- Email: [email protected]
Why do olfactory neurons have unspecific receptive fields?
BioSystems 67 (2002) 229 /238 www.elsevier.com/locate/biosystems
Why do olfactory neurons have unspecific receptive fields? Manuel A. Sa´nchez-Montan˜e´s a,*, Tim C. Pearce b a
E.T.S. de Informa´tica, Universidad Auto´noma de Madrid, Madrid 28049, Spain Department of Engineering, University of Leicester, Leicester LE1 7RH, UK
b
Accepted 22 August 2002
Abstract Biological olfactory neurons are deployed as a population, most responding to a large variety of chemical compounds, that is, they possess unspecific receptive fields. The question of whether this unspecificity results from some physical constraint placed upon chemical transduction, or on the other hand, is beneficial to system performance is unclear. In this paper we employ the notion of Fisher information to address this question by quantifying how both the distribution and the tunings of the receptive fields within olfactory receptor populations affect the optimal estimation performance of the system. Our results show that overlapping sensory neuron tunings that respond to common chemical compounds have better estimation performance than perfectly specific tunings. Our results suggest two phenomena that might represent general principles of organization within biological sensory systems responding to multiple stimuli: maximization of the diversity of tunings and homogeneity in the distribution of these different receptive fields across the stimulus space (independent of the statistics of the input stimuli). Our model predicts that a local randomized mechanism controlling receptor specificities generates optimal multidimensional stimulus estimation, for which there is some experimental evidence from the biology. # 2002 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Population coding; Fisher information; Olfaction; Noise; Stimulus representation
1. Context The mammalian main olfactory bulb relies on input from a large population of unspecific olfactory neurons, the signals from which provide a population coded representation of complex odor stimuli. The degree of specificity of the olfactory receptors underlying perception of gen-
* Corresponding author. Fax: /34-913-482-235 E-mail address: [email protected] (M.A. Sa´nchez-Montan˜e´s).
eral odors in mammals was vividly demonstrated by Sicard and Holley after taking 74 olfactory neurons at random from the olfactory epithelium of the frog and exposing them in vitro to a series of single chemical compounds (Sicard and Holley, 1984). Fig. 1 shows the degree of unspecific tunings observed across the neuron population, the spot size relating to the spiking frequency produced by the cell in response to a single chemical compound. The results showed conclusively how odor perception in mammals is supported by neurons that have overlapping
0303-2647/02/$ - see front matter # 2002 Elsevier Science Ireland Ltd. All rights reserved. PII: S 0 3 0 3 - 2 6 4 7 ( 0 2 ) 0 0 0 8 1 - 3
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M.A. Sa´nchez-Montan˜e´s, T.C. Pearce / BioSystems 67 (2002) 229 /238
Fig. 1. Diagrammatic representation of olfactory sensory neuron activity following stimulation. The spot size is roughly proportional to spike frequency (spike/min). Neurons are identified by a serial number in the left column. Note the mix of specific to unspecific neuron responses to this subset of all possible odors. Importantly, 14 receptors within this randomly selected subpopulation of olfactory sensory neurons failed to respond to any of the odors presented (not shown). ACEacetophenone, ANI */anisole, BUT */n -butanol, CAM */DLcamphor, CDN */cyclodecanone, CIN */1,8-cineole, CYM */ p -cymene, DCI */D-citronellol, HEP */n -heptanol, ISO */isoamyl acetate, IVA */iso-valeric acid, LIM */D-limonene, MAC */methyl-amylketone, MEN */L-menthol, PHE */phenol, PHO */thiophenol, PYR */pyridine, THY */thymol, XOL */cyclohexanol, XON */cyclohexanone. Reproduced with permission from (Sicard and Holley 1984).
unspecific sensitivities of varying degrees to groups of compounds, each here with distinct tunings.
Sicard and Holley later concluded from an analysis of the 60 olfactory neuron responses shown (the other 14 neurons showed no response to the odors tested), that no pair of neurons within this randomly selected sub-population displayed identical tunings to the odors presented. This seemingly implied a bewildering diversity of olfactory neuron tunings suggesting a lack of order in the encoding of odor information. However the level of receptor protein diversity was later quantified by Buck and co-workers who cloned 18 different members of an extremely large multigene G-protein coupled receptor family, thought to be responsible for the transduction of olfactory stimuli (Buck and Axel, 1991). They provided an estimate of the number of receptor protein types based on the sequence homology of this small subpopulation to be between 300 and 1000 in mice. It is now clear that the pattern of activation across a number of such olfactory neurons enables the brain to interpret complex molecular stimuli (Laurent, 1999). Given such enormous diversity of olfactory neuron responses and receptor protein types, combined with their broadly unspecific tunings, a key question that needs to be addressed is, how does the estimation performance of a sensory system responding to many input stimuli depend upon the degree of specificity and on the distribution of receptive fields of the underlying receptor population? We see that for the general odor discrimination task (as opposed to the specialized pheromone detection task) nature deploys largely unspecifically tuned olfactory neurons (Fig. 1), yet what, if any, performance advantage does this provide? In the case of the olfactory system these are complex issues since the stimulus virtually always comprises large numbers of chemical components. In this paper we apply the Fisher information concept to this multi-component chemical stimulus case, in order to quantify any performance advantage. Unlike previous studies examining the role of the receptive field widths on estimation performance (Abbott and Dayan, 1999; Seung and Sompolinsky, 1993; Pouget et al., 1999; Zhang and Sejnowski, 1999; Wilke and Eurich, 2002) we neither impose a particular shape on the receptive
M.A. Sa´nchez-Montan˜e´s, T.C. Pearce / BioSystems 67 (2002) 229 /238
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Fig. 2. A hypothetical statistical estimator takes the response vector from a sensory neuron population and uses this for estimating the multidimensional stimulus. The receptive fields for each of the receptors are represented as tuning parameters to the sensory neuron population. Reproduced with permission from (Sa´nchez-Montan˜e´s and Pearce, 2001).
fields, nor a homogeneous distribution. In fact, it is precisely the shape and distribution of receptor tunings which are of interest in this study.
