Coherent oscillations as a neural code in a model of the olfactory system

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1 Coherent osillations as a neural ode in a model of the olfatory system A. Gutierrez-Galvez and R. Gutierrez-Osuna, Member, IEEE Department of Computer Siene Texas A&M University College Station, TX Abstrat-This paper presents an investigation of two odor oding mehanisms in Freeman's KIII nenrodynamis model. Motivated by experimental evidene that supports the existene of a neural ode based on synhronous osillations, we propose an analogy between synhronization in neural populations and phase loking in KIll hannels. The information arried by the phase is ompared against the onventional amplitude ode in terms of pattern-reovery apabilities. First, the salar invariane of the KIll with respet to phase information is established. Symmetries and redundanies in the assoiative memory matries are then exploited to perform an exhaustive evaluation of patterns on an 8- hannel model. Simulation results show that phase information outperforms amplitude information in the reovery of odor patterns from inomplete or orrupted sensory stimulus. 1. INTRODUCTION Neural oding shemes' based on firing synhrony are the inost ompelling hypothesis for a temporal ode spatially distributed aross large neural populations. These types of odes have been found experimentally in different neoortial areas [I], and play an important role in the integration of visual information [2] and the odifiation of odors [3,4]. The objetive of this paper is to asertain whether this oding mehanism an be exploited in the KIII model, of Freeman et al. [5,6], arguably the most omplete neurodynamis model of the olfatory system. The long term goal of this work is to develop biologiallyplausible omputational models to proess data from hemial sensor arrays, ommonly referred to as the eletroni nose. An e-nose onsists of (1) an array of hemial sensors with broad and overlapping seletivities and (2) a pattern reognition engine apable of assoiating sensor patterns with the orresponding odor labels 171. Biologiallyinspired approahes have been the fous of attention in reent years [8] as an alternative to the statistial pattern reognition proedures ommonly employed in eletroni-nose data proessing [9]. Along these lines, our experiene with the KIII model [ 10,l I] shows that the information provided by the amplitude of the hannels tends to degrade when the input patterns have a signifiant level of overlap, as is oftentimes the ase in eletroni nose data due to the ross-seletivity of the hemial sensors. This raises the question whether additional robust information an be extrated from the output of the model. In partiular, we have observed that the KIII has a tendeny to display similar phases in hannels that enode for the same odor. This observation, along with experimental evidene for a neural ode based on oherent osillations in neural populations, motivates the study presented in this paper. 11. TEMPORAL CODING AND COHERENT OSCILLATIONS Work by Adrian [12,13] more than 75 years ago showed that the firing rate of streth reeptor neurons is related to the fore being applied to the musles. This seminal ontribution led to the widespread belief that firing rate was the ode used by neural systems to transmit information. As a onsequene, early neural network models interpreted the output of artifiial neurons as an abstration of the neural firing rate in their biologial ounterparts. In reent years, this view has been hallenged with ample experimental evidene showing the need to take into onsideration the temporal dimension in neural information proessing. Undisputable evidene for a temporal ode is best illustrated by the work of Thorpe et al. [14,15], who have shown that humans and monkeys are able to respond to a visual ategorization problem in a very short period of time. In their experiments, an image is briefly flashed and the subjet has to deide if it belongs to a target ategory or not. Considering (i) the pathway of the visual signal as it propagates through the brain, (ii) the minimum time required for a neuron to generate an ation potential and (iii) the response time of the subjets in these experiments, it is possible to determine that there is time for only one spike to he generated at every relay station in the visual pathway. This result lashes with a frequeny-rate oding hypothesis, and learly points to the existene of a temporal dimension. A number of possible temporal oding mehanisms have been proposed, inluding inter-spike interval odes, time of arrival (lateny) odes and synhrony odes Among these, the synhronous osillation of ensembles presents the most empirial evidene. Synhronization has been proposed as a potential mehanism to orrelate information from different senses or different parts of the brain [I]. It has also been found to play a role in visual feature integration [2]. Of partiular interest to our work, Lauren1 et al. [3,4] have found that synhronous osillations of neuron populations in insets is used as an odor enoding mehanism. Their work has shown that different odors evoke oherent osillations in different but usually overlapping ensembles of neurons in the olfatory system /03/$ IEEE 341

