NEURONAL PERIODICITY CODING AND PITCH ...

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Zoological Institute, Technical University of Darmstadt. Schnittspahnstr. 3, D-64287 Darmstadt, Germany e-mail: [email protected]. ABSTRACT.
NEURONAL PERIODICITY CODING AND PITCH EFFECTS

Gerald Langner Zoological Institute, Technical University of Darmstadt Schnittspahnstr. 3, D-64287 Darmstadt, Germany e-mail: [email protected]

ABSTRACT The central nucleus of the inferior colliculus (ICC) is the major nucleus in the auditory midbrain and plays an important role in periodicity coding. A neuronal model implementing temporal coding, neuronal oscillations, temporal integration and coincidence detection is able to explain a variety of response properties observed in periodicity tuned neurons in the ICC. Since the periodicity range coded in the ICC seems to cover the existence range of periodicity pitch, the temporal tuning of neurons in the ICC may be the functional basis of pitch coding. The evidence for this hypothesis comes from the fact that the neuronal model describing the response properties on the ICC neurons may also account for a variety of pitch effects. INTRODUCTION In harmonic signals the envelope periodicity corresponds to the fundamental frequency and reflects periodic vibrations of sound sources. Signals with the same envelope frequency (or modulation) may produce the same pitch in spite of different waveforms or frequency compositions of their sounds. Harmonic signals are characterized by a well defined periodicity and therefore have the same pitch as long as the period of their envelopes is the same (‘periodicity pitch’, Schouten, 1940, 1970). It is claimed here that periodicity pitch is analyzed in the temporal domain. For the result of the peripheral frequency analysis it is essential that each frequency component of a harmonic sound is an integer multiple of the fundamental frequency and, consequently, also each superposition of two ore more adjacent frequency components has an envelope with the same frequency as the fundamental. Broadband signals are characterized by a superposition of harmonics in various frequency channels of the auditory periphery and will therefore elicit neural responses synchronized to the fundamental in these frequency channels. According to the present theory, intrinsic neuronal oscillations and temporal integration in the cochlear nucleus and coincidence detection in the ICC contribute to the analysis of periodicity information (Langner, 1992). At the level of the midbrain temporal information is degrading, while periodicity is represented orthogonal to frequency information (Schreiner and Langner, 1988). An orthogonal representation of frequency and pitch was also demonstrated in the human auditory cortex (Langner et al., 1997). Note, that orthogonality implies that the parameters represented are largely independent variables. The idea is, that frequency is related mainly to the timbre and periodicity mainly to the pitch of a (harmonic) sound. In psychophysical

