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Digital software model of the human peripheral auditory system Alexandre, Eric Ernest 1971

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A DIGITAL SOFTWARE MODEL OF THE HUMAN PERIPHERAL AUDITORY SYSTEM by ERIC E. ALEXANDRE B.Sc. Un i v e r s i t y of Calgary, Alberta, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard Research Supervisor Members of the Committee Head of the Department « Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA September, 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced deg ree a t t he U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r ee t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f ELECTRICAL ENGINEERING The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date September 24, 1971 ABSTRACT A model of the human peripheral auditory system is presented which is based on the anatomical and physiological data and mathematical presentations of the principal workers in the field. The physiological data come from human cadavers and cats in vivo. Since not a l l sections of the ear are understood with equal certainty, not a l l sections of the model are presented with the same degree of confidence. The model is used as a tool in carrying out experiments relating psychological performance with the underlying mechanisms. One interesting example is the localization of the mechanism producing combination tones within the human ear. Since the model is also capable of producing combina-tion tones with a l l but one nonlinearity within the model eliminated, the primary auditory neuron is shown to be the nonlinearity causing the formation of combination tones. A plausible theory for this effect is presented and compared to psychological data. The model was programmed in Fortran IV and run on the IBM 360/67. Because of better I/O and display facilities for acoustic input, i t is desirable to implement the model on a smaller computer such as the PDP-12. The large amount of storage for the present model prohibits this but simpli-cations are suggested so as to enable the implementation. The major discrepancy between the data produced by the model and physiological data results because the neural frequency selectivity of the model is inadequate. An inhibitory scheme to sharpen the selectivity is proposed. i i TABLE OF CONTENTS _ Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES v i LIST OF ILLUSTRATIONS v i i ACKNOWLEDGEMENT x i 1. INTRODUCTION 1 1.1 Why Model the Auditory System? 1 1.2 An Outline of the Anatomy and Physiology of the Ear 2 ,1.2.1 The External Ear. 2 1.2.2 The Middle Ear 2 1.2.3 The Inner Ear 2 1.3 Review of the L i t e r a t u r e on Auditory Modelling 6 1.4 B r i e f D e s c r i p t i o n of the Software Model 7 2. MATHEMATICAL BASIS OF THE MODEL 10 2.1 The External and Middle Ear 10 2.2 Hydrodynamics of the Cochlea 10 2.3- Hair C e l l L i m i t e r 16 2.4 Primary Auditory Neuron 16 2.5 Assumptions of the Model 17 3. SOFTWARE REALIZATION OF THE MODEL 1 8 3.1 External and Middle Ear i 8 3.2 Hydrodynamics of the Cochlea.. 18 3.3 Hair C e l l and Noise Generator 20 3.4 Primary Auditory Neuron 20 3.5- O v e r a l l Action , 22 i i i Page 4. UTILIZATION OF THE MODEL 23 4.1 A Comparison with P h y s i o l o g i c a l Data 23 4.1.1 The Outer and Middle Ear Transfer Function 23 4.1.2 B a s i l a r Membrane Displacement f o r Selected Frequencies. 23 4.1.3 B a s i l a r Membrane Stapes Spark Response 23. 4.1.4 B a s i l a r Membrane C l i c k Travel Time 28 4.1.5 Cochlear Amplitude Factor 28 4.1.6 Cochlear Perilymph Impedance 31 4.1.7 Spontaneous Neural A c t i v i t y 31 4.1.8 "Standard" C l i c k Response 34 4.2 The Combination Tone Phenomena 37 4.2.1 L o c a l i z a t i o n of Combination Tones 38 4.2.2 A Hypothesis to Account f o r Combination Tone Formation. 41 4.3 Recommended Extensions 44 4.3.1 In t e r a c t i o n at the Hair C e l l - Primary Auditory Level.. 44 4.3.2 AdaptatLonof the Model to a Smaller Computer: The PDP-12 46 5. CONCLUSION . 48 APPENDICES . . . . ' . " - / , -Appendix I Flow chart of the main program to c a l c u l a t e b a s i l a r membrane frequency response.... 51 D e f i n i t i o n of flow chart v a r i a b l e s 52 Appendix II Flow chart of the subroutine to add bandlimited noise to a s i g n a l 53 D e f i n i t i o n of flow chart v a r i a b l e s 54 i v Page Appendix III Flow chart of the subroutine to simulate a primary auditory neuron 55 Definition of flow chart variables 56 Appendix IV Flow chart of the main program for producing PST histograms at a l l sections 57 Definition of flow chart variables.. 58 REFERENCES . 59 v LIST OF TABLES Page 2.1 Boundary values of the hydromechanical parameters 15 2.2 Boundary values f or the c i r c u i t parameters of the hydrodynamic model 15 v i LIST OF ILLUSTRATIONS Page 1.1 a. The human ear 3 b. Schematic of the human ear with the cochlea unfurled 3 1.2 a. Cross-section of the cochlea............... 5 b. The organ of Corti 5 1.3 Block diagram of the interconnection of the subsystem models... 8 2.1 The transfer function of the outer and middle ear 11 2.2 a. Idealized model of the cochlear hydrodynamics 12 b. Ladder network representation of the hydrodynamics 12 3.1 Typical operation of the primary auditory neuron 21 4.1 Envelope and phase of the basilar membrane displacement 24 4.2 Basilar membrane displacement for a spark input at the stapes.. 25 4.3 First spatial partial derivative of the basilar membrane displacement for a spark input at the stapes 26 4.4 The characteristic frequency at various points along the basilar membrane. 27 4.5 Propagation time of clicks along the basilar membrane 29 4.6 Frequency dependency of the bolume displacement of the stapes/maximum cochlear partition displacement 30 4.7 Frequency dependency of the cochlear perilymph impedance 32 4.8 a. Interval histogram of the spantaneous activity of the model primary auditory neuron 33 b. Interval histogram of the spontaneous activity of a primary auditory neuron of cat 33 4.9 ' Spatial partial derivative of the basilar membrane response to a "standard" click 35 v i i Page 4.10 a. A t y p i c a l model PST histogram i n response to a "standard" c l i c k 36 b. A t y p i c a l PST histogram f or a s i m i l a r input to the cat 36 4.11 PST histograms of responses i n the s i n g l e auditory nerve f i b r e of the cat to dual tone stimulation 39 4.12 PST histograms of the model responses to dual tone sti m u l a t i o n 40 4.13 . Hair cell-primary auditory neuron i n h i b i t o r y interconnection. 45 v i i i ACKNOWLEDGEMENT I wish to express my thanks to the National Research Council f o r f i n a n c i a l support under Grant #67-3290, a postgraduate scholarship, and bursary. I am g r a t e f u l to my supervisor, Dr. M.P. Beddoes, for h i s ideas, enthusiasm, and constant a v a i l a b i l i t y . I am thankful to Dr. R.W. Donaldson f o r reading the manuscript and o f f e r i n g h e l p f u l suggestions. I am thankful to Miss Linda Morris f o r typing the thesis and to Mr. Herb Black f o r the photographic work. I am indebted to fellow students Miss Deb Reidlinger, Mr. Dave Agnew, and Mr. Jim Yan f o r t h e i r peerless proofreading. ix 1 1. INTRODUCTION 1.1 Why Model the Auditory System? Interest in producing models of the subsystems of the auditory chain probably began with von Bekesy's cochlear model in 1928. Finding direct observations within the cochlea of human cadavers beset with anato-mical d i f f i c u l t i e s , von Bekesy designed a hydromechanical model to reproduce the actual relations as closely as possible. This eliminated lengthy periods of anatomical preparation and allowed him to carry-out experiments on a model of dimensions larger than the actual cochlea. Even though the characteristics of the outer and middle ear and the hydrodynamics of the cochlea can be studied by work on cadavers, neural f i r i n g data must be obtained in vivo. Because of the obviously necessary restrictions placed on gathering this data from humans, data from lower animals must be used. By taking into account the differences between the lower animals used and the human, a model producing an equivalent to the otherwise unobtainable human neural f i r i n g data can be created. This thesis incorporates physiological data from the cat (9., 16., 21., 22., 23., 27.). The model presented herein was produced to: 1. Provide a standard against which to compare simpler models. 2. Be used as a tool for explaining the results of psychological tests. This use was aptly demonstrated in the localization of the combination tone phenomena. 3. Provide a basis for displaying the information transmitted from the inner ear to the brain. 2 1.2 An Outline of the Anatomy and Physiology of the Ear A b r i e f review of the anatomy and physiology of the auditory system w i l l be presented. The i n t e r e s t e d reader i s r e f e r r e d to the l i t e r a t u r e (5., 7., 13., 22., 26.). 