2 ~ 111. THE U11 MODEL The KIII is a neurodynamis model of the olfatory system developed by Freeman and olleagues [5,6] over the last 30 years. The output of the model reprodues eletroenephalographi (EEG) reordings in the olfatory system by modeling the osillatory behavior of neuron populations. The topology of the KIII, shown in Fig. 1, is based on the physiologial struture of the mammalian olfatory system. Eah node in the KIII represents a population of neurons, modeled by a seond order differential equation, and eah edge models the interation between two populations. The strength of this interation is ontrolled by a weight, whih is positive when the onnetion is exitatory and negative if the onnetion is inhibitory. Odor stimuli are presented to the system as patterns, generally binary, through an input layer of reeptors. Eah reeptor is onneted to a periglomerular ell and a set of two mitral and two glomerular ensembles, forming a hannel. Eah of these hannels an then he assoiated with one bit of the binary input stimulus and the orresponding output pattem. The KIII is able to store previously seen pattems by means of Hebbian lateral onnetions at the MI mitral layer. This allows the model to work as an assoiative memory for reovering inomplete or orrupted stimuli. In the absene of an external stimulus, the MI1 hannels follow an aperiodi osillatory behavior known as a basal state. When an input is presented, the system moves into a global attrator in state spae, whih an also be observed as pseudo-periodi osillations in the output hannels. The amplitude of the osillations at eah hannel depends on the ativation level of its reeptor input, but is also influened by other reeptors as a result of the Hebhian lateral onnetions. The output pattem of the KIII is ommonly assumed to be enoded in the amplitude or RMS of the osillations of eah hannel [17], whih does not exploit the temporal dimension of the KIII dynamis. Considering that the KIII is a model of neuron populations, it is appealing to onsider the phase of the osillations aross hannels as an analogous of the oherent osillation oding sheme in biologial neural systems. Hene, the goal of this paper is to investigate whether this phase information an in fat be used as a oding mehanism and, if so, ompare its pattern-reovery performane against the onventional amplitude ode. A. Phase oding in the KIII model In order to effiiently apture phase information, we onsider the state-spae trajetory of pairs of U11 hannels (GI populations) as a two-dimensional distribution. As shown in Fig. 2, differenes in phase an be related to the orrelation oeffiient of the 2D distribution. Two ideal sinusoidal waveforms with the same phase will lead to a orrelation oeffiient of 1 (Fig. 2(a)), whereas a 180" phase differene result in a orrelation oeffiient of -1. Intermediate phase differenes result in orrelation oeffiients between those two extremes [-1, I].._ I 1. Fig. 1. Struture afthe KIII model (from [5]) The trajetories in Fig. 2 were obtained by training a 32- hannel KIII on two binary patterns, and presenting a distorted version of the first pattern at the inputs. It an be seen that the shape and orientation of the attrators an be assoiated to different types of errors in the input stimulus. It is also important to note that the orrelation oeffiient not only aptures information ahout the orientation of the prinipal eigenvetor but also about the area enlosed by the trajetory (i.e. the shape of the attrator.) - m L U img CO - E m U r m U N - E m L hannel 1 hannel 1 hannel 1 Fig. 2. Extrating phase information from the Klll U 342

