Decoding First-Spike Latency: A Likelihood Approach. Rick L. Jenison

Transcription

1 eurocomuting (00) Decoding First-Sike Latency: A Likelihood Aroach Rick L. Jenison Deartment of Psychology University of Wisconsin Madison WI enison@wavelet.sych.wisc.edu ABSTRACT First-sike latency at the level of auditory corte aears to correlate well with sound source direction. However an unsettled issue with latency as a neural code is the aarent lack of temoral marking for the onset of the stimulus. One ossible resolution is that relative ensemble latencies could be decoded rather than absolute latencies. How an ensemble timing referent might inform a theoretical ideal observer remains largely unelored. A likelihood aroach is derived for decoding sound direction from a samle oulation of cortical neurons in cat auditory corte based on relative first-sike latencies and is evaluated using Fisher information. Key words: neural coding first-sike latency likelihood Fisher information sound localization BIO Rick Jenison is an Associate Professor in the Deartment of Psychology at the University of Wisconsin-Madison. His research includes the study of neural mechanisms of mammalian sound localization with a focus on mathematical models.

2 ITRODUCTIO Latency from stimulus onset to the first evoked action otential (first-sike latency) in the auditory corte aears to correlate well with the direction of sound sources in sace [3 3 4]. Furthermore resonse latency has been shown to carry a significant roortion of the information for the hysical stimulus in addition to sike count [9 7]. An unsettled issue with resect to the use of latency as a neural code for sound direction is the aarent lack of temoral marking for the onset of the stimulus. This roblem is not secific to directional coding in the auditory system but is roblematic for decoding strategies that have been roosed in visual corte as well [7]. The obvious resolution is an internally available global time marker such as a collective oscillatory attern of activity [0]. An alternative resolution is that latencies relative to one another in an ensemble of neurons could be decoded rather than absolute latency [4 5 8]. How an ensemble timing referent might inform a theoretical ideal observer remains largely unelored. Recently we described a formal method for decoding sound location from a oulation of cortical neurons in field AI of the cat based on maimum likelihood estimation from absolute first-sike latencies []. An etension to this aroach is to relace the unknown stimulus onset by the first observed sike in an ensemble cortical oulation. Under this scenario the observer measures subsequent evoked first-sikes within the ensemble relative to the first oststimulus neural event in the ensemble. Although still based on arametric timing this aroach is related to the statistically nonarametric rank-ordered decoding described by Gautrais and Thore [6]. Using a relative referent rather than absolute imlies that all of the ensemble neural events are temorally shifted by some unknown common value relative to the hysical onset of the stimulus. In the following equations reflects the latency of the th neuron in the ensemble relative to that of the first oststimulus event evoked by one of the

3 neurons in the ensemble. Any member of the ensemble could roduce the first sike in the ensemble. By definition the neuron with the shortest absolute latency will be assigned the value of 0 (see Fig. ). is the common latency across the ensemble of neurons and hence it reflects the arameter that binds the ensemble resonses to the stimulus onset. The statistical model of the relative latency can be eressed in terms of the th recetive field model ( ) common latency and error termε as follows ( ) + ε () where the robability density of the latency noise term ε is assumed to be Gaussian with homoscedastic variance based on our rior analyses of sike latency in cortical field AI (Jenison et al. 998). The variance term actually does deend to some degree on the mean latency in our data however for uroses of this study the variance is assumed to be indeendent of the mean. The robability density function can now be written as follows: ([ + ] ( ) ) ( ) ( ) π e () The arameter is resumably not meaningful to the animal er se and its contribution to the model seems almost tautological. But is only inconsequential when the stimulus onset is known by the observer and thus is known. The observer deends on only in that it is necessary for constructing the robability density about the relative decoding of first-sike latency with resect to the first-sike in the ensemble. In this aer an aroach to deriving a likelihood function that removes this deendency on is shown thus allowing the observer to rely on only the ensemble vector of relative resonse latencies. Given that some form of this likelihood function is reresented in cortical circuitry maimization may be ossible by recursive neural rocessing [6]. 3