1.2.1 The External Ear The external ear (Fig. 1.1) consists of an a u r i c l e which funnels impingent sound into the external auditory meatus (ear canal). The meatus i s a resonator about 2.7 cm long having a natural frequency of about 3 kHz. The meatus terminates at the tympanic membrane (ear drum), a t h i n transparent, e l a s t i c membrane stretched t i g h t l y on a bony frame. 1.2.2 The Middle Ear The middle ear consists e s s e n t i a l l y of a group of a i r c a v i t i e s con-t a i n i n g the o s s i c u l a r chain: malleus (hammer), incus ( a n v i l ) , and stapes ( s t i r r u p ) and t h e i r associated muscles. The o s s i c l e s connect between the tympanic membrane and the oval window of the cochlea, acting as an e f f i c i e n t pressure transformer. The stapedius muscle and the tensor tympani are r e f l e x i v e l y activated by h i g h - l e v e l sound. The former r e s t r a i n s the stapes and the l a t t e r exerts tension on the manubrium of the malleus. Together they serve to increase the s t i f f n e s s of the o s s i c u l a r chain thereby attenuating low.frequency transmission to the cochlea. The o s s i c l e s e x h i b i t an i r r e g u l a r resonance phenomena near 1500 Hz (5., p. 1080). 1.213 The Inner Ear The inner ear, c a l l e d the cochlea because of i t s s n a i l s h e l l shape, i s a f l u i d f i l l e d bony canal coupled to the middle ear by two membrane covered openings: the round and oval windows. The canal i s divided into 3 tube Fig. 1.1a The human ear (13., p. 10; reprinted by permission of P r e n t i c e - H a l l , Inc., Englewood C l i f f s , New Jersey). b A schematic of the ear with the cochlea unfurled. (5., p. 1076; reprinted by permission of John Wiley & Sons, Inc.) three f l u i d - f i l l e d compartments (Fig. 1.2a). The middle compartment, the sca l a media or ductus co c h l e a r i s , i s formed by the b a s i l a r and Reissner's membranes and i s f i l l e d with endolymph. The sc a l a v e s t i b u l i i s s i t u a t e d on the Reissner's membrane side of the sc a l a media and the s c a l a tympani on the b a s i l a r membrane side. Both contain perilymph and are joined at the a p i c a l end by a narrow channel, the helicotrema. The helicotrema allows for e q u a l i -zation of s t a t i c pressures between opposite sides of the membranes. The footplate of the stapes covers the oval window at the basal end of the sc a l a v e s t i b u l i . When the stapes moves inward, the b a s i l a r membrane moves toward the sc a l a tympani and the round window, at the basal end of the sc a l a tympani, moves outward. The b a s i l a r membrane motion causes shearing forces to be applied to some of the 25,000 h a i r c e l l s supported on the b a s i l a r membrane (F i g . 1.2) with i n the organ of Corti.' The h a i r c e l l s nearer the points of maximum displacement are subjected to the most force. A pure tone causes one maximum i n the envelope of the b a s i l a r membrane displacement. The higher the frequency of the tone, the nearer the maximum i s to the stapes end of the b a s i l a r membrane. The h a i r c e l l s are thought to be transducer elements which convert the mechanical a c t i v i t y of the b a s i l a r membrane in t o all-or-none a c t i v i t y i n the 30,000 afferent V l l l - n e r v e f i b r e s . The exact mechanism by which the mechanical a c t i v i t y of the b a s i l a r membrane causes the neural a c t i v i t y remains a matter of considerable con-jecture (13., p. 31). According to Engstrb'm (10.), the hairs of the h a i r c e l l act as levers and i n i t i a t e an electrochemical response proportional to the applied force. The electrochemical event i s transmitted down the sides of the h a i r c e l l membrane releasing a chemical transmitter substance at the base. This substance i n i t i a t e s a response i n the primary auditor}' neurons Inner phalangeal cells Outer phalangeal cells b F i g . 1.2a Cross-section of the cochlea. b The Organ of C o r t i . (13., p. 28). D a r r e l l E. Rose, Editor, Audiological Assessment (c) 1971. Reprinted by permission of P r e n t i c e - H a l l , Inc., Englewood C l i f f s , New Jersey. adjacent to the h a i r c e l l . Each primary auditory neuron i s innervated by a number of h a i r c e l l s and each h a i r c e l l innervates a number of primary auditory neurons. The outer h a i r c e l l s outnumber the inner h a i r c e l l s by about s i x to one (26.). The outer h a i r c e l l s are stimulated maximally by force i n the l o n g i t u d i n a l d i r e c t i o n while the inner h a i r c e l l s are stimulated maximally i n the r a d i a l d i r e c t i o n (6.). The spike a c t i v i t y of the VHI-nerve f i b r e s appears to be inherently s t o c h a s t i c i n that the d e t a i l e d pattern of f i r i n g s v a r i e s unpredictably upon repetition of i d e n t i c a l s t i m u l i , whereas appropriate averages show a s t a t i s t i c r e g u l a r i t y (31.). The primary auditory units f i r e "spontaneously" i n the absense of any c o n t r o l l e d or measurable acoustic s t i m u l i (22.). The thres-hold ( s t i m u l i l e v e l to cause 20% increase i n f i r i n g rate over the spontaneous rate) of d i f f e r e n t primary auditory units varies over approximately 80 db for cats (22., p. 91) when measured at the unit's c h a r a c t e r i s t i c frequency (the c h a r a c t e r i s t i c frequency i s that frequency f o r which the unit i s most s e n s i t i v e ) . 1.3 Review of the L i t e r a t u r e on Auditory Modelling Most of our basic auditory knowledge comes from the extensive experiments by von Bekesy (2., 3., 4., 5., 6., 7.). He investigated the anatomy of the e n t i r e ear and provided much p h y s i o l o g i c a l data from the actual subsections of the human ear and p h y s i c a l models of them. In the early 1950's, Peterson and Bogert (28.), Zwislocki (36.) and Fletcher (12.) provided the d i f f e r e n t i a l equations modelling the hydromechanical action of the cochlea. As Zwislocki (37.) points out i n h i s review of the theories, most are s i m i l a r attempts to f i t von Bekesy's data by d i f f e r e n t approximations and with d i f f e r e n t i n t e r p r e t a t i o n s of the constants and boundary conditions. Later, Flanagan (11.) provided a simpler means of computing the b a s i l a r membrane respnse by taking advantage of the r e l a t i v e l y i n v a r i a n t shape of the impulse response at d i f f e r e n t points along the b a s i l a r membrane. Kiang et a l (21., 22., 23.) have produced an abundant supply of c l i c k and tone burst neural data from the cat. Siebert (31., 32., 33.) has provided far-reaching i n t e r p r e t a t i o n s of t h i s data. Goldstein (15., 16., 17.) has been involved i n l o c a l i z i n g and determining the nature of the ear's n o n l i n e a r i t i e s . He has provided both psychological and p h y s i o l o g i c a l evidence that the n o n l i n e a r i t y causing combination tones occurs within the inner ear. Weiss (34.) and K l a t t (24., 25.) have presented comprehensive, models of the auditory system. Only recently have these models been used with speech input (18.). 1.4 B r i e f Description of the Software Model The ear i s modelled as a number of d i s t i n c t but consecutively arranged subsystems ( F i g . 1.3): 1. The outer and middle ear 2. The hydrodynamics of the cochlea 3. The h a i r c e l l transducers 4. The primary auditory neurons The outer and middle ear are modelled as a l i n e a r low-pass f i l t e r with a broad resonance around 2 kHz. The cochlear model i s based on the mathematical development of Zwislocki (36.). The hydrodynamic action i s s i m p l i f i e d to that of a ladder Inner Ear Acoustic Input Outer and Middle Ear Low-pass F i l t e r Hydro-dynamic A c t i v i t y Bandpass F i l t e r s Hair C e l l s Amplitude Limiters M + <•—• n N Additive Bandlimited Gaussian Noise Primary Auditory Neurons Threshold Detectors Pulse Coded Output F i g . 1.3 Interconnection of the subsystem models. oo network using the methods of Klatt (24.). A 110-section passive ladder network is used. Each capacitor voltage in the first 100-sections is proportional to the displacement in the z-direction of the basilar membrane at a point along the 35 millimeter length of the membrane. The latter ten sections represent the boundary condition due to the helicotrema (24.). The hair cells are modelled as amplitude limiters in the time domain. The primary auditory neurons are modelled as variable-threshold amplitude-sensing pulse position modulators. The model presented herein will be referred to as the software model throughout the thesis. 10 2. MATHEMATICAL BASIS OF THE MODEL 2.1 The External and Middle Ear The pressure transformation accomplished by the external and middle ear can be represented by a system of l i n e a r d i f f e r e n t i a l equations or by an equivalent passive e l e c t r i c a l model (35.). In t h i s study the important aspect of the external and middle ear response i s i t s low-pass property and broad tuning e f f e c t ( F i g . 