- Email: [email protected]
Neural code for sound localization at low frequencies
Neurocomputing 38}40 (2001) 1443}1452
Neural code for sound localization at low frequencies Petr Marsalek * Mind/Brain Institute, Johns Hopkins University, 338 Krieger Hall, 3400 N. Charles Street, Baltimore, MD 21218, USA Department of Pathological Physiology, Charles University Prague, U nemocnice 5, Praha 2, CZ-128 53, Czech Republic
Abstract Several nuclei in the auditory pathway perform sound localization. At lower sound frequencies, they compute the interaural time di!erence. This di!erence is transduced by the dedicated neuronal circuit into a labeled line di!erence. The neurons of the circuit "re only when synaptic inputs from both ears arrive within a short time window. We have shown previously the minimum timing di!erence detectable in the linear model and in the Hodgkin}Huxley-type equations. To better describe the spike code for the sound localization we de"ne a generalized vector strength for point processes. Based on calculations in single cell models, we show how the neural code is transformed depending on the main sound frequency. We conclude that the performance limits of the low-frequency system are set by the frequency of the sound and therefore by both the vector strength and the probability of spike emission in the single unit. This shows the necessity for the existence of two di!erent mechanisms for sound localization (1) one based on the interaural time delay, operating at low sound frequencies, and (2) another based on the interaural intensity di!erence, operating at high frequencies. The two systems then converge in higher-order neural relay stations. 2001 Elsevier Science B.V. All rights reserved. Keywords: Directional hearing; Auditory pathway; Interaural time and intensity di!erences; Coincidence detection; Vector strength
1. Introduction For lateral sound localization, animals use the interaural di!erence of sound loudness at higher sound frequencies and the interaural time di!erence (ITD) at lower * Corresponding author. E-mail address: [email protected] (P. Marsalek). 0925-2312/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 0 1 ) 0 0 5 2 4 - 0
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sound frequencies. Without a spike train which is phase locked to the auditory stimulus, sound localization by timing di!erences would not be possible. Some problems after more than half a century of e!ort [5}7,28] still remain to be solved, in particular, what is the shortest di!erence in timing of synaptic inputs detectable by their target neuron [4]? In mammals and in birds, two corresponding neural systems in auditory nuclei perform the task of the sound localization. Both the systems consist of two circuits (1) one calculating with ITD, and another (2) calculating with the interaural intensity di!erence (IID). 1.1. ITD circuit (1) In mammals, the ITD detecting pathway starts from the anteroventral cochlear nuclei and then projects to the medial nucleus olivaris superior (MSO) [5]. (2) In birds, nuclei magnocellulares project to nucleus laminaris (NL) [2,9,15]. Their computation can be informally illustrated as a logical gate: < "I (AND)I . In 1.')# '.1' !-,20 words: one or none spike voltage < is generated, when unitary synaptic currents, I, arrive from ipsi- and contra-lateral sides within short time window. Neurons in these auditory circuits have the unique property that they do not contain adaptive currents. Therefore, they are well suited for their physiological function: phase lock their output with respect to their synaptic input and therefore transfer phase locked spike trains to higher neuronal relays. 1.2. IID circuit (1) In mammals, the IID detection pathway leading from cochlear nuclei projects to the lateral nucleus olivaris superior (LSO). The ipsilateral branch projects its synapses there directly, and the contralateral branch "rst projects to the inhibitory interneurons in the medial nucleus of the trapezoid body, which in turn projects to the LSO. These neurons create the spiking substrate for calculating the di!erence by adding spikes from ipsilateral side and subtracting spikes from contralateral side [14,17]. (2) In birds, nuclei angulares input to one of the lemniscal nuclei, where both ipsilateral and contralateral inputs are combined [9]. This computation balancing inputs can be informally illustrated as f "I (MINUS)I . In words: net $'0',% '.1' !-,20 "ring frequency f is pulled forth by the net input current, I, from ipsi-lateral side and pushed back by the net input from contra-lateral side. In response of these neurons, sound level is favored before the stimulus lock of the spike train and yet the inputs from both sides are equalized to get just the di!erence of them at output cells. The information about the sound localization is implicit and multiplexed in activities of ganglia spirale of both sides. The signals are de-multiplexed into two streams, (1) ITD and (2) IID, as the next processing stage. In these two streams successively the spike timing code (in ITD) and mean "ring di!erence code (in IID) are converted into the codes of labeled lines. After the processing, the information becomes explicit and readable as "ring of cells of these labeled lines. Finally, this is multiplexed again in higher neuronal relays, which are colliculi inferiores and medial geniculate nuclei of thalamus, to yield a uni"ed signal about the incoming sound source.
