Finding time

Running head: FINDING TIME Finding Time Peter R Killeen* Department of Psychology, Box 1104, Arizona State University, Tempe, AZ 85287-1104, United States Killeen, P R (2013) Finding time Behavioural Processes In press; http://dx.doi.org/10.1016/j.beproc.2013.08.003 Tel: +1 480 967 0560 Email: Killeen@asu.edu Running head: FINDING TIME Abstract We understand time through our models of it These are typically models of our physical chronometers, which we then project into our subjects A few of these models of the nature of time and its effects on the behavior of organisms are reviewed New models, such as thermodynamics and spectral decomposition, are recommended for the potential insights that they afford In all cases, associations are essential features of timing To make them, time must be discretized by stimuli such as hours, minutes, conditioned stimuli, trials, and contexts in general Any one association is seldom completely dominant, but rather shares control through proximity in a multidimensional space, important dimensions of which may include physical space and time as rendered by Fourier transforms Keywords: Boltzmann’s time, Caesar’s time, Fourier’s time, Minkowski’s time, Newton’s time, Temporal lenses, Time as motion, Spectra, Weber’s law Page Introduction The senses provide us not with a picture of the external world but with a model of it They serve not to achieve verisimilitude (whatever that might be) but to facilitate our interaction with it (Treisman 2006, p 222) Time and Association are the central concepts of this symposium Both carry heavy burdens of connotative meaning Their meanings are entangled For events to be associated they must share, it is said, a certain relationship in time For time to manifest, it must be associated with events A respected approach to untangling difficult conceptual issues, such as waveparticle duality, is to adopt the school of thought called “shut up and calculate” In behavioral psychology, its operant avatar is “shut up and reinforce” The reaction to this paper from some readers will be an even more parsimonious school of thought, “shut up and go away” This is because its contents will be most easily understood as non-sense, to those whose sense of sense has been schooled in the schools from which we all have matriculated Those who take time to consider the ideas are welcomed as associates on this strange voyage in search of time 1.1 Mathematical vs common time If you are asked to define time, your definition would probably resonate with Newton’s postulate: “true and mathematical time flows equably without regard to anything external, and by another name is called duration” This is our culture’s default model of time Its conjugate, relative or common time, is known directly, by sensible externals, moves unequably, and with duration measured in terms of the change in location of objects—in terms of motion To which time does the title of this symposium refer, mathematical time, or common time? If the former— absolute, true, and mathematical time—where is it to be found? In mathematical equations? In those, time is the argument; infinitesimal changes in it are the denominators of physics’ multitude of differential equations Time is always the thing with-respect-to-which change is Page evaluated; always the stage, seldom the play, and never the actor, for things only happen in time, not because of time Nothing in physics says that time flows, except Newton’s singular postulate that it does But what are the banks and bed of that river? Which way does it flow? Does it meander? Nothing in basics physics requires t2 to come after t1 (although the principle of least action does require smooth monotonic flow) We arrange things that way on a graph, whenever that makes them look simpler Physics does not tell us how to get from t1 to t2: “Just wait, and it will surely happen” is the best any physicist might Perhaps we can find true time in the vibration of cesium atoms, humming in unison at the national bureau of standards There, on an office wall we may find a graph and caption that looks like that in Figure Fig Reproduced from (Allan, Gray, and Machlan 1972), shows the best contemporary approximation of common time to mathematical time Reproduced with IEEE’s permission and payment Page Three things are notable about these data: Time is measured by means of recurrent motion; The motions are not equable: their periods support error bars, and drift over days; Measurement does not include allowance for the linear frequency drift of the ensemble Our best current standards thus measure common time What then is mathematical time, and is it relevant to psychology? Mathematical time is a hypothetical construct, which, along with absolute space, constitutes the framework of classical mechanics, wherein “time is defined so that motion looks simple” (J A Wheeler, cited in Doughty 1990, p 29) In its modern guise, however, it is less commonsensical than common time This is because we commonly think of true time as linear; or if not linear, at least a monotone dimension along which we can order events But Einstein showed that mathematical time does not itself flow equably: The period of a clock dilates with the clock’s velocity relative to a stationary observer He showed that the concept of “simultaneous” is incoherent across distance; and that the temporal sequence of two events can be