2.1). The tr a n s f e r function presented below was used to represent the outer and middle ear: 10(l+jaj/u )(l+j(o/w ) G(u>) = 1 2 (l-(u/o) 3) 2+j2Co)/u 3) (l+jw/a>4) Optimum values were determined f o r the parameters: oo^ , "u^j a)3» w ^ a n < * £• 2.2 Hydrodynamics of the Cochlea The hydrodynamics was i d e a l i z e d to the f l u i d dynamics of two p a r a l l e l c y l i n d e r s separated by an e l a s t i c membrane of varying width (F i g . 2.2). The p a r t i a l d i f f e r e n t i a l equations expressing t h i s a c t i v i t y , developed a f t e r the method of K l a t t (24., pp. 119-123), are presented below: 3[A(x)u 1(x)] 2 dx = - -r b(x)dxu (x,t) 8x 3 z S f P ^ x . t ) ] -2pdx 3[A(x)u (x)] dx = 9x A(x) 3t 2m(x) 9 [V x» t ) ] • P8tap< t> + V*^ - 3bt09 , 2 d t b(x) at b(x) V x , t ' Fig. 2.1 The pressure transformation between the external auditory meatus and the stapes. (7., p. 100) Fig. 2.2a. Idealized model of the cochlear hydromechanics. (24., p. 17). b. Ladder network representation of the hydromechanics. (24., p. 128) 13 u-.(x,t) = u (x,t) = - V L (x,t) 1 ' ves ' tym ' P. (x,t) = P (x,t) - P. (x,t) 1 ves tym ' A(x) = A (x) = A „ (x) ves tym where: A(x) i s the cross-section area of the s c a l a (cm ) X i s the distance from the stapes (cm) u (x,t) ves i s the p a r t i c l e v e l o c i t y of the perilymph i n the sc a l a v e s t i b u l i i n the z - d i r e c t i o n (cm/sec) u (x,t) tym i s the p a r t i c l e v e l o c i t y of the perilymph i n the s c a l a tympani i n the z - d i r e c t i o n (cm/sec) b(x) i s the width of the b a s i l a r membrane (cm) u z ( x , t ) i s the p a r t i c l e v e l o c i t y at the center l i n e of the b a s i l a r membrane i n the z - d i r e c t i o n (cm/sec) i s the 2 pressure i n the s c a l a v e s t i b u l i at point x(dyne/cm ) P t y m ( x » t > i s the 2 pressure i n the sc a l a tympani at point x(dyne/cm ) P i s the density of the perilymph (gm/cm ) P «. (t) stap i s the 2 pressure at the stapes (dyne/cm ) m(x) i s the mass per unit length of membrane (gm/cm) D z(x,t) i s the displacement of the b a s i l a r membrane i n the z - d i r e c t i o n (cm) k(x) i s the 2 s t i f f n e s s per unit length of membrane (dyne/cm ) By representing the above equations i n di f f e r e n c e equation form i n the Laplace domain and comparing them to the d i f f e r e n c e equations f o r the ladder network of F i g . 2.2b which are shown below: [I (s) - I (s)] = -I fe) n-rl n pn [E (s) - E (s)] = - s L I ti n - l n n E. (s) + E (s) = [sL + R + - i - ]I (s) i n n pn pn pn ^ The following equivalences may be noted: = [A(x ) U l n 1 • I C b ( X n ) Ein< s> = P s t a P ( s ) «„(.) - P ^ . s ) T 7.0 P L n NA(x ) n L m(x )K - A. n n pn " 3 b ( x n ) R f ( x )K n n pn b(x^) b(x n ) . pn 8k(x )K r n n K I (s) r> t \ n pn Vx>s) —%— 3N K n b(x ) n The boundary conditions of the model parameters b(x), A(x), m(x), f ( x ) , and k(x) (Table 2.1; 24.) were used to c a l c u l a t e the corresponding boundary conditions f o r the ladder network (Table 2.2). The parameters were r e s t r i c t e d to exponential functions i n x. The c i r c u i t parameters were ca l c u l a t e d using equations of the form: L n = A n t i l o g [ l o g ( b ( x 1 ) ) - l o g ( b ( x 1 ) / b ( x N ) ) x ( n - l ) / ( N - l ) ] The stapes pressure, P i n ( t ) , i s the input and i s represented as E£ n ( t ) . The output i s the b a s i l a r membrane displacement at d i s c r e t e p o s i t i o n s along the membrane and i s represented as the time i n t e g r a l of the p a r a l l e l currents, l p n C t ) j l n t n e ladder network. Because the outer h a i r c e l l s predominate, only they are considered. K l a t t (24., p. 68) shows that since the outer h a i r c e l l stimualation occurs maximally for force i n the l o n g i t u d i n a l (x) d i r e c t i o n , the shearing force on these h a i r c e l l s i s proportional to the f i r s t s p a t i a l p a r t i a l d e r i v a t i v e .8D (x,t) of the b a s i l a r membrane displacement, . These s i g n a l s , from each 3 X section of the model, are used as inputs to the h a i r c e l l transducers. TABLE 2.1 Boundary values of .the hydromechanical parameters (48.0, p. 26) Parameter Base value Apex value Units b(x) 0.01 0.04 cm A(x) 0.02 0.008 cm2 m(x) 2.5 x 10~ 5 0.01 gm/cm 2 f(x) 3.2 0.8 dyne-sec/cm k(x) 3.0 x 10 5 20 dyne/cm2 TABLE 2.2 Boundary values f o r the c i r c u i t parameters of the hydrodynamic model Parameter Base value Apex value Units L n 3.5 8.75 H L pn 7.13 1.80 H R pn 1.37 2.12 n C pn 9.78 x I O - 1 3 2.22 x 10~ 7 F < 16 2.3 Hair C e l l L i m i t e r The input to the h a i r c e l l l i m i t e r s i s assumed to be the f i r s t s p a t i a l p a r t i a l d e r i v a t i v e of the b a s i l a r membrane displacement. In order to bring about a closer correspondence between p h y s i o l o g i c a l and model data i n response to a c l i c k , Weiss (34., p. 168) suggested incorporating a nonlinear transducer function located at the h a i r c e l l s : G(y) = y[k/(k + 1y|>] where y i s the input to the h a i r c e l l k i s the l i m i t i n g constant The use of such a n o n l i n e a r i t y i s supported by Siebert's rate versus i n t e n s i t y curves f o r more or les s t y p i c a l u n i t s (31., p. 210) which appear to be l i m i t e d by a s i m i l a r n o n l i n e a r i t y . 2.4 Primary Auditory Neuron The primary auditory neuron u t i l i z e d i s a type of pulse p o s i t i o n modulator having a non-constant threshold. I t i s based on Harmon's (19., 20.) a r t i f i c i a l neuron as modified by Weiss (34.). Consider the input s i g n a l , l ( t ) , K t ) = S Q ( t ) + N(t) where S Q ( t ) i s the output from the h a i r c e l l l i m i t e r N(t) i s bandlimited Gaussian noise The noise i s added to account f o r the spontaneous a c t i v i t y of the neuron and the stocha s t i c nature of the f i r i n g s within the VHI-nerve f i b e r s . A threshold function, T(t) i s defined to be i n the range 0 < T. $ T(t) S T w 17 where T. i s the absolute threshold A T i s the r e f r a c t o r y threshold K The threshold function i s defined as follows: T(t) = T A + ( T R - T A ) e " T / T R where T i s the time since the l a s t f i r i n g x D i s the r e f r a c t o r y time constant The neuron obeys an all-or-none r u l e f o r f i r i n g with the output being dependent upon the input and threshold. The threshold i s reset to the r e f r a c t o r y threshold maximum a f t e r each f i r i n g ; i t then follows an exponential decay towards the absolute threshold. 2.5 Assumptions of the Software Model The model includes a number of s i m p l i f y i n g assumptions which are considered below: The external and middle ear are assumed l i n e a r with the r e f l e x muscles of the middle ear i n a c t i v e . The hydrodynamics i s assumed to be adequately represented i n two dimensions by the a c t i v i t y w i t h i n the ladder network. Only the outer h a i r c e l l s are modelled and they are assumed activa t e d by the f i r s t s p a t i a l p a r t i a l d e r i v a t i v e of the b a s i l a r membrane displacement. Along the b a s i l a r membrane, a point-to-point s p a t i a l r e l a t i o n s h i p between the displacement of the membrane and h a i r c e l l shearing i s assumed: each h a i r c e l l stimulates only one neuron and each neuron i s stimulated by only one h a i r c e l l . The e f f e c t of eff e r e n t f i b r e s on the spike a c t i v i t y i s ignored. No mechanism to account f o r act i v e neural i n h i b i t i o n i s included i n the model. 