Inferring Representation Type from the Fractal Dimension of Biological Communication Waveforms

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Digital representation is based on an arbitrary relation between physical signal structure and assigned meanings; analog repmntation uses a proportionality between changes in meaning and changes in the physical structure of the signal that npreaents the meaning. For recursive information, the physical structure of the analog signal wilIalsoberecu&ve_ Sii a digital signal will be physically kbitrary for both IXUE& and non-recur&e i~o~ti~, a measure of physical recursion can provide an index for ~og/~~~ ~B~SBKZS. A simple model for this rektion is examined by computational experiment for spectral density, and by analytic exp&n for Kolmogorov complexity. It is exam&d by empirical measure of fractal dimension for samples of a cape& music, poetry, and prose, in a comparison of rap vs. reggae music, and for cetaoean vocali&ons. Gener&xations of this analysis for other communication systems are discussed.

1. Introduction

Analysis of the representation of information in biological systems has received a powerful set of tools from the recent advances in nonlinear dynamics. The most common application has been in cases where a single parameter time series is used as the basis for recovering information about the global state spaot of the system dynamics. Measures that establish upper bounds for this state space, or define aspects of the geometry of attractive subsets, have shown strong correlations with biological processes (see Degn et al., 1986). In those applications, Tractal dimension” refers to measurement in phase space.

This essay will not make use of any phase space reconstructions; it will instead use fractal dimension measures of the time series waveform, as studied by Mandelbrot (1988, Voss (1988), and others. IIere fractal dimension provides a dire& measure of the “roughness” or “irregularity” of the waveform structure, and is often used to investigate “flicker” or “l/P’ noise in natural phenomena. In his initial investigations of l/F noise,

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Voss (1978) described consistent relations between the fractal dimension of various human acoustic waveforms and the type of repr~ntation or isolation coding used. This essay will provide a more formal expression for Voss’ conclusions, extend his empirical observations to other categories of acoustic communication, and suggest some generalizations to other biological signals.

Since life is fundamentally a self-organizing process, we should not be surprised that it is characterized by self-similar forms. Not only do fractal shapes abound throughout physiology and ecological st~ctu~s, but the time dynamics of processes that govern these structures also exhibit fractal scaling behavior-that is, we expect that living processes will tend to have more energy in long term correlations (thus frequency as the reciprocal of power, or “l/m. When we turn to biological communication systems however, this characterization is complicated by the presence of symbolic codes, or digital representation. Unlike parametric (analog) signals, in which the signal parameter changes in proportion to the meaning it represents, digital codes have a more arbitrary relation between the physical signal and its meaning. This arbitrary relation will diminish the correlation lengths of the physical waveform in the digital case, despite the presence of long correlations in the information being represented. Since analog signals will reproduce the informational correlations in their physical waveform, a measure of relative correlation lengths (such asfractal dimension) can be used to discriminate between these representation types.

The ~~og~~~~ dichotomy is often confused with other dual&s. The same terms are used by engineers to describe the continuous/discrete dichotomy, and by cognitive scientists to discuss “reasoning by analogy” vs. inductive analysis. Here I am using the terms in their original sense, where they were used to discriminate between analog and digital computers (see Dewdney, 1985, for an informal introduction to this distinction).

Just as analog computers represented isolation by their physical structure, rather than through symbol ma~p~ation, analog commu~c~ion embeds information in the structure of its waveform. It is based on a proportionality between a physical parameter and an semantic parameter. In an analog thermometer for example, the height of mercury is proportionate to the temperature. One familiar biological example would be a smile, where (to some extent) the curvature of the lips are in proportion to the degree of affect expressed. A more precise biolo~c~ example occurs in the frog retina, where some neurons have a fring rate which (for most of the range) is in linear proportion to the angular velocity of small moving images (Grusser 8z Grusser-Cornehls, 1976). Of course the proportionality in some analog representation is non-linear; the nature of the proportionality does not matter as long as it is shared by the communicator and receiver.