different for different observers In sum, mathematical time is intrinsically dependent on the position and motion of an observer, and thus different for every observer t2 does not always come after t1 Common time is based on motion Sometimes the motion is linear, as sand or water through apertures But most often time is abstracted from periodic motion, from oscillations Indeed, even the hourglass must be periodically inverted, and water clock periodically refilled; and it is at those turns that information is concentrated, associations made The rotation of the earth gives days, the revolution of the moon gives months, and the revolution of the earth around its common center of gravity with the sun gives seasons and years But gravitational interactions among these bodies affect earth’s periods The standard of ephemeris time is computed in such a way to minimize the errors in imputing a common underlying time to all these oscillations; time is thus derived as the principle component of celestial oscillations Division is always more difficult than addition The length of Roman hours, based on the sundial’s division of the day, varied with the seasons Division of the year into months consistent with lunar cycles proved impossible, various attempts yielding the various calendric systems of extended time By counting the oscillations of a pendulum, medieval church bells enforced a common, fallible time for all within earshot In recent ages, divers communities kept reliable Page local times, often at odds with the time of other nearby communities In modern times reliability has been vastly improved by finding ever-faster oscillators and more accurate counters—and improved again by averaging ensembles of cesium oscillators Common time is now the principle component of atomic oscillators When it drifts sufficiently out of calibration with mundane yardsticks, the definition of one or the other—the year or the second or the clock—is then adjusted Such common time has become essential for commerce, transportation, communication, and geodesy Because of the expansion of the universe, with all other celestial bodies moving at different velocities than our solar system, it is disjoint from the time measured in other galaxies Common time is an artifact, no less than mathematical time But it is a directly, if diversely, measured artifact “Absolute, true time” is inferred for use as the denominator for the rate equations of physics and chemistry, stipulated to make the mathematics consistent What is important about common time is not that it approximates an absolute, true and mathematical time On the contrary What is important about it is that it is common It is close to us, our perceptions, our behavior, and close to our community It is what we evolved to sense as the stage upon which stance unfolds into action Common time is common to you and me Is it common to you and your rats? Does that matter? Not if there is an absolute true time that all organisms engage, each according to their abilities But there is no such time Even common time was commonly different between human communities until the railroad forced consistent time schedules, with telegraphy enabling the necessary synchronization Time is measured by motion, but motion changes the speeds of clocks’ pacemakers (even if only noticeably so at relativistic speeds) Likewise, in the study of animal chronometry, we find that operations such as conditioning and extinction change clock speeds (e.g., Killeen, Bizo, and Hall 1999; Morgan, Killeen, and Fetterman 1993) Unless their conditioning history and current status are identical, individuals—humans and rats alike—are likely to be marching to the beat of different pacemakers (Killeen 1991) Associating motion with meaning: Counting Analog watches are attractive because they show the time at a glance, without requiring enumeration If I asked you for the time, however, you would not hold your arms radially, in Page semblance of the hands on your watch The hands of most watches point to numbers; digital watches drop their hands and tell only numbers; and in either case, it is the numbers that you would tell in response to my request My only clue to your chronograph might be if you told me it was “twenty to ten” rather than “nine-forty”; that is, if you gave me the present time as a prediction of the next hour rather than as a trace of the last These reports of how far the oscillators have gotten since 12:00, to or from a particular number vector, (H, M) are mathematically equivalent, even though they might elicit different emotional responses from someone “running late” They mark the distance between oscillators of slower periods—hours— in terms of faster oscillators—minutes—and between those in terms of still faster oscillators The phases of your oscillator are meaningless until tallied—accumulated and converted to numbers The ratchets and dials and counters of clocks, and of the slower wall-clocks called calendars, the addition and enumeration modulo a basis When they run out of numbers—at 60 or 12 or 24 or 31—they start over In the process, they may increment a counter on another register—smart watches watch days and months as well as minding minutes and seconds Each of the registers is meaningful, giving meaning independently of (as well as in conjunction with) the values of the higher and lower registers That minutes have meaning beyond their