FONTAINE, B

PEREMANS, H

Schrauwen, Benjamin

English

Ghent University Academic Bibliography

Bat echolocation modelling using spike kernelswith Support Vector Regression.Fontaine Bertrand1, Peremans Herbert1 and Schrauwen Benjamin2 ∗†1-Antwerp University, APL, Prinsstraat 13, 2000 Antwerpen, Belgium2-Ghent University, ELIS, St-Pietersnieuwstraat 41, 6000 Gent, BelgiumAbstract. From the echoes of their vocalisations bats extract informa-tion about the positions of reflectors. To gain an understanding of howtarget position is translated into neural features, we model the bat’s pe-ripheral auditory system up until the auditory nerve. This model assumesmultiple threshold detecting neurons for each frequency channel where theinter-spike times are linked to the location of the reflector. To show thatthis coding process can be reversed we compute the kernel product of thespike trains using a non-binned spike kernel function. This approach allowsdoing regression on azimuth and elevation using Support Vector Machines.1 IntroductionBats use their biosonar to navigate through foliage and to localize and recog-nize prey. They emit ultrasound calls and from the modified received echoesextract information regarding the geometry of their environment. Experimentshave shown that bats use spectral clues to determine the spatial position of thereflecting target [4]. This direction dependent filtering is to a large extent causedby diffraction of the sound waves around the particular shape of the bat outer earand head [5], corresponding with the Head Related Transfer Functions (HRTF)studied in human hearing research. Below, we will show that these spectral cuesare preserved/enhanced by cochlear processing [2] and can be used to extractreflector position information from the neural code present at the level of theauditory nerve.Although it is customary in audition research to assume that the informationat the level of the auditory nerve is represented using rate coding, we proposethat in the case of bat echolocation a time coding perspective is more appropri-ate. Indeed, due to the very short durations of both the calls and the echoes,2-10 ms, only 1 or 2 spikes are generated in the Inferior Colliculus i.e., auditorymidbrain, of the bat [7] when presented with a single call-echo pair. Further-more, experiments have shown that single call-echo pairs are sufficient for thebat to base behavioral decisions upon. Hence, as firing rates loose their meaningwhen only single spikes are present we conjecture that the inter-spike times of thegenerated spike burst code for the relevant information. See [3] for a thoroughdiscussion of time vs. rate coding issues.∗This research is funded by the EC, IST-FET Initiative as part of the CILIA project.†We would like to thank C. Moss and A. Murat for making available the Eptesicus FuscusHRTF measurements.Because of the time coding, existing spike kernels [6] cannot be used to embedthe processing of these spike trains in standard Support Vector Machine theoryas they require the computation of firing rates. Hence, a new spike kernel,introduced recently in [9], that keeps the precise timing information intact wasused to investigate to what extent classical regression on azimuth and elevationusing SVM can be efficiently done on spike burst codes generated in the contextof echolocation.2 Peripheral auditory system processing of echoesCall The emitted ultrasound is based on the call of a particular FM bat i.e.,Eptesicus Fuscus. It consists of a downward 2ms FM sweep whose fundamentalranges from 50kHz to 20 kHz.Reflecting objects The reflectors are simulated as a set of point reflectors. Foreach observation a new, random, reflecting object is generated. Hence, whentransmitting the signal s(t), the filtered echo signal z(t) received from such acomposite reflector (point reflectors at (ri, θi, ϕi) with i = 1 . . . Nrefl) can bewritten asz(t) =Nrefl∑i=1hHRTF (t; θi, ϕi) ∗ ais(t− 2rivsound)(1)with the speed of sound given by vsound, the distance by r, the azimuth byθ, the elevation by ϕ and the angular dependent filtering (HRTF) denoted byhHRTF(.). The number of point reflectors (Nrefl) is drawn uniformly between 1and 5. Once this number is known, one of the point reflectors is set to havea reflection coefficient ak = 1 and the ratio of the others reflection coefficientsai with respect to the latter are uniformly distributed between 0.01 and 0.4,reflecting coefficient range we extracted from our measurements. The delaytimes (τi = 2ri/vsound) between point reflectors are generated from a Poissonprocess (simple model for a cloud reflector) whose mean depends on the numberof reflectors. From Eq. 1 we conclude that even when disregarding the angulardependent filtering hHRTF(.) whenever the target is no longer a single pointreflector the echo signal r(t) will be a filtered version of the transmit pulse s(t).Head Related Transfer Function The HRTF