Modeling and Simulation of the Auditory Pathway

Transcription

1 Modeling and Simulation of the Auditory Pathway INTERIM REPORT: Computational Models for the Study of Hearing and Language Impairement in Children Alok Bakshi School of Industrial Engineering Purdue University, West Lafayette, IN Aditya P. Mathur Department of Computer Science Purdue University, West Lafayette, IN January 20, 2007 This work is supported by NSF Award

2 CONTENTS 2 Interim Report: Modeling the Auditory Pathway Contents Abstract 4 1 Introduction Progress summary Model: granular or aggregate? Limitations of the cellular approach to modeling Background Auditory pathway Ear External ear Middle ear Inner ear: the cochlea Auditory neuron Cochlear nucleus Primary-like responses Onset responses Chopper responses Pauser and buildup responses Inhibitory responses Superior olivary complex Medial superior olive Lateral superior olive Inferior colliculus Medial geniculate body Encoding of sound characteristics Modeling Hodgkin-Huxley model Models with stochastic Na + channels Auditory neurons Cochlear nucleus Bushy cell Fusiform cell Octopus cell Pyramidal cell Stellate cell Progress Octopus cell model Bushy cell model Fusiform cell model

3 CONTENTS 3 Interim Report: Modeling the Auditory Pathway 5 Related work Auditory Neuron Cochlear Nucleus Bushy Cell Octopus Cell Future Work 36 Acknowledgements 36 References 37

4 1. Introduction 4 Interim Report: Modeling the Auditory Pathway Abstract The objective of the work reported here is to develop a detailed cell-level computational model of the human auditory pathway is under development. The model, once fully developed and validated against experimental data, will assist in the study of neural plasticity observed in the central auditory pathway as a consequence of auditory training in children with learning and attention deficit disorders. Researchers have quantified the effect of auditory training by the brainstem evoked auditory potential, which the proposed complete computational model is expected to reproduce. Specifically, a complete and validated computational model will be used as a tool to assist in understanding the effect of (a) non-intrusive treatments in children with learning disabilities, and (b) the fault tolerance of the pathway to time varying defects in its cellular substance and structure. This report summarizes the progress made towards the stated objective. Keywords: Auditory pathway, auditory training, computational model, dyslexia, brainstem evoked auditory potential, learning disorders in children. 1 Introduction The objective of our exploratory research is to construct and validate a computational model that (a) mimics experimental results of auditory processing tasks in children diagnosed with auditory processing disorders and learning disabilities and (b) experiment with the validated model to understand the impact of treatments on children with auditory disorders and learning disabilities. Rather than assume any apriori knowledge about the nature and/or location of the malfunction in auditory processing in these children, we will explicitly perform parametric manipulation of models of different parts of the auditory system and compare the model response to that observed in experiments. Parameters under our control relate to individual cells and their interconnections (see Section 3). Note that the long term focus of our research is on children who suffer from various disorders related to the auditory pathway. While one set of researchers study the cause and effect of such disorders [3, 17, 21, 22], we propose to use the observations reported by these researchers to validate our computational models and in turn use the validated models to ask and answer what if questions. Our belief is that the proposed approach to modeling and validation against the observed behavior, e.g. the brainstem evoked auditory potential, will lead to a better understanding of what defects in the auditory pathway lead to dyslexic behavior and how treatments, that lead to changes in the cellular parameters and structure, might bring the behavior back to normality. 1.1 Progress summary Towards our objective, we have completed the following tasks: (a) study the anatomy and physiological nature of various stages in the auditory pathway, (b) select, implement and validate models of the auditory neurons, bushy cell, fusiform cell, and the octopus cell. Several researchers have focused on the development of models of various types of cells found at different stages in the auditory pathway. Rather than build our own model for each cell, we decided to study and then select one model of each cell type, when available.

5 1.2. Model: granular or aggregate? 5 Interim Report: Modeling the Auditory Pathway Thus so far our contribution has been an integration of the different models in the cochlear nucleus with that of the auditory neurons. We implement in MATLAB each model, test it, and then integrate it with the models for other cells in the auditory pathway. We plan to follow this approach until we have an integrated cellular model of the entire complex. 1.2 Model: granular or aggregate? A key question in modeling a complex system such as the auditory pathway, is How granular should the model be? The following paragraphs dwell on this question. The human auditory pathway leading from the cochlea to the auditory complex is a marvelously naturally engineered subsystem in the brain. Researchers have been studying for decades its anatomy and physiological behavior. A variety of computational models have also been developed with the goals of reproducing various aspects of the behavior of the auditory pathway. For example, Travis has reported a computational model of one pathway in the cat subcortical auditory system [20]. Travis has argued that While traditional neural networks have made inroads to understanding cognitive functions, more realism (in the form of structural, dynamic and connectivity constraints) is likely required to explain processes such as vision and audition. Travis models a basilar membrane nerve fiber unit and the subcortical auditory nuclei. His model is stochastic in that inter-neuron connectivity is determined randomly, while accounting for physiologic constraints. Components of the pathway, e.g the Medial Geniculate Body ventral division, are modeled using a set of coupled first order differential equations. While Travis model is detailed in that it treats each neuron as a distinct entity, aggregate behavioral models have also been proposed. For example, Wrigley and Brown have proposal a computational model of auditory selective attention [23]. Their model is motivated by the finding that attention plays a key role in the formation of auditory streams. The model itself consists of a network of neural oscillators where the attentional interest is modeled as a Gaussian distribution in frequency. Works by Travis, and Wrigley and Brown, are similar in that both focus on modeling portions of the auditory pathway. They are quite different, however, in their granularity. While Travis model is cellular and hence highly granular, Wrigley and Brown s model is aggregate and phenomenological. Our approach to modeling the auditory pathway is similar to that proposed and implemented by Travis. We are interested in constructing a detailed cellular model of the entire auditory pathway from the output of the cochlea to the auditory cortex including all the stages described in Section 2. We believe that a detailed cellular level model will aid in reproducing the observed behavior at different stages of the pathway thereby allowing us to study the impact of minor and major faults in the cellular substance and structure. 1.3 Limitations of the cellular approach to modeling We face two major problems when using the modeling approach described above. One is related to the computational effort required to solve the cellular model and the other is the lack of, or difficulty in finding, the cell inter-connnection data, especially along the descending auditory pathway. So far our work has focused on modeling individual cells and connecting the models, also for single cells, across the cochlea and the cochlear nucleus. In the next phase of our research we plan to replicate cell models and incorporate intercellular connections. The computational

