Inferring population statistics of receptor neurons sensitivities and firing-rates from general functional requirements
IInferring population statistics of receptor neurons sensitivitiesand firing-rates from general functional requirements
Carlo Fulvi Mari Abstract
On the basis of the evident ability of neuronal olfactory systems to evaluate the intensity of an odorous stimulus and atthe same time also recognise the identity of the odorant over a large range of concentrations, a few biologically-realistichypotheses on some of the underlying neural processes are made. In particular, it is assumed that the receptor neuronsmean firing-rate scale monotonically with odorant intensity, and that the receptor sensitivities range widely acrossodorants and receptor neurons hence leading to highly distributed representations of the stimuli. The mathematicalimplementation of the phenomenological postulates allows for inferring explicit functional relationships between somemeasurable quantities. It results that both the dependence of the mean firing-rate on odorant concentration and thestatistical distribution of receptor sensitivity across the neuronal population are power-laws, whose respective exponentsare in an arithmetic, testable relationship.In order to test quantitatively the prediction of power-law dependence of population mean firing-rate on odorantconcentration, a probabilistic model is created to extract information from data available in the experimental literature.The values of the free parameters of the model are estimated by an info-geometric Bayesian maximum-likelihood inferencewhich keeps into account the prior distribution of the parameters. The eventual goodness of fit is quantified by meansof a distribution-independent test.The probabilistic model results to be accurate with high statistical significance, thus confirming the theoreticalprediction of a power-law dependence on odorant concentration. The experimental data available about the distributionof sensitivities also agree with the other predictions, though they are not statistically sufficient for a very stringentverification. Furthermore, the theory suggests a potential evolutionary reason for the exponent of the sensitivity power-law to be significantly different from the unit. The power-law dependence on concentration is consistent with thepsychophysical Stevens Law.On the whole, from the formalisation of just a few phenomenological observations a compact model is derived thatmay fit experimental findings from several levels of research on olfaction.
Keywords:
Neural coding, Distributed representation, Olfaction, Receptor sensitivity, Affinity distribution,Concentration invariance, Power-law.DOI: 10.1016/j.biosystems.2020.104153 Ref.: C. Fulvi Mari,
BioSystems (2020) 104153. c (cid:13)
1. Introduction
The sensory system of an animal must generally be ableto evaluate the intensity of a stimulus as well as to recog-nise its identity, or possibly determine that it is not amongthose already learned, for reasons of survival of the indi-vidual or of the species (e.g., food search by odorant con-centration gradient and friend/foe discrimination). Thisability is in fact common to all sensory modalities of mostanimal species, including humans, and has long been ofinterest to psychophysics and neurophysiology, but deter-mining its functional underpinnings has been a challengingtask. In these regards, the olfactory system lends itself to the most extensive and accurate investigations because ofthe considerable simplicity and consistency of its architec-ture and basic functions across many species.In the olfactory systems of insects and vertebrates theprocessing stages are compartmentalised and well localisedin a few small regions, the connectivity between themis quite simple, and their neuronal populations are rela-tively easy to access in electrophysiological and fluores-cence imaging experiments, also in vivo and simultaneous.Additionally, the phenotype of the olfactory system, es-pecially in its front end, that is, the olfactory receptorneurons, is determined by the genotype in such a directway that the system is very amenable to genetic manipu-lations as in, for example, gene knockout and optogeneticsexperiments. All this, together with advances in physi- Preprint submitted to Elsevier Submitted 7 February 2020 – Revised 7 April 2020 a r X i v : . [ q - b i o . N C ] M a y logy methods and technologies, has allowed for detailedexperimental studies not only on the structure of the sub-systems but also on their respective functions, in partic-ular concerning the encoding of stimuli by the neuronalpopulations.The main focus of the present article is on the statis-tical properties of the olfactory system front end, namelyreceptors and receptor neurons, and on the neural codingof odour intensity. It will be shown how both the statis-tical distribution of receptor affinity and the dependenceof receptor neurons mean firing-rate on odorant intensitycan be mathematically derived from just three simple phe-nomenological assumptions. The predictions of the math-ematical theory will be verified against experimental datathat are already available on highly reputable publicationsand then used to propose directions of further experimen-tal investigations, also involving evolutionary elements.