qualification of hours is hinted in national news programs where you will hear: “At five minutes before the hour we bring you …” It doesn’t matter what hour they approach, or what it is that they carry; at five minutes before it they will bring it to you On many campuses classes end 20 minutes after “the hour” Saturdays are great whatever the week, Mondays trying whatever the year If this is so with modern chronometry, might it not also be so with simpler systems of timing in simpler biological systems? The meaning given by an hour’s number is real but scant, until it is associated in its turn with an important event: the 6am alarm, the noon lunch, the 5pm cocktail, the ‘date’ at The hum of cesium gives time by associations, first to cumulating numbers and thence to reinforcers There would be no utility in timing without the last association; and without the last, there is typically no timing Watches and clocks are so familiar and so important in signaling rewards and punishers—in focusing those biologically significant events on the stream of our behavior— that it is hard to see them for what they are: a temporal lens Page Lensing Hold a large sheet of paper up to a window, and light from all of the reflective elements in the scene will illuminate all of the parts of the paper Hold a magnifying or eyeglass over any part of that paper, at the right distance, and the scene will reconstruct over a portion of the paper Move the glass at that distance over various parts of the paper, and the scene moves with it, morphing ever so slightly as it is moved Each point of the paper receives information from all the sources of light in the scene It is a hologram The information may be recorded by replacing the paper with a photographic plate, and the source with light reflected from the scene illuminated by a monochromatic beam and a reference beam, creating interference patterns Then the record of the scene may be revealed by a different kind of lens: Transillumination of the plate with the original reference beam creates a three-dimensional image In either case, by distorting the information arriving at the paper, we may reconstruct different versions of the scene But all of the information was lambent on every part of the paper or plate, slightly different versions, depending on their angle, across the paper as a whole An eye uses a lens to reconstruct holographic scenes projected onto the retina Two eyes use the slight differences in the information they receive to help reconstruct a three-dimensional visual image, an analog of the movement of the magnifying glass, or the interaction of reference and image beams in holography Now open the window and exchange the lens for a microphone Move it over the parts of the paper and it will receive very similar sounds at all points Of course it is not receiving sounds, is transducing acoustic vibrations into electrical waveforms, ones whose voltage is correlated with the rarefactions and compressions of the lambent air If we could measure the slight vibrations at different parts of the paper, they would be highly but imperfectly correlated; they would form a hologram of the auditory scene Our ear is also a transducer, a different kind of lens The cochlea does not focus the incoming waves like the eye’s lens, but resonates differentially along its length to the different frequencies implicit in the waveform According to Ohm’s acoustical law, it conducts a Fourier analysis in real time (only a first approximation to the whole story) The two ears use, among other cues, the slight differences in the information Page they receive to reconstruct a three-dimensional acoustic image, constituting an analog of the 3-D visual reconstruction The correlation of those analogs, and those with touch, create our sense of space All of the information, in original unfocussed format, is available in the electromagnetic and acoustic wave fronts impinging on the subject, and on the environment around her No one version is more veridical than the other But we have evolved the lenses of our eyes and ears because their version of the information—the stories that they told—was more useful in our survival as a species than the unfocussed wave fronts The eye and ear effect additional tweaks, by adding secondary properties They use lateral inhibition to enhance the edges of visual and acoustic objects Most humans colorize scenes, with the hues corresponding to wavelengths-inthe-context of others The color of a focal patch of light typically does not have a close correspondence with its wavelength (Land, 1986) Instead, the hue is determined by the interaction with nearby patches Thus the lens of color vision grants color constancy by denying wavelength veridicality Different creatures have more or fewer eyes, eyes with more facets or more or different wavelength sensitivities Are these integrated for a unique image, or processed separately as survival found useful? Do the chemical sensors of a snake’s tongue, the thermoreceptors on its face and the visual input to its eyes interact before scene reconstruction, or after? If the senses of vision and hearing and touch each have their own kind of lens, what kind of lens does the sense of time have? What distortions of time have nature found condign to survival? How is time edge-enhanced, contextualized, and colorized? Is there temporal synesthesia? If we could reconstruct time at