compositional hierarchy. If nestednessalso implies an overlapping of events, the logic of the subsumptivehierarchy is required, in addition, in order to describe the emergentphenomena which arise as a result of this overlapping. This onlyhappens when we place an observer-participant into the nestedhierarchy, whose extended Now allows for the interpretation of thepast and the anticipated future in terms of the present moment andthus creates a narrative. The emergent property is the extension ofthe temporal perspective: Just as bifocal vision emerges when weslightly overlap two simultaneous images and thereby create depth,our temporal perspective also creates a new dimension as a resultof overlapping perceptions. This new dimension manifests assimultaneity and delivers a framework against the background ofwhich otherwise uncorrelated successive events are related. Anexample is our ability to perceive a tune rather than a succession ofuncorrelated notes. Here, our anticipation of the next note overlapsthe memory of the note, which still lingers on in our extended Now.

External measurements of temporal intervals would focus onone level or a number of levels in the sense of Salthe’s composi-tional hierarchies. The sum of themeasured parts would not exceedthe whole. The participating individual’s perspective, however,would not only encompass perception on all levels, including his orher internal ones, but also be able to relate them, thus forming anarrative. By contrast, external measurements disregard the factthat all theories they are based on are anthropocentric in essence(Pöppel, 1989, 2000; Vrobel, 2007) and therefore do not allow foremergent structures such as narratives. But it is emergence (such asthe perception of a tune) that we are seeking to explain.

So, how can we measure a temporal perspective - that is to say,an extended Now? My Theory of Fractal Time allows us to do so bymeasuring succession and simultaneity in the temporal quantitiesof the length, depth and density of time (Vrobel, 1998).

4. Fractal time: the extended Now

Successive events e that is to say, events which do not overlapwithin a specified observer perspective - can be imagined asoccurring on one level of description, with those events being linedup like beads on a string.

If the events do not overlap, this means that at no point do theyoccur simultaneously in a given observer perspective. In order tomeasure successive events, we first have to specify the level ofdescription. This may be equated with the level of resolution e thatis, the degree to which we zoom into the structure. To measuresimultaneous events, we need to count the number of nestings/embeddings of levels of description (i.e. the number of availabletemporal resolutions) within a specific observer perspective.

If our perceptions of events overlap, the relationship betweensuccession and simultaneity is not merely one of containment inSalthe’s sense, because the sum of the successive overlapping partsexceeds the sum of successive parts on one level of description. Thislast claim may require further elucidation.

For mere containment, we can measure a non-overlappingstructure in terms of its fractal dimension. Imagine a mathemat-ical line, such as the Koch-curve, which has a topological dimensionof 1, but, as wiggles around in space, it becomes verymuch longer if

S. Vrobel / Progress in Biophysics and Molecular Biology 113 (2013) 92e9694

we look at a measurement other than its topological dimension.Mandelbrot defined such a wiggly line as a fractal, because itsHausdorff Dimension of 1.2618 . exceeds its topological dimen-sion (Mandelbrot, 1982). The former is measured by covering astructure with balls or spheres. At each level of resolution, smallerand smaller balls cover the structure. This way, we would measure,at the same time, the lengths from larger curves to those of thesmallest wiggles. And by measuring nested structures in terms oftheir fractal dimension, their density becomes comparable (wemayeven distinguish the density of infinite structures this way). Merenestedness would not allow such differentiation.

If our anticipated perceptions overlap with our memories ofperceptions within the observer-participant’s Now, our temporalperspective expands. It is both external stimuli as well as our in-ternal differentiation which allow us to mould not just notes into atune but any cross-modal perceptions into a gestalt.

Gestalt-formation facilitates the navigation of complex beingsthrough a complex world. As Bruce West has pointed out, our de-gree of internal complexity, as measured in the shape of our rangeof possible responses, determines our ability to adapt to the outsideworld (West, 2006). Apart from the fractal structure of an extendedNow, our fractal physiology displays nested overlapping rhythms,whose variability displays statistical self-similarity. The occurrenceof scaling structures in human physiology points to long-rangecorrelations, which are absent in non-scaling structures (West,2006). Stanley Salthe has suggested that correlations betweenscaling events may form the basis of anticipation (Salthe, 1997).Here, however, it shall suffice to provide a measure for describingthe temporal complexity of the extended Now.

Let me first define the dimensions of fractal time (Vrobel, 1998):

� Dtlength is the length of time, i.e. the number of incompatibleevents (for a specific observer perspective) on a specific level ofdescription. It manifests as succession.

� Dtdepth is the depth of time, i.e. the number of compatibleevents (for a specific observer perspective) on nested levels ofdescription. It manifests as simultaneity.