18 3. SOFTWARE REALIZATION OF THE MODEL 3.1 External and Middle Ear The transfer function presented in Section 2.1, and repeated-below for convenience, was optimized at the given data points by the least sum of squares criterion using Monte Carlo techniques: 10(l+ju/u )(l+ja)/io ) G(u)). = — y-± 2 (1- (w/a>3) V j 2Cco/w3) (1+jco/u4) The parameters co^, u^* u ^ a n d C were restricted to reasonable ranges and the values were selected randomly and sum of squares calculated. New values were then selected and again the sum of squares calculated. The least sum of squares value was determined and i t was printed out in con-junction with the parameter values producing i t . The optimum values were found to be: 6 ^ = 2TT * 312 o>2 = 2TT * 472 o>3 = 2TT * 2365 6 1 . = 2TT * 489 4 C = 0.241 3.2 Hydrodynamics of the Cochlea Modelling the hydrodynamics was simplified to modelling the ladder network equivalent. A LTD-section model was used with the latter ten sections simulating the action of the helicotrema. A number of time domain solutions were attempted including some elementary recursive methods and the state-variable approach using Runge-Kutta integration. Because the differential equations representing the ladder network are s t i f f due to the large parameter v a r i a t i o n , these methods proved e i t h e r unstable or too time consuming. The f e a s i b i l i t y of constructing a hardware ladder network model of the cochlea to be used i n conjunction with the PDP-12 was also studied. But due to the p r e c i s i o n and high Q required of the components, cost ruled out t h i s method. In software simulation the main problem a r i s e s from the fac t that the high-frequency response nature of the stapes end of the network r e s t r i c t s the sampling i n t e r v a l to orders of magnitude lower than that necessary f o r the important speech frequencies (40Hz - 4kHz) ; yet the f i l t e r i n g and delay action of these i n i t i a l sections i s required. A frequency -domain s o l u t i o n s i m i l a r to one presented by K l a t t i n obtaining b a s i l a r membrane displacement for constant frequencies was used. The flow chart of t h i s method i s shown i n Appendix 1. I t i s based on the frequency domain current t r a n s f e r function G^(in) : G (a>) = I (OJ)/I (u) n n+I n Since G.7(u)) = 0, G (CJ) f o r n = N - l , ...1 can be consecutively c a l c u l a t e d using N n the r e l a t i o n presented above. The d i s c r e t e frequency domain b a s i l a r membrane displacement and i t s s p a t i a l d e r i v a t i v e are ca l c u l a t e d at each s e c t i o n of the model at l i n e a r frequency i n t e r v a l s . The input = 1 i s constant f o r a l l frequencies thereby simulating a voltage (pressure) impulse at the stapes i n the time domain. The frequency domain s p a t i a l p a r t i a l d e r i v a t i v e of the 2 impulse responses were stored on data c e l l . This method was not r e s t r i c t e d by the p a r a s i t i c time constants 1 . - •• • ..... : For frequency response versus section number see Figure 4.4 2 The data c e l l i s a bulk storage device included i n the 360/67 system. of the early sections as were the time-domain solutions first attemped. The sampling rate was set at more than twice the characteristic frequency of the most basal section from which basilar membrane data was required. 3.3 Hair Cell and Noise Generator The hair cell limiting action was applied in the time domain to the spatial derivative output of the basilar membrane. Its implementation was tr i v i a l . The bandlimited Gaussian noise added to the signal at the input to the primary auditory neuron was generated by the subprogram flow-charted in Appendix 2. Using the UBC IBM 360/67 library subprogram for generating normally distributed random numbers of known standard deviation, a signai was created. Then, using the system subroutine for calculation of discrete Fourier transforms (8.), the noise signal was transformed to the frequency domain. Frequencies above the specified upper bandlimit were zeroed and the inverse discrete Fourier transform was performed by similar methods. The noise signal was then restored to its initial standard deviation and added to the input signal. The correlation between two similarly generated noise signals was checked and found to be insignificant. 3.4 Primary Auditory Neuron The input to the primary auditory neuron is the output from the hair cell limiter plus bandlimited Gaussian noise ; the output is a -type of pulse position modulated signal, as illustrated in Fig. 3.1. As shown in the flow chart in Appendix 3, the amount by which the 21 Fig. 3.1 T y p i c a l operation of the primary auditory neuron. Input is the "standard" c l i c k . SF = 70 kHz, BLF » 3.5 kHz, 0 „ = 0.41, T. = 1, T_ = 10, T_ = 0.3 msec. N A R R threshold i s to be decremented each time step i s ca l c u l a t e d at the beginning. Within the major loop the threshold i s decreased i f i t exceeds the minimum, T . I f the s i g n a l exceeds the threshold, a pulse i s reg i s t e r e d , i t s time stored, the b i n f o r that time i s incremented, and the threshold i s reset. Because the v a r i a b l e , BPA, containing the number of pulses which occurred i n each time i n t e r v a l i s not zeroed a f t e r each c a l l , repeated c a l l s to t h i s subroutine produces a post stimulus time (PST) histogram. A PST histogram i s a histogram of the number of f i r i n g s i n each s p e c i f i e d i n t e r v a l of time a f t e r the onset of the stimulus. 3.5 Ov e r a l l Action The s p a t i a l p a r t i a l d e r i v a t i v e frequency data and the tr a n s f e r function f o r the outer and middle ear were created f o r 4096 time points at 7kHz sampling rate and stored on data c e l l . As shown i n the flow chart i n Appendix 4, an input s i g n a l sampled at 7 kHz for 4096 points i s created i n t e r n a l l y or read i n v i a the PDP-12 using paper tape. The d i s c r e t e Fourier transform of the input i s taken. This i s m u l t i p l i e d point by point i n the frequency domain by the outer and middle ear t r a n s f e r function and the s p a t i a l p a r t i a l d e r i v a t i v e of b a s i l a r membrane displacement at the required section. This s i g n a l i s then trans-formed back to the time domain and run through the h a i r c e l l amplitude l i m i t e r s . A PST histogram i s then calculated by repeatedly presenting the output s i g n a l from the h a i r c e l l l i m i t e r to the primary auditory neuron subroutine with a d i f f e r e n t noise s i g n a l added each time. The PST data i s then stored or p l o t t e d . 23 4. UTILIZATION OF THE MODEL 4.1 Comparison with Physiological Data 4.1.1 Outer and Middle Ear Response The optimized transfer function for the outer and middle ear provides a reasonably good f i t to the data of von Bekesy as shown i n Figure 2.1. 4.1.2 B a s i l a r Membrane Displacement for Selected Input Frequencies The b a s i l a r membrane displacement envelope and phase response for selected frequencies are compared with the data of von Bekesy (Fig. 4.1). The general envelope shape and skew of the model data i s s i m i l a r to von Bdkesy's data; the frequency s h i f t for the 200 Hz and 300 Hz inputs are shown i n better perspective i n Figure 4.4 where the x's represent the same data of von Bekesy. As found by K l a t t (24.), the model shows excessive phase lag for the low frequencies. Siebert (30.) has c r i t i z e d von Bekesy's phase measurements. But, unfortunately, his calculations are based on the same assumptions as the model. Because t h i s error seems inseparable from the model, data at frequencies below 100 Hz w i l l be suspect. 4.1.3 B a s i l a r Membrane Stapes Spark Response The b a s i l a r membrane stapes spark responses of two sections and th e i r s p a t i a l p a r t i a l derivatives are shown i n Figures 4.2 and 4.3 respec-t i v e l y . Although there i s no direct physiological data to compare with i n th i s instance, the responses are of r e l a t i v e l y invariant shape as predicted by Flanagan (11.). Von Bekesy measured the rate of decay at the natural frequency for various locations along the membrane and found the amplitude 300 Hz 200 Hz 100 Hz 50 Hz _ j , , j 20 25 Distance from 3 Q 3 5 stapesf m.m.) Fig. 4.1 Normalized amplitude envelope and phase with respect to stapes pressure of the basilar membrane displacement for selected input frequencies. (B£k6sy: 7., p. 462). 25 Fig. 4.2 Basilar membrane displacement at two different points along the cochlear model for a condensation spark at the stapes. The input is given in the lower right of the diagram. 26 Fig. 4.3 The f i r s t spatial partial derivative of the basilar membrane displacement at two different points along the basilar membrane. The input i s the same as in Figure 4.2. Mathematical Model after. K l a t t (24.,p 27) Data from Bekesy: G> 3., p 246 X 7., p 462 Data from K l a t t : A 24., p 42 Software Model T 20.0 40.0 60.0 SECTION NUMBER o.o BO.o 100. Fig. 4.4 The c h a r a c t e r i s t i c frequency at various points along the b a s i l a r membrane. 28 ratio to lie between 1:4 and 1:5. The amplitude ratio for the data from section 60 lies within this range (1:4.5) but the ratio at section 40 (1:3.5) indicates too high a Q. Klatt (24., p. 42) presented similar responses for his hardware model which had amplitude ratios of 1:6.3 and 1:4 prespectively in the tails, but the first cycles exhibited an unpredicted irregular response similar to the spatial partial derivatives of the software model (Fig. 4.3). 4.1.4 Stapes Click Travel Time Along the Basilar Membrane . Von BeTce"sy performed an experiment to determine the travel time of a click along the basilar membrane. He attached a small mirror on the cochlear partition and activated the stapes by a spark click. A similar experiment was performed with the software model. The input to the stapes used was an approximation of a spark click (Fig. 4.2). The data of von B£kdsy, the results of Klatt's hardware model (24., p. 43), and the 100% rise time of the software model are shown in Figure.4.5 to follow the same trend. The increased delay time shown for the models is most likely due to the fact that von Bekesy was measuring something less than the 100% rise time but he was not explicit. 