its source, would we find Einstein’s, or Newton’s, or Neanderthal’s time? Or none at all? The answer is addressed by first considering how humans have measured time, and subsequently used those measurement designs as analogies for how humans and other animals cope when tasked to measure time without access to those very artifacts Temporal scene reconstruction 4.1 Clepsydra’s flow Page 10 A clepsydra is a water clock—a bowl with a hole Clepsydras attained some complexity in the ancient world, some accessorized with counters made of elegant turnstiles, others with counters made of indigent humans Some Roman Clepsydra involved nested arrays bowls (Fig 2) Clepsydras manifested, and may have inspired, the metaphor of the flow of time They also may have inspired the leaky-sieve model of Staddon and Higa (1996; 1999) Their clepsydra passes water from one unit to the next not when each has filled, but continually with each input, with the amount passed depending on how empty each was This interesting inversion of the water clock constituted a bank of low-pass filters, and accounted for many of the effects of periodic stimulation Fig A chaotic clepsydra Water enters the top bucket, which, when full, tips over into the larger one below, ad seriatim Killeen and Taylor (2000) likened this clepsydra to a chain of neurons each of which required multiple inputs to fire, and whose output was successfully transmitted with probability p The probability that each increment succeeds is pi The wavefront of this system advances as a logarithmic function of time Another exploitation of the nested clepsydra array involved a cascade of poorly engineered buckets of geometrically increasing size, with the effluent of each having probability p of landing in the one below (Fig 2) The dynamics of this clock are chaotic (Killeen and Page 17 Fourier analysis generates a dual of any function in the time domain in a corresponding frequency domain, preserving all the information in the original signal Conversely, and with astounding symmetry, another Fourier transform of that information from the frequency domain recovers the original signal precisely The original and the transform are called Fourier pairs, or conjugates 4.5.1 Postulates Consider the following hypothesis: Organisms’ temporal lens transforms signals in the time domain to signals in the frequency domain: It performs a spectral analysis In particular: • Association requires proximity of stimuli in a multidimensional character space, with a key dimension of that space being frequency • Events will be positively associated to the extent that they have amplitudes sufficiently in phase at similar frequencies; • Events will be negatively associated to the extent that they have amplitudes sufficiently out of phase at similar frequencies • Frequencies are driven by recurrent events of biological significance, including: o Circadian rhythms; infradian rhythms, such as seasonal; and ultradian rhythms, including hormone, gene, and cortical rhythms o Ultradian rhythms may be exploited, amplified, or created to focus the temporal location of recurrent events such as food, shock, and other biologically significant stimuli (Boulay et al 2011; Bonnefond and Jensen 2012; Marchant and Driver 2012; Freestone and Church 2010) o All together these form a polyrhythm that regulates associative conditioning Page 18 o Researchers have adduced evidence for such oscillations (as early as Treisman, Faulkner, and Naish 1992 for cortical rhythms, earlier for other types), and suggest that they may play important roles in many sensori-motor and cognitive activities (e.g., Brown, Preece, and Hulme 2000; Giraud and Poeppel 2012; Romei, Gross, and Thut 2010) Matell and Meck (2000) provided an excellent review of the map between theories of timing and corresponding brain substrates at the end of the 20th century The maps have expanded exponentially in the 21st (Meck, Doyère, and Gruart 2012) • The guiding polyrhythm is also conditioned by its components’ sources in visual or acoustic or endogenous space (Wilkie 1995; Roussel, Grondin, and Killeen 2009; Grondin 2003) It is a multidimensional time-space map, sensitive to order, phase, and interval (Carr and Wilkie 1997; Uttal 2008) • The resolution of the temporal lens is limited by its instrument or pointspread function—a Gaussian, Cauchy, or other similar distribution 4.5.2 The instrument function Transducers, human and non-, are never perfect The error in transduction may be modeled by what is called an instrument function or point-spread function In the time domain it may correspond to the generalization gradient, the delay of reinforcement gradient, the Shepard function, and so on It affects the representation in the frequency domain by an operation called convolution, effected by taking the product of the Fourier transforms of the signal and the instrument function If the instrument function is Gaussian, and the signal is a periodic pulse (a Dirac function), their conjugates are also Gaussian and periodic pulses, and the resultant representation is one of small Gaussian distributions along the frequency spectrum, arrayed like bumps on a log This is what we would expect for the representation of a periodic feeding schedule Page 19 There is an inverse relation between periods in the time domain and frequencies in the conjugate domain Dirac functions recurring at period t (a Dirac