ABSTRACT. Human perception of sound arises from the transmission of action-potentials (APs) through

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1 Repint fom Poceeding of the 7 th Autalian Confeence onneual Netwok, 996, Canbea ACT A Multi-Thehold Neual Netwok fo Fequency Etimation L. S. Ilicht and Ian C. Buce and G.M.Clak The Bionic Ea Intitute, 384{388 Albet St., E. Melb. VIC 3, Autalia li@mail.medoto.unimelb.edu.au ABSTRACT Human peception of ound aie fom the tanmiion of action-potential (AP) though a neual netwok coniting of the auditoy neve and element of the bain. Analyi of the epone popetie of individual neuon povide infomation egading how featue of ound ae coded in thei ing patten, and hint a to how highe bain cente may decode thee neual epone patten to poduce a peception of ound. Auditoy neuon die in the fequency of ound to which they epond mot actively (thei chaacteitic fequency), in thei pontaneou (zeo input) epone, and alo in thei onet and atuation thehold. Expeiment have hown that neuon with low pontaneou ate how enhanced epone to the envelope of complex ound, while be with highe pontaneou ate epond to the tempoal ne tuctue. In thi pape, we detemine an expeion fo the Came-Rao bound fo fequency etimation of the envelope and ne tuctue of complex ound by goup of neuon with paameteied epone popetie. The etimation vaiance ae calculated fo ome typical etimation tak, and demontate how, in the example tudied, a combination of low and high thehold be may impove the etimation pefomance of a ctitiou `ecient' obeve. Alo, thehold comination may impove the etimation pefomance of neual ytem, uch a biological neual netwok, which ae baed on the detection of dominant intepike time.. Intoduction The auditoy ytem fom a emakably ecient neual netwok fo the poceing of ound. An undetanding of how thi ytem can pefom tak uch a the epaation of imultaneou ound, the ecient poceing of peech, and the identication of peake, will lead to advance in the deign of aticial neual netwok fo imila tak, and alo aid in the deign of heaing pothee uch a cochlea implant. Identication of the mechanim by which the auditoy ytem code popetie of ound i a t tep to uch an undetanding. The auditoy ytem can functionally be boken up into two majo ection. The t ection tanduce ound wave into neual ing patten, and compie the oute, middle and inne pat of the ea. The ound peue wave caue vibation of the eadum which ae then tanmitted via the oicle of the middle ea to uid within the cochlea (inne ea). Thi eult in tavelling wave which popagate along the uface of the baila membane (BM). Analogou to a continuou lte bank, the mechanical tuctue of the BM caue each Dept. of Otolayngology, Univeity of Melboune tavelling wave toeach it maximum amplitude at a poition detemined by it fequency. Each auditoy neve be i excited by the vibation of anaow egion of the baila membane, and i conequently tuned to a pecic fequency, temed it `chaacteitic fequency'. The econd ection i a multi-laye neual netwok which undetake the bulk of the poceing. It conit of the auditoy neve itelf, and the auditoy aea of the bain tem and cotex. In thi pape, we will mainly be inteeted in the epone popetie of the input laye of the netwok - the auditoy neve (AN). Thee ae appoximately 3, be in the auditoy neve. When uciently timulated, an inceae in a be' membane' pemeability to odium ion i initiated, and the coeponding in- ux of odium caue a udden jump in it tanmembane potential known a a pike o action potential (AP). Since thee pike ae lagely identical, it i geneally accepted that ound popetie ae coded by the place (the chaacteitic fequency of the neuon on which apike occu) and the timing of the pike. Thu, any theoy of neual ound coding mut explain how the tempoal (time-peiod)