4.1.5 Cochlear Amplitude Factor Figure 4.6, containing the frequency dependency of the volume displacement per maximum cochlear partition displacement shows only a qualitative agreement with von Beke*sy's data as does the data of Klatt. The same trend is evident; the magnitude difference may be due to the fact that von B£ke"sy measured the displacement of Reissner's membrane and assumed 29 10" Fig. 4.5 Propagation time of clicks along the basilar membrane. 30 M C N >< a o M H W P H C O W H H C O P H < W u o o w P H W C O l-H-o > Q M P H 10 -1 © from Bekesy (7.,p 455) * from software model A from K l a t t (24.,p 35) 10 -2 10" 10 -4 100 200 400 800 F R E Q U E N C Y ( H Z ) 1600 3200 F i g . 4.6 Frequency dependency of the volume displacement of the 2 stapes per maximum displacement of the cochlear p a r t i t i o n (cm ). 3] the d e f l e c t i o n of the b a s i l a r membrane equal to i t . The discrepancy here i s not considered s i g n i f i c a n t as i t i s j u s t a matter of amplitude s c a l i n g . 4.1.6 Cochlear Perilymph Impedance Figure 4.7 shows a comparison between the frequency dependency of the model's cochlear perilymph impedance as compared to data obtained by von B£k£sy from human cadavers and data from K l a t t ' s model. The magnitude data follows the same trend; the software model i s s i g n i f i c a n t l y c l o s e r to von Bekesy's r e s u l t s than i s the data from K l a t t . For lower frequencies the impedance of the perilymph acts as a resistance while f o r higher frequencies the e f f e c t of i n e r t i a i s greater. The software model does not show as extreme a change as von Bekesy's data but the tendency i s correct. The phase trend i n K l a t t ' s data i s reversed. 4.1.7 Spontaneous Neural A c t i v i t y The parameters of the neural model and the standard deviation of the i n t e r n a l noise were varied i n order to produce an i n t e r v a l histogram of spontaneous a c t i v i t y s i m i l a r to the histogram presented by Kiang (22., p. 99) f o r a unit with a medium discharge rate. Noise was presented to the model neuron and a histogram of the times between f i r i n g was produced. A sampling frequency of 12.8 kHz was used so that groups of eight consecutive samples constituted the histogram i n t e r v a l as used by Kiang, 0.625 msec. Band-limited Gaussian noise with an upper frequency cut-off of 3.5 kHz and a standard deviation a^= 0.410, with neural constants x R = 0.3 msec, T^ = 1.0, T = 10.0 gave excellent r e s u l t s ( F ig. 4.8). These same constants K were retained f or use i n the production of PST histograms but were e f f e c t i v e l y 32 F i g . 4.7 Cochlear perilymph impedance as a function of frequency. 500 200 i o 100 O ct UJ ca I 50 20 10 500 200 3 100 rr u o tr ui <r> 2 2 50 20-.V, V 10 20 30 40 50 60 70 80 INTERVAL IN MSEC 0 10 20 30 40 SO 60 70 60 INTERVAL IN MSEC. a. b. F i g . 4.8a. I n t e r v a l histogram of the spontaneous a c t i v i t y of the model auditory neuron. Model parameters are given i n Figure 3.1. Bin width = 0.625 msec. Spontaneous rate = 44 spikes/second. Three minutes of data are represented, b. I n t e r v a l histogram of a cat's primary auditory unit with a medium rate of discharge (22., p. 99). Bin width = 0.625 msec. Spontaneous rate = 45.5 spikes/sec. Three minutes of data are represented. OJ OJ 34 scaled down by a factor of 2.0 * 10 by increasing the amplitude of the spatial derivative output by that factor. 4.1.8 "Standard" Click Response In order to facilitate comparison with neural data from cats, the definition of a "standard" click as used by Kiang et al (22., pp. 7,20) was adopted. The standard click is a rarefaction click caused by a 100 usee electric pulse at -50 db re lOOv input to a condenser earphone. Figure 4.9 depicts the acoustic click used as an approximation to the standard click and the resulting spatial partial derivative response of the software model at two sections. Using the spatial partial derivative at section 60 (CF = 640 Hz) for a standard click and noise as input to the model neuron, a PST histogram was produced. This is compared to a PST histogram from a neural unit in a cat in Figure 4.10 (22.). The histograms are alike in that the i n i t i a l pulse is highest and narrowest, the spontaneous activity is suppressed in intervals between peaks, the amplitude ratio of the first two peaks is approximately the same and the peaks have similar spacing. There are two major discrepancies: the delay time before the onset of the first peak is too short and the subsequent peaks f a l l off too rapidly. Some of the added delay time may come from the fact that no chemical transit time was accounted for at the point of stimulation of the neurons by the hair cells. The latter discrepancy is most likely due to the sharpening caused by lateral inhibition which was omitted in the model. A scheme to allow implementation of such interaction is presented in section 4.3.1. Fig. 4.9 The f i r s t s p a c i a l p a r t i a l d e r i v a t i v e of the b a s i l a r membrane displacement at two points along the membrane i n response to a standard c l i c k (according to Kiang: 22., p. 20) at the outer ear. The approximation of the standard c l i c k i s given in the center. a • • ' - ... b. F i g . 4.19a. A typical PST histogram in response to a "standard" click. The response of section 60 in Figure 4.9 i s the input to the hair c e l l limiter. Gaussian noise of upper bandlimit =3.5 kHz and OJJ = 0.205 * 10-6 i S added at the input to the primary auditory neuron. T^=0.5 *l0-6~t TR=0.5 *10-5, T r =0.3 msec. Bin width«0.0315 msec. CF=625Hz. b. A typical PST histogram for a similar input to the cat. CF = 650 Hz. A number of PST histograms synchronized to selected frequencies for multiple tone stimuli are presented in the following section along With the equivalent physiological data. The purpose of this presentation is not only to provide further data comparison, but to present evidence as to the location of the mechanism producing combination tones within the human auditory system. 4.2 The Combination Tone Phenomena Combination tones are tones perceived by a human listener when presented with a complex acoustic stimulus. For a two tone input at frequencies f^ and f^, the combination frequencies are f(n) = f^-n(f2 - f ^ ) • In only a few cases are combination tones of pitch greater than the stimulus tones perceived. This is at least partially caused by the fact that lower frequency tones mask higher frequency tones much more than vice versa (29.). The most easily heard combination tone i s of frequency 2 -where f^ < f^. Knowledge of the presence of combination tones has existed since their discovery in 1714 by Tartini (17.). Only recently has Helmholtz's theory that such tones were present only at high levels due to overloading of the middle ear been refuted. Goldstein (15.) in 1967 showed by tone cancellation methods that the combination tones were present at relatively low sound levels and were relatively unaffected by stimulus level. Goldstein suggested that combination tones are formed in the inner ear but he discounts that the r e c t i f i e r action occuring there is the cause because such a nonlinear transformation would generate only even order combination frequencies. 38 4.2.1 Localization of the Cause of Combination Tones Goldstein and Kiang (16.) in 1968 showed that the combination tone Was present in the output of the primary auditory neuron of the cat (Fig. 4.11). Two stimulus frequencies f^ < were presented to the outer ear of a cat. One-minute samples of data from a neuron with characteristic' frequency of approximately the same frequency as the stimulus tones were synchronized to the two different stimulus frequencies and 2 - and 2 f.^ - f.^ for three cycles. The periodicity within the PST histograms shows that the combination tone 2 - is present at almost a l l levels and the tone 2 - is present for some intermediate levels. A similar experiment was run with the model. Two sampled pure tones were generated internally and presented to the model. PST histograms synchronized to the stimulus frequencies and combination frequencies were produced from the output of section 36 (CF = 3.38 kHz). Sixty runs of data approximately 1/60 sees, long were used to generate the histograms. As seen in Figure 4.12 the two primary tones are present for a l l stimulus levels. The combination tone 2 - is easily apparent at the -30 db and -50 db levels but are more subdued at the -70 db level."'" This is consistent with the Goldstein - Kiang data (Fig. 4.11). For both sets of data, the histograms synchronized to 2 - f^ show a lesser degree of regularity. The fact that the combination tones were shown to be present even though a linear model of the outer and middle was used, gives further evidence against Helmholtz' theory that the middle ear is the cause. The •*"The results for the model (Fig. 4.12) are less clearly defined than those of Figure 4.11 at least partially because the samples used in the model were 1/60 the length of those used by Goldstein-Kiang. Cost prohibited the use of one minute samples. 