� The measure resulting from the length and depth of time isDtdensity e the density of time. It is the fractal dimension, whichquantifies the density of time e that is to say, the temporalcomplexity of an observer perspective.

So while Dtlength counts temporally incompatible rhythms ata specific temporal resolution, Dtdepth adds up the number ofsimultaneous rhythms at various temporal resolutions in a nestingcascade.

The human body is a temporal yardstick which measures andtranslates between internal and external rhythms (Lefebrexe,1992). The observer-participant adapts to physical and biologicalconstraints as well as social and linguistic conventions (Vrobel,1999). As Henri Lefebre has pointed out, our body needs to trans-late between biological and social rhythms: External rhythms set byour work and social life increasingly determine our biologicalrhythms of wakeesleep cycles or food intake (Lefebrexe, 1992). Ourembodied temporal yardstick measures both internal and externalrhythms simultaneously in Dtdepth.

Any nesting of overlapping temporal intervals creates simulta-neity and thus increases Dtdepth and, in its wake, Dtdensity. So does anoverlapping of nested levels in the Now if an observer-participant isinvolved. Nesting without an observer-participant generatescontainment in the sense of Salthe’s compositional hierarchy,whereas the presence of an observer-participant generates a narra-tive, namely, his or her history. This usually involves acquiredrhythms in Lefebre’s sense which are simultaneously internaland social (Lefebrexe, 1992). However, generating Dtdepth does not

necessarily require interactionwith the outsideworld. Itmay also allhappen within: Recursive recollection of past events or of expectedfuture events will form a nesting cascade, too. The overlapping ofpostdictionandanticipationexpandour timebubble (Vrobel, 2011a).

An example is the so-called ‘missing fundamental’. This phe-nomenon consists of a completion process by means of which wefurther expand our Now. When we hear a number of overtonesfromwhich the fundamental frequency has been removed, we willstill hear this missing frequency e it emerges in the listener’s Now,although it is physically non-existent.

Overtones aremulti-layered signalse that is to say,musical noteswhich are played simultaneously. The least complex frequency ratiobetween two musical notes is 2:1. It defines the interval betweenthem as an octave. So if we played, for instance, the note A on anoboe, we would produce a frequency of 440 Hz. The next A e anoctave higher ewould correspond to a frequency of 880 Hz, and soon. And since theovertones are integermultiples of the fundamentalfrequency (into which they are embedded), the sine waves aretranslatable into each other in terms of Dtlength and Dtdepth. We arethus facing a self-similar temporal pattern, inwhich the structure ofthe (shorter and shorter) nested sine waves is identical to those ofthe embedding frequencies (albeit of different extension in Dtlength).Self-similarity and translatability make the multi-leveled temporalstructures commensurate to one another (Vrobel, 2007). And it isbecause all these nested intervals simultaneously compose our Now,that an emergent level such as the ‘missing fundamental’ arises and,in the wake of its emergence, further expands our Now.

This phenomenon is used in telecommunications, where itsuffices to transmit higher frequencies, as the listener constructsthe lower ones spontaneously. It is also applied in car stereos whosespeakers cannot produce the low-frequency bass sounds. In bothcases, the listener generates the embedding, lower frequencies. Asthese span a larger interval in Dtlength, the listener expands his orher Now by accommodating the lower frequencies by anticipatingthe entire (longer) sine wave.

A further distinction should be made as to whether the rhythmstaken into account when we measure Dtlength, Dtdepth and Dtdensityare purely internal or both internal and external. As we shall see,this task is not as simple as it may seem at first sight, as the internalperspectivewould not only generate a higher fractal dimension, butalso vary with the observer-participant’s extension.

5. Temporal interfaces: endo- vs. exo-perspectives andassignment conditions

Otto Rössler has pointed out that we need, apart from initialconditions and natural laws, a further notion in order to describe adynamical system, namely, the assignment conditions (Rössler,1998). These state what belongs to the observer-participant andwhat belongs to the rest of the world. As it turns out, the boundarybetween inside and outside is negotiable (Clark, 2008; Vrobel,2010a).

If a human incorporates, through frequent use, a tool such as awalking stick or a stick for reaching, neural plasticity manifests it-self in the brain, mapping far space onto near space (Clark, 2008).The interface shifts outwards if, for instance, the boundary betweenthe observer-participant and the rest of the world is no longer atthe point where the hand grasps the stick, but at the end of thestick, which is in contact with the ground. The observer-partici-pant’s sphere of influence has widened and part of the context hasbeen incorporated.

The observer-participant, however, would be blind to thisboundary shift, as his or her movements would continue smoothly.Only another observer-participant who is watching from anexternal vantage point could make out the two nested interfaces:

S. Vrobel / Progress in Biophysics and Molecular Biology 113 (2013) 92e96 95

one at the point where the hand grasps the stick and onewhere thetip of the stick touches the ground or any other object.