6 2. Background 6 Interim Report: Modeling the Auditory Pathway requirements of the resulting interconnections of cellular models will likely force us to move the computation to a supercomputer. Whenever we fail to find anatomical details of intercellular connections, we will explore using a Travis-like approach of (intelligently) establishing random connections and select and validate a suitable connection structure against the physiologically observed behavior. 2 Background This section covers background material necessary for an understanding of the computational models described subsequently. The material in this section is heavily based on two books: The Central Auditory System by Ehret et. al. [2] and An Introduction to the Physiology of Hearing by Pickles [14]. Numerous models of the individual neurons exist. These are shown to have specific functions in the auditory pathway, such as decoding of pitch and sound localizaton information. Nevertheless, we lack a complete model of the auditory pathway, from which we can understand exactly how the information is decoded and relayed to the brain. There are models which attempt to simulate the entire nuclei, without going down to the level of individual neurons. Though such models capture few properties of the auditory pathway without much computation effort, we do need a detailed computational model of the auditory pathway. The detailed model can be used for computing a correlation between defect types in the nuclei and malfunction of the auditory pathway. Such correlation in turn will likely be useful in the diagnosis and treatment of auditory disorders in children. Next we explain the pathway as a whole and the individual nuclei (group of nerve cell bodies), as well as nerve cells which process and relay the auditory information in the pathway: 2.1 Auditory pathway The central auditory pathway transfers auditory signals from the ear to the auditory cortex. The auditory pathway can be categorized broadly into the following two parts. Ascending Auditory Pathway Auditory information received through the ear is sent to the auditory cortex along this pathway that contains the intermediate nuclei. Descending Auditory Pathway This pathway is responsible for the feedback signals that emanate from higher nuclei (group of nerve cell bodies) back to the lower nuclei. The feedback mechanism from higher to lower nuclei is not well understood. Hence currently we model only the ascending pathway; the descending pathway will be ignored in the current phase of our research. The impact of this simplification will be reviewed after the computational model is built and validated. As shown in Figure 1, the ascending auditory pathway consists of a series of nuclei connected by fiber tracts (axons of the nerve cells). There are specialized cells in the nuclei which process specific auditory information encoded in the form of nerve impulses, while few cells in the nuclei simply relay the information to the higher nuclei without any processing. The sound, which we hear is the travelling pressure wave in the air, which is picked up by the outer ear. The outer ear transfers the vibration to tympanic membrane (eardrum). The middle ear transmits vibrations from tympanic membrane to oval windows in cochlea by three bones

7 2.1. Auditory pathway 7 Interim Report: Modeling the Auditory Pathway Figure 1: Ascending auditory pathway. known as ossicles. These vibrations cause displacement of basilar membrane resulting from the movement of cochlear fluids. The high frequency and low frequency tones generate vibrations, which peak at the base and apex of the cochlea respectively. The inner and outer hair cells fire because of movement of basilar membrane and their response is carried by auditory neurons to the auditory pathway. The auditory neurons provide input to brainstem nuclei known as the cochlear nucleus. The cochlear nucleus encodes the input data in various forms and transfers the information to contralateral superior olivary complex and ipsilateral lateral lemniscus. From there the neural response goes to higher centers, namely, the inferior colliculus, medial geniculate body, and ultimately to auditory cortex. The brainstem nuclei in the auditory pathway decode information, i.e., sound location, while the auditory cortex interprets its meaning. Brainstem auditory evoked potential (BAEP) is the electrical potential recorded from the scalp after presenting the stimulus. The abnormal brainstem auditory evoked potential (BAEP) reflects the faults in either functioning of the ear or in the various nuclei present in the auditory pathway [9]. The BAEP is the resultant of the response generated by all the nuclei present in auditory pathway. The peaks labeled I through V in Figure 2 are generated by the auditory nerve fibers and nuclei till lateral lemniscus, which come in the ascending auditory pathway [11]. The effect of each nuclei in auditory pathway on the BAEP is shown in Figure 2.

8 2.2. Ear 8 Interim Report: Modeling the Auditory Pathway Figure 2: Brainstem auditory evoked potential. 2.2 Ear The ear converts the sound waves, which it receives in the form of waves from the external ear, into the neural impulses of the AN fibers in innner ear. We will now describe briefly the three significant parts of the human ear External ear The human pinna amplifies high frequency sound depending on its angle of incidence. The brain interprets these changes as direction of the sound source. Thus it serves to localize the auditory signal Middle ear Vibrations are transmitted effectively into the fluid of the cochlea from the eardrum by the middle ear. It requires turning a large amplitude vibration in air into smaller amplitude vibration without a significant loss of energy. The large area of the eardrum, as well as the lever action of three small bones, helps in achieving the reduction of amplitude Inner ear: the cochlea The cochlea is filled with liquid that actuates in response to the vibration of the middle ear. There are specialized cells called hair cells in the cochlea, that transform the mechanical movement of the fluid into the electrical signals of auditory neurons via neurotransmitters. The auditory neurons have cell bodies in the spiral ganglion and carry the impules from the cochlea to the higher nuclei in the auditory pathway.