39 PST HISTOGRAMS SYNCHRONIZED TO: f, (3.46 kHi) f 2 ( 3 9 ? k H « ) 2f, - f 2 (3-33 kHi) 2f2 -f, (4.32kHx) • 8 0 * - 70dB Z o < U J o z - 6 0 * 3 2 Fl*F8-70 SYW CS1 8 WT ni*ooeuttc 003*79 MP ow: OM): . . . CM: rvia-? pi.rs-«o tm rt i i m ou'ooeusrc amanr 100: «»S-T PI-PS-40 JYH PS tt PST M-OOSUSCC 0039W PSP wo: >.«-? ri.rc-tfo S T M COT . PS1 M'OMUSU 0 0 3 M » « » loo : K*S-7 PI«P8-00 tTM PI ts m tu-owuac «wui» too: - SOdB ow. rv<l-7 P I . P9-«O fm n u m au'ocx usee O M N H P too: C.i.2-7 Pl.PS-80 SYN COT 6 P S I wooousfc ootsiontr 100 Uu ^ -^ii Ml'7 PI«T8-^ tYM Pl 13 PIT or " — •oo: 40dB oool r*.a-7 P l^a^o STN PS 17 m tuiOOOUSCC OOMSOH* loo: nts-? P I - P I - * O rm err • *V*1-7 M-F2-30 ilM FI 11 PIT W009IBCC OOtMfefrSP too M-JS-T FI*F8-» SYM F8 11 PST MI'OOB USCC 00*302 MP 100' - 30dB <*o< f S S i - 7 F l ^ i - S v SYH F I Iff » ^ 2 - 7 f 1^2-20 SYM F2 10 PSI ftU'OOSUttC 003319 MP 100 I t . * . ' . - ' Fj*ri ' -?0 SYM CfrTIO bU 'OOSUSK 00v22%P$P I W -20dB 0 060-J»«S-P Pi.«-ro rm cos a MT tlHOWUttC 009*79 *SP> KM9-7 PfTS-00 tYM COS 9 pn diioosinec O O M T S I O P too: )• : • oso. . . . . 080 I 060 yjyjy yyjy KM3-7 F1-F8-S0 SYN CB2 «. PtT BU<008USeC OMStflRSP too: Krt2-7 Fl*F2-*0 SYM C02 9 PST 6U>00SUS£C 0OV4O2BS? too: »"»«.2-7 F1-F2-50 SYN C62 tf PST woos usee OMSWRSP 100 H V « - 7 f 1-F2-20 SYN C82 7 PST WOOSUttC 0031<»»RSP 100" F i g . 4.11 PST histograms of responses i n a sin g l e auditory-nerve f i b r e i n the cat (CF = 3.20 kHz). The acoustic stimulus to the cat consisted of two tones with frequencies f-^ and f2« The histograms were synchronized with four reference frequencies. The responses to three cycles of the reference s i g n a l are shown i n each histogram. The data for each row of histograms are one-minute samples of the same record, each histogram i n a row being synchronized with a d i f f e r e n t reference frequency. The histograms i n each column represent data taken at seven d i f f e r e n t l e v e l s of the stimulus. The le v e l s given are for each tone i n the acoustic stimulus. The data i s from Goldstein-Kiang ..(16. , p. 984). PST HISTOGRAMS SYNCHRONIZED TO: 40 40 ' 5 0 d b yLk JAJL 40 F i g . 4.12 PST histograms at section 36 (CF - 3.38 kHz). As i n Figure 4.11, the histograms are synchronized with four d i f f e r e n t reference frequencies. Stimulus tones for the top three rows are f-^ and f2 at the l e v e l indicated at the f a r l e f t . The bottom row includes a stimulus at 3.33 kHz, -63 db. The data for each row represents approximately 1/60 second samples repeated s i x t y times with d i f f e r e n t noise added each time. The maximum ordinate value i s repeated to the upper l e f t of each histogram. The maximum abscissa value i s i n each case one mil l i s e c o n d . model was also run without the hair cell amplitude limiter nonlinearity and the periodicity was s t i l l present for the histograms synchronized to the combination tones (not shown). The.bottom line of Figure 4.12 shows histograms for which a tone was added to cancel the combination tone at 2 - f^. The stimuli consisted of tones at f^, at -50 db and 2 f^ - at -63 db. The level and phase of the third tone were varied to produce the flattest 2 f^ - synchronized histogram. This is similar to the method used by Goldstein in his psycho-_ logical investigation of combination tones and Goldstein-Kiang in their physiological methods. According to Goldstein (15., p. 680) for a frequency ratio ^2^1 = a s u s e d here, the cancellation tone should be down approxi-mately 14 db from the level of the stimulus tones (cf 13 db herein). The fact that the combination tone phenomena can be shown to be present in a model incorporating only one nonlinearity: that of the primary auditory neuron suggests very strongly that this is the element causing the tones within the human peripheral auditory system. 4.2.2 A Hypothesis to Account for Combination Tone Formation One plausible hypothesis for explaining the phenomena is proposed in this thesis. It is based on the fact that the primary auditory neuron is essentially an amplitude sensing device. Consider an input, f^(t), consisting of two pure tones: f i(t) = a^cos 2irf^t + a 2 cos 2iTf2t a 2 < a^ f l < f 2 which can be represented as follows: c l - 3.« f±(t) = [~Y cos 2irf ; Lt + -y cos 2Trf 2t+ -y cos 2 7 r(2fj-f 2>t] a l a2 a2 +[-y cos 2 i r f 1t + -y cos 2 iTf 2t -r — cos2iT&f^-f^)t] = f ± 1 ( t ) + f . 2 ( t ) where f-j^Ct) a n c * r e P r e s e n t e a c n of the two l i n e s on the r i g h t hand side r e s p e c t i v e l y . By trigonometric s u b s t i t u t i o n , a l f ^ ( t ) = -y cos 2irf^t + a 2 cos 2 'iTf^t cos 2v{f^-i^)t ~ A, cos 2iTf t + A„ cos 2irf t cos 2irf t 1 c 2 c m where A^ = a^/2, A 2 = a 2 = [1 + m cos 2iTf tjA^cos 2Trf t m 1 c A2 where m = — , £ c = f v f f f l = f 2 - f± The l a t t e r form of t h i s equation shows ' f ^ C t ) t o represent an amplitude modulated s i g n a l with a c a r r i e r frequency = f ^  and a modulating cosine s i g n a l of frequency ( f 2 - f^) . By s i m i l a r trigonometric s u b s t i t u t i o n , a f i 2 ( t ) = -y cos 2Trf 1t - a 2 s i n 2-ir^t s i n 2 i r(f 2 - f ^ t = A-cos 2irf t - A_sin 2Trf t s i n 2nf t 1 c 2 c m where A^ = a^/2, A 2 = a 2 = A., cos 2irf t - A. k £ /"cos 2irf t dt s i n 2irf t 1 e l f ' m c A2 where kr = -.— 2irf f A^ m The l a t t e r form of t h i s equation shows f - ^ C t ) to represent a narrowband frequency modulated s i g n a l with a c a r r i e r frequency f ^ and a modulating / 43 cosine s i g n a l of frequency ( f ^ - f ^ ) • An amplitude sensing system w i l l y i e l d an output, f Q ( t ) j which i s pro p o r t i o n a l only to the AM portion of the s i g n a l : a- a~ a-f (t) = -^-cos 2irf.t + -TT cos 2Trf-t •+- -x- cos 2ir(2f. - f„)t O z L A A A I Z The output would therefore contain the two stimulus frequencies f ^ , and the combination frequency 2 f - f ^ . That the amplitude of the combina-t i o n tone 2 - f j has an amplitude equal to that of the stimulus frequencies at the same point on the b a s i l a r membrane i s substantiated by the p h y s i o l o g i c a l data of Figure 4.11. The combination tone hypothesis p r e d i c t s that the c a n c e l l a t i o n tone l e v e l w i l l r i s e l i n e a r l y when both stimulus si g n a l s are increased i n loudness. Goldstein (15., p. 677) presents data from two subjects i n order to te s t the e f f e c t of the primary stimulus l e v e l upon the c a n c e l l a t i o n tone l e v e l . The l i n e a r increase i s substantiated within 10 db over a 50 db stimulus l e v e l range. The hypothesis also p r e d i c t s that i f the l e v e l ofone stimulus tone i s held constant while the l e v e l of the other i s var i e d at a lower l e v e l , the l e v e l of the c a n c e l l a t i o n tone w i l l r i s e approximately l i n e a r l y with the lower l e v e l tone u n t i l the stimulus tones become equal (100% modulation). Goldstein presented two sets of c o n f l i c t i n g r e s u l t s i n t h i s case. The hypothesis i s consistent with one set of data. Goldstein's data i s sparse although h i s methods are good; further psychological experi-ments a f t e r h i s methods are suggested i n order to furth e r test the hypo-thesis and model. 4.3 Recommended Extensions of the Model 4.3.1 Interaction at the Hair Cell-Primary Auditory Neuron Level It is well known that the primary auditory neuron selectivity is much greater than that of the basilar membrane alone. This suggests sharpening by some form of inhibition possibly similar to that found to give good results for a visual receptor network (1.). The fact that each hair cell is involved in the innervation of a number of primary auditory neurons and each primary auditory neuron is influenced by a number of. hair cells suggests that the inhibition takes place at this level. The software model could be extended to include this effect by modifying the hair cell-primary auditory neuron interconnection to that shown in Figure 4.13. The corresponding equation is: N x. = e. - £ k. . e. 1 1 J-l 1 3 3 where x_^  is the output of the summing junction and the input to the primary auditory neuron e^ is the output from the hair cell limiters and the input to the summing junction k^ is the "inhibition" constant specifying the effect of the jth section of the output of the ith section Unlike the simulator used for the visual receptor network, the signals x^ and e^ are in sampled analogue form instead of pulse rates and therefore can take on negative values. The values of the constants k.. should be determined experimentally using the rate of decay of the standard click results of Kiang as the target data. Because the inhibition Primary Auditory Neuron: r+1 r+1 Fig. 4.13 Three hair c e l l and primary auditory neurons with inhibition interconnection. The "inhibition" constant, k ., determines the effect of hair c e l l j on primary auditory neuron i . 