2. Fisher information Consider a population of noisy sensory neurons. We model the input as a vector s of which component j is the level of concentration of the single chemical compound j (j/1,. . ., N ). When a multi-component stimulus, s; is exposed to the system, each neuron i responds with firing rate ri following some probability distribution p(ri ½s) defined by its tuning to the stimulus and its intrinsic noise. An estimator is attempting to reconstruct the stimulus s from the population response r (Fig. 2). An optimal estimator that uses the population response r for reconstructing the stimulus s should give the real stimulus concentration values on the average over a large number of repeated presentations, that is, the mean estimate for repeated presentations of the same stimulus s should be equal to s: We call this type of estimator an ‘unbiased estimator’. Moreover, the estimate should be as close as possible to the applied stimulus when the presented stimulus is fixed (minimum variance; Deneve et al., 1999). The entries of the Fisher information matrix (FIM), Jjk/(s); are defined as (Cover and Thomas, 1991)
@ dr p(r½s) ln p(r½s) @sj @ ln p(r½s) : @sk
Jjk (s)
g
(1)
where j, k /1,. . ., N . Then for every unbiased estimator that uses the population response r for reconstructing the stimulus s : var(sˆj ½s)](J 1 (s))jj ;
(2)
where ‘var’ means variance, and sˆi is the estimation of the component j of s; j/1,. . .N . Note that the variance of an unbiased estimator is just its squared error. This well-known result is the ‘Crame´r/Rao bound’ (Cover and Thomas, 1991) and limits the performance of the best unbiased estimator we can build. Then, using Eq. (2), we can calculate the minimum estimator variance across all of the stimulus components: var(s½ˆ s)
N X j1
var(sˆj ½s)]
N X (J 1 (s))jj
(3)
j1
so the performance of the best unbiased estimator we can build is defined by the entries of the FIM, Jjk . Importantly, the psychophysical discriminability across a range of individual stimulus components in the animal can be linked directly to the global reconstruction error defined by this equation (Dayan and Abbott, 2001).
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3. Methods In order to experiment with different receptive field distributions and their effect on the global reconstruction error, we consider a sensory system consisting of an arbitrary population of 100 neurons. The input to the system is a combination of many single odor components (individual chemical compounds). Hence, the dimension of the input space is usually high. As such, we consider the stimulus to be multidimensional as opposed to previous work where the stimulus was considered to be scalar (Abbott and Dayan, 1999; Seung and Sompolinsky, 1993; Pouget et al., 1999). We now model the response of the ith sensory neuron to s: For simplicity, we approximate this as linear: ri aTi sbi hi ;
(4)
where ri is the firing rate of the neuron i , ai is the vector of sensitivities of this neuron to the different single chemical compounds (which we call the ‘receptive field (RF)’), bi is its spontaneous firing rate, and hi is its zero-mean noise. Note that linear simplification is equivalent to requiring that the firing rate of the neuron is scaled with the stimulus intensity, which is a reasonable assumption for moderate concentrations above detection threshold. Also note that here we are not imposing any constraints about the shape of the RFs themselves, which are determined uniquely by the sensitivity vectors ai :/ Finally, we make the approximation that the noise within each neuron is Gaussian and independent. This is a reasonable assumption since the receptor neurons have no lateral connections at the epithelium level and the noise processes are likely to be local to the neuron. Then the FIM of the system can be calculated as: J
R X 1 i1
s2i
ai aTi
(5)
where s2i is the noise variance in neuron i, and R is the number of receptors in the population. We assume each neuron within the population generates the same independent and identically dis-
tributed noise of variance s2. Eq. (5) gives the sum of the independent contributions to the Fisher information from each sensor. Our goal is to find the set of receptive fields (/ai ; i/1... R ) that minimizes the trace of J1, which bounds the optimal reconstruction error of the system (Eq. (3)). The free parameters to be optimized are then the individual sensitivities, aij (i /1. . . R ; j /1. . . N ) of the population. Note that aij is the sensitivity of receptor i to single chemical compound j. Therefore, we do not impose any particular distribution on the RFs, as opposed to previous work where it is usually assumed a priori an homogeneous distribution within the population. Hence we can study which distribution of RFs is optimal in terms of overall system estimation performance. Note that the numerical value of s2 is irrelevant for the optimization since it represents a constant factor multiplying the global function (Eq. (5)). If there are no additional constraints on the system, this function has no global minimum, since tr(J1)0/0 as j/ai/j 0/ and jJj"/0. Therefore, we should bound our search space in order to find the optimal configuration. This is defined by the physical constraints placed on our system, which we take as /c 5/aij 5/c . This could be interpreted as a system where each neuron can have olfactory receptor proteins interacting with intracellular currents of any type (all excitatory, all inhibitory, or mixed) and which sensitivities cannot be arbitrarily large. The biological plausibility of these constraints are discussed in the Section 5. Because the RFs are linear, the value of c has no effect on the optimal configuration, so we take c / 1. The optimal error is then calculated as a function of the number of single odor components (‘input dimension’). The global optimization is done using a standard genetic algorithm (Levine, 1998). The default parameters are used with the algorithm (see the user manual at Levine, 1998).
4. Results The results show that each neuron within the optimal system configuration can be influenced by
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Fig. 3. Properties of the optimal OSN configuration in the search space /1 5/aij 5/1. (A) odor sensitivities of an arbitrary neuron, input dimension/9. (B) number of neurons in the population with the same number of positive sensitivities (input dimension/9). In this plot we have taken into account that a receptive field ai is totally equivalent toai for the FIM (Eq. (5)). Therefore, if ai has more than four positive sensitivities, it is multiplied by /1. Diamonds: theoretic distribution considering that the probability for a sensitivity to be 1 or /1 is 0.5 (Poisson distribution). (C) Number of different tuning curves as a function of the input dimension (N ). Dashed: number of theoretically different tuning curves assuming that each sensitivity can be arbitrarily either 1 or /1. Since a and a are considered as the same receptive fields, this number is 2N1. (D) Number of different tuning curves per sensor as a function of the input dimension. Vertical bars indicate the standard deviation. Dashed: theoretical line assuming an homogeneous distribution of the different receptive fields.
any individual compound (Fig. 3a), since no neurons assume zero sensitivity to any of the input dimensions. Note that some stimuli can excite a given neuron (positive sensitivities) while others can inhibit it (negative sensitivities). In any case, the sensitivities have maximum gain, which is 1 within the constraints imposed. The distribution of
RFs across the population shows an exact Poisson distribution (Fig. 3b). That is, the probability for a given sensitivity assuming a value of 1 or /1 is independent of the values of the other sensitivities for each neuron, and is 0.5. This leads to the system having a mixture of all kinds of receptive fields (Fig. 3b). However, through chance, RFs
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Fig. 4. (A) Minimum expected squared error in the reconstruction of s in units of noise variance. Crosses: specific RFs case. Circles: unspecific RFs case. (B) specific squared-error to unspecific squared-error ratio in the reconstruction of s: The psychophysical discriminability across a range of individual stimulus components in the animal can be linked directly to the global reconstruction error defined by this equation (see Section 2).