9 2.3. Auditory neuron 9 Interim Report: Modeling the Auditory Pathway 2.3 Auditory neuron The auditory neurons receive input from hair cells and carry it to the cochlear nucleus. The soma of AN fibers lie in the Spiral Ganglion. The properties of AN fibers can be summarized as follows. 1. Most AN fibers have a moderate spontaneous activity that appears to be abolished by hair cell destruction. AN fibers are characterized in the following three categories based on SA (Spontaneous Activity): High spontaneous activity Medium spontaneous activity Low spontaneous activity 2. There is a single excitatory best frequency to which the fibers respond at the lowest intensity, which is called characteristic frequency of that fiber (see Figure 3).The fibers have a simple V- or U-shaped excitatory tuning curve surrounding the characteristic frequency. 3. The fibers display a monotonic increase in their respective firing rate up to a maximal level as the intensity of the preferred tone is increased and then show a constant or slightly reduced firing rate. Figure 3: Tuning curve of fibres. 2.4 Cochlear nucleus The cochlear nucleus is the first mid-brain nucleus of the ascending auditory pathway. It is morphologically differentiated into following three sub-regions: Anteroventral Cochlear Nucleus (AVCN) Posteroventral Cochlear Nucleus (PVCN)

10 2.4. Cochlear nucleus 10 Interim Report: Modeling the Auditory Pathway Dorsal Cochlear Nucleus (DCN) AVCN and PVCN are also collectively known as Ventral Cochlear Nucleus (VCN). These three different divisions of the cochlear nucleus have different response properties as well. The AVCN neurons have similar response properties as that of the AN fibers and therefore it acts as a simple relay for higher nuclei. While DCN neurons have complex response properties, and they project directly to the lateral lemniscus and inferior colliculus bypassing the superior olivary complex altogether. The properties of PVCN neurons intermediate of the properties of AVCN and DCN neurons. Figure 4: Tonotopic organization (from Ryugo and Parks, 2003). Each AN fiber bifurcates while entering the cochlear nucleus sending one branch to AVCN and the other branch to PVCN and DCN. Each subregion of cochlear nucleus has orderly arrangement of afferent AN fibers according to their characteristic frequency and thus preserve tonotopicity. For example the fibers originating from the cochlear base, which encode higher frequency projects to DCN. Whereas the fibers from cochlear apex, which encode low frequencies projects to the VCN. The fibers having intermediate frequencies end up in between these two regions as shown in Figure 4. Both type I and type II AN fibers project to the cochlear nucleus thus constituting the two parallel pathways. The first pathway, consisting of type I AN fibers, originate from inner hair cell of the cochlea and project to the central part of cochlear nucleus. The second pathway, consisting of type II AN fibers, originate from outer hair cells of the cochlea and project to the peripheral part of cochlear nucleus. Thus type I AN fibers transfer the main auditory input to cochlear nucleus, while type II AN fibers provide modulation for the acoustic data. The cells of the cochlear nucleus can be differentiated morphologically or by the reponse pattern. The cochlear nucleus neurons can be classified into different categories based on the

11 2.4. Cochlear nucleus 11 Interim Report: Modeling the Auditory Pathway response to the short tone bursts delivered just above the threshold and at the characterisic frequency of the neuron. The following response types are observed in cochlear nucleus: Primary-like Responses Onset Responses Chopper Responses Pauser and Buildup Responses Inhibitory Responses Phase Locked Responses Primary-like responses Such response pattern is observed throughout in ventral cochlear nucleus, in particular in AVCN area. The spherical bushy cells of AVCN display this kind of response. As the name suggests, the response is similar to that of auditory nerve fibers, with a peak at the onset and then gradual decrease to a lower response for tones. There are no inhibitory sidebands in response and it resembles the tuning curve for AN fibers. The cells showing primary-like response receive secure synaptic input, with short synaptic delay and hence relay the information from AN fibers to higher auditory nuclei. Primary-like responses divide into two subtypes: Primary-like response Primary-like-notch response The shape of PSTH (post stimulus time histogram) is similar for both these responses, but in notched response there is short disruption of firing after the initial peak of maximum firing, which is followed by the return to normalcy in firing. Spherical bushy cells generate primary like response while globular bushy cells show notched response. The primary like response is observed in anterior of AVCN while the notched response is observed more in posterior AVCN Onset responses Onset response is characterized by the sharp peak in PSTH at the beginning of tone bursts and a low level of sustained acivity thereafter. This response is oberved in all over the cochlear nucleus, but more frequently in ventral cochlear nucleus. The ocopus cell area in VCN shows only onset response and thus the octopus cell are known to have such response pattern. The response is thought to be generated by the presence of excitatory input and then a delayed inhibitory input. The octopus cell area produces only onset responses and the area consists almost entirely of octopus cells. Thus octopus cells in this region generate onset responses. It might be supposed that there is an excitatory input and then a delayed inhibitory input for these cells. This response can be further categorized into following responses: OI Onset responses which shows transient activity are called OI. This response shows a single action potential per tone burst, which is given just above threshold at the characteristic frequency.

12 2.4. Cochlear nucleus 12 Interim Report: Modeling the Auditory Pathway OL Onset responses which show sustained activity after initial peak are called OL. OC The multipolar cells of posteroventral cochlear nucleus generate such response. This response is characterized by the initial 2 4 peaks, with decreasing amplitudes at the onset. OG This response is found in few cells of dorsal cochlear nucleus. It is characterized by the graded decreasing of firing after the initial peak of response. The onset response encodes the temporal coding of onset of stimulus which is necessary for sound localization by the comparison of arrival-time differences of sound at the two ears Chopper responses The chopper responses is observed throughout the cochlear nucleus, predominantly in the posteroventral cochlear nucleus and the polymorphic cell layer of the dorsal cochlear nucleus. Chopper response is characterized by the repetitive firing during a sustained tone burst with a rate independent of the period of the stimulus waveform. The PSTH, therefore has number of peaks for this response. Chopper response can be further differentiated into following three types: Responses having long intervals between peaks. This response is shown mainly by the cells in the dorsal cochlear nucleus. Responses having short intervals in between the peaks, which also have significant decrement in peak amplitude with the duration of the stimulus. Responses which shows chopper characteristic only during the onset of stimulation. The chopper response is shown by the stellate cells present in the cochlear nucleus. As the chopper units lose the temporal information, so a likely role for such units is in the encoding of the intensity of stimulus Pauser and buildup responses Pauser response is characterized by the initial onset of the response, then complete disruption of discharge for some time and a gradual resumption to normal discharge. Buildup response are similar to pausup response except that these units don t show the onset of response. The pyramidal cells present in the cochlear nucleus show these kind of responses along with the polymorphic cell layer of the dorsal cochlear nucleus. Oftenly the cells displaying such response show other kinds of response if input parameters are changed. This is because of the interneuronal circuits present in the DCN, which may change response of the cell from pauser to buildup response based on the intensity of the sound Inhibitory responses In this response, there is only inhibition in the discharge rate whenever a stimulus is provided, and this inhibition continues as long as the stimulus is provided. So far, this kind of response is not attached to any type of cell in cochlear nucleus.