Digital representation is equally common; it is based on a code or look-up table of waveform connation {symbols or symbol clusters) and their assigned meanings (e.g., a dictionary, a genetic code). Unlike the analog thermometer, the physical structure of the waveforms produced by a digital thermometer are not in any kind of proportional relation to the information they represent. In an analog thermometer, a mercury rise of seven length units indicates a proportionate temperature rise of of seven. In a digital thermometer, the physical shape of a numeral “7” has no structural relation to the numeric value seven. This is the sense in which a digital signal can be said to be physically “arbitrary:” the relation between meaning and physical structure has been assigned by convention rather than determined by a physical or natural proportionality. There are, of course, instances where convention happens to use a structurally analogous waveform-in the

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numeral “3” for example, the physical signal is indeed structured according to the information it represents-and such exceptions will be considered in the models which follow this section.

One familiar human example of digital representation would be sign language, in which some agreed-upon convention has assigned a gesture to a meaning (and indeed we find that sign languages from different cultures can use completely different conventions). A more precise example cau be found in the motor neurons that open the crayfish claw. Here a specific temporal ftig pattern (off~n~n~~ switches on a defense reflex augmentation (Wilson & Davis, 1965). This temporal pattern is independent of the firing rate.

Neurons are a particularly good example because they help to distinguish the analog/ digital contrast in the sense used here from its association with the continuous/discrete dichotomy: the neural signals are always discrete but sometimes analog. We should also be clear that the use of analog and digital to ~~~ different kinds of notation is not applicable; the diitinction as used here is only a difference in the way information is physically represented. [Editor’s Note 1: Popular references to analogic vs. digital information are in most cases likely shorthands for analogically vs. digitally encoded information.-PLI In the two previous examples, neurons used analog representation for visual i~o~tion and digital representation for motor notation, but we also find digital ~p~n~tion in some visual system neurons (Lestienne et al, 1990) and analog representation in some reflex motor neurons (Burrows & Pfluger, 1988).

Although this distinction between analog and digital representations can be applied to a wide variety of bio-behavioral signals, it is rarely clear-cut. Typically there is some complex waveform that varies with many different biological processes, and neither the information content of the signal nor the ways in which different coruscation channels are embedded in this signal are entirely understood. But even with such limited knowledge, we may be able to use the generic properties of analog and digital representation to discern their presence.

The following sections will examine the analog/digital distinction in speech waveforms. Here both the information content and the various communication channels have been well explored. A simple rna~~ti~ model will allow the difference between analog and digital representation to be measured by computational experiment for spectral density plots, by analytic expression for Kolmogorov complexity, and in spectral measures of fractal dimension for empirical studies of human and non-human vocalizations. The discussion section will return to a generalization of this analysis for a variety of biological communication systems.

2. AnaIog and DigItal Representation in Acoustic Communidoa

Both analog and digital communication are found in the change of frequency over time in human speech. Figure la shows a graph of frequency over time for spoken American English, with energy as a grey scale (darker = higher energy). In this graph the word “cat* is pronounced three times with increasing Ievels of excitement (“cat” “cat!” “cat!!“). Those features that are invariant constitute the diital component of the word, and those that vary due to the change in emotional emphasis are the analog component.

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Watching a real-time display of these graphs can give an appreciation for how subtle semantic differences-happy excitement vs. curious excitement, for example-are encoded by different changes in the acoustic waveform. There has been a great deal of research on the correlations between such paralinguistic variation and its semantic content (see Traunmuller, 1988). Often these descriptions have a rather digital flavor-e.g., researchers ask what acoustic features are a “sign of” anger. If we are to examine this as analog representation, we would do better to phrase this question in terms of finding the proportionalities that relate changes in semantic parameters to changes in acoustic parameters. Most importantly, we want to consider the fluctuations in acoustic and semantic parameters over many different time scales, not just the single word or sentence.

This characterization underscores the progressive nature of the variation we see in the three utterances of “cat”: the analog parameter (frequency) rises in proportion to the increase of the semantic parameter (excitement). In Figure lb the phrase “cat dog cat dog cat” is graphed. Here affect is held constant, and the difference in frequency is only due to the digital variation. In this digital code the relation between semantic content and the physical signal is arbitrary. The frequencies for “cat” are higher than those of “dog,” but that convention does not imply anything about the differences in meaning-and indeed we find this convention reversed in other languages, such as Spanish (see Figure lc).