comb) are represented in the frequency domain by Dirac pulses spaced at 1/t (In general, if F(t) is conjugate to φ(f), then F(at) is conjugate to |1/a|φ( f /a).) If the period t is increased by the factor a, then the representation on the frequency domain will be crowded as f/a, or 1/(at) Imagine the complementary accordioning of adjacent instrument functions in the frequency domain as the period of the signal is changed in the temporal domain If the variance of the instrument function σ Hz is constant, as a is increased the crowding of the means in the frequency domain as 1/(at) will force increased perceptual overlap In particular, it makes the coefficient of variation of such judgments increase proportionately with the interval judged This is inconsistent with Weber’s law, and thus invalidates such a mapping In the case of pitch, the primary tonotopic maps on the basilar membrane (Greenwood 1990), and on the auditory cortex (Romani, Williamson, and Kaufman 1981) are logarithmic, as is the timing cascade of Fig If the instrument functions are constant on such axes, perceptual judgments and confusions on the temporal axis will perfectly accord with Weber’s law, aka the scalar property As the frequency representation is bounded on one end by the upper limit of resolution of ensembles of neural volleys, and on the other end by the ability of the nervous system to sustain very slow oscillations, this becomes a generalized Weber Law, with relative sensitivity deteriorating quickly at the very shortest intervals, and more slowly at the longer (Gibbon et al 1997; Bizo et al 2006) These profiles may be used to characterize the transfer function of the temporal lens In their striatal beat-frequency model of timing Oprisan and Buhusi (2011, p 1; 2013) provide converging evidence for such instrument functions, in that “parameter variability (noise) – which is ubiquitous in the form of small fluctuations in the intrinsic frequencies of neural oscillators … – seems to be critical for the time-scale invariance of the clock and memory patterns” The variance is always finite, even in the finest of oscillators (see, Fig 1) 4.5.3 Ringing If animals learn to anticipate certain frequencies of feeding, we might expect anticipatory behavior to continue when feeding is discontinued—to see “ringing” of behavior Sometimes we Page 20 Monteiro and Machado (2009) provide an excellent review of the issues, along with spectral analyses of new data Apparently animals must learn both when to respond, and when not to respond This can be trained with the peak interval procedure, constituting an interleaving of feeding at periods of p and np, with various additional stimuli and fixed delays The period np is typically or times the base period If the animals conditioned in real time, we would expect to see ringing at an average (or alternation, if the mixing is not random) of the two periods—the modulation of one frequency by the other—, which these authors found Animals can readily adjust to some anisochronies in period, and only poorly to others (Staddon, Chelaru, and Higa 2002; Higa, Thaw, and Staddon 1993) The analysis of these dynamics in the frequency domain will provide good exercise for the student 4.5.4 Spectral distances Consider the following experiment: Replicate a stimulus train, F(t), called US, with another train identically distributed in time, but shifted earlier with respect to the US, F(t – a) Call the leading signal the CS train The Fourier conjugate of the US train F(t) is φ(f); the conjugate of the CS train is the same as that of the US train, plus a projection on the imaginary axis, φ ( f ) e−2 π ifa (the fundamental shift theorem of Fourier analysis) If a is zero, the original US signature is recovered As the two stimuli move increasingly out of phase, they become increasingly different in their projections on the imaginary frequency axis (the relative sine component) Notice that the projection changes exponentially with the lag, a So also does the delay of reinforcement gradients for trace conditioning of responses (Killeen, 2011) If association requires proximity of stimuli in a multidimensional character space, and a key dimension of that space is frequency, then as a increases, so does dissimilarity in the relative phase dimension There will be a generalization decrement between the stimulus trains Notice that in the frequency space, a multiplies frequency, which we may rewrite as a/t We expect similar decrements in conditioning when this ratio is the same This is also the case for the ratio of trial to inter-trial durations in autoshaping (Gibbon et al 1977) When between 90° - 270° out of phase, they may become negatively associated Generalization does not require strict contiguity (Killeen and Pellón 2013), nor does it require periodicity of the US train The shift, Page 21 however, cannot be random (Neuringer and Chung 1967) If the similarity function is the Euclidean metric, then the distance of the CS and US in this space is the square root of the sum of the squares of the projection of the imaginary, sine, axis, and the projection on the cosine axis This is how one computes the spectrum of the signal Thus, the effects of delay of reinforcement may reflect in differences of the spectra in the frequency domain Instrument functions that constitute the temporal generalization gradients could be reflected as exponentials (Shepard 