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1.3. Dexnitions We will at several places use the exponential type sigmoidal function, de"ned as: F " : F (x, X, K)"(1#exp((!x!X)/K))\ with parameters X and K, K is called 1 1 the slope coe$cient. Translating for X"0 this function by 1/2 along the y-axis, we get an odd function: F (x, 0, K)!1/2"!(F (!x, 0, K)!1/2). We will use these func1 1 tions as approximate substitutes at several places, where there is a sigmoidal like relation between experimentally obtained quantities. This function captures for example activation of transmembrane current through voltage activated membrane channels [8], or maximal response of the leaky integrator model to two inputs delayed in time [10]. Moreover, this function can be used as an approximation to the distribution function to the normal probability density function (2)\ exp(!x/2), where the actual distribution function cannot be written in a closed analytical form. If we restrict our concept of input signal in the auditory pathway to the "rst auditory neuron (of ganglion spirale) and to next neuronal relays along the pathway, then to describe the transmitted signal it is su$cient to use a discrete measure of point events (action potentials) only. Such a measure is the vector strength. Vector strength is the degree of locking of output time events (spikes) to input (stimulus) events. We introduce more general de"nition, than in [4,5,10], to cope with non-periodic point processes. Times of input events are denoted t (t (2, times of output events ' ' t (t (2, their phase time transform as in [27]: - - t!t 'G , for t3[t , t " : 2m#2 ], and i31,2,2, m, 'G 'G> t !t 'G> 'G the vector strength is then as in [4,5,10]: r" : (1/m(( cos(t ))#( sin(t )), -G -G where the sum runs for all i, i31, 2, 2, m. For periodic stimuli, the meaning of phase time transform is obvious. For non-periodic stimuli, the vector strength measures, how well the output train is locked to the input train. We call this that the train is stimulus locked, because the term phase lock implies periodicity. In [10] we have shown how the time accuracy of the spike generating system can be achieved. We have shown that the system using the Hodgkin}Huxley-type dynamics is the most plausible biophysical solution. In the next section we show some limits imposed to the coincidence detection system by increasing sound frequency.
2. Results 2.1. Low-frequency system In [10] we compared the leaky integrator (LI) model [22,23,18,26] with the model using the Hodgkin}Huxley (HH) mechanism [18,26]. We have shown that the best
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response of the LI model can be written as <(t)"1/C(1#exp(!t/)),
(1)
where < is a dimensionless maximal voltage attained in response to two pulses delayed by t. From max <"<(0)"1 we got dimensionless capacitance C"2. Time constant moves the in#ection point along the x-axis time scale. Considering t as the only input variable is a simpli"cation desired for our purpose. This function has a sigmoidal shape. Of course any response of the coincidence detecting mechanism should be a decreasing function of t. In the HH model, the voltage can be normalized as well, <"K <#K , where < is the original membrane potential. (In the numerical example in [10], K "0.0066 and K "0.33; similar normalization was used for example by Kepler and others in [8]. The same relationship of < to t in the HH models can be approximated by the function <(t)"(1#exp((log t!K )/K ))\,
(2)
where K is proportional to log and K is the slope constant. The exact values of K and K can be obtained (1) either by "tting them to the experimental data from brainstem slices [12,13], (2) or by "tting them to a numerical solution of the detailed HH model [10]. In both cases the slope given by K is steep enough and leaves the exact value of K (here K "0.1) less important for the frequency related results. Crucial for the coincidence detection (CD) is the value of . In the case of Eq. (1), the time constant of the passive circuit equivalent to neurons in phase locking auditory nuclei was in experimental recordings as "2 ms. Even though this is lower than in other central neurons, passive circuit properties do not explain the accuracy of the CD mechanism. The passive time constant in the HH model is the same, however the e!ective time constant [4,10,20] attained by the system is almost 100 times lower. The value of this e!ective is in#uenced by the morphology and ion channels of neurons. Its values in Eq. (2) therefore do not depend on the tuning of neurons to particular sound frequency. Functions in Eqs. (1) and (2) are shown in Fig. 1. 2.2. Towards higher frequencies As the main sound frequency gets higher, the vector strength r of the spike train lock to the stimulus is compromised. Once the period of sound is less than the single cell's refractory period, a single "ber is unable to lock in such a high frequency. Comparing experiments in several animals, in [10] we proposed that r has to have a value lower than inversely proportional to the mean sound frequency, fM , r)fM \, "1 ms. 1 1 Let us divide the sound frequencies into two ranges: (1) from 16 Hz to 1 kHz and (2) from 1 to 16 kHz. The function r"fM \ gives the upper bound of r in the range (2). By 1 de"nition, r)1. To extend the bounding curve to the range (1), let us use the function r"(1#exp(( fM !f )/K))\, passing through the point 1 [ f , r]"[1.1 kHz, 0.9]. This limit for the r in dependence on fM is shown 1
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Fig. 1. Responses of the LI and the HH systems to two synaptic inputs delayed by t. These are the maximal responses of the LI and the HH systems to two synaptic inputs delayed in time by t. The maximal voltage response of the HH system is normalized as shown in the text. The LI system cannot distinguish coincidences below approximately 1 ms and the HH system those below approximately 20 s. Since all real neurons in the auditory nuclei are equipped with a particular set of ion channels, these results for single cells are not dependent on the input frequency.