4 METHODS We have recently described a methodology by which cortical recetive fields that deend on the location (azimuth and elevation ) of a virtual sound source in sace (see Fig. A) can be functionally modeled []. The methods used to collect the single neuron resonses that constrain the recetive field models are described in [3] and are briefly summarized here. Barbiturate-anesthetized cats were fitted with hollow earieces sealed into the eternal ear canals through which virtual acoustic sace stimuli [5] were resented. Single neurons were recorded etracellularly in the high-frequency region of the left AI field of auditory corte. The latency of the first-sike in resonse to transient virtual acoustic sace stimuli was used as the rimary resonse variable for von Mises basis function aroimation [] (see Fig. B). The von Mises model for the recetive field is a continuous sherical function of azimuth and elevation ( ) b e{ κ ( sin sin β cos( α ) + cos cos β ) } b. (3) + The shortest latency corresonds to the best resonse of the cortical neuron and its location is defined by α and β. κ secifies the width or radius of the recetive field and the vector b contains weights whose dimensionality could etend to accommodate a linear combination of basis functions. Although we have found these basis functions to yield accurate models of satial recetive fields the analysis described here does not deend on any secific recetive field model. The latency noise within the neural ensemble is assumed to be indeendent such that the resulting likelihood function across neurons can be written as ( ) ( ) ( ) L. (4) Eanding the likelihood function and collecting the term yields 4

5 5 ( ) ( ) ( ) ( ) ( ) + + e e π L (5) which factors out into a second eonential term. In the statistical literature would commonly be referred to as a nuisance arameter and indeed several methods eist for removing nuisance arameters []. Under conditions where there are a small number of fied nuisance arameters maimum likelihood (ML) estimation of each nuisance arameter and backsubstitution of the estimates into the original likelihood function offers one solution. Although the nuisance arameters can be removed from the likelihood function there are still consequences when the nuisance arameters are nonorthogonal with resect to the arameters of interest. Other aroaches more Bayesian in nature include integrating out the nuisance arameters but these are not considered in this study. Deriving the ML estimator for entails taking the derivative of the log-likelihood function and setting it equal to zero ( ) 0 log L then solving for. For this articular case ( ) ( ) log + L (6) and therefore ( ) ˆ. (7)

6 Back-substituting the ML estimate ˆ for the arameter of Equation 5 yields L ( ) ( π ) e ( ( )) e ( ( )) (8) which is no longer deendent on. The second eonential term now contains the mean over the resonse latencies and the mean over recetive fields ( ) at locations and. This likelihood function is now a function of only and. This new function is commonly referred to as the rofile likelihood [4] and denoted L ( ). The rofile likelihood can now be used by the observer to obtain oint estimates of the arameters of interest and by whatever means of maimization. In the following eamles is assumed to be fied (or known) in order to simlify the analysis as well as for uroses of dislay. An ensemble of 65 modeled neurons was emloyed in the analysis. The satial distribution of the samle is shown in Fig. C with contours corresonding to the iso-intensity at 50% u from the shortest latency located at coordinates α and β (center circles). A cross-section as a function of azimuth at 0 elevation is shown in Fig. D. In this study the radius of the recetive fields were fied across the ensemble. The likelihood functions are derived from an additional 65 neurons that are a reflection of the original samle of 65 neurons about the midline in order to account for the contribution of neurons in the oosite cortical hemishere. RESULTS To investigate the imact of decoding sound direction from first-sike latency timed off the ensemble referent first-sike latencies were samled from a secific sound direction. Then the rofile likelihood function was evaluated and comared against the likelihood function without 6

7 the introduction of that is assuming the stimulus onset to be known. Fig. 3 shows the two log transformed likelihood functions for four azimuthal ( ) sound directions on the left side of the animal assuming that is fied at 0. The maimum of the log-likelihood functions would rovide the ML estimate of given the observed resonses. As can be seen both log-likelihood functions have nearly the same maimum. However the imact of introducing the binding arameter can be observed in the slight flattening of the rofile log-likelihood function near the global maimum in Fig. 3B (-95 ). The consequence of the flattening is that estimates of the maimum would be somewhat more variable. However differences near the maimum aear to be small at the other sound source directions shown in Fig. 3(A) -45 (C) -40 and (D) 0 suggesting that the degree of flattening is directionally deendent. The elanation for the flattening of the likelihood function can be found by deriving the Fisher information matri for the full likelihood (including as a arameter) and comaring it to the Fisher information for estimating and under the condition where the stimulus onset is known (ecluding as a arameter). The Fisher Information matri reflects the amount of information available for estimating each arameter from the data (neural resonses ) and is related to the curvature of the loglikelihood function near its maimum []. Again is assumed to be fied and therefore will not contribute to the information matri in this study. The full matri of course needs to be considered however the imact of is more difficult to interret with the full 3 by 3 matri. The Fisher information matri for ust two of the arameters can be defined as 7