with similar numbers of positive and negative sensitivities are more common than RFs with a dominating sign in the sensitivities (Fig. 3b). The number of potentially different receptive fields, assuming each sensitivity can be arbitrarily either 1 or /1, is shown in Fig. 3c (dashed line) as a function of the input dimension. The optimal system configuration reaches this limit with low input dimensions (Fig. 3c). For higher input dimensions, the number of potentially distinct RF configurations is greater than the number of neurons. In this case the number of RFs saturates the maximum allowable vertices of the search space reachable by the system (Fig. 3c). In other words, there is maximum diversity in the RFs configuration for every input dimension. Moreover, these different RFs are homogeneously distributed across the population (each different configuration is used by the same number of neurons as the others), see Fig. 3d. The question of how much better is this unspecific configuration compared with a specific configuration is addressed in Fig. 4. In the specific configuration each neuron is configured to respond to only one single odor component with maximum gain; that is, all the sensitivities are null but one. Furthermore, each single chemical component is assumed to be detected by an equal number of specific neurons (this configuration can
be shown to be the one which minimizes the optimal expected error given that the receptive fields are specific). The optimal estimation errors are calculated by (Eq. (3)). Fig. 4a shows that the unspecific configuration is much better than the specific one, and this difference increases linearly with the input dimension (Fig. 4b). To summarize, our results show that the optimal configuration is a mixture of receptive fields of greatest possible diversity, and their distribution across the population follows the maximum diversity and maximum homogeneity principles, obtaining an estimation performance much better than the specific configuration.
5. Discussion In this work we have studied the optimal configuration of a population of sensory neurons using the Fisher information1. The advantage of unspecific tuning over specific tuning (each neuron responds to just a single chemical compound) across the population was shown to increase linearly with the number of stimulus dimensions. 1 The relationship of this concept with mutual information in the context of population coding has been previously studied (Brunel and Nadal, 1998).
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Although clearly these results are appropriate to the case of olfaction, where the stimulus dimensionality is very high, they can be equally applied to any sensory system based upon a population code which is simultaneously coding for many distinct stimulus components. 5.1. Unspecific tuning and conferred biological advantage Our results show that the optimal configurations follows the maximum diversity and maximum homogeneity principles in the receptive field distribution, in other words the mixed tuning case produces the best performance. That is, the different receptive fields that belong to the optimal configuration have the same probability of being observed in any given neuron. However, as shown in the results, specific tunings would be unusual for high input dimensionalities. Hence this answers the question of why olfactory neurons have unspecific receptive fields. Because the distribution that maximizes Shannon’s entropy is homogeneous (Cover and Thomas, 1991), we can say that the optimal configuration of our system maximizes the entropy of the receptive field distribution. The unspecific tuning case has important biological relevance. The optimal configuration of unspecific tunings produce better sensing performance than the imposed specific tuning case for any given dimensionality of stimulus, clearly an important performance criterion for any sensory system. Such an arrangement might be preferred by nature in circumstances where sensitivity to each of a large number of stimuli is important, which we describe as the general odor discrimination task. This has the added benefit that the system is able to respond to unseen or even entirely novel stimuli and so is extremely broadly tuned to its environment (for example see Laurent, 1999 for a discussion). 5.2. Distribution of RFs and conferred biological advantage It is perhaps surprising that the RF tuning widths for the results, defined here as the fraction
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of stimuli to which each neuron responds positively, are not uniform across the population. This suggests that a mix of broad and narrowly tuned receptive fields is the optimal arrangement for a population coded system responding to multidimensional stimuli, where there is no a priori preference to any of the dimensions. Specifically, our results show a Poisson distribution of neuron tunings across the population, predicting in the biology equiprobable excitatory and inhibitory responses of single olfactory neurons to each specific dimension of the stimulus, P /0.5. There is evidence to suggest that the distribution of neuron tunings in the biology may approach the Poisson case. Although one must be careful when interpreting olfactory neuron responses when stimulus concentration plays an important role, Duchamp-Viret et al. recently recorded responses from 91 randomly selected olfactory sensory neurons from the rat to 16 pure odorants at a number of different concentration levels. The distribution of responses across the population was categorized into those neurons responding to 1, 2, 3, 4, 5, and 6 of the 16 compounds (Duchamp-Viret et al., 1999). Exactly 12, 17, 10, 7, 22, 32% of the population responded to these numbers of odors within the test set, respectively, suggesting a distribution not dissimilar to that shown in Fig. 3B (note that in their study 16, not nine, dimensional stimulus is considered). It is important to mention that this optimal arrangement of Poisson distributed olfactory neuron tunings is straightforward to implement biophysically, since this could rely upon a local randomized mechanism producing either inhibitory or excitatory responses to each stimulus component in any given neuron with equal probabilities. Although such a mechanism would imply utter disorder in the tunings when considered locally, across the population it would appear to represent the optimal tuning configuration for stimulus estimation, for any given noise level. The mixed RFs scenario observed in our results presents the intriguing possibility of clustering of sensitivities towards groups of compounds of interest to the animal that might arise phylogenetically. Superclusters of similar sequence homology in olfactory 7-transmembrane receptors
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observed from recent phylogenetic studies might reflect this aspect of our model (Zozula et al., 2001). 5.3. Bipolar sensitivities and conferred biological advantage It might be expected that permitting bipolar sensitivities in the neurons to each of the stimuli (as opposed to purely excitatory or inhibitory responses) produces the best overall estimation performance. As Schild and Restrepo (1998) state: ‘Differential stimulation or suppression of olfactory neurons by odors could be used by the olfactory system as a mechanism for contrast enhancement. In addition, the responses resulting from simultaneous stimulation and inhibition of different neurons by one odorant could be contrasted in the olfactory bulb in such a way that low odorant concentrations could be detected at signal levels that could not be resolved from noise in a purely excitatory system’. Interestingly, such suppressive or inhibitory effects of particular odors on olfactory sensory neurons has only recently been observed in mammals, whereas it is much more common in amphibians. For this reason the role of inhibitory receptor responses across species is only now becoming clear and the prediction of our model for requiring both forms of sensitivity exist is difficult to verify from the published experimental data. Even so, it is clear that excitatory OSN responses are far more common. In any case, the conclusions obtained from our model do not depend on bipolar sensitivities since the case with only excitatory sensitivities distributed following a Poisson distribution (each sensitivity has a chance of 0.5 of being 0, and 0.5 of being 1) is still better than the specific case, but worst than the bipolar optimal configuration (data not shown). There is direct evidence for such bipolar sensitivities in biological olfactory sensory neurons that might result from multiple second messenger signaling pathways (mediated by cAMP and IP3 for example) within the neuron and perhaps even
multiple receptor types within a single neuron. Fig. 5 clearly shows both inhibitory and excitatory responses to different single compounds observed in a single neuron. More recent evidence for bipolar sensitivities is given by Sanhueza et al., who observed both an excitatory cAMP-dependent current and inhibitory Ca2-dependent current in a single olfactory sensory neuron of the rat (Sanhueza et al., 2000).