in Fig. 2. This way we divided the sound frequencies according to the vector strength value. Analogously, we can divide the sound frequencies into other two ranges: (1) from 16 to 256 Hz and (2) from 256 Hz to 16 kHz. This time we use the criterion, whether or not every pulse in stimulus train is followed by the output spike of given cell. Note the di!erence*the previous division was according to stimulus locking. Let us de"ne p , $ the probability that given cell responds to the input spike by "ring output spike. p "fM /fM , where fM is cell "ring averaged over time. $ $ 1 $ Now let us estimate, how long it could take to a receiver connected to just the single coincidence detection unit before it gets a spike signaling coincidence detection. The probability that the units gets spikes from both sides at the same time is clearly p and $ we can understand the vector strength r like another probability, which gets us to the term rp . When rewriting the probability as a uniform probability density function in $ time, we get p(t) " : rp , for t3[t , t #r\p\], p(t) " : 0 otherwise. In other words $ $ the time for single "ber would be r\p\ time units, where one time unit is f \. $ 1
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Fig. 2. The best r for corresponding sound frequencies f . The curve shows physical limits for the best vector 1 strength r attained by the neuronal spike relaying system in given sound frequencies, f . All experimental 1 data (in dog, hen, and barn owl) showing the decrease of r with the increase of f lie below this curve. For 1 experimental data see for example Fig. 6(A) in [13].
The information for the time delay t"0 attained at the appropriate position in MSO (MSO in mammals, NL in birds) is conveyed by several tens of "bers. In this case the population coding supplements the low "ring probability in one sound cycle unit and makes the time for the decision shorter. For a quantitative estimate about a size of the population and decision time we refer to [4], which is in agreement with our preliminary results. 2.3. Both low- and high-frequency systems Now, that we have noted some constraints for the coincidence detection system, we can go back and draw the whole picture of the sound localization system. (1) The same argument that holds for coexistence of both frequency and tonotopic coding in cochlea holds for the sound localization: At low frequencies more directional information is extracted from the timing delay. However, the coincidence detection system fails to detect interaural time delay (ITD) at higher frequencies. Therefore, higher interaural intensity di!erence (IID) is more pronounced in higher than in lower
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frequencies. (2) As noted in Section 1, sound localization pathways diverge into two specialized systems, a system for ITD detection and a system for IID detection. They calculate the sound localization independently and at some higher relay, they have to converge again to yield uni"ed information about the sound source.
3. Discussion Sound localization systems have unique properties which make them suitable as a model system for neuronal coding investigation. This paper is the next part of work started in [10]. We have shown before: (1) analytical calculations, which constrain the properties of biophysically plausible coincidence detector in auditory nuclei and (2) numerical example using the Hodgkin}Huxley-type simulation of point neuron based on experimental values. We were somewhat surprised that we were able to raise the Hodgkin}Huxley-like properties of ion channels one level higher*to the level of the description of neural circuit. Therefore, this paper is the next part of the attempt to describe the neural circuit and to comment on some of its properties, needed for population coding. Further part of the work will be focused on items, where single neuron performance di!ers from the performance of the whole population. As an example consider the maximal "ring frequency f with a 90% (r"0.9) stimulus lock. For a single unit it is several $ hundred Hz (we here used 256 Hz). For the whole nervus acusticus the maximal f reached for r"0.9 is 10 times higher, several kHz. This property of the population $ enables to divide e!ectively the task of transmitting stimulus evoked spikes among several cells. The key role in distributing this task is played by noise and a probabilistic cell "ring caused by the #uctuations of membrane potential. Therefore, the timing jitter, which apparently limits the performance in medial superior olive and its analogs, in lower auditory nuclei can be bene"cial, helping to `hasha spikes among cells. The timing jitter changes from one nucleus to another, as documented in [7,16]. Joris and others show there that in ganglion spirale, as in a more general sound encoder, timing jitter is higher, compared with anteroventral cochlear nucleus, as in a specialized stimulus locked encoder, where jitter is lower. An important point is that the latter nucleus is just the higher relay along the auditory pathway, and yet the jitter is lower, measured by r. Assumed there is a certain convergence onto a single cell, timing jitter can be lowered, preserved, or become higher [11,3,16]. Once becomes the implicit information about sound localization explicit (it means encoded), its encoding into a labeled line code saves the resources of the system. This is because in the lower order nuclei all cells "re in stimulus locked regime without adaptation. In contrast, in the labeled line system, only those "bers belonging to the speci"c location "re, and other "bers remain silent, conserving the energy resources. The concept of neuronal representation of outer space was studied at various relays of sensory pathways. In lower relays, like colliculi inferiores, it is relatively well documented [14,17,28]. Less known is: how the spatial information from di!erent