2 and/o patial ing patten ae decoded to poduce auditoy pecept. Mot do thi by popoing that peceptual infomation i coded in one o anothe apect of the neual ing patten, uch a the pike ate o the ditibution of the intepike time, meaued aco eithe a ingle neuon o a population of neuon. Howeve, auditoy neve neuon die in moe than jut thei chaacteitic fequencie. They alo die in thei pontaneou ing ate, and in thei epone thehold. Thee epone diffeence ugget that auditoy ound coding could be baed on moe than jut the CF of the neuon. In fact, phyiological expeiment demontate that when timulated by a complex ound, be with low pontaneou ate pedominantly epond to the envelope, and thoe with high pontaneou ate to the ne tempoal tuctue of the ound [, ]. The neual epone thehold ae highly coelated to the pontaneou ate [3, 4], but eaonably independent ofchaacteitic fequency. Thu the bain tem eceive infomation fom be which may appoximately be paameteied in a twodimenional epone pace - whee one paamete epeent chaacteitic fequency, and the othe epeent thehold. Much wok ha been done to undetand how the epone of be with diffeent chaacteitic fequencie ae yntheied fo the tak of fequency etimation [5, 6,7],butvey little analyi ha been applied to undetanding the ole that be with dieent thehold play in the ame tak. In thi pape we invetigate the impotance of having a multi-thehold ytem. Thi i achieved by geneating a model of neual epone to a complex ound, and invetigating via Came-Rao bound [8], and via the ditibution of intepike time, the accuacy to which infomation about the fequencie within the ound may be etimated, baed eithe on the output of high-thehold and/o low-thehold be.. Signal and Netwok Model.. Signal Model Conide a common etimation tak pefomed by the auditoy ytem: the analyi of the fequency component of a peech ignal. Such ignal ae compoed of complex ound which exhibit a numbe of eonance (fomant), all modulated by a voicing pitch. Peceptual expeiment how thatif the actual fundamental of the voicing pitch i miing fom the pectum, then the etimated voice pitch coepond to the mallet dieence between the hamonic peent. Thi i a well noted auditoy phenomenon known a the \miing fundamental" [, 9]. Hee we exploe the etimation of the voice pitch fom the tempoal chaacteitic of neual epone fo a ignal whee two hamonic of the voice pitch ae peent, but not the fundamental. The input to the neual netwok i taken to be the ound peue wave paed though a linea lte, the cochlea. The lte chaacteitic of the cochlea to a 7 Hz tone i hown in Figue. Thu, the lteed ignal (t) i expeed a: Magnitude (db) (t) =+ X i= A i in(f i t + i ) () whee f and f ae the hamonic component of the voice pitch peent in the lteed ignal. 7 Hz pue tone Fequency (khz) Fig. : Filte chaacteitic of the cochlea to a 7 Hz tone. The tak i to etimate the voice pitch, f ; f, and it hamonic, f and f. Thi could be done eithe by etimating f and f imultaneouly and calculating the dieence, o by intoducing the f ; f component to the ignal via a nonlineaity and etimating the voice pitch diectly fom the modied ignal... Neual Netwok Model The epone of neuon of the auditoy neve may be modelled by an inhomogeneou Poion poce [], whee the intenity (epone ate) i decibed by mean of a compeively nonlinea (igmoidal) function, which i bought about by a numbe of nonlineaitie involved in AP theholding and geneation. Fom the fom of input-output cuve deived fom phyiological data [] we take tanh(:) to be a uitable igmoidal function. Although pontaneou ate i outinely ued to claify be epone, a thehold hift can bette explain the dieing epone []. Thu the Poion ate of the n th neuon, n (t), may be decibed by: Repint fom Poceeding of the 7 th Autalian Confeence on Neual Netwok, 996, Canbea ACT