4 6 / i s only l o c a l , the constants k_^_, w i l l be nonzero only when t h e i r indices conform to the following r e s t r i c t i o n : | i - j | < m where m - 3. This interconnection scheme i s not unlike the forward shunting i n h i b i t o r y model of Furman and Frishkopf(14.). 4.3.2 Adaptation of the Model to a Smaller Computer: The PDP-12 The model produces reasonable r e s u l t s but i s unsuitable f o r a small computer because of the large amount of storage space required to contain the frequency data of each section of the hydrodynamic model. For t h i s reason, a recommended extension would be the modi f i c a t i o n of the model to use the s i m p l i f i c a t i o n proposed by Flanagan (11.). This proposal takes into account the f a c t that the shape of the displacement impulse responses along the b a s i l a r membrane are nearly i n v a r i a n t . Only the delay time before the onset of the response and the na t u r a l frequency vary. Storing only one impulse response and the delay time and natural frequency of a l l sections would diminish the storage requirements of t h i s part of the program by a f a c t o r of nearly two orders of mangitude f o r a 100-section model. The data of the modified model could be compared with the o r i g i n a l model's data to j u s t i f y the modification. Weiss (34.) has used Flanagan's proposal with good r e s u l t s . The saving i n core requirement would allow implementation on a smaller computer such as the PDP-12 provided that a fa s t access bulk storage device were available"*". The bulk storage device i s necessary i n 1 ' A disk u n i t of at l e a s t 500 K 12-bit words would be necessary. order to store the output s i g n a l from each section. The PDP-12 has the advantage over a l a r g e r computer i n that i t has better I/O f a c i l i t i e s f o r auditory input^ and di s p l a y . The display and photographic arrangement a v a i l a b l e on the PDP-12 make possible the d i s p l a y i n g of the output of the per i p h e r a l auditory system PST histogram data i n speech spectrogram form. This i s d e s i r a b l e i n that comparison of such a spectogram with the input 2 speech spectogram would give a good v i s u a l i n d i c a t i o n of the processing and d i s t o r t i o n s of the pe r i p h e r a l auditory system when a complex input s t i m u l i such a§- speech i s used. The author designed and aided i n the construction of the present audio i n t e r f a c e and clock f o r the PDP-12. The author has already produced the software f o r creating the input speech spectogram on the PDP-12. 48 5. CONCLUSION A d i g i t a l software model of the p e r i p h e r a l auditory system has been presented. The model consists of a number of subsystem models: the external and middle ear, the hydrodynamics of the cochlea, the h a i r c e l l l i m i t e r , and the primary auditory neuron. The external and middle ear are represented by a t r a n s f e r function having low pass c h a r a c t e r i s t i c s with a broad resonance near 2 kHz. The hydrodynamics of the cochlea are based on the anatomical studies of von Bekesy, the mathematics of Zwislocki, and the s i m p l i f i c a t i o n s u t i l i z e d by K l a t t i n h i s hardware ladder network of the cochlea. The h a i r c e l l l i m i t e r and the primary auditory neuron are based on the work of Weiss with the l a t t e r element based on Harmon's model neuron. The data from the software hydrodynamic model and K l a t t ' s hard-ware hydrodynamic model produced s i m i l a r r e s u l t s as expected since they have the same ba s i s . Any di f f e r e n c e s were usually i n the software model's favour when compared to von Bekesy's r e s u l t s . Comparison with Kiang's neural data from the cat showed that f o r PST histograms of a standard c l i c k , the data of the software model produced data that was s i m i l a r i n a number of respects: the i n i t i a l pulse i s highest and narrowest, the spontaneous a c t i v i t y i s suppressed i n the i n t e r v a l s between peaks, the amplitude r a t i o of the f i r s t t*?o peaks have s i m i l a r spacing. The most important discrepancy i s that the l a t e r peaks f a l l o f f too r a p i d l y . Since t h i s i s most l i k e l y due to the f a c t that i n t e r a c t i o n at the h a i r c e l l - primary auditory neuron i n t e r f a c e was omitted a method f o r extending the model to include t h i s i n t e r a c t i o n i n the form of l a t e r a l i n h i b i t i o n has been included. 49 The p h y s i o l o g i c a l experiments of Goldstein and Kiang (16.) on combination tone production are duplicated using the model. The model produced combination tones i n the presence of dual tone s t i m u l i and was able to a i d i n the l o c a l i z a t i o n of the phenomena producing them. With no other n o n l i n e a r i t i e s but that of the primary auditory neuron amplitude sensor, the combination tones were s t i l l evident. This strongly suggests that t h i s i s the mechanism which produces combination tones i n the human auditory system. A p l a u s i b l e hypothesis for the production of combination tones i s suggested. The r e s u l t s predicted by the hypothesis compared favourably with the psychological data of Goldstein (15.). Because the I/O f a c i l i t i e s of a smaller computer are more suited to acoustic data, i t would be de s i r a b l e to implement the model on a computer such as the PDP-12. This can not be done with the model i n i t s present form because of the large amount of core storage required. U t i l i z i n g the s i m p l i f y i n g assumption of Flanagan (11.) i n s t o r i n g only one b a s i l a r membrane response and varying the propagation time and frequency of t h i s response to s u i t d i f f e r e n t sections, the requirements for the e n t i r e program could be at l e a s t halved. This Would enable the implementation on the PDP-12 provided that a f a s t access bulk storage device of approximately 500 K 12-bit words were a v a i l a b l e to store intermediate r e s u l t s . The PST histograms of d i f f e r e n t sections represent an amplitude v a r i a t i o n with time for a p a r t i c u l a r frequency: the c h a r a c t e r i s t i c f r e -quency of the neuron producing i t . This form lends i t s e l f to the creation of a speech spectogram of the information being relayed from the inner ear to the c e n t r a l nervous system. Such a spectogram, when compared to spectogram of the same s t i m u l i at the outer ear should give a good v i s u a l i n d i c a t i o n as to the processing and d i s t o r t i o n s of the p e r i p h e r a l auditory system. APPENDIX I Main program to c a l c u l a t e B a s i l a r Membrane Frequency Response Read: NTP.SF 7 C a l c u l a t e : Ln>Lpn>%>n,Cpn,Kn  NFP = NTP/2 + 1 , DF = SF/NTP Do m = 1,...NFP Do n = 1,...NS Z n-= j ( 2 i r F m L n ) Zpn = R p n + j ( 2 T T F m L p n - 1 / ( 2 t f F m C p n ) GNS = 0 Do n = NS-1,. Do n = ?, . NS In I. pn G , I . D l = ( Im(I p l) - j R e ( I p l ) ) K X / (2TrF m) Do n — 1, NS-1 D n = ( I m ( I p n ) - j R e ( I p n ) ) ^ / (2ifF n) p n,m = < D n " Dn+1 > / DX Write,: P n m o n D a t a C e l 1 DEFINITION OF VARIABLES FOR THE MAIN PROGRAM TO CALCULATE BASILAR MEMBRANE DISPLACEMENT NTP contains the number of time points SF contains the sampling frequency L ,L ,R ,C contain the c i r c u i t parameter values n pn pn pn NFP contains the number of frequency points DF contains the frequency i n t e r v a l Z^ contains the s e r i e s branch impedance (complex) Z contains the p a r a l l e l branch impedance (complex) G contains the forward current t r a n s f e r r a t i o (G = I L 1'/I ) (complex) n n n+1 n r E ^ contains the input voltage I contains the se r i e s branch currents (complex) I contains the p a r a l l e l branch currents (complex) D contains the b a s i l a r membrane displacement at the nth section n P^ .m contains the complex frequency domain s p a c i a l p a r t i a l d e r i v a t i v e of the b a s i l a r membrane displacement at the nth section f o r frequency F m APPENDIX II Subroutine to Add Bandlimited Noise to a Signal ENTER SIG,SIGWN,NTP,GNSTD,BLF,DT NFP = NTP/2 + 1 ; DF = l./(DT * NTP) Do n = 1....NTP  TNSIG = GNSTD * FRANDN(O.O) n Calculate STD. DEV. of noise: SDEV1 Calculate DFT of noise TNSIG ---> FNSIG F ,= -DF Do m = 1,...NFP  F = F+DF FNSIG = 0 1 X, 1 1 Calculate IDFT of FNSIG --Bandlimited noise > TNSIG Calculate STD. DEV. of noise: SDEV2 Do n = l r...NTP SIGWN = SIG„ + TN n n SIG