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sensory inputs is integrated in both neocortex and hippocampus? At the present stage of knowledge in tracking the labeled line code gathered from both ITD and IID systems we are left with the notion that output from these nuclei converges at the next higher relay [14,17]. The spiking mechanism enabling coincidence detection has to switch o! after the time window for adding inputs from two sides expires. We want to stress that showing limits of the LI model in [10] and here is not in the contradiction with [4]. We found the equation of Gerstner, Kempter and others very insightful: d< "!G
(3)
where the term with G is the rectifying potassium current which should serve as the 0 above-mentioned o! switch. For the study [4] with synaptic learning this term was not crucial, and "nally this equation stayed only in the unpublished supplement to the paper [4]. The computational solution to the biophysical problem, which was formulated to generate spike in a time window as short as possible, was obtained in [4] by learning synaptic weights in the LI model. This solution reaches the goal of "nding the plausible CD system [19,21]. In [10] such a solution was found using one point Hodgkin}Huxley model and in [1] such a solution contained Morris}Lecar-type model [18,24,25] and reconstruction of the dendritic morphology of CD cells, to give just two more examples out of many [7]. One of the purposes of the work presented here is to consider it as an attempt to "nd a method for solving the following problem. We assume that the discussed circuit is serving such a speci"c purpose (sound localization) and that it has to have a speci"c design [2]. Several physical properties constrain the way the population should calculate the direction of incoming sound. Putting such constraints together has its methodical value. Of course, particular frequency ranges shown here are not sharp boundaries. In these regions one type of coding continuously and smoothly changes into another. In this sense boundaries of frequency regions are not exact numbers, we used here 16 Hz, 256 Hz, 1.1 kHz just as a prototype numbers. There is a straight demand for further work. The picture of population coding in both ITD and IID systems can be re"ned further with the estimate of number of spiking units. This could make possible to calculate the ratio, in which contributes one of the two systems to the localization sensation, and then ultimately to verify it by disabling one of the two systems in experiment.
Acknowledgements Supported by the Research Initiative of the Charles University no. 206023/1999, by the GAC[ R grant C 305/00/P101 (Charles University reg.C 201352/00) and by the NATO-NSF postdoctoral fellowship. Thanks to Chris DiMattina and Jir\ mH KofraH nek.
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References [1] H. Agmon-Snir, C.E. Carr, J. Rinzel, The role of dendrites in auditory coincidence detection, Nature 393 (1998) 268}272. [2] C.E. Carr, M.A. Friedman, Evolution of time coding systems, Neural Comput. 11 (1999) 1}20. [3] J.F. Feng, Behaviors of spike output jitter in the integrate-and-"re model, Phys. Rev. Lett. 79 (1997) 4505}4508. [4] W. Gerstner, R. Kempter, J.L. van Hemmen, H. Wagner, A neuronal learning rule for sub-millisecond temporal coding, Nature 383 (1996) 76}78. [5] J.M. Goldberg, P.B. Brown, Response of binaural neurons of dog superior olivary complex to dichotic tonal stimuli: some physiological mechanisms of sound localization, J. Neurophysiol. 32 (1969) 613}636. [6] L.A. Je!ress, A place theory of sound localization, J. Comp. Physiol. Psychol. 41 (1948) 35}39. [7] P.X. Joris, P.H. Smith, T.C.T. Yin, Coincidence detection in the auditory system: 50 years after Je!ress, Neuron 21 (1998) 1235}1238. [8] T.B. Kepler, L.F. Abbott, E. Marder, Reduction of conductance-based neuron models, Biol. Cybernet. 66 (1992) 381}387. [9] M. Konishi, Similar algorithms in di!erent sensory systems and animals, in: Cold Spring Harbor Symposia on Quantitative Biology, CSH Laboratory, New York, pp. 575}583, 1990. [10] P. Marsalek, Coincidence detection in the Hodgkin}Huxley equations, BioSystems 58 (2000) 83}91. [11] P. Marsalek, C. Koch, J. Maunsell, On the relationship between synaptic input and spike output jitter in individual neurons, Proc. Natl. Acad. Sci. USA 94 (1997) 735}740. [12] A.D. Reyes, E.W. Rubel, W.J. Spain, Membrane-properties underlying the "ring of neurons in the avian cochlear nucleus, J. Neurosci. 14 (1994) 5352}5364. [13] A.D. Reyes, E.W. Rubel, W.J. Spain, In-vitro analysis of optimal stimuli for phase-locking and time-delayed modulation of "ring in avian nucleus laminaris neurons, J. Neurosci. 