One exception to this arbitrariness is onomatopoeia: words which sound like the thing they describe (e.g., “buzz”,). The time scale for this analog component is, however, rarely longer than a single word. If I get increasingly excited while speaking, my fundamental frequency may be slowly rising through hundreds or thousands of words. There are a few other exceptions, but they also have few instances where the correlation between semantic parameters and physical parameters exceeds the length of a word.

This arbitrariness is increased by homonyms and synonyms, since they prevent unique mappings between meanings and waveforms (“degenerate coding” in information theory terminology). For example, “cat dog cat dog cat” is not only a periodic semantic sequence, but its physical waveform is also periodic. The sequence “cat dog feline canine” is also semantically periodic, but its physical waveform is not a repetition.

In addition to the arbitrary relation between meaning and individual words, longer term correlations in digital representation are disrupted by syntax, which allows very similar strings of words to have very dissimilar meanings, and vice-versa.

3. Spectral Density Comparisons of Analog and Digital Representations

We can formally model this difference by considering a signal in semantic space and its corresponding communication signal in physical space. Thus “ball” and “sphere” would be close together in semantic space but not in physical space; and vice-versa for “ball” and “brawl.” Figure 2 illustrates a onedimensional time series in semantic space and corresponding time series in physical space for both analog and digital representations. For the analog case I have assumed a linear proportionality, as we might find over a restricted range for the excitement-fundamental frequency relation. For the digital case I have assumed a simple code in which values in the semantic signal are represented by arbitrarily (randomly) assigned values in the physical signal, as we might find for the average frequencies of individual words in a sentence.

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Figure 2. Periodic analog and digital time series.

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We can examine the differences in these signals by looking at the energy levels over a range of correlation lengths. Figure 3 shows a spectral density plot (by FFT) for the analog and digital waveforms in Figure 2 (here with more sample points). The analog signal’s spectrum resembles the Fourier transformation for a unit ramp function (which, in fact, it is). Three different examples of arbitrary codes for the digital representation are shown; although their frequency composition varies, they all show the same white noise profile.

The portion represented in speech is not, of course, always new-in fact, we expect it will always include some repetition. If we consider a unique digital coding for repetition of the same information-as would occur for the same sentence repeated several times-then we would expect something like Figure 4. Figure 5 shows the spectral density plots for these waveforms, with three different codings of the digital representation. The digital signal shows the same sharp low-frequency peak, due to the repetition, in each case. The numero~ high frequencies here are partly due to Gibbs p~no~non, from the large number of discontinuities. In summary, both the analog and digital representations (in unique repetition) show a large low-frequency component, but this peak is smeared out across a wider band in the digital signal.

Finally, we can add the effects of non-unique digital coding-homonyms and synonyms for individual words, and syntax for strings-for the same repetition. Figure 6 shows the same periodic example with a completely different coding for each cycle in the digital case. The spectral density plots for these signals, with three different examples for the digital case, are shown in Figure 7. No large low-frequency component appears for the digital signals in this case. Of course, real communication will not use a completely different coding each time, and we can expect some mixture of unique and non-unique coding.

4. Complexity Comparbon~ of Analog and Digitnl Reprat&ations

The difference between analog and digital representations can also be quantified by complexity. Kolmogorov (1968) has defirked the complexity of a symbol string as the sire of the smallest program needed to calculate the string; for random strings the complexity is equal to the length of the string. Here we will use the grammar complexity defmed by Jimenez-Montano (1984):

1. The program consists of a context-free grammar that generates one unique string. 2. The complexity of the program is the sum of the lengths of strings comp~

the right-hand side of each production rule. 3. An exception to (2) are those strings consisting of only one symbol; their

complexity is equal to l+logr(length).