2. Essentials of an olfactory system
The basic functional anatomy of olfactory systems isquite simple and largely shared across very many species,from insects to mammals. In this Section, only a gen-eralised qualitative description is presented, focusing onthe main common features, and therefore it should onlybe considered as minimal information for the justifica-tion of the model assumptions; much more informationis provided in the cited references and reviews (Grabe andSachse, 2018; Su et al., 2009; Hildebrand and Shepherd,1997) along with the references therein, the relevant liter-ature being very conspicuous. An effort, likely only par-tially successful, was also made to cite the precursors of theexperimental or theoretical work here explicitly instanced,including those that have been historically followed by sev-eral other publications of ever increasing accuracy and so-phistication as further developments.The front end of a typical olfactory system is made of olfactory receptor neurons (ORNs), whose cilia are more orless directly exposed to the flow of the medium in whichthe animal lives. Each cilium expresses olfactory recep-tors (ORs) to which the odorant molecules in the mediummay bind. When a ligand binds with a receptor, a cas-cade of biochemical reactions inside the ORN is triggeredthat leads to the opening of membrane ion channels. Asa consequence, influx and efflux of various types of ionstake place, hence altering the difference of electric poten-tial between the inside and the outside of the cell, whichat rest is negative. In most cases the effect is one of de-polarisation (excitatory); if the relative magnitude of theelectric potential is brought up to a certain threshold, anaction potential, or spike , is generated that propagates for-ward through the axon of the ORN to downstream neuralstages. In a minority of cases, the effect of the bindingis one of hyperpolarisation (inhibitory), so that the usu-ally present spontaneous activity of the ORN is reducedor even completely suppressed. The activation cascadeis immediately followed by a negative feedback sequence that quickly compensates for the consequences of the for-mer (Kurahashi and Shibuya, 1990; Firestein et al., 1990;Zufall and Leinders-Zufall, 2000).The affinity between any OR and odorant may dif-fer greatly across ORs and odorants. It has been clearlyshown that generally each ORN expresses one only typeof OR on its cilium, directly defined by a gene, and thatseveral types of ORs are expressed across the populationof ORNs (Buck and Axel, 1991). Because of all this, anyodorant will normally evoke a pattern of distributed gradedactivity across the population of ORNs (Malnic et al.,1999; Rubin and Katz, 1999; Duchamp-Viret et al., 1999),therefore providing a means to produce a different repre-sentation for any of a large number of odorants over a widerange of concentrations.Each ORN projects its axon into only one of the down-stream glomeruli that lie within the olfactory bulb in ver-tebrates or the antennal lobe in insects. The axons of theORNs of the same class, that is, of the ORNs that expressthe same OR type, converge into the same glomeruli, andeach glomerulus only receives afferents from ORNs of thesame class. Each glomerulus is innervated by the apicaldendrite of a relay neuron , which does not innervate anyother glomerulus, so that the pattern of spiking activityof the relay neurons faithfully reflects the ORs activation,then conveying information about the odorant to the cen-tral nervous system.When the population of ORNs is exposed to an odor-ant, a pattern of spiking activity is evoked. As the con-centration of the odorant pulses is increased, already ac-tive ORNs increase their respective firing-rates (FRs), un-less they are already at saturation, and more and moreORNs are recruited as they are activated to the point ofgenerating spikes. This of course changes the populationoutput pattern (Ma and Shepherd, 2000; Stopfer et al.,2003; Wachowiak et al., 2002) and therefore some kindof mechanism for concentration invariance has to act ina downstream system in order to maintain the identityof the odorant, at least for an ecologically relevant range.The simplest way that may be conceived to normalise con-centration dependence is naturally some form of divisiveinhibition whose amplitude scale monotonically with themean FR of the afferent pattern. Indeed, it was discov-ered that normalising divisive inhibition of relay neuronsexists in the antennal lobe of insects (Olsen et al., 2010);as the structure of the glomerular network in vertebratesis anatomically and functionally very similar to that of in-sects, it seems most likely that divisive normalisation takesplace there too. It is also known that at least some odor-ants may be identified differently if they are presented at The ORNs cilia are usually immersed in a watery layer; therefore,the concentration that matters is the one of the odorant moleculesthat, after absorption and diffusion, are present within such mucusrather than the concentration in the carrier medium. When measur-ing the concentration in the medium, affinity should be intended asan effective affinity.