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experiments this independence (or orthogonality) of timbre and pitch has been shown by Plomp and Steeneken (1971). The present neuronal model is able to explain a variety of response properties observed in neurons of the ICC. Some of them are: -the tuning to a certain best modulation frequency (BMF), -multiple tuning of many neurons, especially to harmonically related modulation frequencies, -a change of BMF with the carrier frequency of sinusoidal amplitude modulated sinusoids (AM) -oscillatory spike patterns of responses to AM-cycles -interval patterns of onset responses -adaptation and facilitation in responses to different AM-signals. Moreover, some of these response properties are adequate to explain certain perceptual effects, like the perception of the missing fundamental, the first and second pitch-shift effect and the perception of harmonics. Finally, the model is capable of predicting new percepts, like the onset pitch-shift described below. TEMPORAL ANALYSIS IN THE AUDITORY SYSTEM In a correlation analysis a delayed and an undelayed version of the signal are multiplied ‘point by point’ and the result is summed up. The result is maximal when the delay equals a major period of the signal. Consequently, in order to analyze periodicity, delayed and undelayed neuronal responses to signal envelopes are required which then converge on neurons functioning as coincidence detectors. The model presented in Figure 1 suggests that each cycle of a modulated signal triggers a rapid and short intrinsic oscillation, that is, a neuronal response including short delays only. At the same time an integrator neuron responds to the same cycle with a longer (integration) delay. The coincidence unit receives input from the integrator and the oscillator. In spite of their different delays, an input from the integrator and from the oscillator (responding to the following cycle) will arrive at the same time, provided the integration delay is equal to the period of the signal. More precisely, the coincidence unit will be activated when the different delays of its inputs are compensated by the periodicity of the signal. The result is that the coincidence unit is tuned to a certain periodicity and therefore is appropriate to code the corresponding pitch. It is obvious that a multitude of oscillators, integrators, and coincidence detecting units covering the relevant range of periodicities is required for the perception of the total pitch range. Evidence from electrophysiological studies of neuronal properties in the brainstem are in line with this theory (Langner, 1983). Auditory nerve fibers show synchronized responses for frequencies up to about 5.kHz and to periodic envelopes with modulation frequencies even above 800 Hz (Palmer, 1982). The temporal filter curves of neurons in the cochlear nucleus more or less resemble those of bandpass filters (Frisina et al., 1990). Responses highly synchronized to envelope periodicities may be obtained from onset, chopper and pauser units (Rhode and Greenberg, 1994). For the present model it is assumed that these units provide the temporal information appropriate for the tuning of the coincidence units. Onset neurons are known to provide very precise temporal representations of AM envelopes (Moller, 1970). In the present model such a property is suitable to synchronize the action of the oscillator and integrator. In the model, chopper and pauser neurons introduce delays in the temporal representations of periodic signals: chopper neurons by means of their well-known regular firing patterns (intrinsic oscillations; Pfeiffer, 1966) and pauser neurons by means of temporal integration. Neuronal recordings in the auditory brainstem of various mammals and birds have revealed intrinsic oscillations with intervals in the millisecond range (for an overview, see Langner, 1992). In the present model, chopper neurons were assumed to produce intrinsic oscillations with periods which are multiples of a synaptic delay of 0.4 ms. The reason for introducing such preferred periods in the theory was that evidence for such preferred oscillation intervals was 1

found in guinea fowl, cat and also in pitch-shift experiments in human subjects (Langner, 1981, 1983, 1992; Langner and Schreiner, 1988). Some characteristics of pauser neurons including their inputs from the auditory nerve and from inhibitory interneurons make them likely candidates for integrator units of the model. The response properties of coincidence units suggest that the delay of the integrator is preferentially a multiple of the carrier period. As a consequence, it is assumed that the integration process requires always the same number of spikes to drive the membrane potential of the integrator unit to threshold. At a level more than 30 - 40 dB above threshold auditory nerve fibers are saturated and the number of spikes is indeed independent of the signal amplitude. Therefore, the small dynamic range of the auditory nerve is an essential prerequisite of the present model. As a result of phase coupling to the signal fine-structure (e. g. the carrier frequency of AM) and the time constant of their cell membrane, the integrator neurons generate action potentials with delays corresponding to a certain integer multiple of the carrier period. For higher frequencies this is only possible by use of the ‘volley principle’, that is a parallel coding of nerve fibers (Wever, 1949). However, if the activity of the auditory nerve varies, for example during the onset response, the integration interval may vary too. Finally, for frequencies higher than 5 kHz, the integrator could function in the same way, but the resulting delay could not be correlated with the carrier period. As required for the present model, chopper and pauser neurons are known to project directly to the ICC (Moore and Osen, 1979). The coincidence of oscillator and integrator responses with different delays may explain why neurons in the ICC respond best to a particular modulation frequency. Neurons tuned to a certain periodicity were indeed found in the ICC of many different animals (Langner, 1992). Coincidence neurons respond best, when their inputs are synchronized. The activation of coincidence neurons indicates that a certain periodicity of the acoustic signal is adequate to synchronize their input neurons and to compensate for oscillation and integration delays of their inputs. In consequence, the modulation period τm which drives a coincidence unit may be computed from a linear equation of the involved periods, that is the periods τc of the carrier, and τk of the intrinsic oscillation: m τm = n τc - τk (periodicity equation), where m and n are small integers (Langner, 1983). For a given neuron the preferred τc would be defined by its center frequency (CF = 1/ τc) due to peripheral frequency analysis, while n and τk are defined by the temporal characteristics of its input neurons (integration delay and oscillation period, respectively). The parameter m takes into account that some coincidence neurons do respond also to a harmonic (m>1) of its best modulation frequency (m=1). Such responses to higher harmonics require that the integrator is not disturbed by the modulation once it is triggered, while the oscillator is triggered by each modulation cycle. The occurrence of such responses in the ICC indicates that this kind of temporal analysis tends to confuse harmonically related sounds, or, positively expressed, indicates harmonic relationships.