5.4. Model properties In order to make the global search of the RF space feasible, some simplifications have been made to the model. For example, the neurons have been approximated as linear elements. It is reasonable to assume a linear model for low concentrations where the firing rate is approximately dependent linearly on the number of sites filled on the OSN (olfactory stimulus neuron) and there are sufficient sites relative to molecules such that no competition occurs for sites between compounds. It is important to remark that we have avoided describing the olfactory system as a scalar, as has been done in previous work (Abbott and Dayan, 1999; Seung and Sompolinsky, 1993; Pouget et al., 1999). Using the representation we have chosen it is possible to account for both the odor’s intensity as well as its chemical composition. In addition, our choice is able to account for the case when several simultaneous stimuli are present, which can not be described by just a scalar. In this work the receptive fields of all the neurons are subject to optimization. Therefore, we neither impose a functional form on the RFs nor that these are homogeneously distributed, as opposed to previous papers (Abbott and Dayan, 1999; Seung and Sompolinsky, 1993) where these assumptions lead to mathematical simplifications. The optimization is carried out using a numerical technique. An analytical approach is also possible here, but these approaches depend critically on the particular constraints placed upon the sensitivities. We have chosen an arbitrary population size of 100 neurons but additional experiments show that our results are independent of population size as
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Fig. 5. Extracellular recordings of a single olfactory sensory neuron of channel catfish, Ictalurus punctatus , to six odor stimuli and a water control. (a) no significant change from spontaneous activity to water control. (b) Inhibitory response to 10 4 M methionine (Met). (c) Inhibitory response to 10 4 M alanine (Ala). (d) Excitatory response to 10 4 M arginine (Arg). (e) Excitatory response to 4 10 3 glutamic acid (Glu). (f) Excitatory response to 3 / /10 M MBS (sodium salts of cholic acid, taurocholic acid, and taurolithocholic acid each at 10 4 M). (g) Excitatory response to 10 4 M ATP. Vertical dotted line indicates beginning of neural responses as defined by onset of simultaneously recorded electrolfactogram response (local field potential). Reproduced with permission from (Schild and Restrepo, 1998).
long as there is a sufficient number of neurons (data not shown). In conclusion, our results show that in the case of multi-dimensional stimuli where we allow both excitatory and inhibitory responses, the minimum reconstruction error is obtained when, (a) all of the input dimensions are coded by each of the neurons in the population; (b) maximum allowable gain of the sensitivities occurs in each case; (c) the distribution of tunings across the population follows a Poisson distribution; (d) the diversity of tunings across the population is maximum; and (e)
the spread of tunings across the stimulus space is homogeneous.
Acknowledgements We thank Ramo´n Huerta useful discussions about this work. We are also grateful to the comments made be the referees of this paper which have helped to improve its clarity. MAS-M was supported by grant BFI2000-0157 from MCyT. TCP was supported by grant IST-2001-33066 from
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the European Commission and GR/R37968/01 from the United Kingdom Engineering and Physical Sciences Research Council.
References Abbott, L.F., Dayan, P., 1999. The effect of correlated variability on the accuracy of a population code. Neural Comput. 11, 91 /101. Brunel, N., Nadal, J.P., 1998. Mutual information, Fisher information, and population coding. Neural Comput. 10, 1731 /1757. Buck, L., Axel, R., 1991. A novel multigene family may encode odorant receptors */a molecular-basis for odor recognition. Cell 65, 175 /187. Cover, T.M., Thomas, J.A., 1991. Information Theory. Wiley, New York. Dayan, P., Abbott, L.F., 2001. Theoretical Neuroscience. MIT Press. Deneve, S., Latham, P.E., Pouget, A., 1999. Reading population codes: a neural implementation of ideal observers. Nat. Neurosci. 2, 740 /745. Duchamp-Viret, P., Chaput, M.A., Duchamp, A., 1999. Odor response properties of rat olfactory recepter neurons. Science 284 (5423), 2171 /2174. Laurent, G., 1999. A systems perspective on early olfactory coding. Science 286 (5440), 723 /728.
Levine, D., 1998. PGAPack Parallel Genetic Algorithm Library. http://www.fp.mcs.anl.gov/CCST/research/reports_pre1998/comp_bio/stalk/pgapack.html. Pouget, A., Deneve, S., Ducom, J.C., Latham, P.E., 1999. Narrow versus wide tuning curves: what’s best for a population code. Neural Comput. 11, 85 /90. Sa´nchez-Montan˜e´s, M.A., Pearce, T.C., 2001. Fisher information and optimal odor sensors. Neurocomputing 38 /40, 335 /341. Sanhueza, M., Schmachtenberg, O., Bacigalupo, J., 2000. Excitation, inhibition, and suppresion by odors in isolated toad and rat olfactory receptor neurons. Am. J. Physiol. Cell Physiol. 279, C31 /C39. Schild, D., Restrepo, D., 1998. Transduction mechanisms in vertebrate olfactory receptor cells. Physiol. Rev. 78 (2), 429 /466. Seung, H.S., Sompolinsky, H., 1993. Simple models for reading neuronal population codes. Proc. Natl. Acad. Sci. USA 90, 10749 /10753. Sicard, G., Holley, A., 1984. Receptor cell responses to odorants: similarities and differences among odorants. Brain Res. 292, 283 /296. Wilke, S.D., Eurich, C.W., 2002. Representational accuracy of stochastic neural populations. Neural Comput. 14, 155 / 189. Zhang, K., Sejnowski, T.J., 1999. Neuronal tuning: to sharpen or broaden. Neural Comput. 11, 75 /84. Zozula, S., Echeverri, F., Nguyen, T., 2001. The human olfactory receptor repertoire. Genome Biol. 2 (6), 1 /12.