1987), possibly asymmetric in temporal direction, and thus be consistent with observed delay of reinforcement gradients for trace stimuli Because the point spread functions lie in the frequency domain, with phase represented on the imaginary frequency axis, the importance of proximity of a signal to reinforcement (determining the sine component, represented on the imaginary axis) will interact with the base period of the stimuli (Gibbon et al 1977) 4.5.5 How delay debases spectral correlations A CS may extend in time, constituting delay conditioning rather than trace conditioning Such a rectangular signal in the time domain of width a has as its Fourier conjugate the sinc function, sin(πfa)/πfa (see the visual abstract) The height of the sinc distribution changes inversely with the duration of the CS For simplicity of illustration, treat the US as a rectangular pulse of width centered at t = 0, with the CS centered on it Call the delay from the leading edge of the CS to that of the US d As d increases, the overlap of signal area in the frequency domain decreases as approximately 1/(1+kd) A more precise comparison is found by asking for the similarity in the frequency domain Figure shows the cross-correlations of the power spectra of US signals with those of CSs that lead it by one through units of time Power spectra are the root-mean-square of the cosine and sine coefficients at each frequency, so include information about phase Such hyperbolic decreases in associability with increases in CS duration have been well documented, even while the gradients for trace signals decay exponentially (Killeen 2011), presumably as they move out of the ambit of the instrument function Page 22 Fig As a signal is extended to lead the US, its signature in the frequency domain becomes increasingly dissimilar to that of the US The squares are the cross-correlations of the spectra of the CS and US signals The curve is the inverse linear function called hyperbolic Whereas conjugate pairs can represent any signal profile, they should be especially useful for rhythmic or repeating stimuli There is a large literature on beat, or synchronization timing (Repp and Su 2013; Teki, Grube, and Griffiths 2011) and its neural substrates (Grahn 2012; Matell and Meck 2004) Analysis in the frequency domain should clarify certain findings For instance, each of teeth in the Dirac comb has conjugates on the frequency domain, each with its own instrument function If these are independent, as the number of teeth increase, the standard error of measurement of the period should decrease as the square root of their number If the JND for differences in tempo at an optimal tempo were 4.4% for a single stimulus, we would expect 3.1, 2.2, and 1.8% for 2, 4, and teeth Drake and Botte (1993) performed the experiment, and found JNDs of 4.4, 3.0, 2.2, and 1.6% In the frequency domain, this is seen as a narrowing and heightening of the energy at the frequency of the tempo They also found that JNDs increased with irregular tempos For instrument functions such as the Gaussian, this is expected, as more will be lost due to crowding of some teeth than is gained from separating others; the variance of the stimuli will need to be added to that of the instrument functions In the Page 23 frequency domain, energy will be lost to harmonics that are not at the base tempo See McAuley (2010) for a summary of recent work in this area 4.5.6 Aliasing When brief non-overlapping stimuli are placed close enough in the time dimension, their frequency conjugates will exceed the upper limit of resolution of the temporal lens When this happens, an effect called aliasing occurs The frequency domain picks up different parts of the signal, increasingly out of phase This causes the familiar apparent backward rotation of wagon wheels in old western movies If such an effect occurs for the temporal lens, we should see apparent reversals of order for sufficiently brief stimuli Such effects, if found, will provide an excellent analytic tool for determining the transfer function of the temporal lens 4.5.7 Bases Fourier analysis uses trigonometric functions for good reasons, but other bases may be more useful for the study of associations in time Gabor functions constrain the abscissae more realistically, and wavelet analysis is a more general approach, of greater potential use in the study of human behavior in general, and temporal conditioning in particular Indeed, George Zweig invented the continuous wavelet transform for his analysis of acoustic responses Wavelet analysis permits a larger set of basis functions (e.g., rectangular signals), and localizes signals in both frequency and time It thus constitutes a kind of confocal lens on epochs of times, past or future Other types of basis functions are found in the oscillations of registers comprising centuries, decades, years, and months; and on successive registers of a binary counter The brain need not follow the logic of addition-with-carry in using these registers—each bank could act as a discriminative stimulus in its own right, one that combines with the others in multidimensional association space Simulations under this assumption provide some verisimilitude to behavioral data (Killeen and Taylor 2000; Killeen 2002) Crystal and associates (Cheng and Crystal 2008; Crystal 2006) have documented multiple points of peak sensitivity to time, which presumably correspond