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Figure 1. A hypothetical neuronal circuit for a periodicity analysis in the auditory brainstem. It is assumed that the coincidence neuron is located in the ICC and that all other neurons of this circuit correspond to cells of the cochlear nucleus and the ICC. The spike trains at the bottom of the figure indicate trigger, oscillator, and integrator responses synchronized to the signal envelope. As expressed by the periodicity equation, the delay due to the integration time of the integrator circuit must be compensated by the period of the signal for the coincidence unit to be activated. Therefore the integration time of the integrator is crucial for the best modulation frequency of the circuit. In addition, a more or less small contribution takes account of the oscillator delay. These conditions may be expressed by τm = n τc - τk (periodicity equation).

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Time constants characterizing temporal synchronization, intrinsic oscillations, and neuronal integration all range in the order of a few milliseconds and restrict coding of signal envelopes mostly to frequencies below about 1 kHz. This may explain, why the envelope periodicities coded by most neurons in the ICC as well as the fundamental frequencies of human speech sounds and of most musical sounds are restricted to that frequency range. As a result of coincidence detection temporal information is translated into a rate code, while synchronization especially to higher modulation frequencies is diminished in the auditory midbrain and, consequently, also in the cortex (Langner and Schreiner, 1988, Schreiner and Langner, 1988b). Another interesting aspect of the temporal analysis is the emergence of a periodicity map at the level of the ICC. While the result of the frequency analysis is arranged in frequency bands related to psychophysical critical bands and organized in anatomical laminae (Schreiner and Langner, 1997), the result of the temporal analysis is represented in the ICC, orthogonal to the tonotopic organization (Schreiner and Langner 1988; Langner, 1992; Heil et al., 1995). Neuronal maps are topographic arrangements of information bearing signal parameters. By providing relative positions and continuous shifts of excitations, such maps may be useful for processing relations and variations in the corresponding signal space. Also, in the forebrain of mynah bird envelope periodicity was found to be represented roughly orthogonally to the frequency gradient of the tonotopic map (Hose et al, 1987). In the cat cortex corresponding evidence was now found with optical recordings (Langner et al. 1997). Finally, in accordance with the topographic arrangements found in animals, magnetoencephalographic recordings from the human auditory cortex showed that cortical responses to pure and harmonic tones are arranged orthogonally (Langner et al., 1997b).

PROPERTIES OF COINCIDENCE NEURONS AND PITCH EFFECTS A harmonic signal elicits the same pitch as its fundamental frequency, even when the fundamental frequency is not a physical component of the signal. Even a small part of its frequency spectrum with two or three harmonics may be sufficient to elicit the percept of the ‘missing fundamental’ (Schouten, 1970). Already Schouten (1940) suggested that this effect may be explained by a neuronal periodicity analysis. In the present theory, the best frequency range for the analysis of the periodicity of a broadband signal is characterized by certain demands for the integrator circuit: some neurons should be able to resolve the harmonics while for the triggered oscillations others should be tuned broadly enough and synchronized to the envelope. For a given center frequency, neurons with different thresholds may be appropriate to code both signal aspects (Langner, 1992). Because the theory predicts optimal results when the envelope periodicity as well as the temporal fine structure of the signal are coded, it predicts also an optimal frequency range for the perception of periodicity pitch. Such an optimal frequency range was indeed found around the 4th harmonic and was called the ‘dominance region’ (Ritsma, 1967).