16 (1996) 993}1007. [14] M.J. Zigmond, F.E. Bloom, S.C. Landis, J.L. Roberts, L.R. Squire (Eds.), Fundamental Neuroscience, Academic Press, San Diego, CA, USA, 1999. [15] Y. Albeck, M. Konishi, Responses of neurons in the auditory pathway of the barn owl to partially correlated binaural signals, J. Neurophysiol. 74 (1995) 1689}1700. [16] P.X. Joris, L.H. Carney, P.H. Smith, T.C.T. Yin, Enhancement of neural synchronization in the anteroventral cochlear nucleus. I. Responses to Tones at the Characteristic Frequency, J. Neurophysiol. 71 (1994) 1022}1036. [17] E.R. Kandel, J.H. Schwartz, T.M. Jessell, Principles of neural science, Prentice Hall Intl., London, UK, 1991. [18] C. Koch, Biophysics of computation: information processing in single neurons, Chapter on Phase Space Analysis, Oxford University Press, New York, 1999. [19] C. Koch, OG . Bernander, R.J. Douglas, Do neurons have a voltage or a current threshold for action potential initiation? J. Comput. Neurosci. 2 (1995) 63}82. [20] C. Koch, M. Rapp, I. Segev, A brief history of time (constants), Cerebral Cortex 6 (1996) 93}101. [21] P. KoK nig, A.K. Engel, W. Singer, Integrator or coincidence detector*the role of the cortical neuron revisited, TINS 19 (1996) 130}137. [22] L. Lapicque, Recherches quantitatifs sur l'excitation des nerfs traitee comme une polarisation, J. Physiol. Paris 9 (1907) 620}635. [23] L. Lapicque, L'excitabilite en fonction du temps, Presses Universitaries de France, Paris, 1926. [24] C. Morris, H. Lecar, Voltage oscillations in the barnacle giant muscle "ber, Biophys. J. 35 (1981) 193}213. [25] J. Rinzel, B. Ermentrout, Analysis of neural excitability and oscillations, book Methods in Neuronal Modeling, 2nd edition, C. Koch, I. Segev (Eds.), MIT Press, 1998, pp. 250}291. [26] H.C. Tuckwell, Introduction to theoretical neurobiology, Vols. 1}2, Cambridge University Press, New York, 1988. [27] D.S. Reich, J.D. Victor, B.W. Knight. The power ratio and the interval map: Spiking models and extracellular recordings. J. Neurosci. 18 (1998) 10090}10104.
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[28] J.E. Rose, N.B. Gross, C.D. Geisler, J.E. Hind. Some neural mechanisms in the inferior colliculus of the cat which may be relevant to localization of a sound source. J. Neurophysiol. 29 (1966) 288}314.
Petr Marsalek completed two disparate courses at the Charles University of Prague, Czech Republic: medicine, with MD, and computer science and biophysics, with Ph.D. He was a lecturer of physiology at the First Medical Faculty, Charles University, from 1993. In years from 1995 to 1997 he was a NIMH postdoctoral fellow at the California Institute of Technology at the group of Christof Koch. In the 2000 he won a NATO-NSF postdoctoral fellowship to work at his present position at the Johns Hopkins University at the group of Ernst Niebur.
Neural code for sound localization at low frequencies Petr Marsalek * Mind/Brain Institute, Johns Hopkins University, 338 Krieger Hall, 3400 N. Charles Street, Baltimore, MD 21218, USA Department of Pathological Physiology, Charles University Prague, U nemocnice 5, Praha 2, CZ-128 53, Czech Republic
Abstract Several nuclei in the auditory pathway perform sound localization. At lower sound frequencies, they compute the interaural time di!erence. This di!erence is transduced by the dedicated neuronal circuit into a labeled line di!erence. The neurons of the circuit "re only when synaptic inputs from both ears arrive within a short time window. We have shown previously the minimum timing di!erence detectable in the linear model and in the Hodgkin}Huxley-type equations. To better describe the spike code for the sound localization we de"ne a generalized vector strength for point processes. Based on calculations in single cell models, we show how the neural code is transformed depending on the main sound frequency. We conclude that the performance limits of the low-frequency system are set by the frequency of the sound and therefore by both the vector strength and the probability of spike emission in the single unit. This shows the necessity for the existence of two di!erent mechanisms for sound localization (1) one based on the interaural time delay, operating at low sound frequencies, and (2) another based on the interaural intensity di!erence, operating at high frequencies. The two systems then converge in higher-order neural relay stations. 2001 Elsevier Science B.V. All rights reserved. Keywords: Directional hearing; Auditory pathway; Interaural time and intensity di!erences; Coincidence detection; Vector strength
1. Introduction For lateral sound localization, animals use the interaural di!erence of sound loudness at higher sound frequencies and the interaural time di!erence (ITD) at lower * Corresponding author. E-mail address: [email protected] (P. Marsalek). 0925-2312/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 2 3 1 2 ( 0 1 ) 0 0 5 2 4 - 0