Why do olfactory neurons have unspecific receptive fields? Manuel A. Sa´nchez-Montan˜e´s a,*, Tim C. Pearce b a
E.T.S. de Informa´tica, Universidad Auto´noma de Madrid, Madrid 28049, Spain Department of Engineering, University of Leicester, Leicester LE1 7RH, UK
b
Accepted 22 August 2002
Abstract Biological olfactory neurons are deployed as a population, most responding to a large variety of chemical compounds, that is, they possess unspecific receptive fields. The question of whether this unspecificity results from some physical constraint placed upon chemical transduction, or on the other hand, is beneficial to system performance is unclear. In this paper we employ the notion of Fisher information to address this question by quantifying how both the distribution and the tunings of the receptive fields within olfactory receptor populations affect the optimal estimation performance of the system. Our results show that overlapping sensory neuron tunings that respond to common chemical compounds have better estimation performance than perfectly specific tunings. Our results suggest two phenomena that might represent general principles of organization within biological sensory systems responding to multiple stimuli: maximization of the diversity of tunings and homogeneity in the distribution of these different receptive fields across the stimulus space (independent of the statistics of the input stimuli). Our model predicts that a local randomized mechanism controlling receptor specificities generates optimal multidimensional stimulus estimation, for which there is some experimental evidence from the biology. # 2002 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Population coding; Fisher information; Olfaction; Noise; Stimulus representation
1. Context The mammalian main olfactory bulb relies on input from a large population of unspecific olfactory neurons, the signals from which provide a population coded representation of complex odor stimuli. The degree of specificity of the olfactory receptors underlying perception of gen-
* Corresponding author. Fax: /34-913-482-235 E-mail address: [email protected] (M.A. Sa´nchez-Montan˜e´s).
eral odors in mammals was vividly demonstrated by Sicard and Holley after taking 74 olfactory neurons at random from the olfactory epithelium of the frog and exposing them in vitro to a series of single chemical compounds (Sicard and Holley, 1984). Fig. 1 shows the degree of unspecific tunings observed across the neuron population, the spot size relating to the spiking frequency produced by the cell in response to a single chemical compound. The results showed conclusively how odor perception in mammals is supported by neurons that have overlapping
0303-2647/02/$ - see front matter # 2002 Elsevier Science Ireland Ltd. All rights reserved. PII: S 0 3 0 3 - 2 6 4 7 ( 0 2 ) 0 0 0 8 1 - 3
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Fig. 1. Diagrammatic representation of olfactory sensory neuron activity following stimulation. The spot size is roughly proportional to spike frequency (spike/min). Neurons are identified by a serial number in the left column. Note the mix of specific to unspecific neuron responses to this subset of all possible odors. Importantly, 14 receptors within this randomly selected subpopulation of olfactory sensory neurons failed to respond to any of the odors presented (not shown). ACEacetophenone, ANI */anisole, BUT */n -butanol, CAM */DLcamphor, CDN */cyclodecanone, CIN */1,8-cineole, CYM */ p -cymene, DCI */D-citronellol, HEP */n -heptanol, ISO */isoamyl acetate, IVA */iso-valeric acid, LIM */D-limonene, MAC */methyl-amylketone, MEN */L-menthol, PHE */phenol, PHO */thiophenol, PYR */pyridine, THY */thymol, XOL */cyclohexanol, XON */cyclohexanone. Reproduced with permission from (Sicard and Holley 1984).
unspecific sensitivities of varying degrees to groups of compounds, each here with distinct tunings.
Sicard and Holley later concluded from an analysis of the 60 olfactory neuron responses shown (the other 14 neurons showed no response to the odors tested), that no pair of neurons within this randomly selected sub-population displayed identical tunings to the odors presented. This seemingly implied a bewildering diversity of olfactory neuron tunings suggesting a lack of order in the encoding of odor information. However the level of receptor protein diversity was later quantified by Buck and co-workers who cloned 18 different members of an extremely large multigene G-protein coupled receptor family, thought to be responsible for the transduction of olfactory stimuli (Buck and Axel, 1991). They provided an estimate of the number of receptor protein types based on the sequence homology of this small subpopulation to be between 300 and 1000 in mice. It is now clear that the pattern of activation across a number of such olfactory neurons enables the brain to interpret complex molecular stimuli (Laurent, 1999). Given such enormous diversity of olfactory neuron responses and receptor protein types, combined with their broadly unspecific tunings, a key question that needs to be addressed is, how does the estimation performance of a sensory system responding to many input stimuli depend upon the degree of specificity and on the distribution of receptive fields of the underlying receptor population? We see that for the general odor discrimination task (as opposed to the specialized pheromone detection task) nature deploys largely unspecifically tuned olfactory neurons (Fig. 1), yet what, if any, performance advantage does this provide? In the case of the olfactory system these are complex issues since the stimulus virtually always comprises large numbers of chemical components. In this paper we apply the Fisher information concept to this multi-component chemical stimulus case, in order to quantify any performance advantage. Unlike previous studies examining the role of the receptive field widths on estimation performance (Abbott and Dayan, 1999; Seung and Sompolinsky, 1993; Pouget et al., 1999; Zhang and Sejnowski, 1999; Wilke and Eurich, 2002) we neither impose a particular shape on the receptive
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Fig. 2. A hypothetical statistical estimator takes the response vector from a sensory neuron population and uses this for estimating the multidimensional stimulus. The receptive fields for each of the receptors are represented as tuning parameters to the sensory neuron population. Reproduced with permission from (Sa´nchez-Montan˜e´s and Pearce, 2001).
fields, nor a homogeneous distribution. In fact, it is precisely the shape and distribution of receptor tunings which are of interest in this study.