3 n (t) = +tanh( n [(t) ; n ]) () whee (t) i the cochlea lteed ignal dened in the peviou ubection..3. Filteing Popetie of the Neual Model Changing the teepne and poition of the igmoidal tanh(:) allow the imulation of a ange of neual epone with vaiou onet and atuation thehold, and thee nonlinea epone will attenuate o magnify vaiou component of the ound pectum. Fouie analyi can be ued to nd the thehold value that minimie a cot function which meaue the elative magnitude of a pecic fequency component at the output of the igmoid. Such a cot function can include a penalty function which pevent the abolute magnitude of the majo component fom being ovely attenuated. Fo the etimation tak decibed in Section., we ae inteeted in the component at fequencie f and f, and the miing fundamental of the voice pitch, f ; f, and conequently pefom the analyi decibed above to detemine thehold which accentuate each of thee component. The elative magnitude of the two voice-pitch hamonic peent in (t) will depend on thei magnitude in the ound peue wave and on the lte chaacteitic of the cochlea at the place of the be' input. It i theefoe poible to have a ange of modulation depth in the ignal. Hee we invetigate two ignal with magnitude choen abitaily to poduce a lightly modulated (t) (Example ) and a highly modulated (t) (Example ). Example : Slightly modulated (t) Conide the ignal and neual epone: (t) = + 6 in(6t)+ 5 6 in(7t) = + tanh (((t) ; T )) whee T i a thehold hift. Example : Highly modulated (t) Conide a neual epone the ame a fo Example, but with the ignal: (t) =+ in(6t)+ in(7t) Fo both example, Fouie analyi of the output of each igmoid wa ued to maximie the elative ize of it component at the fequencie 6, 7 and Hz fom among the paameteied igmoid function: =+tanh (((t) ; T )) The optimal thehold value, T, ae hown in Table. 6 7 Ex. : Optimal T Ex. : Optimal T Table: : Optimal Thehold fo Example and The igmoid ae hown in Figue, and the input and output of the igmoid and thei Fouie tanfom ae hown in Figue 3 and 4. The implication of the lteing popetie of the igmoidal nonlineaitywillbeinvetigated in the next ection. Ex. : Sigmoid with T =.73; Optimal fo 6 Hz Ex. : Sigmoid with T =.97; Optimal fo 6 Hz.5.5 Ex. : Sigmoid with T = ; Optimal fo 7 Hz.5.5 Ex. : Sigmoid with T =.75; Optimal fo Hz Ex. : Sigmoid with T = ; Optimal fo 7 Hz.5.5 Ex. : Sigmoid with T =.68; Optimal fo Hz.5.5 Fig. : Left: Ex. - Slightly modulated (t). Sigmoid with optimal thehold fo 6, 7 and Hz (top to bottom). Right: Ex. - Highly modulated (t). Sigmoid with optimal thehold fo 6, 7 and Hz (top to bottom). (t) Input to igmoid 4 6 Output fom igmoid with T = Output fom igmoid with T = 4 6 Output fom igmoid with T = S(f) FFT of input to igmoid FFT of output fom igmoid with T= FFT of output fom igmoid with T= FFT of output fom igmoid with T= Fig. 3: Ex. : Slightly modulated (t). Output value of the igmoid with optimal thehold fo 6, 7 and Hz..4. Came-Rao Bound fo Neual Etimation of Fequency The auditoy ytem take the epone of ome 3 auditoy neve neuon, and can poduce etimate of the amplitude, A i, and fequencie! i Repint fom Poceeding of the 7 th Autalian Confeence on Neual Netwok, 996, Canbea ACT