13 2.5. Superior olivary complex 13 Interim Report: Modeling the Auditory Pathway 2.5 Superior olivary complex Superior olivary complex is the first nucleus along the auditory pathway that receives information from both the ipsilateral and contralateral side. This nucleus plays an important role in the localization of sound. The localization is done based on the following two cues: Interaural intensity difference (IID) This is the difference in intensity of sound perceived by the ear due to the location of the sound. Interaural time difference (ITD) This is the difference in arrival time at both ears when the sound source is not equidistant from the ears. This nucleus is divided into the following three sub-regions: Medial superior olive Lateral superior olive Medial nucleus of the trapezoid body The tonotopicity is preserved in all of these three nuclei.medial nucleus of the trapezoid body (MNTB) provides the LSO and MSO inhibitory inputs, which represent the contralateral ear, providing thus the cues for sound localization Medial superior olive The medial superior olive helps in sound localization by comparing difference in the arrival time of sounds of both the ears. The cells in MSO are most responsive to low frequency sound. It projects to ipsilateral inferior colliculus ipsilaterally, while MSO projects bilaterally to the inferior colliculus. It receives fibers mainly of the low frequency Lateral superior olive The lateral superior olive helps in sound localization by comparing difference in the intensity of sound on both the ears. It receives direct high frequency inputs from the ipsilateral AVCN(spherical Bushy cells) and indirectly from the globular cells of the contralateral ventral cochlear nucleus, which pass through the nucleus of trapezoid body. The LSO neurons are excited by the direct connections i.e. from ipsilateral AVCN while the indirect input is inhibitory. 2.6 Inferior colliculus The inferior colliculus is a nucleus situated in midbrain, which is present in both ascending as well as descending auditory pathway. It consists of following subregions: Central Nucleus Pericentral Nucle External Nucleus

14 2.7. Medial geniculate body 14 Interim Report: Modeling the Auditory Pathway Central Nucleus, which is a principal subregion, consists of principal neurons, characterized by flat dendritic tree and multipolar neurons, which are interneurons and thus modulate the local circuit. Central Nucleus receives contralateral ascending inputs from cochlear nucleus and superior olive complex, while it receives ipsilateral ascending inputs from lateral superior olive and medial superior olive. The lateral superior olive projects high CF neurons to contralteral inferior colliculus and low CF inhibitory input to ipsilateral olivary complex. 2.7 Medial geniculate body The medial geniculate body contains thalamic auditory relay, which receives inputs from inferior colliculus and projects to the auditory cortex. There are three subregions for MGB, which are connected to the inferior colliculus from three parallel pathways. The subregions of MGB are: ventral MGB It receives projection from central nucleus of inferior colliculus, which form the pathway named tonotopic pathway. The pathway is named so because of the spatial arrangement of neurons according to their frequency. dorsal MGB The projection from inferior colliculus to this sub-region lack tonotopic organization, so they form the pathway named diffused pathway. medial MGB It receives projections from lateral part of inferior colliculus, which form polysensory pathway. The pathway is named so because lateral part of inferior colliculus receives projection outside from the auditory pathway. There exists some overlap as well as interconnections between these three parallel pathways. 2.8 Encoding of sound characteristics The mechanism by which basic properties of the sound, for example pitch and localization, is given below in brief: Earlier two seperate theories were proposed for the mechanism by which frequency of the auditory information is decoded. The current theory is somewhat a combination of the following two: Frequency theory The basilar membrane vibrates in sync with a sound causing the auditory nerve axons to produce action potentials with the same frequency. Place theory Each area of the basilar membrane is tuned to a specific frequency and vibrates if that frequency is present. The current thoery postulates the following encoding of frequency: 1. Frequency of action potentials in the auditory nerve directly correlates with the frequency of the sound in case of low frequency sound. 2. Volleys of responses across many receptors can lead to the encoding of sounds of frequency up to 5000 Hz in the whole auditory nerve even though no individual axon can fire with that frequency.

15 3. Modeling 15 Interim Report: Modeling the Auditory Pathway 3. Higher frequency sounds are characterized by area of greatest response along the basilar membrane due to location of peaks of the travelling wave in the basilar membrane. The sound is localized by following two different mechanisms: 1. Loudness difference between ears is used to detect the sound location for high frequency sound. 2. Differences in phase between ears is used to detect the sound location for low frequency sound. 3 Modeling Next discuss models already built and verified for the individual neurons by other researchers. In the first two subsections, we discuss the generic models, which we use whenever the specific model of a neuron is not available. The first generic model is the Hodgkin Huxley model, which simulates the neuron through voltage gated ion channels, while the others model the cell behavior using stochastic ion channels. 3.1 Hodgkin-Huxley model The model [5] computes different types of ion current, for example Na +, K +, and a leakage current consisting mainly of Cl ions. Specific voltage dependent ion channels, one for Na + and another one for K +, control the flow of those ions through the cell membrane. The leak current takes care of other channel types which are not described explicitly. The circuit diagram for the model is shown in the Figure 5. Here is the brief description of the variables and constants which are present in the differential equation setup for the model: I The total ionic current across the membrane m The probability that 1 of the 3 required activation particles has contributed to the activation of the Na gate (m 3 : the probability that all 3 activation particles have produced an open channel) h The probability that the 1 inactivation particle has not caused the Na gate to close G N a Maximum possible Sodium Conductance (about 120 mohms 1 /cm 2 ) E Total membrane potential (about -60 mv) E N a Na membrane potential (about 55 mv) n The probability that 1 of 4 activation particles has influenced the state of the K gate. G K Maximum possible Potassium Conductance (about 36 mohms 1 /cm 2 ) E K K membrane potential (about -72 mv) G L Maximum possible Leakage Conductance (about.3 mohms 1 /cm 2 )