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sound frequencies. Without a spike train which is phase locked to the auditory stimulus, sound localization by timing di!erences would not be possible. Some problems after more than half a century of e!ort [5}7,28] still remain to be solved, in particular, what is the shortest di!erence in timing of synaptic inputs detectable by their target neuron [4]? In mammals and in birds, two corresponding neural systems in auditory nuclei perform the task of the sound localization. Both the systems consist of two circuits (1) one calculating with ITD, and another (2) calculating with the interaural intensity di!erence (IID). 1.1. ITD circuit (1) In mammals, the ITD detecting pathway starts from the anteroventral cochlear nuclei and then projects to the medial nucleus olivaris superior (MSO) [5]. (2) In birds, nuclei magnocellulares project to nucleus laminaris (NL) [2,9,15]. Their computation can be informally illustrated as a logical gate: < "I (AND)I . In 1.')# '.1' !-,20 words: one or none spike voltage < is generated, when unitary synaptic currents, I, arrive from ipsi- and contra-lateral sides within short time window. Neurons in these auditory circuits have the unique property that they do not contain adaptive currents. Therefore, they are well suited for their physiological function: phase lock their output with respect to their synaptic input and therefore transfer phase locked spike trains to higher neuronal relays. 1.2. IID circuit (1) In mammals, the IID detection pathway leading from cochlear nuclei projects to the lateral nucleus olivaris superior (LSO). The ipsilateral branch projects its synapses there directly, and the contralateral branch "rst projects to the inhibitory interneurons in the medial nucleus of the trapezoid body, which in turn projects to the LSO. These neurons create the spiking substrate for calculating the di!erence by adding spikes from ipsilateral side and subtracting spikes from contralateral side [14,17]. (2) In birds, nuclei angulares input to one of the lemniscal nuclei, where both ipsilateral and contralateral inputs are combined [9]. This computation balancing inputs can be informally illustrated as f "I (MINUS)I . In words: net $'0',% '.1' !-,20 "ring frequency f is pulled forth by the net input current, I, from ipsi-lateral side and pushed back by the net input from contra-lateral side. In response of these neurons, sound level is favored before the stimulus lock of the spike train and yet the inputs from both sides are equalized to get just the di!erence of them at output cells. The information about the sound localization is implicit and multiplexed in activities of ganglia spirale of both sides. The signals are de-multiplexed into two streams, (1) ITD and (2) IID, as the next processing stage. In these two streams successively the spike timing code (in ITD) and mean "ring di!erence code (in IID) are converted into the codes of labeled lines. After the processing, the information becomes explicit and readable as "ring of cells of these labeled lines. Finally, this is multiplexed again in higher neuronal relays, which are colliculi inferiores and medial geniculate nuclei of thalamus, to yield a uni"ed signal about the incoming sound source.
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1.3. Dexnitions We will at several places use the exponential type sigmoidal function, de"ned as: F " : F (x, X, K)"(1#exp((!x!X)/K))\ with parameters X and K, K is called 1 1 the slope coe$cient. Translating for X"0 this function by 1/2 along the y-axis, we get an odd function: F (x, 0, K)!1/2"!(F (!x, 0, K)!1/2). We will use these func1 1 tions as approximate substitutes at several places, where there is a sigmoidal like relation between experimentally obtained quantities. This function captures for example activation of transmembrane current through voltage activated membrane channels [8], or maximal response of the leaky integrator model to two inputs delayed in time [10]. Moreover, this function can be used as an approximation to the distribution function to the normal probability density function (2)\ exp(!x/2), where the actual distribution function cannot be written in a closed analytical form. If we restrict our concept of input signal in the auditory pathway to the "rst auditory neuron (of ganglion spirale) and to next neuronal relays along the pathway, then to describe the transmitted signal it is su$cient to use a discrete measure of point events (action potentials) only. Such a measure is the vector strength. Vector strength is the degree of locking of output time events (spikes) to input (stimulus) events. We introduce more general de"nition, than in [4,5,10], to cope with non-periodic point processes. Times of input events are denoted t (t (2, times of output events ' ' t (t (2, their phase time transform as in [27]: - - t!t 'G , for t3[t , t " : 2m#2 ], and i31,2,2, m, 'G 'G> t !t 'G> 'G the vector strength is then as in [4,5,10]: r" : (1/m(( cos(t ))#( sin(t )), -G -G where the sum runs for all i, i31, 2, 2, m. For periodic stimuli, the meaning of phase time transform is obvious. For non-periodic stimuli, the vector strength measures, how well the output train is locked to the input train. We call this that the train is stimulus locked, because the term phase lock implies periodicity. In [10] we have shown how the time accuracy of the spike generating system can be achieved. We have shown that the system using the Hodgkin}Huxley-type dynamics is the most plausible biophysical solution. In the next section we show some limits imposed to the coincidence detection system by increasing sound frequency.