We can convert the point sampled waveforms into symbol strings by using zeros to represent the height of each point and ones to separate the zeros. For a ramp function eight points long we have the string: 101~1~1~1~1~1~1~. A periodic wave made of eight of these ramps, as displayed in the last example, can be generated by the following one-string grammar-

Fmctals in Biological Gnnmunication - 309

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which gives a complexity K~=(l+logz8)+24=28. For the same information with non- unique digital representation, the first rule S-e* cannot be applied-a different production rule must be used for each cycle. In general, given the same average complexity within each cycle, N cycles would give Ko=N+25 for this digital case. For the analog case Kc=(log~N)+25. Thus the complexity for non-unique digital representation will increase as a linear function of the length of the signal, but the analog representation of the same information will increase only as the log of the length of the signal. Again, real communication will not give a purely non-unique coding for each repetition, and empirical measures are required to determine the extent to which the arbitrary aspects of digital coding can diminish signal correlation lengths.

5. Comparisons of An&q and Digital Representations for Empirical Measures of Fractal Dimension

Both spectral density functions and Kolmogorov complexity can be related to fractal dimension. We need to understand that “fractal dimension of a time series” is not used here to designate the dimension of a phase-space object reconstructed from the time series. It is the fractal dimension of the time series waveform structure. Just as one can characterixe the irregularity of a mountain or cloud by treating it as the large scale portion of an infinite geometric recursion, time series waveforms can also be characterixed by their scaling properties (Mandelbrot, 1982). For the simple deterministic fractals generated by recursive replacement of a seed shape of N parts by reduction ration r, the “similarity dimension” is given by lo@/ log( 1 / r). This value can also be estimated by finding the number of boxes of size r needed to cover the curve (“box dimension”), or the length produced when N rulers of size r are used to traverse the curve (“compass dimension”). The type of curves produced by the time series used in this study are more complex however, in that the vertical (amplitude) and horizontal (time) coordinates scale differently. Such “self-time” curves will produce two different values for box dimension and compass dimension, and cannot be defined by a similarity dimension (Mandelbrot, 1985). They are best characterized by scaling properties of their probability measures.

In terms of probability, fractals can be seen as a compromise between order and chance. A shape is “most fractal” when the fractional part of its dimension is near 0.5. For less random shapes the dimension will increase towards the next highest whole number, for more random shapes the dimension will decrease toward the next lowest whole number. For time series this can be measured by the spectral density function. The fractal dimension of a white noise is 1.0, and the dimension increases for less random signals (i.e., lower Kolmogorov complexity). For continuous spectra with log dependence on frequency F and -2CslopeC0, the fractal dimension can be measured by beta in the spectral density function 1 over ?. Here beta is in linear proportion to the fractal dimension &according

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to D&2 - beta)/2 (see Gardner, 1978, for an informal account, and Voss, 1988, for a more rigorous approach).

As mentioned in the Introduction, we can expect fractal waveforms-that is, waveforms with beta=l-to dominate the information content of biological (and social) communications. But the physical representation of this information will have )a spectral density function that depends on its representation type. Since many representations will be somewhere between analog and digital, we can think of 0~ (in this context of l/F information distributions) as an index of position on an analog-digital continuum.

The initial studies of Voss (1978) suggest a confirmation of this relationship between the analog-digital distinction and fractal dimension. The communication categories used by Voss were instrumental music, music with few vocals, music with many vocals, and speech. The physical parameters were average instantaneous frequency (heard as pitch) for all categories, and an additional test of amplitude (heard as loudness) for speech and instrumental music. Since the frequency of speech contains contributions from both analog and digital components, we can expect long-term frequency time series of speech to have a fractal dimension (0~) between one and 1.5 (i.e., between white noise and l/F noise). The frequency of music, in contrast, produces long term fluctuations that mimic the fluctuations of affect (Meyer, 1956). As the orchestra swells, so do our feelings. In characterizing music frequency as analog representation, we can expect a fractal dimension around 1.5. By adding vocals to music, we can expect fractal dimension to change accordingly-the more vocals, the lower the dimension. In the case of amplitude time series, we can expect a different result for speech. Long term changes in speech amplitude reflect emotional intonation (“paralinguistic features”); as analog representation, we can expect a tendency to D~1.5. For music, loudness is modulated in analog relations similar to those of pitch, and thus we expect amplitude time series of music to also approximate D,=lS.