2. Fisher information Consider a population of noisy sensory neurons. We model the input as a vector s of which component j is the level of concentration of the single chemical compound j (j/1,. . ., N ). When a multi-component stimulus, s; is exposed to the system, each neuron i responds with firing rate ri following some probability distribution p(ri ½s) defined by its tuning to the stimulus and its intrinsic noise. An estimator is attempting to reconstruct the stimulus s from the population response r (Fig. 2). An optimal estimator that uses the population response r for reconstructing the stimulus s should give the real stimulus concentration values on the average over a large number of repeated presentations, that is, the mean estimate for repeated presentations of the same stimulus s should be equal to s: We call this type of estimator an ‘unbiased estimator’. Moreover, the estimate should be as close as possible to the applied stimulus when the presented stimulus is fixed (minimum variance; Deneve et al., 1999). The entries of the Fisher information matrix (FIM), Jjk/(s); are defined as (Cover and Thomas, 1991)
@ dr p(r½s) ln p(r½s) @sj @ ln p(r½s) : @sk
Jjk (s)
g
(1)
where j, k /1,. . ., N . Then for every unbiased estimator that uses the population response r for reconstructing the stimulus s : var(sˆj ½s)](J 1 (s))jj ;
(2)
where ‘var’ means variance, and sˆi is the estimation of the component j of s; j/1,. . .N . Note that the variance of an unbiased estimator is just its squared error. This well-known result is the ‘Crame´r/Rao bound’ (Cover and Thomas, 1991) and limits the performance of the best unbiased estimator we can build. Then, using Eq. (2), we can calculate the minimum estimator variance across all of the stimulus components: var(s½ˆ s)
N X j1
var(sˆj ½s)]
N X (J 1 (s))jj
(3)
j1
so the performance of the best unbiased estimator we can build is defined by the entries of the FIM, Jjk . Importantly, the psychophysical discriminability across a range of individual stimulus components in the animal can be linked directly to the global reconstruction error defined by this equation (Dayan and Abbott, 2001).
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3. Methods In order to experiment with different receptive field distributions and their effect on the global reconstruction error, we consider a sensory system consisting of an arbitrary population of 100 neurons. The input to the system is a combination of many single odor components (individual chemical compounds). Hence, the dimension of the input space is usually high. As such, we consider the stimulus to be multidimensional as opposed to previous work where the stimulus was considered to be scalar (Abbott and Dayan, 1999; Seung and Sompolinsky, 1993; Pouget et al., 1999). We now model the response of the ith sensory neuron to s: For simplicity, we approximate this as linear: ri aTi sbi hi ;
(4)
where ri is the firing rate of the neuron i , ai is the vector of sensitivities of this neuron to the different single chemical compounds (which we call the ‘receptive field (RF)’), bi is its spontaneous firing rate, and hi is its zero-mean noise. Note that linear simplification is equivalent to requiring that the firing rate of the neuron is scaled with the stimulus intensity, which is a reasonable assumption for moderate concentrations above detection threshold. Also note that here we are not imposing any constraints about the shape of the RFs themselves, which are determined uniquely by the sensitivity vectors ai :/ Finally, we make the approximation that the noise within each neuron is Gaussian and independent. This is a reasonable assumption since the receptor neurons have no lateral connections at the epithelium level and the noise processes are likely to be local to the neuron. Then the FIM of the system can be calculated as: J
R X 1 i1
s2i
ai aTi
(5)
where s2i is the noise variance in neuron i, and R is the number of receptors in the population. We assume each neuron within the population generates the same independent and identically dis-
tributed noise of variance s2. Eq. (5) gives the sum of the independent contributions to the Fisher information from each sensor. Our goal is to find the set of receptive fields (/ai ; i/1... R ) that minimizes the trace of J1, which bounds the optimal reconstruction error of the system (Eq. (3)). The free parameters to be optimized are then the individual sensitivities, aij (i /1. . . R ; j /1. . . N ) of the population. Note that aij is the sensitivity of receptor i to single chemical compound j. Therefore, we do not impose any particular distribution on the RFs, as opposed to previous work where it is usually assumed a priori an homogeneous distribution within the population. Hence we can study which distribution of RFs is optimal in terms of overall system estimation performance. Note that the numerical value of s2 is irrelevant for the optimization since it represents a constant factor multiplying the global function (Eq. (5)). If there are no additional constraints on the system, this function has no global minimum, since tr(J1)0/0 as j/ai/j 0/ and jJj"/0. Therefore, we should bound our search space in order to find the optimal configuration. This is defined by the physical constraints placed on our system, which we take as /c 5/aij 5/c . This could be interpreted as a system where each neuron can have olfactory receptor proteins interacting with intracellular currents of any type (all excitatory, all inhibitory, or mixed) and which sensitivities cannot be arbitrarily large. The biological plausibility of these constraints are discussed in the Section 5. Because the RFs are linear, the value of c has no effect on the optimal configuration, so we take c / 1. The optimal error is then calculated as a function of the number of single odor components (‘input dimension’). The global optimization is done using a standard genetic algorithm (Levine, 1998). The default parameters are used with the algorithm (see the user manual at Levine, 1998).
4. Results The results show that each neuron within the optimal system configuration can be influenced by
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Fig. 3. Properties of the optimal OSN configuration in the search space /1 5/aij 5/1. (A) odor sensitivities of an arbitrary neuron, input dimension/9. (B) number of neurons in the population with the same number of positive sensitivities (input dimension/9). In this plot we have taken into account that a receptive field ai is totally equivalent toai for the FIM (Eq. (5)). Therefore, if ai has more than four positive sensitivities, it is multiplied by /1. Diamonds: theoretic distribution considering that the probability for a sensitivity to be 1 or /1 is 0.5 (Poisson distribution). (C) Number of different tuning curves as a function of the input dimension (N ). Dashed: number of theoretically different tuning curves assuming that each sensitivity can be arbitrarily either 1 or /1. Since a and a are considered as the same receptive fields, this number is 2N1. (D) Number of different tuning curves per sensor as a function of the input dimension. Vertical bars indicate the standard deviation. Dashed: theoretical line assuming an homogeneous distribution of the different receptive fields.
any individual compound (Fig. 3a), since no neurons assume zero sensitivity to any of the input dimensions. Note that some stimuli can excite a given neuron (positive sensitivities) while others can inhibit it (negative sensitivities). In any case, the sensitivities have maximum gain, which is 1 within the constraints imposed. The distribution of
RFs across the population shows an exact Poisson distribution (Fig. 3b). That is, the probability for a given sensitivity assuming a value of 1 or /1 is independent of the values of the other sensitivities for each neuron, and is 0.5. This leads to the system having a mixture of all kinds of receptive fields (Fig. 3b). However, through chance, RFs
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Fig. 4. (A) Minimum expected squared error in the reconstruction of s in units of noise variance. Crosses: specific RFs case. Circles: unspecific RFs case. (B) specific squared-error to unspecific squared-error ratio in the reconstruction of s: The psychophysical discriminability across a range of individual stimulus components in the animal can be linked directly to the global reconstruction error defined by this equation (see Section 2).