8 8 ( ) ( ) ( ) ( ) ( ) ( ) log log log log log log E E E E J r r r r r r. (9) The cell in the uer left is the Fisher information for estimating and the cell in the lower right is the Fisher information for estimating the nuisance arameter for the rofile likelihood. The off-diagonal cells reflect the cross-information between and and will evaluate to zero if the two arameters are orthogonal. Evaluating the derivatives and eected values of each cell in Eq. 9 yields ( ) ( ) ( ) J. (0) The cells of the Fisher information matri contain the gradients of the recetive field models ( ) and the latency noise variance (assumed to be constant across the ensemble). The Cramer-Rao lower bound (CRLB) corresonds to the diagonal of the inverse of the Fisher information matri ˆ ˆ J K and reresents the lower bound on the covariance matri of any unbiased estimator. The elanation for the flattening of the rofile likelihood is revealed by eamining the lower bound of the variance about the estimate ˆ that results from the full matri inversion of J

9 ˆ ( ) ( ). () Contrast Eq. with the variance lower bound under conditions when the stimulus onset is known ˆ ( ) () which is a simle scalar inverse of the Fisher information for estimating. The difference between Eqs. and is due to the non-zero cross-information term which due to its squaring must necessarily inflate the lower bound in Eq. with resect to Eq.. The magnitude of this inflation deends on the gradients of the sace recetive fields for a articular sound direction. To better illustrate this comarison lots of the resective CRLBs are shown for values of (azimuth). First it is interesting to note the general trend for both functions. The minimum of both CRLB functions hence the best acuity is found at 0 azimuth - a finding that arallels sychohysical eormance in most animals. Second a bum in both CRLB functions occur in the region lateral to the midline where the recetive field gradients tend toward zero that is where the first-sike latency is at its minimum. Of course this is where Fisher information declines as well. From these lots we can also infer that the greatest imact on estimation variance or error from using the ensemble first-sike as the timing referent is in this region. However the relative difference between the two CRLB functions is actually quite small for all directions suggesting that the imact of rofiling over is rather small. 9

10 DISCUSSIO In this aer we have shown how an ideal observer given the likelihood function for measured first-sike latencies timed to the first evoked sike in an ensemble of neurons can reliably estimate sound direction. When the stimulus onset is unknown the likelihood function is deendent on a common latency that must be eliminated or rofiled out in order to estimate the arameters of interest in this case and. Given our modeled ensemble based on measured sace recetive fields in the cat the imact of rofiling out the nuisance arameter is small. This suggests that decoding first-sike latency using the first evoked sike in the ensemble is a viable aroach and one that a neural system could emloy quite successfully without knowledge of when the stimulus was turned on. Furthermore the rofile likelihood is not sensitive to shifts in ensemble first-sike latencies that result from changes in sound source intensity. The ower of analyzing a secific statistical model articularly when the neural recordings are available is that the Cramer-Rao lower bound informs us as to the best eormance ossible from any unbiased estimator. So whatever neural circuit might eorm sound direction estimation it is bounded by the CRLB. Fisher information further informs us as to what asects or characteristics of the recetive fields account for the observed variations in estimation error. ACKOWLEDGEMETS This work was suorted by IH grants DC03554 and DC006. 0