n * SDEV1/SDEV2 54 DEFINITION OF VARIABLES FOR SUBROUTINE TO ADD BANDLIMITED GAUSSIAN NOISE TO A SIGNAL ON ENTRY: SIG (NTP) contains the input s i g n a l SIGWN (NTP) i s undefined NTP contains the number of time points i n SIG and SIGWN GNSTD contains the standard deviation of the noise BLF contains the upper frequency l i m i t of created noise ON EXIT: A l l inputs are unchanged SIGWN contains the input s i g n a l with bandlimited Gaussian noise added INTERNAL: NFP DT DF TNSIG FNSIG SDEV1 SDEV2 contains the number of frequency points contains the sampling period of SIG contains the frequency i n t e r v a l of points i n the frequency domain representation contains the time domain representation of the generated noise contains the complex frequency domain representation of the generated noise contains the standard deviation of the non-bandlimited noise contains the standard deviation of the bandlimited noise APPENDIX I I I Subroutine to Simulate a Primary Auditory Neuron c ENTER SIG,NTP,TA,TR,RTC,PT,NOP,BPA THM = EXP( -NOP THRF = FRAND(O.O)' DT / RTC ) = 0 *<TR-TA)/5.0 + T A Do n = 1,...NTP THRP = THRF Yes THRP = T A Yes NOP = NOP + 1 PT(NOP) — n' >"DT BPA - B P A n + 1 THRF = TR THRF = ( THRP - TA)*THM + T A No DEFINITION OF VARIABLES FOR SUBROUTINE TO SIMULATE A PRIMARY AUDITORY NEURON ON ENTRY: SIG (NTP) contains the input s i g n a l NTP contains the number of time points i n SIG TO contains the absolute threshold T l contains the amount by which the maximum threshold exceeds the absolute threshold RTC contains the r e f r a c t o r y time constant BPA i s undefined but must be zeroed before the f i r s t entry ON EXIT: A l l inputs are unchanged PT (NOP) contains the pulse times NOP contains the number of pulses generated i n t h i s c a l l BPA contains the number of responses i n each of NTP bins from t h i s and a l l previous c a l l s INTERNAL: THM contains the threshold m u l t i p l e THRF contains the future threshold ( i e . f o r n+l) THRP contains the present threshold (ie, f o r n) FRAND i s a c a l l to system random number generator (uniform d i s t r i b u t i o n between 0 and 1) 57 APPENDIX IV Main Program for Producing PST Histograms at a l l Sections \ Read P n > m from Data C e l l / Read or create input signal: INSIG(NTP) Read: TFOAME from Data C e l l Calculate DFT of INSIG INSIG --- INSIGF Do n = 1,.. .NS  Do m = 1,.... NFP  SIGF„ „ = INSIG * TFOAME„ * P n,m m m n Calculate IDFT of SIGF Limit SIGT by hair c e l l subprogram Do k --= 1 NREP Add noise to signal Calculate primary auditory neuron response Store or plot post stimulus time histogram data 58 DEFINITION OF VARIABLES FOR PRODUCING PST HISTOGRAMS AT ALL SECTIONS P contains the s p a t i a l p a r t i a l d e r i v a t i v e of the frequency domain n,m c . impulse response of the b a s i l a r membrane model at the nth section f o r frequency F^ INSIG (NTP) contains the input s i g n a l i n the time domain INSIGF (NTP) contains the input s i g n a l i n the frequency domain (complex) TFOAME (NTP) contains the tr a n s f e r function of the outer and middle ear SIGT contains the time domain output s i g n a l from the h a i r c e l l s SIGF contains the frequency domain output s i g n a l from the h a i r c e l l s NREP contains the number of r e p e t i t i o n s used i n formulating the PST histogram output 59 REFERENCES 1. Beddoes, M.P., Connor, D.J. and Melzak, Z.A., "Simulation of a visual perceptor network". IEEE Transactions On Bio-Medical Engineering, Vol. BME-12, #3 and 4, pp. 136-138, July-October, 1965. 2. Bekesy, G. von, "The variation of phase along the basilar membrane with sinusoidal vibrations". J. Acoust. Soc. Am., Vol. 19, p. 452, 1947. 3. Be3c6sy, G. von, "On the resonance curve and the decay period at various points on the cochlear partition". J. Acoust. Soc. Am., Vol. 21, pp. 245-254, May 1949. 4. Be3cdsy, G. von, "The vibration of the cochlear partition in anatomical preparations and in models of the inner ear". J. Acoust. Soc. Am., Vol. 21, p. 233, 1949. 5. Bekesy, G. von and Rosenblith, Walter A, "The mechanical properties of the ear" from Handbook of Experimental Psychology, Stevens, S.S., editor. New York: John Wiley and Sons, Inc., 1951, pp. 1075-1115. 6. Bekesy, G. von, "Description of some mechanical properties of the organ of corti". J. Acoust. Soc. Am. , Vol. 25, #4, pp. 770-785, July 1953. 7. Bekesy, G. von, Experiments in Hearing. New York: McGraw-Hill Book Co., Inc., 1960. 8. Cooley, J.W. and J.W. Tukey, "An algorithm for the machine calculation of complex Fourier Series". Math. Comput., Vol. 19, pp. 297-301, April 1965. 9. Elliot, D.N., Stein, L. and J.J. Harrison, "Determination of absolute-intensity threshold and frequency-difference thresholds in cats". J. Acoust. Soc. Am., Vol. 32, pp. 380-384, 1960. 60 10. Engstrbm, Hans, Harlow W. Ades and Joseph E. Hawkins, Jr., "Structure and functions of the sensory hairs of the inner ear". J. Acoust. Soc. Am., Vol. 34, No. 8, Part 2, pp. 1356-1363, September 1962. 11. Flanagan, J.L., "Computational model for basilar membrane displacement". J. Acoust. Soc. Am., Vol. 34, pp. 1370-1376, 1962. 12. Fletcher, Harvey, "On the dynamics of the cochlea". J. Acoust. Soc.  Am., Vol. 23, #6, pp. 637-645, November 1951. 13. Fletcher, Samual G., "Anatomy and physiology of the auditory system". From Audiological Assessment, Rose, Darrell E., editor. Englewood Cliffs, New Jersey: Prentice-Hall Inc., 1971. 14. Furman, G.G. and C.S. Frishkopf, "Model of neural inhibition in the mammalian cochlea". J. Acoust. Soc. Am., Vol. 36, pp. 2194-2201, 1964. 15. Goldstein, J.L. "Auditory nonlinearity", J. Acoust. Soc. Am., Vol. 41, #3, pp. 676-689, March 1967. 16. Goldstein, J.L. and Nelson Y.S. Kiang, "Neural correlates of the aural combination tone 2f^ - i.^ '• Proceedings of the IEEE, Vol. 56, #6, pp. 981-992, June 1968. 17. Goldstein, J.L. "Aural combination tones". Symposium on Frequency Analysis and Periodicity in Hearing, Driebergen, The Netherlands, June 1969. 18. Goldstein, Alland J. and J. Ryland Mundie. "Response of the primary auditory neuron to human speech". Presented to the 81st Meeting of the Acoustical Society of America, Washington, D.C, 20-23, April 1971. 19. Harmon, L.D. and R.M. Wolfe, "An a r t i f i c i a l neuron". Bell Telephone Laboratories. Technical Memorandum 57-133-24, 57-133-21, July 1, 1957. 61 20. Harmon, L.D., "Studies with ar t i f i c i a l neurons, I: properties and functions of an a r t i f i c i a l neuron". Cybernetic, Vol. 1, No. 1, p. 89, January 1961. 21. Kiang, N.Y.S., "Stimulus coding in the auditory nerve and cochlear nucleus", Acta Oto-Laryngolog, Vol. 59, pp. 186-200. 22. Kiang, N.Y.S., Watanabe, T., Thomas, E.C. and L.F. Clark, "Discharge patterns of single fibres in the cat's auditory nerve". Research  Monograph #35, Cambridge, Mass." M.I.T. Press, 1965. 23. Kiang, N.Y.S., Sachs, M.B. and W.T. Peake, "Shapes of tuning curves for single auditory-nerve fibres", J. Acoust. Soc. Am., Vol. 42, #6, pp. 1341-1342, 1967. 24. Klatt, Dennis H., "Theories of Aural Physiology". Ph.D. thesis, University of Michigan, November 1964. 25. Klatt, Dennis H. and Gordon E. Peterson, "Reexamination of a model of the cochlea". J. Acoust. Soc. Am., Vol. 40, No. 1, pp. 54-61, 1966. 26. Littler, T.S. The Physics of the Ear. Vol. 3. Oxford, England: Pergamon Press Ltd., 1965. 27. Neff, W.D. and J.E. Hind, "Auditory thresholds of the cat". J. Acoust.  Soc. Am., Vol. 27, pp. 480-483, 1955. 28. Peterson, L.C. and B.P. Bogert, "A dynamical theory of the cochlea". J. Acoust. Soc. Am., Vol. 22, No. 3, pp. 369-381, May, 1950. 29. Plomp, R. "Detectability threshold for combination tones". J. Acoust.  Soc. Am., Vol. 37, pp. 1110-1123, 1965. 30. Siebert, W.M., "Models for the dynamic behaviour of the cochlea partition". Quarterly Progress Report, #64, Research Lab of Electronics, M.I.T., Cambridge, Massachusetts, pp. 242-258, January 15, 1962. 31. Siebert, William M., "Some implications of the stochastic behaviour of primary auditory neurons". Kybernetik, June 1965. 32. Siebert, William M., "Frequency discrimination in the auditory system place or periodicity mechanisms?" Proceedings of the IEEE, Vol. 58, No. 5, May 1970. 33. Siebert, William M., "On stochastic neural models of the diffusion type". Quarterly Progress Report, No. 94, Research Laboratory of Electronics, M.I.T., Cambridge, Massachusetts. 34. Weiss, Thomas F., "A model of the peripheral auditory system". Kybernetik, Band III, Heft 4, pp. 153-175, November 1966. 35. Zwislocki, J., "Analysis of the middle-ear function, Part I: Input impedance". J. Acoust. Soc. Am., Vol. 34, No. 8, Part II, September 1962. 36. Zwislocki, J., "Theory of the acoustical action of the cochlea". J. Acoust. Soc. Am., Vol. 22, No. 6, pp. 778-784, November 1950. 37. Zwislocki, Josef, "Review of recent mathematical theories of cochlear dynamics". J. Acoust. Soc. Am., Vol. 25, No. 4, pp. 793-751, July 1953. 

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