3 ~ (-: Overlap 1111 Pattern A 1111 Pattern B (b) I Stn"us *....- () Fig. 3. Ovrlap bw& input patterns for a 16-hannel KIll (a). Simulating an inomplete (b) and a orrupted () input pattm. The KIII model with matrix formulation desribed in [6] was employed in this work. The model was implemented in MATLAB using fourth-order Runge-Kutta ODE integration with a time step of 1.O ms. Initial onditions for all variables and their derivatives were set lo zero. KIII parameter settings were borrowed from [ 181. IV. SCALING INVARIANCE Stimulus Invariane of the KlIl model with respet to the number of hannels has been empirially established by Yao and Freeman [5] in terms'of the amplitude and general shape of the osillations observed at eah hannel. To determine if these results an be generalized to phase information, we present a thorough study on three KIII models with 16, 32 and 64 hannels. The study simulates a two-odor lassifiation problem with varying levels of omplexity in terms of (I) the degree of overlap between the two stored patterns and (2) the number of missing or orrupted hannels that are presented at the input. Fig. 3 illustrates the two input patterns for the 16-hannel ase. Eah pattern onsists of four ative hannels, represented by a 1, and 12 inative hannels, represented by a blank bit. Due to the symmetry of the problem, in what follows the KIlI model is always exited with a stimulus from Pattem A. To inorporate different degrees of overlap, three patterns sets are onsidered having 0, 1 and 2 hits of overlap. Eah one of these three sets leads to a unique Hebbian assoiative matrix and, therefore, a separate KIII model. Fig. 3(a) illustrates the situation where the patterns have an overlap of 2 bits. To simulate inomplete patterns (e.g. aused by sensor degradation), 0, 1 or 2 of the ative bits in the stimulus may be set to zero. Fig. 3(b) illustrates the ase where the stimulus for pattern A is inomplete by one bit. Finally, to simulate orrupted patterns (e.g. to due to bakground odors), 0, I and 2 bits not belonging to either pattern may be set to one. Fig. 3() shows a stimulus for Pattern A with one orrupted bit. All these different ombinations lead to 3.3.3=27 possible senarios for a 16- hannel KIII model. Data for the 32- and 64-hannel models is obtained by saling the stored patterns and input stimuli by a fator of 2 and 4, respetively. a, v) 0.51 (4 Ob j.9 Amplitude (mv) -1 Q. il # * *- *a -.. ' d.9 Amplitude (mv) ' Amplitude (mv) Fig. 4. Satter plot afamplihlde vs. phase odes far different inputstimulus~desired-response eases Simulation results are presented in Fig. 4 in the form of bivariate satter plots. Eah point in the satter plots represents the output of one KIII hannel for one stimulus. The absissa axis is the RMS aniplitude of the hannel, whereas the ordinate axis is the phase relative to hannel 0, whih is onsistently ativated and used as a referene. In this work the GI signals are not band-pass filtered in the gamma band (20-80 Hz) as suggested in [19]. Comparable results, however, were obtained on the gamma-band filtered signals. 343

4 ~._ *.". 1+0 i i *k C._.. 0 U) m.*+.o+ 1. 'e. 030.f e Amplitude Fig. 6. Rdution of a 4-hannel model to I1 non-isomorphi graphs. The index abov eah graph denotes the number ofaliv Hebbian onnetions. Phase Fig. 5. Univariat density funtions for th amplirud and phas ods Four different ases of input-stimulus3desired-response are onsidered in the study, whih orrespond to two orret and two erroneous input stimuli:. 1 3 I (no error): stimulus in a hannel that enodes for pattem A 030 (no error): no stimulus in a hannel that does not enode for pattem A (inomplete pattem): missing stimulus in hannel that enodes for pattem A = 130 (orrupted pattem): noisy stimulus in a hannel that does not enode for pattem A The results in Fig. 4 show that, although a higher number of hannels yields a more detailed struture of the amplitudeiphase lusters, these satter plots have a similar struture regardless of the number of hannels. This result leads to the onlusion that the KIII model is not only sale invariant with respet to amplitudes, a result previously established by Yao and Freeman [5], but also with respet to phase information. A. Pattern reovery: preliminay results An important onlusion an also be extrated from these satter plots. Fig. 5 shows the univariate distribution of the amplitude and phase odes in the 64-hannel model for the eah of the four input-stimulus3desired-response ases. A Gaussian distribution has been assumed for visualization purposes. It an be observed that the amplitude ode is able to reover either inomplete or orrupted bits, but not both, sine the 031 and 130 densities lie on opposite sides of their desired response. This result indiates that the amplitude of a partiular hannel tends to be driven primarily by the input stimulus rather than by the lateral onnetions in the Hebbian matrix. In the ase of a phase ode, a simple threshold an he obtained to orret the majority of the inomplete or orrupted bits, indiating a higher sensitivity to lateral onnetions. Pattem reovery is, therefore, more reliable using phase information. This key result is further explored in the next setion. V. SYMMETRY OF THE ASSOCIATIVE MEMORY MATRIX One the sale invariane of the KlIl model with respet to phase information has been