16 3.2. Models with stochastic Na + channels 16 Interim Report: Modeling the Auditory Pathway Figure 5: Hodgkin Huxley Model [5]. E L Leakage membrane potential (about -50 mv) The voltage of the membrane depends on the ion channels by following relation C m de dt = m3 hg Na (E E Na ) + n 4 G K (E E K ) + G L (E E L ) The probability associated with each gate i at any given instant, depends in turn on the membrane potential at that time. So here a I s and b i s depend on the membrane potential and the ion gate probability satisfied following differential equations: dm dt = a m(1 m) b m m dn dt = a n(1 n) b n n dh dt = a h(1 h) b h h 3.2 Models with stochastic Na + channels Nonlinear deterministic models such as Hodgkin Huxley model fails to take account of the spike timing and threshold fluctuations. The fluctuations of Na + current accounts for the stochastic nature of spike timing and threshold. The stochastic nature of Na + ions can be modelled as continuous time, discrete state Markov jumping processes. The Na + channel has three activating gates(denoted by m) with four different states, and one inactivating gate(denoted by h)with two distinct states according to the Hodgkin Huxly model. Therefore, the resultant markov process has 4 2 = 8 states, and 20 transition states as shown in Figure 6. Such models are computationally intensive owing to their biophysical complexity. The algorithms for stochastic simulation can be grouped into 1. Approximation algorithms of the differential equation using Langevin s equation (F algorithm)

17 3.3. Auditory neurons 17 Interim Report: Modeling the Auditory Pathway 2. Exact algorithms, which can be further subdivided into two categories: Channel state tracking (CST) algorithms This algorithm tracks the state of each channel and superimposes individual channel currents corresponding to the states in order to generate sodium channel current fluctuations. This algorithm is computationally very intensive, and is implemented in SD algorithm, R algorithm. Channel number tracking (CNT) algorithms This algorithm (CW algorithm) tracks the number of channels in each state, assuming that all the channels are independent and memoryless. It has much greater efficiency compared to the other one. Figure 6: Kinetic scheme for stochastic models. Comparing the algorithms [4], we find that the CNT algorithm (CW) is computationally most efficient and robust. 3.3 Auditory neurons A phenomenological model of the Auditory Neurons is implemented [24], which describes the responses of high SA auditory neuron fibers, including several non linear properties. The general scheme of AN model implementation is shown in the Figure 7. The model takes the instantaneous pressure waveform of the stimulus, without taking into account the effect of external and middle ear. As shown in the figure, non linear filtering section of the model are the signal path and the feedforward control path, so that the model gives same type of non linear response as observed in the experiment. Several AN response properties were not included in the model. For example, the model does not incorporate the tails of tuning curve, the effects of efferents on the rate and timing of AN discharges, and low and medium spontaneous activity. The implication of these limitations will be assessed after the completion of model. 3.4 Cochlear nucleus The cochlear nucleus is the first site of the neuronal processing of the newly converted digita data from the inner ear. The information is brought via the auditory nerve. The lower frequency axons innervats the ventral portions of the dorsal cochlear nucleus and the ventrolateral portions of the anteroventral cochlear nucleus. In contrast, the axons from the higher frequency organ

18 3.4. Cochlear nucleus 18 Interim Report: Modeling the Auditory Pathway Figure 7: Block diagram of the AN Model [24]. of corti hair cells project to the dorsal portion of the anteroventral cochlear nucleus and the uppermost dorsal portions of the dorsal cochlear nucleus. The mid frequency projections end up in between the two extremes, in this way the frequency spectrum is preserved. The different types of cells present in the cochlear nucleus are listed in Table 1. There are three major projections from the cochlear nucleus. Through the medulla, one projection bifurcates, and shoots to the contralateral the superior olivary complex via the trapezoid body, whilst the other half shoots to the ipsilateral SOC( Superior Olive Complex). Another projection rises above the medulla into the pons where it meets the nucleus of the lateral lemniscus Bushy cell Bushy cells of anteroventral cochlear nucleus (AVCN) receive inputs from AN fibers and phace lock to the characteristic frequency thus preserving the temporal information. Bushy cells pass information to the superior olivary complex (SOC), which is the center for sound localization. For sound localization, inter-aural time difference is used, the accuracy of which is ensured by Bushy cells because the temporal information related to input waveform is preserved by it.

19 3.4. Cochlear nucleus 19 Interim Report: Modeling the Auditory Pathway Table 1: Types of cells in the Cochlear Nucleus. Cell Origin Location Projection Activity Temporal Receptive Types Patterns Fields Spherical AVCN Rostral LSO(ipsi) Excitatory Primary-like Type I Bushy MSO(bil) VNTB(contra) LNTB(bil) VNLL(contra) Globular AVCN Caudal MNTB(contra) Excitatory Primary-like Type I Bushy VNTB(contra) and Periolivary N. Primary-like (bil) notch Stellate\ PVCN Anterior Periolivary N. Inhibitory Chopper Types I, Multipolar PVCN (bil.) and III, I/III AVCN Excitatory O C Posterior VNLL(contra) PVCN IC(contra) Octopus PVCN Anterior Periolivary N. Excitatory O I Types PVCN (bil.) and I/III, IV VNLL(contra) O L IC(contra) Stellate DCN Molecular and Pyramidal Cells Inhibitory Chopper Types III Pyramidal Cartwheel Cells and I/III, II cell layers Stellate Cells O C Pyramidal DCN Pyramidal VNLL and IC Excitatory Buildup-Pauser Types III, cell layer (contra) and Chopper IV Here we have taken the model of Bushy cell [16] which successfully simulates the behavior of Spherical Bushy Cells. The model consists of a single compartment including three voltage sensitive ion channels. The model is adendritic and anaxonal, in which only the soma of Bushy cell is modeled. The membrane conductances in the soma are: G B G K Slow low-threshold potassium conductance Fast high-threshold potassium conductance G L Passive leakage conductance G I G E Inhibitory synaptic conductance Excitatory synaptic conductance The cell potential V is described by the following equation: C S dv dt + G B(V E K ) + G K (V E K ) + G Na (V E Na ) + G L (V E L ) + G I (V E I ) + G E (V E E ) = I ext