The data produced by Voss (1978) matches these predictions. Voss found that loudness in both music and speech produces a spectral density close to one over I;’ (i.e., D~1.5). The spectral density of frequency variations in music depended, however, on the amount of vocals in the music-the more vocals, the lower the beta. For speech alone, the spectral density approached one over I;” (DF~). Voss drew conclusions from this data that were similar to the distinctions I have discussed here.

I conducted three sets of experiments to extend these findings of Voss. The samples of the fint set compared a reading of prose, a reading of poetry, and a capella song. Voss did not examine song with all vocals and no instrumental music, but we can expect that this will have a much stronger analog component than normal speech, since the frequencies in singing are at least partly determined by the tune. Poetry nicely situates itself between speech and music. In poetry, words are picked as much for their sound as for their symbolic meaning. Here the normally accidental, short term associations such as onomatopoeia are carefully orchestrated, together with rhyme, meter, and other physical aspects of the waveform they are used to produce long-term physical signal correlations. Thus we can expect poetry to be positioned midway between analog and digital. The first five acoustic samples (of three categories each) were produced by an English speaker with a California Jewish-American accent (the author). The sixth sample was produced by an English speaker with a California African-American accent. The seventh sample was produced by a Mandarin Chinese speaker.

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The second set of experiments compared rap music with reggae music. The samples chosen for both types of music were heavily vocal, but differed in the way the vocals were sung. Rap music is notorious for a flat, non-melodic vocal style. I expected that rap would have a lower fractal dimension than reggae.

A third experiment examined cetacean communication. In my previous work (Eglash, 1984), a review of behavioral and neurological evidence indicated that cetacean communication was primarily analog rather than digital. Research on killer whales (Kirkevold & Lockard, 1986; Bain, 1988) indicates that their communication may have some digital representation as well as analog, whereas the “songs” of humpback whales (Payne, 1983) appear more purely analog. This experiment compared a 15 minute sample from Payne’s Song of the Humpback whale record with a seven-minute sample of killer whale vocalizations, recorded in the Johnstone strait by students from the University of California-Santa Cruz Long Marine Lab.

Initial recordings were made with a Sony TCD-SM and transferred to a RAYCAL for slowdown. Short term plots on an HP545 1 C spectrum analyzer indicated that aliasing distortion could be kept below six percent for sampling rates of 48 Hz for the human and humpback whale samples, and 192 Hz for killer whale samples. An HP5427 frequency counter was used to transform the audio recordings into a frequency time series (with the above sampling rates), which were then collected by an Apple IIe. The time series were transferred to an IRIS workstation, where an FFT produced spectral density plots. The spectra were truncated at a fluctuation rate of about I Hz to reduce error due to aliasing and Gibbs distortions, and this slope was calculated by least squares regression.

5. Experimental Results

A graph of the spectral densities from one of the samples of the author’s voice from the first experiment set is shown in Figure 8. The increase in the long-term correlations (amount of power in the low frequencies) for more analog signals can be observed. Figure 9 shows a graphical summary of all seven samples from this first experiment. Figure 10 shows a graphical summary of the samples in the second experimental set. Sources for samples from the first and second sets are summarixed in Figures 1 la and 1 lb. Figure 12 shows spectral density plots from the cetacean vocalizations.

As in the experiments by Voss, the presence of scaling structure in the spectral density is poorest for short wavelength fluctuations, and better for the low frequency changes. Even at low frequencies, there is considerable deviation from the log scaling predicted by a fractal model (standard error of estimate varied from 3.0 to 4.0; some of this was avoided in Voss by use of a lowpass filter). The theory presented here does not claim that fractal geometry is a good model for these waveforms, however; it claims only that the variation in fractal dimension co-varies with differences in representation type. The significant digits in the fractal dimension estimate are justified in terms of the precision of waveform resolution (sampling at 48 Hz for measurements starting at 1 Hz) but not in terms of the accuracy of geometric modeling.