validated, the study an now be foused on a lower dimensional model where an exhaustive evaluation of every possible ombination of input stimulus and pattem sets is omputationally feasible. Although 2- and 4-hannel models ould he used to this effet, the results in Fig. 4 show that more hannels lead to higher resolution. For this reason, an &hannel model is hosen for the final study. In order to avoid exploring redundant ombinations, the symmetries in the KlIl assoiative-memory matrix will he exploited. Following [5], the mitral-mitral lateral onnetions an be omputed as: w.1 = Crb, pr) (1) vi where pi is the orret input pattem for the i-th odor lass, andf(9 is a threshold funtion so that the elements in W, are binary, either HIGH or LOW (diagonal elements in the matrix are set to zero.) Thus, a HIGH element in the Hebbian matrix represents two KlIl hannels that are simultaneously ative for at least one odor pattern. Sine different pattem sets an lead to the same Hebbian matrix, an exhaustive evaluation of every possible pattern set an he redued to the study of all possible matrix onfigurations. A representation of the Hehhian matrix as an undireted graph will he used to illustrate the existing symmetries. For simpliity, assume a 4-hannel model with only one onnetion between hannels. Sine the hannels are symmetri, the behavior of the system will be the same regardless of where this onnetion is loated. Generalizing this idea, graphs with the same number of onnetions and the same topology (i.e. isomorphi graphs) have to he onsidered just one. As a result, a 4-hannel model with Z6 = 64 possible onfigurations is redued to the 11 nonisomorphi graphs in Fig. 6. Similarly, a 8-hannel model an he redued from 228 = 2,7.108 to 1,192 ases, for a signifiant savings in CPU time. VI. PATTERN RECOVERY: FINAL RESULTS The final omparison of the two oding shemes will be performed on an 8-hannel KIII model, for a total of 1,

5 non-isomorphi ases. As opposed to the preliminary study in setion IVA, where performane was studied as a funtion of degrees of overlap in the pattern set and distortions in the input stimulus, the pattern reovery apabilities in this final study an only he evaluated in terms of the properties of eah graph sine the relationship between pattern sets and graphs is many-to-one. Two observations will greatly simplify this analysis. First, any two neighboring nodes (those that are onneted diretly by an edge) represent hannels that are simultaneously ative for at least one stored pattern. Conversely, any two nodes that are more than one edge away represent hannels where two or more patterns overlap. Therefore, it is possible to analyze the aross-fiber patternreovery performane of the phase and amplitude odes by omparing the ativation between nodes onneted by a single edge (whih represents a true pattern) against the ativation between nodes onneted by a. multi-edge path (whih represents an undesirable overlap). This idea is illustrated in Fig. 7(a). For a given graph, a single input stimulus is applied to the hannel with the highest number of lateral onnetions (do in the figure), and the ativity on the remaining hannels is analyzed. Nodes within one edge from do (denoted by dl) are part of a pattern. Nodes two edges away (denoted by d2) are the effet of an overlap between two or more patterns. The remaining nodes (d3 through dm) an he negleted sine they involve higher-order overlaps between patterns. Fig. 7(b) illustrates the situation where the response of do and dl nodes.is 'linearly separable from the rest, indiating that a simple ;threshold funtion ould he used to reover an inomplete pattern from an input stimulus having a single ative hannel. Fig. 7() illustrates the opposite situation, where the response of do and dl nodes is not linearly separable from the rest and, as a result, the inomplete pattern annot be reovered. (b) (4 1 ode linearly separable non separable Fig. 7. (a) Partm and overlap bits for a given Hehhian graph. (b, ) Pattern reovery as a linear separability problem... Amplitude *. U* Overlap (# d2 onnetions) Fig. 8. Ovrall performane of the phase and amplitude odes as a funtion of patrem overlap The proedure outlined in Fig. 7 is repeated individually for eah of the 1,192 non-isomorphi ases in the 8-hannel KlII in order to measure the pattern reovery apabilities of the amplitude and phase odes. The results are presented in Fig. 8 as a funtion of the number of d, onnetions in eah graph, whih an he related to the overlap (e.g. the omplexity) of the orresponding pattern sets. Eah point in the plot represents the perentage of graphs where the linear separability in Fig. 70) is ahieved, relative to the total number of graphs. This result shows that the