4 (t) Input to igmoid 4 6 Output fom igmoid with T = Output fom igmoid with T = 4 6 Output fom igmoid with T = S(f) FFT of input to igmoid FFT of output fom igmoid with T= FFT of output fom igmoid with T= FFT of output fom igmoid with T= Fig. 4: Ex. : Highly modulated (t). Output value of the igmoid with optimal thehold fo 6, 7 and Hz. of the ound (t). Exactly how thi i achieved i lagely unknown, howeve tatitical method can yield infomation about the ability ofanypo- poed neual tuctue to etimate popetie of the ound. In tun, thee abilitie help hed light on likely mechanim fo the infomation poceing capabilitie of the auditoy ytem. One method of analying the ability of popoed mechanim to code paamete i via the application of the Came-Rao Bound [8]. Thi pemit alowe-bound to be given fo the vaiance of any unbiaed etimato fo the paamete in quetion. Of coue, uch an optimal etimato may not exit, o even be compatible with the tuctue of the auditoy ytem. Such an analyi i till ueful, howeve, becaue it can ule out mechanim which do not convey the equied infomation. The following Lemma i baed on calculation pefomed in [3], fo the etimation of a pue tone. Lemma. Conide an obevation fo duation T of an inhomogeneou Poion poce with ate (t f A). Then the Came-Rao inequality can be expeed a: ^ f Z T (t f f dt Thi eult can be extended to dene the Fihe Infomation Matix I(), fo the etimation of the unknown vecto paamete = [A! A! ], whee A i! i i ae the paamete of (t) a decibed in Equation. Lemma. Conide obevation of a numbe of inhomogeneou Poion pocee, with ate n (t ). In thi cae, the Came-Rao inequality can be expeed a: [I n ()] ij = Z n (t n (t ) dt n (t j n( i ) I ; n ii (3) Remak : A tandad eult of Came-Rao theoy, how that the infomation matix of the combined eult of independent expeiment equal the um of the infomation matice of each individual expeiment. Thu, unde the aumption of conditional independence of auditoy neve epone, a calculation of the Fihe Infomation Matix of the output of two o moe neuon can be achieved by umming the individual matice. Thi facilitate eay compaion of the output of vaiou goup of be, and the ability to take the output of one be, and elect the be which minimie the etimato vaiance baed on the combined infomation of both be. Thu, the evaluation of Lemma. whee the epone ate ae taken fom the neual and ignal model of Section. and. enable calculation of bound on etimato pefomance baed on the output of a numbe of neuon. In the cae of a igmoidal epone function (), and inuoidal ignal model (), the integal of (3) doe not appea to be analytically tactable, and conequently it i not olvable fo genealied condition. Howeve, it i numeically olvable fo any given paamete, and in a late ection we numeically invetigate etimato vaiance fo ome epeentative ituation..5. Intepike-Time Analyi Although the mechanim by which the auditoy ytem code fequency ae till lagely unknown [4], it ha been hypotheied that one method may be via the detection of dominant time-inteval between neual epone - eectively picking the peiod of the epone wavefom. Thi could be achieved via a eie of delay line and coincidence detecto [5, 6]. How would the theholding neuon eect thi kind of ytem? Although the Came-Rao bound of the peviou ection can limit the vaiance of etimato baed on neual epone, they can not indicate the degee to which the auditoy ytem' vaiance follow the optimal bound, and conequently ae not neceaily an accuate meaue of how ueful the output of a elected neuon i to the auditoy ytem. To invetigate thi quetion, we utilie the ditibution of inte-pike time. Lemma.3 Conide an Inhomogeneou Poion Poce with ate (t), ove the time inteval [ T]. Repint fom Poceeding of the 7 th Autalian Confeence on Neual Netwok, 996, Canbea ACT