20 3.4. Cochlear nucleus 20 Interim Report: Modeling the Auditory Pathway Table 2: Values of model parameters at temperature 30 o C. C S = 23 pfd G L = 5.2 ns G B = 86.6 ns G K = ns G Na = ns E E = 10 mv E L = 2.8 mv E I = 66.5 mv E K = 77 mv E Na = 55 mv Here E Na, E K, E I, and E E are reversal potentials for the corresponding ions, while C S is the membrane capacitance and G L is leakage potential, the values of whom are given in the Table 2. Because all the rate constants wre defined for 22 o C, So Q 0 factor is used, which scales them back to normal body temp(38 o C). (T 22)/10 T f (Q 10 ) = Q 10 The three membrane conductances G B, G K, and G Na are modeled with activation variables w, n, and m, and inactivation variable h, which themselves depend on membrane potential and time through α and β α w = T f (3) 1 + exp { (V + 33)/13.1} β w = T f (3) exp { (V + 30)/30.3} α n = T f (3) (V + 9) 1 exp {1 (V + 9)/12} 6 T f (3) β n = 6 T f (3) exp { (V + 144)/30} exp (V + 62) α m = 0.36 T f (3) (V + 49) 1 exp { (V + 49)/3} β m = 0.4 T f (3) (V + 58) 1 exp {(V + 58)/20} α h = 2.4 T f (3) 1 + exp { (V + 68)/3} T f (10) 1 + exp { V } 3.6 T f (3) β h = 1 + exp { (V + 21)/10}

21 3.4. Cochlear nucleus 21 Interim Report: Modeling the Auditory Pathway Auditory Neuron inputs are modeled by excitatory synaptic conductance G E. If spikes arrives at the time t n the conductance reaches its peak value of A E at time t n +t p. As suggested by the paper [16], t p was chosen as constant 0.5 ms. G B = G B w G K = G K n G Na = G Na m 2 h dη dt = α η(1 η) + β η η for η = w, n, m, and h G E = A E t t n t p exp { 1 t t } n t p for t > t n Fusiform cell The fusiform cells in Dorsal Cochlear Nucleus (DCN) are responsible for the pauser and buildup response. At present, the synaptic organization and intrinsic membrane properties of these cells is not known, so a Hodgkin-Huxley based model [8] is taken which reproduce the response pattern of cell without taking into account the calcium based conductances. As shown in the Figure 8 the cell is modeled by an electrical network where current source represents the synapse input signal, and each of the branch represents nonlinear conductance. The conductances are of following types: Delayed rectifier potassium conductance Inward/anomalous rectifier potassium conductance A transient outward A type potassium conductance Transient, rapidly inactivating sodium conductance Persistent, noninactivating sodium conductance Figure 8: A model of the fusiform cell [8].

22 3.4. Cochlear nucleus 22 Interim Report: Modeling the Auditory Pathway Table 3: Parameter values for the Fusiform Cell model. E ion g max τ A τ B d A d B g ion,type (mv) (ns) x y (ms) (ms) c A c B (mv) (mv) g K,DR g K,DR g K,A g Na,T g Na,P g L The potential of soma is given by the equation below, where I s is the input current for the model and E ion represents the equilibrium potential for corresponding ion. Also g ion,type represents the conductance for a particluar ion of a particular type. de(t) dt = 1 { [E K E(t)][g K,IR (t) + g K,A (t) + g K,DR (t)] C + [E Na E(t)][g Na,P (t) + g Na,T (t)] } + [E leak E(t)]g leak + I s (t) The conductances in the model, depend on the voltage by following equations: g K,DR (E, t) = g K,DR,max A x K,DR(E, t)b y K,DR (E, t) da K,DR (E, t) dt db K,DR (E, t) dt = A K,DR (E, ) = = B K,DR (E, ) = 1 τ AK,DR [A K,DR (E, ) A K,DR (E, t)] exp [c AK,DR (d E AK,DR )] 1 τ BK,DR [B K,DR (E, ) B K,DR (E, t)] exp [c BK,DR (d BK,DR E)] Here τ N denotes the time constant for N, while denotes the steady state value for the corresponding parameter. A ion,type and B ion,type denote the activation and inactivation component for the corresponding conductances. c N s and d Ns give slope and position for the N versus potential graph, where N represents the activation/inactivation component of the conductance. The parameter values for the model are given in the Table 3. The membrance capacitance C was chosen as 210 pf, while resting potential was taken mv Octopus cell Octopus Cell is found at the caudal and medial region of the PVCN. These cells are characterized by spiking activity which happens precisely at the tone bursts over broad frequency range.

23 3.4. Cochlear nucleus 23 Interim Report: Modeling the Auditory Pathway This cell has been associated with O I and O L response, however it has been demonstrated that Octopus cells are not the only cells which exhibit these responses. Here we have implemented a computational model of the Octopus Cell [10]. It is a compartmental model in which soma, axon and dendrites are represented as compartments, where each compartment is equipotential. Following simplifications were made while implementing the model. Following Rall s 3/2 power law, four dendrites were represented by single cylinder. Active axon was lumped into the soma. The single cylinder(representing four dendrites) was modeled by 15 compartments while the soma was modeled by two cylinders Dendrites were assumed to lack any active ion channels. As shown in the Figure 9, each compartment of dendrite receives AN of the particular characteristic frequency shown in the figure in khz. The one compartment of soma receives four AN, while the other contains all the active ion channels. Figure 9: A model of the octopus cell [10]. The constants used in the model are summarized in Table 4. The follwoing equation gives the dendritic potential for each compartment j j {1 15} c m dv dj dt = g L (V dj E r ) g syn (t)(v dj E syn ) + 1 r a (V d(j+1) V dj ) 1 r a (V dj V d(j 1) ) The equation written below gives the potential of passive somatic compartment c m dv s1 dt { = g L (V s1 E r ) (V s1 E syn ) 4 g syn (t) j} j=1 + 1 r a (V s2 V s1 ) 1 r a (V s1 V d1 )