2. Results 2.1. Low-frequency system In [10] we compared the leaky integrator (LI) model [22,23,18,26] with the model using the Hodgkin}Huxley (HH) mechanism [18,26]. We have shown that the best
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response of the LI model can be written as <(t)"1/C(1#exp(!t/)),
(1)
where < is a dimensionless maximal voltage attained in response to two pulses delayed by t. From max <"<(0)"1 we got dimensionless capacitance C"2. Time constant moves the in#ection point along the x-axis time scale. Considering t as the only input variable is a simpli"cation desired for our purpose. This function has a sigmoidal shape. Of course any response of the coincidence detecting mechanism should be a decreasing function of t. In the HH model, the voltage can be normalized as well, <"K <#K , where < is the original membrane potential. (In the numerical example in [10], K "0.0066 and K "0.33; similar normalization was used for example by Kepler and others in [8]. The same relationship of < to t in the HH models can be approximated by the function <(t)"(1#exp((log t!K )/K ))\,
(2)
where K is proportional to log and K is the slope constant. The exact values of K and K can be obtained (1) either by "tting them to the experimental data from brainstem slices [12,13], (2) or by "tting them to a numerical solution of the detailed HH model [10]. In both cases the slope given by K is steep enough and leaves the exact value of K (here K "0.1) less important for the frequency related results. Crucial for the coincidence detection (CD) is the value of . In the case of Eq. (1), the time constant of the passive circuit equivalent to neurons in phase locking auditory nuclei was in experimental recordings as "2 ms. Even though this is lower than in other central neurons, passive circuit properties do not explain the accuracy of the CD mechanism. The passive time constant in the HH model is the same, however the e!ective time constant [4,10,20] attained by the system is almost 100 times lower. The value of this e!ective is in#uenced by the morphology and ion channels of neurons. Its values in Eq. (2) therefore do not depend on the tuning of neurons to particular sound frequency. Functions in Eqs. (1) and (2) are shown in Fig. 1. 2.2. Towards higher frequencies As the main sound frequency gets higher, the vector strength r of the spike train lock to the stimulus is compromised. Once the period of sound is less than the single cell's refractory period, a single "ber is unable to lock in such a high frequency. Comparing experiments in several animals, in [10] we proposed that r has to have a value lower than inversely proportional to the mean sound frequency, fM , r)fM \, "1 ms. 1 1 Let us divide the sound frequencies into two ranges: (1) from 16 Hz to 1 kHz and (2) from 1 to 16 kHz. The function r"fM \ gives the upper bound of r in the range (2). By 1 de"nition, r)1. To extend the bounding curve to the range (1), let us use the function r"(1#exp(( fM !f )/K))\, passing through the point 1 [ f , r]"[1.1 kHz, 0.9]. This limit for the r in dependence on fM is shown 1
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Fig. 1. Responses of the LI and the HH systems to two synaptic inputs delayed by t. These are the maximal responses of the LI and the HH systems to two synaptic inputs delayed in time by t. The maximal voltage response of the HH system is normalized as shown in the text. The LI system cannot distinguish coincidences below approximately 1 ms and the HH system those below approximately 20 s. Since all real neurons in the auditory nuclei are equipped with a particular set of ion channels, these results for single cells are not dependent on the input frequency.
in Fig. 2. This way we divided the sound frequencies according to the vector strength value. Analogously, we can divide the sound frequencies into other two ranges: (1) from 16 to 256 Hz and (2) from 256 Hz to 16 kHz. This time we use the criterion, whether or not every pulse in stimulus train is followed by the output spike of given cell. Note the di!erence*the previous division was according to stimulus locking. Let us de"ne p , $ the probability that given cell responds to the input spike by "ring output spike. p "fM /fM , where fM is cell "ring averaged over time. $ $ 1 $ Now let us estimate, how long it could take to a receiver connected to just the single coincidence detection unit before it gets a spike signaling coincidence detection. The probability that the units gets spikes from both sides at the same time is clearly p and $ we can understand the vector strength r like another probability, which gets us to the term rp . When rewriting the probability as a uniform probability density function in $ time, we get p(t) " : rp , for t3[t , t #r\p\], p(t) " : 0 otherwise. In other words $ $ the time for single "ber would be r\p\ time units, where one time unit is f \. $ 1
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Fig. 2. The best r for corresponding sound frequencies f . The curve shows physical limits for the best vector 1 strength r attained by the neuronal spike relaying system in given sound frequencies, f . All experimental 1 data (in dog, hen, and barn owl) showing the decrease of r with the increase of f lie below this curve. For 1 experimental data see for example Fig. 6(A) in [13].
The information for the time delay t"0 attained at the appropriate position in MSO (MSO in mammals, NL in birds) is conveyed by several tens of "bers. In this case the population coding supplements the low "ring probability in one sound cycle unit and makes the time for the decision shorter. For a quantitative estimate about a size of the population and decision time we refer to [4], which is in agreement with our preliminary results. 2.3. Both low- and high-frequency systems Now, that we have noted some constraints for the coincidence detection system, we can go back and draw the whole picture of the sound localization system. (1) The same argument that holds for coexistence of both frequency and tonotopic coding in cochlea holds for the sound localization: At low frequencies more directional information is extracted from the timing delay. However, the coincidence detection system fails to detect interaural time delay (ITD) at higher frequencies. Therefore, higher interaural intensity di!erence (IID) is more pronounced in higher than in lower
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frequencies. (2) As noted in Section 1, sound localization pathways diverge into two specialized systems, a system for ITD detection and a system for IID detection. They calculate the sound localization independently and at some higher relay, they have to converge again to yield uni"ed information about the sound source.