This is where Otto Rössler’s notions of endo and exo come inRössler (1998). In his terms, the endo-perspective is the view fromwithin, whereas the exo-perspective is the external vantage point.In the case of a boundary shift, the endo- and exo-observers wouldcount a different number of interfaces. In the stick example, theexo-observer would be able to see two nested interfaces whereasthe endo-observer would only be able to make out one. Likewise,the emergent missing fundamental would be part of the endo-observer’s Now, while a hypothetical exo-observer would not beable to measure it (as it is physically non-existent).

Elsewhere, I have denoted this relation between Dtdepth of theendo- and Dtdepth of the exo-perspective as the boundarycomplexity (Vrobel, 2010a, 2010b).

It is measured as

DtdepthðendoÞ ¼ 1DtdepthðexoÞ ¼ 2

¼ 0:5

Strictly speaking, it is not really an external observer we aredealing with, but another endo-observer. Therefore, we had betterwrite

Dtdepthðendo1Þ ¼ 1Dtdepthðendo2Þ ¼ 2

¼ 0:5

The temporal analogy to such boundary shifts involves theinsertion or removal of temporal delays. Such delays are likewiseincorporated by the observer-participant, in the wake of which hisor her time bubble expands.

An example would be compensation for a delay in a controlloop. Andy Clark reports an experiment in which a monkey (whocontrolled a cursor on a screen through its neural activity) becameconfused by a delay which was inserted into the control loop (Clark,2008). After a short time, though, the monkey adapted to the newsituation by anticipating the delay and allowing for it. With thiscompensation, the monkey’s temporal interface had shifted.

The same compensatory effect also works in reverse, if anexisting delay is removed from a control loop. An example is re-ported by Stetson et al., who conducted an experiment in whichindividuals were conditioned to seeing a slightly delayed flash on amonitor after pressing a button (Stetson et al., 2005). If the delaywas suddenly shortened and they continued to press the button,they were convinced that the flash had appeared before the buttonwas pressed. This apparent time reversal shows the powerful effectof anticipatory compensation, and the lack thereof, on our temporalperspective.

Delay compensation is a temporal interface, which is trans-parent (in the sense of invisible) from the endo-perspective, butvisible to the external observer who can oversee the entire controlloop (Vrobel, 2010b). Based on Robert Rosen’s seminal book (Rosen,2012), a more general mathematical model of delay compensationhas been developed by Daniel Dubois with his incursion andhyperincursion models of anticipatory regulation (Dubois, 1998).Here, it shall suffice to describe two measures e the dimensions ofFractal Time and boundary complexity e for quantifying theextension of the Now, both from the endo- and exo-perspective.

We do not know how many invisible delays we have incorpo-rated. We become aware of their existence only when we incor-porate new delays or when existing ones are removed. So weshould stop and think whenever we appear to get out of sync,because this may just be the moment when our time bubbles areexpanded or reduced. A loss of trust, too, may hint at a possibleremoval of an incorporated delay, as trust likewise expands our

time bubble and may indeed be our most powerful anticipatoryfaculty (Vrobel, 2011c). It smoothes our navigation through oureveryday lives, because we are blind to our incorporated assump-tions about the rest of the world. As Niklas Luhman has aptly put it,trust is the reduction of social complexity (Luhmann, 2000).

On a speculative note: It is conceivable that, one day, we mightdefine health also in terms of the ability to incorporate or removedelays. A healthy balance may be hard to define. If all delays wereremoved, we would probably die, as we could no longer man-oeuver. If we incorporated all external delays of our universe, wewould turn into an infinitely extended observer-participant. This,however, would entail that we would cease to be an observer andbecome a true participant (Vrobel, 2011b). Would we, in such astate, have cleansed the doors of our perception through amaximum of anticipation?

6. Conclusion

Anticipatory systems exist within an extended Now, whoseextension expands or shrinks with every boundary shift. It can bemeasured in Dtlength, Dtdepth and Dtdensity. The observer-partici-pant’s range of potential internal responses is part of the equation.

An endo-observer will measure a Dtdepth which is different fromthat measured by an exo-observer. This is so because an endo-observer is blind to the expanding or shrinking of his or her timebubble. Only a second endo-observer is able to see that, for the first,two or more nested interfaces have merged into one. The boundarycomplexity is a quantity which links endo-perspectives.

Acknowledgments

I would like to thank Dr. Plamen Simeonov for inviting me tocontribute to this special issue and Professor Stanley Salthe forhelpful comments and pointing out that there is a difference be-tween ‘nested’ and ‘fractal’.

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