5 Then the ditibution of pike occuing with a gap of i: D() = R T; (t)(t + )dt hr T (t)dt i = Remak : Thi ditibution meaue the elative fequency of pike occuing with a time dieence of, egadle of the exitence of pike within the inteval. It i conitent with the type of etimato popoed ealie in thi ection. An altenative expeion fo the ditibution of inte-pike time can alo be geneated. The eult of Lemma.3 ae ued in a late ection to calculate the eect of the igmoidal nonlineaity on the intepike-time ditibution. Simila to the Came-Rao bound, the integal appea analytically intactable, but can eaily be calculated numeically fo pecic example. 3. Reult In thi ection, the eect of vaiou neual thehold on the fequency etimation tak of Section. i detemined fo the cae of an ecient etimato, and alo fo an intepike-time baed etimato. 3.. Etimation by an Ecient Etimato Analytical deciption of the inne tem of the integal given in (3) ae deived fo a ignal of the fom expeed in (). The integal, howeve, appea not to be analytically tactable, and theefoe numeical integation wa implemented uing an adaptive ecuive Newton Cote 8 panel ule. Came-Rao Bound wee calculated fo Example and of Section.3, with all thee poible combination of low-thehold (T = :) and highthehold (T = :8) igmoid pai (L+L, H+H and L+H). The bound wee evaluated ove m (Table ) and m (Table 3). P Ex. T 6 Hz 7 Hz Hz L+L 7.4e-3.73e-4 7.8e-3.5e- H+H 4.3e e-4 5.5e e-3 L+H 5.38e-3.3e e-3.6e- L+L 4.e-4 4.3e-4 9.e-4.7e-3 H+H.3e-3.6e-3.65e-3 5.4e-3 L+H 5.89e-4 5.8e-4.34e-3.5e-3 Table: : Came-Rao Bound fo Example and : m In ome cae the low-thehold igmoid poduced the mallet bound, in othe the highthehold. In none of the example tudied did the combination of thehold (L+H) poduce the P Ex. T 6 Hz 7 Hz Hz L+L 5.59e-5.6e-6 5.7e-5.4e-4 H+H.85e-5 3.4e-6 3.6e-5 6.3e-5 L+H 3.77e-5.67e e e-5 L+L.85e-6.88e e-6.e-5 H+H 8.3e-6 8.7e-6.69e e-5 L+H 4.3e-6 4.7e-6 8.4e-6.67e-5 Table: 3: Came-Rao Bound fo Example and : m mallet eo, howeve when aveaged ove the ignal model tudied the L+H combination had a lowe mean eo (m: 7.5e-3 m: 4.78e- 5) compaed to the L+L (m: 8.45e-3 m: 6.5e-5) and H+H (m: 7.4e-3 m: 4.83e-5) combination. Thu fo the paticula ituation invetigated, an `ecient' etimato will opeate bet when the thehold ae identical, howeve dieent ignal will equie dieent optimal thehold, validating the need fo auditoy neve be with a ange of thehold. 3.. Etimation fom the Intepike-Time Ditibution One poible mechanim fo fequency etimation would be to utilie the intepike-time ditibution a examined in Section.5 to meaue the dominant intepike time obeved between two be [7]. Figue 5 how the elative fequency of occuence of intepike time fo Example and, with a low-thehold and a high-thehold igmoid. Relative fequency Relative fequency.5.5 Ex. : Low thehold (T =.) 5 Intepike time (m).5.5 Ex. : High thehold (T =.8) 5 Intepike time (m) Relative fequency Relative fequency Intepike time (m).5.5 Ex. : Low thehold (T =.) Ex. : High thehold (T =.8) 5 Intepike time (m) Fig. 5: Left: Ex. - Slightly modulated (t). Intepike-time ditibution fo low-thehold (T =.) and high-thehold (T =.8) igmoid (top and bottom). Right: Ex. - Highly modulated (t). Intepike-time ditibution fo lowthehold (T =.) and high-thehold (T =.8) igmoid (top and bottom). Fo both Example and, the intepike time coeponding to the weighted aveage of the peiod eulting fom the 6 and 7 Hz component of the ignal i emphaied by the low-thehold igmoid, and to the peiod of the Hz \mi- Repint fom Poceeding of the 7 th Autalian Confeence on Neual Netwok, 996, Canbea ACT