3. Discussion Sound localization systems have unique properties which make them suitable as a model system for neuronal coding investigation. This paper is the next part of work started in [10]. We have shown before: (1) analytical calculations, which constrain the properties of biophysically plausible coincidence detector in auditory nuclei and (2) numerical example using the Hodgkin}Huxley-type simulation of point neuron based on experimental values. We were somewhat surprised that we were able to raise the Hodgkin}Huxley-like properties of ion channels one level higher*to the level of the description of neural circuit. Therefore, this paper is the next part of the attempt to describe the neural circuit and to comment on some of its properties, needed for population coding. Further part of the work will be focused on items, where single neuron performance di!ers from the performance of the whole population. As an example consider the maximal "ring frequency f with a 90% (r"0.9) stimulus lock. For a single unit it is several $ hundred Hz (we here used 256 Hz). For the whole nervus acusticus the maximal f reached for r"0.9 is 10 times higher, several kHz. This property of the population $ enables to divide e!ectively the task of transmitting stimulus evoked spikes among several cells. The key role in distributing this task is played by noise and a probabilistic cell "ring caused by the #uctuations of membrane potential. Therefore, the timing jitter, which apparently limits the performance in medial superior olive and its analogs, in lower auditory nuclei can be bene"cial, helping to `hasha spikes among cells. The timing jitter changes from one nucleus to another, as documented in [7,16]. Joris and others show there that in ganglion spirale, as in a more general sound encoder, timing jitter is higher, compared with anteroventral cochlear nucleus, as in a specialized stimulus locked encoder, where jitter is lower. An important point is that the latter nucleus is just the higher relay along the auditory pathway, and yet the jitter is lower, measured by r. Assumed there is a certain convergence onto a single cell, timing jitter can be lowered, preserved, or become higher [11,3,16]. Once becomes the implicit information about sound localization explicit (it means encoded), its encoding into a labeled line code saves the resources of the system. This is because in the lower order nuclei all cells "re in stimulus locked regime without adaptation. In contrast, in the labeled line system, only those "bers belonging to the speci"c location "re, and other "bers remain silent, conserving the energy resources. The concept of neuronal representation of outer space was studied at various relays of sensory pathways. In lower relays, like colliculi inferiores, it is relatively well documented [14,17,28]. Less known is: how the spatial information from di!erent
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sensory inputs is integrated in both neocortex and hippocampus? At the present stage of knowledge in tracking the labeled line code gathered from both ITD and IID systems we are left with the notion that output from these nuclei converges at the next higher relay [14,17]. The spiking mechanism enabling coincidence detection has to switch o! after the time window for adding inputs from two sides expires. We want to stress that showing limits of the LI model in [10] and here is not in the contradiction with [4]. We found the equation of Gerstner, Kempter and others very insightful: d< "!G
(3)
where the term with G is the rectifying potassium current which should serve as the 0 above-mentioned o! switch. For the study [4] with synaptic learning this term was not crucial, and "nally this equation stayed only in the unpublished supplement to the paper [4]. The computational solution to the biophysical problem, which was formulated to generate spike in a time window as short as possible, was obtained in [4] by learning synaptic weights in the LI model. This solution reaches the goal of "nding the plausible CD system [19,21]. In [10] such a solution was found using one point Hodgkin}Huxley model and in [1] such a solution contained Morris}Lecar-type model [18,24,25] and reconstruction of the dendritic morphology of CD cells, to give just two more examples out of many [7]. One of the purposes of the work presented here is to consider it as an attempt to "nd a method for solving the following problem. We assume that the discussed circuit is serving such a speci"c purpose (sound localization) and that it has to have a speci"c design [2]. Several physical properties constrain the way the population should calculate the direction of incoming sound. Putting such constraints together has its methodical value. Of course, particular frequency ranges shown here are not sharp boundaries. In these regions one type of coding continuously and smoothly changes into another. In this sense boundaries of frequency regions are not exact numbers, we used here 16 Hz, 256 Hz, 1.1 kHz just as a prototype numbers. There is a straight demand for further work. The picture of population coding in both ITD and IID systems can be re"ned further with the estimate of number of spiking units. This could make possible to calculate the ratio, in which contributes one of the two systems to the localization sensation, and then ultimately to verify it by disabling one of the two systems in experiment.
Acknowledgements Supported by the Research Initiative of the Charles University no. 206023/1999, by the GAC[ R grant C 305/00/P101 (Charles University reg.C 201352/00) and by the NATO-NSF postdoctoral fellowship. Thanks to Chris DiMattina and Jir\ mH KofraH nek.
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Petr Marsalek completed two disparate courses at the Charles University of Prague, Czech Republic: medicine, with MD, and computer science and biophysics, with Ph.D. He was a lecturer of physiology at the First Medical Faculty, Charles University, from 1993. In years from 1995 to 1997 he was a NIMH postdoctoral fellow at the California Institute of Technology at the group of Christof Koch. In the 2000 he won a NATO-NSF postdoctoral fellowship to work at his present position at the Johns Hopkins University at the group of Ernst Niebur.