Results in all three experimental sets matched the expected results, although there was considerable variation within the categories of the first experiment. Statistical significance was measured by the Wiicoxon rank sum test. The poorest significance level

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Figure 8. Spectral density plots of prose, poetry, and song.

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Nolrs: Summmy of rcaults from saxd expimentd set Mean tractal diinsiona are u fohws Rap - 1.216, Rspe - 1.351.

Figure 10. Summary of results from second experimental set.

Frucfals in Bidogical Cormnunkation - 395

Fmctal Dimadon

I.115 1.131 1.130 1.244 1.153 1.200 I.182

poetry hctal Llhension

1.141 1.193 I.250 1.280 1.359 1.228 1.221

SonS Ftacret Dimtnrfon

1.302 1.243 1.201 1.479 1.300 1.529 1.455

Rqp Fhwtal Dinlensrion

1.245 1.219 1.170 1.274 1.259 1.186 1.158

&r@= &zc& Dfmmrion

1.454 1.2% 1.341 1.329 1.285 1.386 1.374

SoUrn

May et al., “J Acoast Sot Am” 85@) 1989 pp. 837-847, English speaker Naimstadt et 1, “D/A Conversions” 1973, pp. l-2, English speaker Hawkes, %ructuralism and semiotics” 1977, pp. l-2, English speaker Spivak, intro to “Grammatology” 1976, p. xix, English speaker Heims, “Von Neumaan and Wiener” 1984, pp. l-2, English speaker Wasser et al,, m tie” 1976, pp. l-2, English speaker PRC, -Geography of BeQmg” 1970, pp. l-2, Mandarin Chinese speaker

source “Custard the Drqon” by Ggden Nash, English speaker “Dispassionate Pagan” by Ogden Nash, Erqlish speaker

‘Linap to be Scribbled.. .” by Ogden Nash, English speaker “The BigTent...” by Ogden Nash, English speaker “Jabbenvocky” by Lewis Carroll, English speaker “The Face of Old Woman” by Ann Cameron, English speaker “Snow” by Mao Tse-Tung, Mandarin Chinese speaker

SocUee

“Party in the Woods” by Jonathan Richmond, English speaker “Back in Your Life” by Jonathan Richmond, English speaker “Buzz Goes the Honeybee” by Jonathan Richmond, English speaker “Ice Cream Man” by Jonathan Richmond, English speaker ‘Bet You?n Goin’ Fishin’” by Taj Mahal, English speaker %a Top of Spagh&i” traditi- English speaker ~~~~~~~0~ ~~S~~

F@#e zzu. Sources for acolutic samples in set 1

SOWCC

"why is that?_ by Bwgie Down Productiolls ‘HoId Your Own” by Rid Frost “Eric B for President” by Eric B “The Bridge” by M.C. Shari “Supersonic” by JJ Fad “Queen of Royal Badness” by Queen Latifah “1% I%” by M.C. Lyte

source “Many Rivers to Cross” by Jimmy Cliff “Treach Town Rock” by Bob Marley “Pressure Drop” by Jiiy Cliff “Rivers of Babylon” by Jimmy Cl@ “You Can Get It” by Jimmy Cliff “Sing Our Own Song- by Judy Mowatt ‘Rock Me” by Judy Mowatt

F&we 116. Sources for acoustic samplea in Set 2.

3%- RON EGLA!SH

a

2

I\

l/l

1

4i

¶’

‘: \ l/f

1

Now Spearal dcmity plots for humpback and kilkr wbak vaaliutionr

Figure 12. Spectral density plots for cetacean vocakations.

Fractals in Biolqicd Communication - 397

was at 0.05, for discrimm ation between prose and poetry. ovation between poetry and song was at 0.025, and prose vs. song and rap vs. reggae better than 0.01. The Mandarin Chinese carried the same relation among the three categories as did the English samples. Humpback whale vocalizations had a higher fractal dimension than killer whale vocalizations, and both were higher than average human prose speech.