24 3.4. Cochlear nucleus 24 Interim Report: Modeling the Auditory Pathway Table 4: Octopus Cell Parameters Description Symbol Value Dendrite Number of compartments 15 Diameter d d 17.6 µm Length l d 238 µm Peak synaptic conductance density g syn 1.2 ms/cm 2 Soma Number of compartments 2 Diameter d s 25 µm Length l s 25 µm Peak K + conductance density g K ms/cm 2 Peak Na + conductance density g Na 60.0 ms/cm 2 Specific membrane resistivity R m 0.5 kωcm 2 Specific axial resistivity R a 150Ω cm Specific membrane capacitance C m 1 µf/cm 2 Leakage conductance reversal potential E r 65 mv Synaptic input reversal potential E syn 45 mv Na + reversal potential E Na 45 mv K + reversal potential E K 90 mv Maximum synaptic conductance density g syn 0.3 ms/cm 2 Compartment membrane resistance r m R m /πdl kω Compartment axial resistance r a 4R a l/πd 2 kω Compartment capacitance c m C m πdl µf Dendritic characteristic length λ d 383 µm Somatic characteristic length λ s 456 µm The equation written below gives the potential of active somatic compartment c m dv s2 dt = g L (V s2 E r ) g Na (t)(v s2 E Na ) g K (t)(v s2 E K ) 1 r a (V s2 V s1 ) The voltage dependent conductance are modeled by following differential equations: g Na (V m, t) = A c g Na m 3 h g K (V m, t) = A c g K n 4 dm dt = α m(1 m) β m m dh dt = α h(1 h) β h h dn dt = α n(1 n) β n n

25 3.4. Cochlear nucleus 25 Interim Report: Modeling the Auditory Pathway α m (V m ) = 0.64(V m E r 13) e (Vm Er 13) 4 1 α n (V m ) = 0.064(V m E r 15) e (Vm Er 15) 5 1 α h (V m ) = 0.256e (Vm Er 17) 18 β h (V m ) = β m (V m ) = 0.56(V m E r 40) e Vm Er β n (V m ) = 1.0e (Vm Er 10) e (Vm Er 40) With one input spike at t = 0, the synaptic conductance varies with time by following differential equation: t g syn (t) = A c g syn tp e t tp If other spikes also occur in between, then the total conductance was taken as simply the sum of individual conductances for each spike Pyramidal cell The pyramidal cell model [7] is represented by a single compartment, in which, membrane capacitance (C m ) is connected in parallel with voltage and time varying ionic conductances. The membrane voltage depends on the voltage dependent ionic conductances and leakage conductance by following relation: dv m dt = 1 ( I KIF (V m, t) + I KIS (V m, t) + I KNI (V m, t) + I Na (V m, t) C m ) + I h (V m, t) + I L (V m, t) The rate of change of gating variable x(v, t) is given by: dx(v, t) dt = x (V, t) x(v, t) τ x (V ) The kinteic equation for Na + current is given below: I Na (V m, t) = g Na m 2 Nah Na (V m V N a) m Na (V m ) = (1 + e [(Vm+38)/3]) 1 h Na (V m ) = (1 + e (Vm+43)/3) 1

26 3.4. Cochlear nucleus 26 Interim Report: Modeling the Auditory Pathway The kinetic equation for fast and slow inactivating potassium currents I KIF and I KIS is given below: I KIF (V m, t) = g KIF m 4 F h F (V m V K ) m F (V m ) = (1 + e [(Vm+53)/25.8]) 1 h F (V m ) = (1 + e (Vm+89.6)/6.7) 1 I KIS (V m, t) = g KIS m 4 Sh S (V m V K ) m S (V m ) = (1 + e [(Vm+40.9)/23.7]) 1 h S (V m ) = (1 + e (Vm+38.4)/9) 1 A third potassium current I KNI, which is noninactivating is: I KNI (V m, t) = g KNI m 2 N(V m V K ) m N (V m ) = (1 + e [(Vm+40)/3]) 1 Hyperpolarization activated potassium current I h behaved according to following differential equations: I h (V m, t) = g h m h n h (V m V h ) m h (V m ) = n h (V m ) = (1 + e (Vm+68.9)/6.5) 1 ( τ mf (V m ) = 0.15 e (Vm+57)/ e [(Vm+57)/10]) ( τ hf (V m ) = e (Vm+87)/ e [(Vm+87)/20]) ( τ ms (V m ) = 0.15 e (Vm+40)/ e [(Vm+40)/10]) τ hs (V m ) = 200 τ mh (V m ) = (1 + e (Vm+183.6)/15.24) 1 τ nh (V m ) = (1 + e (Vm+158.6)11.2 ) (1 + e (Vm+75)/5.5 ) 1 The leakage current representing resistive losses over the membrane is described by Ohm s law: I L (V m ) = g l (V m V l )

27 Figure 10: A model of the stellate cell [1] Stellate cell The stellate or mulipolar cells are found in the AVCN. These cells exhibit the chopper response pattern if given the short CF tone bursts. In the model, which we have adopted [1], the electrical network is used to represent the cell. The whole dendritic tree of the cell is collapsed into a single cylinder following Rall s 3/2 power law. Because of lack of information, it was assumed that dendrites don t have any active ion channels. As shown in the Figure 10, the single dendritic cylinder is modeled by ten compartments. Soma and axon are also modeled with single compartment. Each compartment, in turn is modeled by an electrical circuit. The inactivating sodium conductance g Na depends on membrance potential V and time t by the relation g Na = g o Nam 3 (V, t)h(v, t)s

28 3.4. Cochlear nucleus 28 Interim Report: Modeling the Auditory Pathway Table 5: Stellate cell model parameters (continued in the next table). Name Symbol Value Anatomical Dendrite Diameter d d 2.2µm Dendrite Length l d 600µm Soma Diameter d s 25µm Axon Diameter d a 3.0µm Axon Segment Length l a 70µm Electrical Membrane Resistivity R m 10KΩ cm 2 Axial Resistivity R i 150Ω cm Membrance Capacitance/Area C m 1.0µF/cm 2 Dendritic Space Constant λ d = d d R m 4R i 600µm Dendritic Electronic Length L d = l d λd 1.0 Equivalent Cylindrical Equivalent Dendrite Diameter d eq = (6(d d ) 3/2 ) 2/3 7.3µm deqr Dendritic Space Constant λ eq = m 4R i 1100µm Dendritic Electronic Length L eq = L d 1.0 Compartmental Number of Compartments N 10 Electronic Length Z i = I eq /N 0.1 Metric Length l i = λ eq /N 110 Membrane Capacitance c i = πd eq l i C m µf Conductance between i to j g ij = πd2 eq 4R il i S Excitatory Reversal Potential E e 0mV Inhibitory Reversal Potential E i 68mV where S is the membrane surface area of the soma or axon, and m, h are the activation and inactivation variables respectively. The activation variable m follows following differential equation: τ m dm dt + m = m