6 ing fundamental" by the high-thehold igmoid. Thi ugget that fo a ytem meauing intepike time, a combination of low-thehold and high-thehold be i ueful to etimate all thee fequency component when uing thi etimation technique, paticulaly if futhe lteing i ued to extact only one fequency pe be (eg. [7]). 4. Concluion The neuon of the input laye of the auditoy ytem (the auditoy neve) may be paameteied in tem of thei epone popetie including the fequency of ound to which they bet epond, and thei epone thehold. Fo the tak of fequency etimation, we have meaued the impotance of combining the output of auditoy neve neuon with dieing thehold. The eulting Came- Rao bound pemit the calculation of fequency etimation vaiance fo any given neual paamete and how, fo the calculated example, that an `ecient' obeve of the output of two neuon may benet fom a mixing of dieently theholded neuon. It ha been hypotheied that auditoy fequency etimation i lagely baed on the detection of action potential with uitably dened delay, and we alo demontate how combining the output of dieently theholded neuon may impove the fequency etimation capabilitie of uch a ytem. Due to the numeical natue of the calculation, thee eult ae not calculated paametically, but ae demontated fo a numbe of pecic example. An open quetion i the extenion of thee eult to a geneal cae - theeby pecifying condition unde which thecombined thehold epone ae moe ueful than ingle thehold epone, and condition unde which theyaenot. Refeence [] G. Langne, \Peiodicity coding in the auditoy ytem," Hea. Re., vol. 6, pp. 5{4, 99. [] J. W. Hot, E. Javel, and G. R. Faley, \Coding of pectal ne tuctue in the auditoy neve. I. fouie analyi of peiod and intepike inteval hitogam.," J. Acout. Soc. Am., vol. 79, pp. 398{46, Febuay 986. [3] M. C. Libeman, \Auditoy neve epone fom cat aied in a low noie chambe," J. Acout. Soc. Am., vol. 63, pp. 44{455, 978. [4] I. M. Winte, D. Robeton, and G. K. Yate, \Diveity of chaacteitic fequency ateintenity function in guinea pig auditoy neve be," Hea. Re., vol. 45, pp. 9{, 99. [5] M. I. Mille and M. B. Sach, \Repeentation of voice pitch in dichage patten of auditoyneve be," Hea. Re., vol. 4, pp. 57{79, 984. [6] D. O. Kim and K. Paham, \Auditoy neve patial encoding of high-fequency pue tone: Population epone pole deived fom d' meaue aociated with neaby placealong the cochlea," Hea. Re., vol. 5, pp. 67{8, 99. [7] P. Sulovicz and J. L. Goldtein, \A cental pectum model: A ynthei of auditoy-neve timing and place cue in monaual communication of fequency pectum," J. Acout. Soc. Am., vol. 73, pp. 66{76, Apil 983. [8] S. M. Kay, Fundamental of Statitical Signal Poceing - Etimation Theoy. Pentice Hall Signal Poceing Seie, New Jeey: PTR Pentice Hall, 993. [9] E. Javel and J. B. Mott, \Phyiological and pychophyical coelate of tempoal pocee in heaing," Hea. Re., vol. 34, pp. 75{ 94, 988. [] M. J. Penne, \Neual o enegy ummation in a Poion counting model," J. Math. Pychol., vol. 9, pp. 86{93, 97. [] M. B. Sach, R. L. Winlow, and B. H. A. Sokolowki, \A computational model fo atelevel function fom cat auditoy-neve be," Hea. Re., vol. 4, pp. 6{7, 989. [] J. W. Hot, E. Javel, and G. R. Faley, \Coding of pectal ne tuctue in the auditoy neve. II: Level-dependent nonlinea epone," J. Acout. Soc. Am., pp. 656{68, Decembe 99. [3] W. M. Siebet, \Fequency dicimination in the auditoy ytem: Place o peiodicity mechanim?," Poc. IEEE, vol. 58, pp. 73{ 3, 97. [4] J. O. Pickle, An Intoduction to the Phyiology of Heaing. London: Academic Pe Inc. Ltd, 98. [5] J. C. Licklide, \`Peiodicity' pitch and `Place' pitch," J. Acout. Soc. Am., vol. 6, p. 945, 954. [6] G. M. Clak, L. S. Ilicht, and T. D. Cate, \A neual model fo the time-peiod coding of fequency fo acoutic and electic timulation," Peented at the 6th Annual Meeting of the Autalian Neuocience Society, Januay 996. [7] D. Au, I. Buce, L. Ilicht, and G. M. Clak, \Co-be intepike inteval pobability ditibution in acoutic timulation: A compute modelling tudy," Ann. Otol. Rhinol. Layngol., vol. 4 - Supplement 66, pp. 346{349, Septembe 995. Repint fom Poceeding of the 7 th Autalian Confeence on Neual Netwok, 996, Canbea ACT