with similar numbers of positive and negative sensitivities are more common than RFs with a dominating sign in the sensitivities (Fig. 3b). The number of potentially different receptive fields, assuming each sensitivity can be arbitrarily either 1 or /1, is shown in Fig. 3c (dashed line) as a function of the input dimension. The optimal system configuration reaches this limit with low input dimensions (Fig. 3c). For higher input dimensions, the number of potentially distinct RF configurations is greater than the number of neurons. In this case the number of RFs saturates the maximum allowable vertices of the search space reachable by the system (Fig. 3c). In other words, there is maximum diversity in the RFs configuration for every input dimension. Moreover, these different RFs are homogeneously distributed across the population (each different configuration is used by the same number of neurons as the others), see Fig. 3d. The question of how much better is this unspecific configuration compared with a specific configuration is addressed in Fig. 4. In the specific configuration each neuron is configured to respond to only one single odor component with maximum gain; that is, all the sensitivities are null but one. Furthermore, each single chemical component is assumed to be detected by an equal number of specific neurons (this configuration can
be shown to be the one which minimizes the optimal expected error given that the receptive fields are specific). The optimal estimation errors are calculated by (Eq. (3)). Fig. 4a shows that the unspecific configuration is much better than the specific one, and this difference increases linearly with the input dimension (Fig. 4b). To summarize, our results show that the optimal configuration is a mixture of receptive fields of greatest possible diversity, and their distribution across the population follows the maximum diversity and maximum homogeneity principles, obtaining an estimation performance much better than the specific configuration.
5. Discussion In this work we have studied the optimal configuration of a population of sensory neurons using the Fisher information1. The advantage of unspecific tuning over specific tuning (each neuron responds to just a single chemical compound) across the population was shown to increase linearly with the number of stimulus dimensions. 1 The relationship of this concept with mutual information in the context of population coding has been previously studied (Brunel and Nadal, 1998).
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Although clearly these results are appropriate to the case of olfaction, where the stimulus dimensionality is very high, they can be equally applied to any sensory system based upon a population code which is simultaneously coding for many distinct stimulus components. 5.1. Unspecific tuning and conferred biological advantage Our results show that the optimal configurations follows the maximum diversity and maximum homogeneity principles in the receptive field distribution, in other words the mixed tuning case produces the best performance. That is, the different receptive fields that belong to the optimal configuration have the same probability of being observed in any given neuron. However, as shown in the results, specific tunings would be unusual for high input dimensionalities. Hence this answers the question of why olfactory neurons have unspecific receptive fields. Because the distribution that maximizes Shannon’s entropy is homogeneous (Cover and Thomas, 1991), we can say that the optimal configuration of our system maximizes the entropy of the receptive field distribution. The unspecific tuning case has important biological relevance. The optimal configuration of unspecific tunings produce better sensing performance than the imposed specific tuning case for any given dimensionality of stimulus, clearly an important performance criterion for any sensory system. Such an arrangement might be preferred by nature in circumstances where sensitivity to each of a large number of stimuli is important, which we describe as the general odor discrimination task. This has the added benefit that the system is able to respond to unseen or even entirely novel stimuli and so is extremely broadly tuned to its environment (for example see Laurent, 1999 for a discussion). 5.2. Distribution of RFs and conferred biological advantage It is perhaps surprising that the RF tuning widths for the results, defined here as the fraction
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of stimuli to which each neuron responds positively, are not uniform across the population. This suggests that a mix of broad and narrowly tuned receptive fields is the optimal arrangement for a population coded system responding to multidimensional stimuli, where there is no a priori preference to any of the dimensions. Specifically, our results show a Poisson distribution of neuron tunings across the population, predicting in the biology equiprobable excitatory and inhibitory responses of single olfactory neurons to each specific dimension of the stimulus, P /0.5. There is evidence to suggest that the distribution of neuron tunings in the biology may approach the Poisson case. Although one must be careful when interpreting olfactory neuron responses when stimulus concentration plays an important role, Duchamp-Viret et al. recently recorded responses from 91 randomly selected olfactory sensory neurons from the rat to 16 pure odorants at a number of different concentration levels. The distribution of responses across the population was categorized into those neurons responding to 1, 2, 3, 4, 5, and 6 of the 16 compounds (Duchamp-Viret et al., 1999). Exactly 12, 17, 10, 7, 22, 32% of the population responded to these numbers of odors within the test set, respectively, suggesting a distribution not dissimilar to that shown in Fig. 3B (note that in their study 16, not nine, dimensional stimulus is considered). It is important to mention that this optimal arrangement of Poisson distributed olfactory neuron tunings is straightforward to implement biophysically, since this could rely upon a local randomized mechanism producing either inhibitory or excitatory responses to each stimulus component in any given neuron with equal probabilities. Although such a mechanism would imply utter disorder in the tunings when considered locally, across the population it would appear to represent the optimal tuning configuration for stimulus estimation, for any given noise level. The mixed RFs scenario observed in our results presents the intriguing possibility of clustering of sensitivities towards groups of compounds of interest to the animal that might arise phylogenetically. Superclusters of similar sequence homology in olfactory 7-transmembrane receptors
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observed from recent phylogenetic studies might reflect this aspect of our model (Zozula et al., 2001). 5.3. Bipolar sensitivities and conferred biological advantage It might be expected that permitting bipolar sensitivities in the neurons to each of the stimuli (as opposed to purely excitatory or inhibitory responses) produces the best overall estimation performance. As Schild and Restrepo (1998) state: ‘Differential stimulation or suppression of olfactory neurons by odors could be used by the olfactory system as a mechanism for contrast enhancement. In addition, the responses resulting from simultaneous stimulation and inhibition of different neurons by one odorant could be contrasted in the olfactory bulb in such a way that low odorant concentrations could be detected at signal levels that could not be resolved from noise in a purely excitatory system’. Interestingly, such suppressive or inhibitory effects of particular odors on olfactory sensory neurons has only recently been observed in mammals, whereas it is much more common in amphibians. For this reason the role of inhibitory receptor responses across species is only now becoming clear and the prediction of our model for requiring both forms of sensitivity exist is difficult to verify from the published experimental data. Even so, it is clear that excitatory OSN responses are far more common. In any case, the conclusions obtained from our model do not depend on bipolar sensitivities since the case with only excitatory sensitivities distributed following a Poisson distribution (each sensitivity has a chance of 0.5 of being 0, and 0.5 of being 1) is still better than the specific case, but worst than the bipolar optimal configuration (data not shown). There is direct evidence for such bipolar sensitivities in biological olfactory sensory neurons that might result from multiple second messenger signaling pathways (mediated by cAMP and IP3 for example) within the neuron and perhaps even
multiple receptor types within a single neuron. Fig. 5 clearly shows both inhibitory and excitatory responses to different single compounds observed in a single neuron. More recent evidence for bipolar sensitivities is given by Sanhueza et al., who observed both an excitatory cAMP-dependent current and inhibitory Ca2-dependent current in a single olfactory sensory neuron of the rat (Sanhueza et al., 2000).