29 3.4. Cochlear nucleus 29 Interim Report: Modeling the Auditory Pathway Table 6: Model parameters (continued from the previous table). Name Symbol Value Soma Membrane Surface Area S = πd 2 s cm 2 Membrane Capacitance c s = C m S µf Axial Conductance between g s1 = πd2 eq 4R il i/ S Soma and First Compartment Max. Leakage Conductance/area g1 o 0.090S/cm 2 Leakage Conductance g l = gl o S Leakage Equilibrium Poential E l 53mV Sodium Equilibrium Poential E Na 55mV Potassium Equilibrium Potential E K 80mV Axon Membrane Surface Area S a = πd 2 a cm 2 Membrance Capacitance c a = C m S a µf Axial Conductance between g as = πd2 a 4R il a/ S Axon and Soma Max. Leakage Conductance/Area gl o 0.025S/cm 2 Leakage Conductance g l = gl os a S where τ m = 1/T fac α m + β m α m m = α m + β m α m = 0.1(V M SH { } exp (V +37+MSH ) 10 1 { } (V MSH ) β m = 4exp 18 Similarly inactivation variable h follows the following differential equations: τ h dh dt + h = h τ h = 1/T fac α h + β { } h (V MSH ) α h = 0.07exp 20 1 β h = } exp 1 { (V +32+MSH ) 10

30 4. Progress 30 Interim Report: Modeling the Auditory Pathway The delayed rectifier potassium conductance depends on V, and t as in equation: g K = gkn o 4 (V, t)s where the inactivation variable n is defined as: dn τ n dt + n = n 1/T fac τ n = δ(v )(α n + β n ) α n n = α n + β n 4 Progress α n = 0.01(V { N } SH) exp (V +52+NSH ) 10 1 { } (V NSH ) β n = 0.125exp 80 δ(v ) = (1 0.19)(V E r ) Table 7 summarizes the work done so far. The table lists the neuronal models implemented and ones yet to be implemented. The auditory pathway consists of different nuclei and each nucleus in turn contains different types of neurons having specific response and that communicate among themselves through interneurons. Figure 11 shows the ascending audiory pathway highlighted with the neurons we have implemented so far withing each nuclei. Table 7: Status in modeling auditory pathway. Nuclei AN Fibers Cochlear Nucleus Bushy Cell Fusiform Cell Octopus Cell Pyramidal Cell Stellate Cell Superior Olivary Complex Inferior Colliculus Medial Geniculate Body Status Implemented Partially Implemented Implemented Implemented Implemented Not Implemented Not Implemented Not Implemented Not Implemented Not Implemented Now we discuss the models of each individual nuclei of auditory pathway in detail.

31 4.1. Octopus cell model 31 Interim Report: Modeling the Auditory Pathway INFERIOR COLLICULUS Not Implemented SUPERIOR OLIVARY COMPLEX Lateral Superior Olive Medial Superior Olive Medial Nucleus of the Trapezoid Body Not Implemented COCHLEAR NUCLEUS Pyramidal Cell Stellate Cell Not Implemented Inter-Neurons Bushy Cell Fusiform Cell Octopus Cell Implemented AN Fibers Cochlea Unknown Connection Nucleus Boundary Known Connection Figure 11: Modeled nuclei in auditory pathway. 4.1 Octopus cell model The model for the AN fibers was implemented as described by Zhang et. al. [24]. The octopus cell model by Levy et al. [10] was implemented by solving the differential equations in MAT- LAB given in Section Output from the AN model was fed to the model for the octopus cell. The response characteristic of the octopus cell is shown in Figure 12, in which the potential of the soma (in mv) is shown against time (msec) for the given pattern in which AN fibers innervate the octopus cell. The characteristic frequency of AN fibers which innervate a particular octopus cell has a bandwidth of 1/3rd of the entire range (from 20Hz to 20kHz). Hence all octopus cells can be simulated by this method assuming we know the types and numbers of AN fibers which it receives. There is always some variability in the response of octopus cell due to intrinsic variation in

32 4.2. Bushy cell model 32 Interim Report: Modeling the Auditory Pathway Figure 12: Response observed from a computational model of the octopus cell. the individual octopus cell and the experimental procedure. Therefore, the model was verified by demonstrating that somatic potential lies withing a biologically plausible range, which we found to be so in simulation. As shown in Figure 12, the response curve of the octopus cell implementation (on the right) is compared against that of the model implemented by Levy et. al. [10]. Here the somatic potential is compared for 20 db, 40 db, and 60 db respectively for the CF tone bursts. 4.2 Bushy cell model The model of the bushy cell selected for this research is by Rothman et. al. [16]. The differential equations were implemented in MATLAB as described in Section The response of the bushy cell model depends on the number of auditory neurons it receives as well as on the conductance of each input. These two inputs are modeled using two parame-

33 4.3. Fusiform cell model 33 Interim Report: Modeling the Auditory Pathway Figure 13: Response from a computational model of the bushy cell for different values of A E. ters: N and A E for, respectively, the number of innervating neurons and their conductance. In Figure 13, A F show the response of the implemented model for different amplitude of synaptic condcutances A E, which are 18.2, 27.3, 36.4, 54.6, 109.1, 218.3, and 327.4, respectively. While G shows the response curve for the model as implemented by Levy et. al. The effect of changing the number of inputs N S and amplitude of synaptic conductance A E on the soma potential is shown in Figure 14. In A D, the left part of the figure shows the membrane potential from the implementation while the corresponding right part shows the same from the model implemented by Levy et. al. 4.3 Fusiform cell model The model selected for the fusiform cell is by Kim et. al. [8]. Since the model takes input current as the parameter, it is same as that described by Kim. The implementation was verified as described in [8]