11 REFERECES [] R. E. Blahut. Princiles and ractice of information theory. Addison-Wesley Reading MA 987. [] O. E. Barndof-ielson and D. R. Co. Inference and Asymtotics. Chaman and Hall London 994. [3] J. F. Brugge. R.A. Reale and J. E. Hind. The structure of satial recetive fields of neurons in rimary auditory corte of the cat. J. eurosci. 6: [4] J. J. Eggermont. Azimuth Coding in Primary Auditory Corte of the Cat. II. Relative Latency and Intersike Interval Reresentation. J. eurohysiol. 80: [5] S. Furukawa L. Xu and J. C. Middlebrooks. Coding of sound-source location by ensembles of cortical neurons J. eurosci. 0: [6] J. Gautrais and S. Thore Rate coding versus temoral order coding: a theoretical aroach. Biosystems 48: [7] T. J. Gawne T. W. Kaer and B. J. Richmond. Latency: Another otential code for feature binding in striate corte. J. eurohysiol. 76: [8] P. Heil. Auditory cortical onset resonses revisited: First-sike timing. J. eurohysiol. 77: [9] J. Heller J. A. Hertz T.W. Kaer and B. J. Richmond. Information flow and temoral coding in rimate attern vision J. Com. eurosci. : [0] J. J. Hofield Pattern recognition comutation using action otential timing for stimulus reresentation ature 376:

12 [] R. L. Jenison Models of sound location acuity with broad cortical satial recetive fields. In Central Auditory Processing and eural Modeling (ed. P. Poon and J. F. Brugge) Plenum ew York 998. [] R. L. Jenison R. A. Reale J. E. Hind and J. F. Brugge. Modeling auditory satial recetive fields with sherical aroimation functions. J. eurohysiol. 80: [3] J. C. Middlebrooks A. E. Clock L. Xu and D. M. Green. A anoramic code for sound location by cortical neurons Science 64: [4] S. Murhy and A. W. van der Vaart. On rofile likelihood J. Am. Stat. Assoc. 95: [5] A. D. Musicant J. C. K. Chan and J. E. Hind. Direction-deendent sectral roerties of cat eternal ear: ew data and cross-secies comarisons. J. Acoust. Soc. Am [6] A. Pouget K. Zhang S. Deneve and P. E. Latham P. E. Statistically efficient estimation using oulation codes. eural Comut. 0: [7] M. C. Wiener and B. J. Richmond. Using resonse models to estimate channel caacity for neuronal classification of stationary visual stimuli using temoral coding. J. eurohysiol. 8: FIGURES FIG.. Schematic for decoding first-sike latencies. The vector [... ] latencies with resect to the first evoked sike in the ensemble. r contains the FIG.. A) Virtual auditory sace surrounding the head of the cat. Each osition on the shere reresents the osition of a virtual sound source used to evoke a resonse B) Eamle of a

13 modeled cortical AI recetive field (first-sike latency) with best direction (shortest first-sike latency at 57 azimuth 5 elevation). The recetive field is modeled using a linear combination of sherical von Mises basis functions (Jenison et al. 998) and shown roected onto an equalarea ma of auditory sace C) A reresentative distribution of best directions ( α and AI recetive fields (circles) (from [3]) surrounded by iso-latency contours 50% u from the minimum latency. D) A cross-section across the samle oulation shown in Fig.. B. at 0 elevation. β ) for 65 FIG. 3. Profile likelihood L ( ) with ˆ substitution (solid line) vs. the likelihood ( ) L assuming stimulus onset to be known (dotted line) for true values of azimuth at (A) -45 (B) -95 (C) - 40 and (D) 0. The vertical lines reflect the true ositions of the four sound sources. The directions labeled -80 and 80 are actually the same direction which corresonds to the midline behind the cat s head. FIG. 4. Cramer-Rao Lower Bound (in degrees) as a function of azimuth. CRLB for the rofile likelihood L ( ) with ˆ substitution (solid line) vs. the likelihood ( ) onset to be known (dotted line). L assuming stimulus 3

14 Stimulus Onset euron #5 Resonse euron #3 Resonse euron #3 Resonse Time (ms) Figure

15 A B First-sike Latency (ms) C D FirstSike Latency (ms) Azimuth (deg) Figure Figure

16 Log-likelihood A B Log-likelihood C D Azimuth (deg) Azimuth (deg) Figure 3

17 CRLB (deg) Azimuth (deg) Figure 3 Figure 4