5.4. Model properties In order to make the global search of the RF space feasible, some simplifications have been made to the model. For example, the neurons have been approximated as linear elements. It is reasonable to assume a linear model for low concentrations where the firing rate is approximately dependent linearly on the number of sites filled on the OSN (olfactory stimulus neuron) and there are sufficient sites relative to molecules such that no competition occurs for sites between compounds. It is important to remark that we have avoided describing the olfactory system as a scalar, as has been done in previous work (Abbott and Dayan, 1999; Seung and Sompolinsky, 1993; Pouget et al., 1999). Using the representation we have chosen it is possible to account for both the odor’s intensity as well as its chemical composition. In addition, our choice is able to account for the case when several simultaneous stimuli are present, which can not be described by just a scalar. In this work the receptive fields of all the neurons are subject to optimization. Therefore, we neither impose a functional form on the RFs nor that these are homogeneously distributed, as opposed to previous papers (Abbott and Dayan, 1999; Seung and Sompolinsky, 1993) where these assumptions lead to mathematical simplifications. The optimization is carried out using a numerical technique. An analytical approach is also possible here, but these approaches depend critically on the particular constraints placed upon the sensitivities. We have chosen an arbitrary population size of 100 neurons but additional experiments show that our results are independent of population size as
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Fig. 5. Extracellular recordings of a single olfactory sensory neuron of channel catfish, Ictalurus punctatus , to six odor stimuli and a water control. (a) no significant change from spontaneous activity to water control. (b) Inhibitory response to 10 4 M methionine (Met). (c) Inhibitory response to 10 4 M alanine (Ala). (d) Excitatory response to 10 4 M arginine (Arg). (e) Excitatory response to 4 10 3 glutamic acid (Glu). (f) Excitatory response to 3 / /10 M MBS (sodium salts of cholic acid, taurocholic acid, and taurolithocholic acid each at 10 4 M). (g) Excitatory response to 10 4 M ATP. Vertical dotted line indicates beginning of neural responses as defined by onset of simultaneously recorded electrolfactogram response (local field potential). Reproduced with permission from (Schild and Restrepo, 1998).
long as there is a sufficient number of neurons (data not shown). In conclusion, our results show that in the case of multi-dimensional stimuli where we allow both excitatory and inhibitory responses, the minimum reconstruction error is obtained when, (a) all of the input dimensions are coded by each of the neurons in the population; (b) maximum allowable gain of the sensitivities occurs in each case; (c) the distribution of tunings across the population follows a Poisson distribution; (d) the diversity of tunings across the population is maximum; and (e)
the spread of tunings across the stimulus space is homogeneous.
Acknowledgements We thank Ramo´n Huerta useful discussions about this work. We are also grateful to the comments made be the referees of this paper which have helped to improve its clarity. MAS-M was supported by grant BFI2000-0157 from MCyT. TCP was supported by grant IST-2001-33066 from
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the European Commission and GR/R37968/01 from the United Kingdom Engineering and Physical Sciences Research Council.
References Abbott, L.F., Dayan, P., 1999. The effect of correlated variability on the accuracy of a population code. Neural Comput. 11, 91 /101. Brunel, N., Nadal, J.P., 1998. Mutual information, Fisher information, and population coding. Neural Comput. 10, 1731 /1757. Buck, L., Axel, R., 1991. A novel multigene family may encode odorant receptors */a molecular-basis for odor recognition. Cell 65, 175 /187. Cover, T.M., Thomas, J.A., 1991. Information Theory. Wiley, New York. Dayan, P., Abbott, L.F., 2001. Theoretical Neuroscience. MIT Press. Deneve, S., Latham, P.E., Pouget, A., 1999. Reading population codes: a neural implementation of ideal observers. Nat. Neurosci. 2, 740 /745. Duchamp-Viret, P., Chaput, M.A., Duchamp, A., 1999. Odor response properties of rat olfactory recepter neurons. Science 284 (5423), 2171 /2174. Laurent, G., 1999. A systems perspective on early olfactory coding. Science 286 (5440), 723 /728.
Levine, D., 1998. PGAPack Parallel Genetic Algorithm Library. http://www.fp.mcs.anl.gov/CCST/research/reports_pre1998/comp_bio/stalk/pgapack.html. Pouget, A., Deneve, S., Ducom, J.C., Latham, P.E., 1999. Narrow versus wide tuning curves: what’s best for a population code. Neural Comput. 11, 85 /90. Sa´nchez-Montan˜e´s, M.A., Pearce, T.C., 2001. Fisher information and optimal odor sensors. Neurocomputing 38 /40, 335 /341. Sanhueza, M., Schmachtenberg, O., Bacigalupo, J., 2000. Excitation, inhibition, and suppresion by odors in isolated toad and rat olfactory receptor neurons. Am. J. Physiol. Cell Physiol. 279, C31 /C39. Schild, D., Restrepo, D., 1998. Transduction mechanisms in vertebrate olfactory receptor cells. Physiol. Rev. 78 (2), 429 /466. Seung, H.S., Sompolinsky, H., 1993. Simple models for reading neuronal population codes. Proc. Natl. Acad. Sci. USA 90, 10749 /10753. Sicard, G., Holley, A., 1984. Receptor cell responses to odorants: similarities and differences among odorants. Brain Res. 292, 283 /296. Wilke, S.D., Eurich, C.W., 2002. Representational accuracy of stochastic neural populations. Neural Comput. 14, 155 / 189. Zhang, K., Sejnowski, T.J., 1999. Neuronal tuning: to sharpen or broaden. Neural Comput. 11, 75 /84. Zozula, S., Echeverri, F., Nguyen, T., 2001. The human olfactory receptor repertoire. Genome Biol. 2 (6), 1 /12.