used was taken from measurementson an FM-bat, Eptesicus Fuscus [5]. The reflector positions considered span from-30 degrees to 30 degrees azimuth and the same for elevation. From this spatialdomain we choose 150 non-evenly distributed locations.The HRTF results in spatially dependent filtering of the received echo. Thespectral features of the received echo signal e.g., peaks and notches, introducedby the HRTF can thus code for the position of the reflecting object. The fre-quency dependent nature of the HRTF (see Fig. 1) allows for the ambiguityi.e., multiple locations mapped onto the same echo strength, present in a singlefrequency channel to disappear if multiple frequency channels are combined.Fig. 1: 3dB contours of the monaural HRTF at (a) 40kHz and (b) 68kHz.band-pass filter rectifier low-pass filterecho enters ear time representationcochlear representationt(ms)0 2 4t(ms)0 56auditory nerveFig. 2: The model of the peripheral auditory processingCochlea The joint time-frequency analysis performed on the incoming signal ismodelled on the transduction stage located in the inner ear (cochlea) of the bat.A simple, yet functionally adequate, model of this analysis [2] is a filter bankconsisting of parallel band-pass filters with subsequent envelope extraction (halfway rectification and low pass filter) in each channel. This model is illustratedin Fig. 2. The band-pass filters cover a range from 20kHz to 80kHz and have aquality factor of 25. The overlapping factor is set such that 100 filters cover thefrequency range of interest.Spike Generation As illustrated in Fig. 2 Inner Hair Cells (IHC) are situatedalong the basilar membrane. The IHC convert local motion of the basilar mem-brane into amount of neurotransmitter released. Several spiral ganglion cells(SGC) synapse with the same IHC. The SGC fire an action potential if theirmembrane voltage due to the neurotransmitter released by the IHC exceeds athreshold. Different IHC-SGC synapses have different thresholds.We model a simplified version of this spike generation process using 15 thresh-old neurons per frequency channel that fire deterministically when the signalexceeds their threshold. The thresholds are chosen logarithmically followingpsychoacoustic custom. Each threshold neuron can trigger only once, but thereis not that much information lost as the time scale of the signal is the sameas the refractory period of a neuron. Finally, the spikes coming from the 15threshold neurons per frequency channel are embedded in one spike train.3 Spike KernelTechniques using kernel functions for classification and regression problems havebeen proven to be very efficient. Here we want to use standard Support VectorMachines (SVM) to do regression on azimuth and elevation [1]. However, kernelsare traditionally applied to vectors whereas for our purposes we need a kernelfunction that can be applied to a set of spikes. Kernels specifically developedfor spike data have been described [6], but they require binning thereby losingthe precise timing information of the spike trains. Moreover, as stated before,the occurrence of only 1 or 2 spikes in each neuron’s output does not allowcomputing a firing rate either.The new spike kernel introduced in [9] does allow us to preserve the temporalresolution needed for our application. In this paper, defining a spike train x asthe set of threshold crossing times x = {t1, t2, ...tN}, the linear spike kerneloperating on two spike trains x and y containing N and M spikes respectivelyis defined as:Klinear (x,y) =N∑iM∑j[max(1− λ2|ti − t′j |, 0)](2)where ti is the i-th spike of x and t′j is the j-th spike of y. The λ parameterdetermines the temporal width of the kernel. A large λ leads to a kernel compar-ing only close-by spikes whereas a small one results in a kernel comparing eachspike with every other spike. This kernel comparison could possibly be executedin the Superior Colliculus (SC) which links sensory based spatial informationwith orienting responses [8]. SC receives inputs from multiple auditory centersamong which the cortex.In our echolocation application we use the different spike trains from thedifferent frequency channels as input for our regression system. The kernelmethod has to be slightly extended to take such structured collections of spiketrain data into account. If we have P input spike trains X = (x1,x3...xP ) fromP frequency channels, a new multi spike train kernel K′can be built by takingthe sum of the P individual kernels:K′linear (X,Y) =P∑iK (xi,yi) (3)Using multiplication instead of addition, a single kernel product which yieldszero would cancel out the contributions from all other frequency channels.4 Simulations and resultsThe data set is built as follows: for each position from the 150 considered,10 observations are simulated with the reflecting object complexity being drawnrandomly (see Sec. 2). It gives thus 1500 observations in total. Each