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As expressed by the periodicity equation, the BMF (= 1/τBMF ) of a coincidence neuron in the ICC is also a function of the carrier frequency. Figure 2 shows as an example for such a relationship a unit of the ICC of the cat (updated from Langner and Schreiner, 1988). The best modulation frequency of this unit shifts systematically with the carrier frequency of the AM signal. The right side of Figure 2 shows a computer simulation where the time constants of oscillation and integration were selected to reproduce the BMF-shift of the actual recording. Plotting the best modulation period τBMF over the carrier period reveals that the neuronal response and the simulation may be characterized by the relation τBMF = 9 τc - 0.8ms. For the computer simulation this relation is easily explained: the integrator of the circuit needs a certain number of spikes provided by the neuronal activity phase coupled to nine carrier cycles in order to reach its threshold and the oscillator introduces a delay of 0.8 ms. The fact that the neuronal results are also matched by this equation, suggests that similar conditions hold also for the actual neuronal recordings. Provided, all coincidence neurons show similar BMF-shifts, a corresponding effect on pitch perception must be expected. Therefore, to a first approximation, the pitch of an amplitude modulated signal must be equal to the pitch of the modulation frequency and, to a second approximation, it must be a function of the carrier frequency. Psychophysical experiments are in line with this hypothesis (de Boer, 1956, Schouten et al., 1962). In principle, this effect could be explained by the so-called ‘fine-structure theory’, which predicts that the pitch period is a multiple of the carrier period. However, periodicity pitch is not a simple function of the carrier frequency alone. For non-harmonic AM signals pitch deviates from that of a simple subharmonic of the carrier. This ‘second effect’ of pitch shift gave rise to the so called ‘pattern models’ of pitch perception, which are able to explain the pitch-shift effect by a correlation analysis in the spectral domain (Terhardt, 1972; Wightman, 1973; Goldstein, 1973). These models are interesting for the present discussion, because a spectral correlation analysis is mathematically equivalent to a temporal correlation analysis and may lead to the same result. In comparison to the present theory, their disadvantage is that the corresponding neuronal mechanisms are unknown. Another difference is, that in the present theory the ‘second effect’ of pitch shift is explained by the contribution of intrinsic oscillations to the periodicity equation. The assumption is that the periodicity equation, which originally describes only the behavior of single coincidence neurons, may be used to approximate also the pitch estimation of the auditory system as a whole. Consequently, the function describing the deviation of pitch for the non-harmonic case was computed in the following way: The period τm of the modulation frequency is related in a linear way to the carrier period τc : τm = α τc - τk , where α now is an average value. In general, in order to match τm, α has to be a non-integer, and several integrators with slightly different delays have to contribute by neuronal averaging to estimate the pitch. τk is the delay introduced by the oscillator and is assumed here to be in the order of 0.8 ms, an oscillation period typically found in coincidence neurons tuned to a modulation frequency of 200 Hz. The equation may be resolved for α: α = (τm + τk ) / τc = τm/τc + τk/τc = nh (1 + τk / τm) = 1.16 nh. It is assumed here that the equation holds also for the special case τm = nh τc (nh integer), where the carrier frequency of the AM signal is an integer multiple (nh) of the modulation frequency (200 Hz). 1

Therefore, the pitch measurements presented in Figure 3 may be approximated by τp =1.16 nh τc - 0.8 ms.