to frequencies that have greater representation in the polyrhythm Banks of such Page 24 oscillators have been proposed in several models of timing (e.g., Church and Broadbent 1990; Miall 1989; Miall 1993) 4.5.8 Componential analyses The separate accountability of the various frequencies in associative conditioning is reflected in the research of Friedman, who found that people caught in an earthquake were much more accurate in reporting its hour than its month: “In linear conceptions of time, cyclic patterns are subsumed by the single chronological sequence But … this subordination is artificial … we can often remember cyclic locations with an accuracy far beyond our sense of linear distance in the past… cyclic time is independently derived in memory for time.” (Friedman 1993, p 55) That is, each of the registers matters, beyond its contribution to the final count The basis functions may be independently conditioned, not subsumed by the aggregate In M R Jones’s (e.g., Large and Jones 1999) theory of dynamic attentional responding, the periods and phases of the attentional rhythm may themselves be conditioned Such conditioning may be effected by the selection of appropriate rhythms to guide that attention— equivalent to the conditioning of the coefficients in the Fourier series The experiments in rhythm perception and production addressed by that theory would be an ideal sandbox in which to develop applications of wavelet theory for time Conclusions The possibility of a severely nonlinear lens on time, such as constituted by a Fourier analysis, has been approached by a number of authors based on their analyses of behavioral characteristics and their potential neurophysiological substrates (e.g., Church and Broadbent 1991; Lewis and Miall 2006) The work of Meck and his associates (e.g., Buhusi and Meck 2005; Matell and Meck 2004) is particularly penetrating The present speculations may be seen as suggestions for additional tools that might prove useful in those research programs The idea that explicit visual or acoustic stimuli should have absolute control over behavior is one that confounds many observers—animals will respond for food in the signaled absence of food, climb over piles of pellets to access pellets of the same food, work for food that is freely available, and begin responding toward the end of an interval during which no food will Page 25 be forthcoming Fear covaries with the time of day as much as time-to-shock Animals respond differently to a stimulus presented in one location on a screen, than to the “exact same” stimulus presented in another location; time-outs from an experiment—the inter-trial intervals—interact, often strongly, with the behavior occurring during the trial Our disappointment in the irrationality of our subjects follows from the implicit model of rationality that we project as a standard for them, one in which the most predictive cue should receive all of the decisional weight—a “supremum metric” But once we recognize that animals have evolved a more balanced, hedged, view—with the information they consider being a weighted average of all cues, including time since start of day (Valentinuzzi et al 2001; Hineline 1972), and start of session, and of zero-crossing of other rhythms—then the above curious behaviors may move into better focus Spectral analysis permits each of the fundamental frequencies to enter their own associations, along with spatial and other representations, to the phylogenetically important events that drive the process of association (Davison and Baum 2010) “If we can learn what happens at specific parts [phases] of patterns of time,” Friedman observed (1993, p 60), “whether it is the migration of a species at a particular time of year or the days of the lunar cycle when one can safely travel at night, we can exploit this knowledge in the future The absolute distance of an event in the linear past is nearly always useless information” It is this prominence of phase information that may have led to the evolution of a spectral representation of time And yet, what is typically studied and reported in special issues such as this one is animals’ ability to “learn what happens” in linear time—in Newton’s time We may only find our subjects’ common time by looking into the frequency domain As Pribram (2013) has repeatedly observed, it is only through the right lens that we will be able to see the form within Page 26 References: Allan, D.W., Gray, J.E and Machlan, H 1972 The National Bureau of Standards atomic time scales: Generation, dissemination, stability, and accuracy IEEE Transactions on Instrumentation and Measurement, 21: 388-391 Bizo, L.A., Chu, J.Y., Sanabria, F and Killeen, P.R 2006 The failure of Weber's law in time perception and production Behav Processes, 71: 201-10 Bonnefond, M and Jensen, O 2012 Alpha oscillations serve to protect working memory maintenance against anticipated distracters Curr Biol., 22: 1969–1974 Boulay, C., Sarnacki, W., Wolpaw, J and McFarland, D 2011 Trained modulation of sensorimotor rhythms can affect reaction time Clin Neurophysiol., 122: 1820-1826 Brown, G.D., Preece, T and Hulme, C 2000 Oscillator-based memory for serial order Psychol Rev., 107: 127-181 Buhusi, C.V and Meck, W.H 2005 What makes us tick? 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If the former— absolute, true, and mathematical time? ??where is it to be found? In