observationconsists of maximum 100 spike trains generated for the 100 frequency channels.A single spike train consists of maximum 15 spikes, less if not all the thresholdsare crossed. Of the 1500 observations, 1000 are used for learning and the rest fortesting. Within those 1000 observations, 200 are used to do a grid search on theparameters of the model using 5 cross-validations. The parameters to optimizeare the cost C, the epsilon of the regression SVM [1] and the scale parameter λof the kernel. Once the parameters are found 5 others cross-validations are donewith the testing data. Azimuth and elevation are estimated independently.First the system takes its input from one frequency channel situated in themiddle of the frequency range (50kHz). Next, the number of frequency channelsis increased by distributing channels evenly over the full frequency range whilekeeping channels used in a previous simulation in the new set (monotonousincrease of information). The results of those simulation are plotted in Fig. 3.0 0.2 0.4 0.6 0.8 1 1.244.555.566.57Jitter [ms]AzimuthElevation1 3 7 15 31 63 100Meanoftheerroronazimuth(degr.)Meanofthetestingerror(degr.)234567891011Number of frequency channels usedFig. 3: (a) Resulting mean testing error (deg.) using a growing number offrequency channels over 5 validations; (b) Mean error (15 frequency channels)in elevation as a function of spike time jitter.5 DiscussionAs expected the mean error goes down as the number of frequency channelsis increased (see Fig. 3(a)), due to a reduction of ambiguity when combiningHRTF’s of different frequencies (see Fig. 1). For the biological system thiswould mean more and more accurate position information is available moving upthe monaural auditory pathway starting from the cochlear nucleus through theinferior colliculus ending in the cortex. This is a consequence of the systematicbroadening of the frequency integration properties of the tonotopically organizedneurons along this pathway.We also notice that the elevation estimate is more accurate than the azimuthestimate for the same number of frequency channels. This is also expected frombehavioral experiments on bats as azimuth estimation relies on binaural cueswhereas elevation estimation can be performed using monaural spectral infor-mation [4, 5]. Furthermore, the actual accuracies achieved are also in accordancewith measured accuracies for elevation in bat echolocation [4]. A previous modelbased on a backpropagation network [10] making use of the same monaural cuesachieved performances of 7.5 for azimuth and 8.9 for elevation. Our modelclearly achieves better performances and is less complex.From Fig. 3(b) we conclude that the kernel indeed makes use of temporal in-formation to estimate the position. On the other hand, the asymptotic behaviorfor larger jitter (>1ms) values shows that even in the absence of precise timinginformation the performance of the elevation estimation is better than chancebehavior.Future work should investigate and use more biologically plausible kernels.For instance, spike kernels should consider the absence of spike as informativewhereas the present used kernel does not.References[1] B. Schokopf and A.J. Smola. Learning with kernels: Support Vector Machines, Regular-ization, Optimization and Beyond, MIT press, 2002.[2] J. O. Pickles, An Introduction to the physiology of hearing, Academic Press, 1982.[3] F. Rieke and D. Warland and R. de Ruyter van Steveninck and W. Bialek, Spikes, MITPress, 1999.[4] J. Wotton, T. Haresign, and J. Simmons, Spectral cues and perception of the verticalposition of targets by the big brown bat, eptesicus fuscus, J. Acoust. Soc. Am., 107(2),1034-1041, 2000.[5] A. Murat, E. Grassi, M. Sahota and C. Moss, The bat head-realted transfer functionreaveals binaural cues for sound localization in azimuth and elevation, J. Acoust. Soc. ofAm., 116, 3594-3605, 2004.[6] L. Shpigelman, Y. Singer, R. Paz and E. Vaadia, Spikernels; Predicting arm movements byembedding population spike rate patterns in inner-product spaces, Neural Computation,17(3), 671-690, 2005.[7] M. Sanderson and J. Simmons, Neural responses to overlapping FM sounds in the inferiorcolliculus of echolocating bats,J. Neurophysiology, 83, 1840-1855, 2000.[8] D.E. Valentine and C.F. Moss, Spatially Selective Auditory Responses in the SuperiorColliculus of the Echolocating Bat, The Journal of Neuroscience, 17(5), 1720-1733, 1997.[9] B. Schrauwen and J. Van Campenhout, Linking Non-binned Spike Train Kernels to Sev-eral Existing Spike Train Metrics. Neurocomputing, doi:10.1016/j.neucom.2006.11.017,2007.[10] J.M. Wotton and R.L. Jenison, A backpropagation network model of the monaural local-ization information available in the bat echolocation system, J. Acoust. Soc. Am., 101(5),2964-2971, 1996.

Bat echolocation modelling using spike kernels with Support Vector Regression

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