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carrier frequency fc [Hz] Figure 3. The pitch of a periodic sound varies as a function of the carrier frequency, in spite of a constant envelope frequency of 200 Hz. Results were read out of figures of Schouten et al. (1962) and Smoorenburg (1970); Sch1-3 correspond to 3 subjects of Schouten et al., Sm1-2 to subjects of Smoorenburg. (In Smoorenburg’s experiment a two-tone complex was used and therefore pitch was plotted above the lower frequency of the signal.) The fitting lines in this figure represent a prediction of the effect of pitch shift from the present model.

In average, the measurements of Schouten and Smoorenburg are indeed quite well matched by this equation. However, the results obtained from some subjects still deviate systematically from the average, especially for higher carrier frequencies. Similar deviations were observed when comparing these results with the predictions from pattern models and attributed to nonlinear distortion products (Goldstein, 1973). In the context of the present theory, these deviations may be explained by the assumption that different oscillation periods may be used by the auditory system in order to account for different integration delays. The idea is, that with higher carrier frequency of the AM-signal the neuronal tuning curves are broader and, consequently, modulation sidebands are less suppressed. As a result, the modulation of the neuronal responses to AM-signals increases with the carrier frequency. This must result in longer integration intervals, which then have to be compensated for by longer oscillations. Note, that this explanation offers also a possible explanation why there are two delay mechanisms in the auditory system, provided by oscillations and integrations. Systematic individual pitch variations as observed in Figure 3 also for lower carreir frequencies may be explained by attempts of the individual auditory systems to match the periodicity of signals, either using integer multiples of the carrier period (corresponding to the fine structure hypothesis) or by using prominent intrinsic oscillation intervals. Evidence for this hypothesis comes from results plotted in Figure 4. The individual pitch measurements from Figure 3 were approximated (divided in portions below 200Hz and above 200 Hz) by use of the 2

periodicity equation τp = α τc - τk. The result is that values for α tend to be integer and values for τk tend to be integer multiples of 0.4 ms. However, the effect is small and obvious only in the distributions of the deviations from such values and for τk < 1.6 ms and α< 16. Since such preferences are unlikely for α and τk simultaneously, the distributions exclude also all values of τk where the corresponding α is nearly integer and all values of α where the corresponding τk is nearly an integer multiple of 0.4 ms. 8

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Figure 4. Pitch values obtained from the measurements demonstrated in Figure 3 may be fitted by a periodicity equation τm = α τc - τk. The distributions show that τk tends to be a multiple of 0.4 ms (left side), while α tends to be integer (right side).

The neuronal model is able to explain also a complex effect of adaptation observed frequently in the ICC. An example from the midbrain of Guinea fowl is given in a point plot in Figure 5 (top). It shows responses to AM signals with modulation frequencies between 135 and 219 Hz and a carrier frequency of 1800 Hz, the centre frequency (CF) of the unit. Several features of the neuronal recording show up in the point plot obtained from a computer simulation ( Figure 5, bottom). Since similar effects are also present in the neuronal recording ( Figure 5, top), the following description are supposed to hold for both, the recording and the simulation. The vertical clusters of points result from 100 repetitions of the same AM signals and indicate strong phase coupling for modulation frequencies around 170 Hz. Often these ‘lines’ seem to split up into double lines, due to an intrinsic oscillation with a period of 1.2 ms. After an initial onset oscillation, consisting of a few spikes only, and a gap of about 6 ms, a slightly stronger response may be observed. This response results from the integrator input which needs about 7.2 ms or 14 cycles of the carrier to reach threshold. The onset gap corresponds to the difference between integration and oscillator interval and, in accordance with the periodicity equation, is a predictor for the best modulation period. Finally, while the synchronization of the responses to the modulation cycles increase in time for modulation frequencies below BMF (e.g. for 143 Hz), it decreases for modulation frequencies above BMF (e.g. for 219 Hz). This asymmetric effect of adaptation is due to an increase of the integration interval in time up to 50 - 100 ms after signal onset. The explanation is, that the integration interval varies with the strength of its inputs from the fast and slowly adapting nerve fibers, reaching threshold fast during the onset response and 3

slower later on. From the periodicity equation it is clear that an increase of the integration interval will result in a increase of the preferred modulation period, corresponding to a decrease of preferred modulation frequency. Note, that this explanation is correct for the computer simulation ( Figure 4, bottom) and, because of the similarity of the simulation and the actual recording, is likely to hold also for the real coincidence neuron ( Figure 4, top).