performane of the amplitude ode dereases dramatially as the degree of overlap inreases, whereas the phase ode degrades in a more graeful manner and always provides higher lassifiation rates. Along with the preliminary data presented in Fig. 5, this result learly demonstrates the remarkable superiority of the phase ode over the amplitude ode. VII. CONCLUSIONS This artile has proposed an analogy between oherent osillations in neural populations and phase loking in the KIII model. We have shown that phase information between two hannels an he effiiently aptured by treating the statespae trajetory as a two-dimensional distribution and omputing its orrelation oeffiient. Sale invariane of the KIII with respet to this phase information has been empirially validated on three models with 16, 32 and 64 hannels. An exhaustive omparison of the pattem reovery apabilities of the proposed phase oding and the onventional amplitude oding has been presented on an 8- hannel model. In order to avoid the ombinatorial explosion, redundant pattern and stimulus ombinations have been eliminated by means of graph isomorphism. Experimental results show that information embedded in the phase of the KlII hannels learly outperforms the amplitude ode. The present study has foused on binary stimuli, but the KIII has also been shown [I91 to work effiiently with 345

6 methods ontinuous inputs. Thus, a natural extension of the work presented in this artile is to onsider ontinuous amplitude and phase representations. Additional information other than orrelation oeffiients (whih relate to phase for sinusoidal waveforms) ould also be extrated from the KIII dynami attrators in two- or higher-dimensional state spaes. These areas onstitute promising diretions for future work. ACKNOWLEDGEMENTS Ping Sun is aknowledged for earlier work on an implementation of the KIII model. REFERENCES [I] M. Ree. Enodieg information in neuronal ativity. in Pulsed Nural Nhvorks, W. Mass and C. M. Bishop, E&. Cambridge: the [2] [3] MIT prss, 2001, pp. I W. Singer, and C. M. Gray, Visual fatur integration and tho tmporal orrlatian hypothsis. Ann Rev Neurosiene, vol. 18, pp, G. Laurent, and H. Davidowitz, Enoding of olfatory information with osillating neural assemblies, Sien, vol. 265, pp , September [4] K. MaLad, and G, Laurent, Distint mehanisms for synhronization and tmporal patterning of ador-enoding neural [SI [6] assemblis. Sin, vol. 214, pp , November Y. Yao, and W. J. freeman, Madl of biologial pattern reognition with spatially haoti dynamis. Neural Ntwork, vol. 3, pp , H. Chang, W. J. Frman, and B. C. Burke, Optimization of olfatory model in softwar to give Ilf power splra reveals numerial instabilities in solutions gavrned by aperiodi (haoti) mators, Neural Ntworks. vol. 11, pp , [7] 1. W. Gardner and P. N. Bartlett, Eleaoni Noses. Priniples and Appliations, New York Oxford Univenily Press, [8] T. C. Peare, Computational parallels behven the biologial olfatory pathway and its analogue The Eletroni Nose : Part I. Biologial Olfation. Biosystems, vol. 41, no. 2, pp , [9] R. Gutinr-Osuna, Pattem Analysis for Mahine Olfation: A Review, in IEEE Sensors Journal, 2(3), ,2002, [IO] R. Gutirrz-Osuna. and A. Gutierrez-Galvez. Habituation in the KIIl olfatory model using gas sensor arrays, in Proeedings of the 9th International Symposium on Olfation and Eletroni Nose (ISOEN 02), Rome, Italy, Sept 29 - Ot 2,2002. [I I] R. Gutirrez-Osuna and P. Sun, A biologially-plausible omputational arhiteture far sensor-based mahine olfation, in Prodings of the 9th International Symposium on Olfation and Eletroni Nose (ISOEN 02). Rome, Italy. SeDt 29 - Ot [I21 E. D. Adrian fie impulses prdued.by &sory ne& ending, J. Physiology, vol. 61, pp , II.. 31 E. D. Adrian Th basis of sensation, W. W. Norton. New York [I41 S. I. Thorpe, A. Delorme, and R. Van Rullen, Spike-based smtegies for rapid proessing, Neual Networks, vol. 14, pp , January [I 51 R. Van Rullen, and S. J. Thorpe, Is it a bird? Is it a plane? Ultra-rapid visual ategorization of nahtral and artifatual objets, Pereption, vol. 30, pp , Deember [I61 P. Cariani, As if time really mattered temporal strategies for neural oding of sensory information. Communiation and Cognition- Artifiial Intelligen, vol. 12, nos. 1-2, pp , Marh [I71 Y. Yao. W. I. Freeman, B. Burke and Q,Yang, Pattern Reognition by a Distributed Neural Ntwork An Industrial Appliation, Neural Nehvorks.vol.4,pp , [I81 H. J. Chang, W. J. Freeman, and B. C. Burke, Biologially modeled nois stabilizing neurodynamis for pattern reognition, Int J Bifurat Chaos, vol. 8, pp , [I91 R. Kama, and W.J. Freeman, Chaoti resonane ~ and appliations for robust lassifiation of noisy and variable pattms, In1 J Bifurat Chaos, vol. I1(6), pp ,