6. Discussion

In addition to unknown parameters, variation within categories may be due to lack of control over word rate, length of acoustic sample, and category definition. An attempt was made to restrict rap songs to those in the classic “talking” style, reggae to strongly melodic styles, and poetry to those with simple meter and rhyme, but these examples were sometimes ambiguous nonetheless. For example, the lowest fractal dimension for any reggae song measured was Bob Marley’s “Trenchtown Rock.” Here a chant style of singing is used to represent Rastafarian ideals of natural&d speech-that is, a more analog version of talking. For rap music, ideals celebrating urban identity lead to artificialized song, that is, a more digital version of singing. Since there is a conscious manipulation of the analog- digital dichotomy in these two expressions of African diasporic identity, the cultural meaning (and hence category) is determined by the context, and not simply the fractal dimension.

Other discourse modalities should also carry analogdigital distinctions. For example, we can expect ordinary limb gestures accompanying speech to be more analog than the gestures of sign language, and within sign language we can expect both poetic (emphatic) and linguistic communication. The fmger gestures for musical keyboards vs. word processing keyboards, and the written trace of handwriting vs. typeface are also good candidates for representation contrast.

An entirely different approach to the area of discourse analysis has been initiated by Li (1989), who has measured correlation fun~io~ for symbol d~t~bution rather than physical parameters. His results suggest that the correlation decrease due to the non-unique mapping discussed here applies to symbol strings and not just physical measures.

Although discourse offers the most complicated communication, other biobehavioral processes can also be analyzed in terms of analog-digital distinctions. For example, the difference between specializations of the right cerebral hemisphere and left cerebral hemisphere has often been characterized as an analog-digital contrast [Editor’s Note 2: See, e.g., McLuhan, M., “The Brain and the Media: The ‘Western’ Hemisphere,” Journal of Communication, 28 (4), Autumn 1978, pp. 5440. But note also that the degree to which specific furor and modes of ~ought~~~ion can localize in the two ~phe~ is controversial.-PLI; and the lateralization of signals from a wide variety of sensory modalities and communication systems has supported this distinction (Bradshaw & Nettleton, 1981). Both behavioral and neurological evidence suggests that cetaceans utilize the right cerebral hemisphere for their analog commu~catio~ (Eglash, 1984).

The lateralixation of abstract visual images has also indicated consistent relations with fractal dimension. Familiar Euclidian shapes attached to specific names (circle, rectangle, etc.) seem lateralixed to the digital left hemisphere, whereas more random shapes are lateral&d to the analog right hemisphere. But the right hemisphere lateralixation decreases

398 - RON EGLASH

as shapes become “too random” (Young, 1983). In other words, the right hemisphere lateralixation seems to co-vary with approximation to l/F spectral distribution for unfamiliar visual images.

Cerebral lateralization also offers us the opportunity to see how these different representations operate systemically. For example, observations of neural signals in the visual cortex of the cat (Lestienne et al., 1990) indicate that digital coding increases gradually with early development. This suggests a parallel development with the change from analog to digital representation in social communication, and right hemisphere to left hemisphere cerebral lateralization, as observed in a variety of non-human animals (Deuenburg, 1981).

Social communication in humans is a much more complex issue; in the matrix of culture the contrast of analog and digital has historically been used to denigrate various social groups. The “primitive” is said to be concrete, physical, and more concerned with emotion and body than with arbitrary symbol systems. The “oriental” is stereotyped as abstract, Arabesque, and more concerned with logic and symbolic systems than emotion. By promoting the sense of analog and digital as representation systems equally capable of conveying any type of information-a claim supported by recent evidence for computational universality in analog systems (Blum et al., 1989)-we can contest some of these denigrations. But more importantly, we should encourage recognition of the complex multiplicity of communication channels existing in a seemingly unitary waveform. Social discourse is always an intricate network of many different kinds of representation; it cannot be reduced to any one position on the analog-digital continuum.

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Acknowledgement

I am grateful for the assistance of the staff, students, and faculty of the Computational Mathematics Lab, Long Marine Lab, and the History of Consciousness Department at University of California-Santa Cruz.

About the Author

Ron Eglasb is currently a Fulbright Research Scholar at the Institut de Technologie Nucleaire Appliqu& University of Dakar, Senegal, where he studies fractal geometry in traditional African material culture.