Figure 5. Point plots of the responses of a neuron in the midbrain of Guinea fowl (top) and of a computer simulation (bottom) to amplitude modulations (carrier frequency = CF = 1800 Hz; 65 dB SPL). Note, the temporal increase of synchronized responses for modulation frequencies below BMF (about 180Hz) and the decreasing synchronization for modulation frequencies above BMF.

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Figure 6. Right: Point plots of the responses of a neuron in the midbrain of Guinea fowl to amplitude modulations As in Figure 5 an increase of synchronized responses for modulation frequencies below BMF (about 800 Hz) and the decreasing synchronization for modulation frequencies above BMF may be seen. Left: Systematic shifts of the phase delay of the responses in relation to a hypothetical coincidence window (between the stippled lines) allow to explain the temporal response shifts by coincidence mechanisms.

In Figure 6 another example (from Langner, 1983) for the same effect is shown. Because the best modulation frequency in this case is quite high (about 840 Hz), best modulation period, intrinsic oscillation period, and integration interval must all be similar and, therefore, cannot be separated in the point plot. However, the average phase of the responses in relation to the zero crossings of the modulation cycles show a time course which is again in line with the explanation given above. The phase curves ( Figure 6, left side) may be attributed to an increasing integration interval which is due to the varying input from the auditory nerve during signal onset and defines the phase delay of the coincidence unit. If one assumes that the coincidence window corresponds to the range between the stippled lines the time course of these responses may be explained even in detail. Note, for example, that the synchronization to signal 7 fluctuates in parallel to time course of the corresponding phase curve crossing several times the upper boundary of the coincidence window. These adaptation effects imply that the coincidence neurons change their tuning slightly during time. Provided the coincidence neurons code pitch, pitch is also expected to change during time after signal onset. Because a given coincidence neuron needs a slightly higher modulation frequency at signal onset for a good response, pitch is expected to be slightly lower. In other words, a pitch effect may be predicted which has a time course corresponding to that of the onresponse in the auditory nerve. By shortening the duration of an AM-signal pitch should decrease and increase again for very short durations. A preliminary measurement of this effect measured in one subject at different intensities is given in Figure 7. The fact that the on-effect is reduced for low intensities, may explain the intensity dependence of this pitch effect. Note, that this pitch-effect was correctly predicted by the theory. However, it turned out that this effect was already found by Metters and Williams (1973). In line with the present interpretation, the authors suggested that the ‘pitch depression’ may be attributed to periodicity detection. (However, their preliminary measurement of the level dependence of this effect is opposite to the one presented here.) 5

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CONCLUSIONS A temporal theory of periodicity analysis is able to explain details of response properties of neurons in the auditory midbrain. According to this theory periodic signals are represented by synchronized neuronal activity and processed by neurons characterized by intrinsic oscillations, temporal integration, and coincidence detection. Neurons in the auditory midbrain act as coincidence detectors and transfer the temporal information into a rate code. Consequently, pitch is arranged topographically, orthogonal to the frequency map. The neuronal theory is able to explain periodicity pitch, the perception of the missing fundamental and the pitch-shift effects. It is also adequate to predict additional pitch effects, like a pitch shift due to the on-response in the auditory nerve. ACKNOWLEDGMENT Supported by the Deutsche Forschungsgemeinschaft, SFB 269. REFERENCES de Boer E. (1956) Pitch of inharmonic signals. Nature 178:535-536. 6

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