3. Model, analysis and predictions
Because the odorant-induced spiking of any ORN isquickly reduced or suppressed by the intracellular nega-tive feedback, the spikes considered in this work are onlythose generated within the first 500ms after the stimulusdelivery, that is, before the negative feedback becomes rel-evant. Also, because of the
OR type ↔ ORN class ↔ relay neuron correspondence, from this point onwards inthis article it will be conventionally assumed that the ORof each type is expressed by one and only one ORN, sothat the total number of ORNs is equal to the total num-ber of OR types (or, more simply, ORs). The sensitivity ofany ORN to any odorant molecule will then be synonymicto the OR affinity to that same molecule, unless explicitlystated otherwise. As it will appear in the following, thissimplification will cause no loss of generality.The mean FR across a population of ORNs for anygiven odorant at concentration c is defined by ν ( c ) . = 1 M M (cid:88) i =1 ν i ( c ) , (1)where M is the number of ORNs and ν i is the FR of ORN i . Having assumed that each ORN expresses a differentOR, every ORN can be labelled by its sensitivity to thepresented odorant. Therefore, the FR of the ORN withsensitivity K to the odorant that is presented at concen-tration c is here indicated with ν ( K, c ).Given an odorant and its concentration, because ofthe existence of the threshold voltage for neuronal action-potential generation, only the ORNs with sufficiently largesensitivity to that odorant will respond to the stimulus.The value of the sensitivity that corresponds to such thresh-old is a function of the odorant concentration here indi-cated with (cid:98) K ( c ). For M sufficiently large, postulating thaton such large scale the possible statistical dependence be-tween sensitivities is of minor importance, one can makethe approximation ν ( c ) (cid:39) (cid:90) + ∞ (cid:98) K ( c ) dx f K ( x ) ν ( x, c ) , (2) where f K is the probability density function (PDF) of thesensitivity.Information on the concentration of an odorant is con-veyed by the mean FR of the ORNs and, as described inSection 2, the downstream systems must also be able tocompensate for changes of concentration in order to main-tain a stable representation of the identity of the odorant.In view of the feasibility of normalisation mechanisms as,e.g., divisive inhibition, it seems reasonable to postulatethat the mean FR scales monotonically with the odorantconcentration: ν ( η c ) = g ( η ) ν ( c ) , (3)where η ∈ R + , g is a differentiable monotonically increas-ing function on R + , and naturally g (1) = 1.The third and last postulate requires that the FR ofany ORN is a function of the sensitivity K and of theodorant concentration c through their product Kc only,that is (with some abuse of notation) ν ( K, c ) = ν ( Kc ) . (4)This assumption would be certainly correct if the stimulus-evoked cross-membrane current I was proportional to thenumber of ligand-receptor bindings as derived from theMass Action Law (MAL) of the reaction L + R (cid:29) LR ,where L and R represent, respectively, a ligand (odorantmolecule) and a receptor on the ORN cilium. However, ithas been known for long (Hill, 1910; Monod et al., 1965)that in biological systems the binding of a ligand to areceptor may modulate (e.g., allosterically) the probabil-ity that another ligand bind to the same receptor or oneclose to it (Prinz, 2010). The (statistical) consequenceof this complex of phenomena at the molecular level isreflected into the Hill coefficient γ in the Hill formula I ∝ / [1 + ( Kc ) − γ ] not being equal to 1, as it would be forthe simple MAL, or any other positive integer, as it wouldbe for the MAL of γL + R (cid:29) L γ R . In fact, the Ca influxcurrent of the ORNs in the experiments by Si et al. (2019)was shown by the authors to be very well described by aHill function with γ (cid:39) .
42. The hypothesis that the de-pendence be on the product Kc may also be derived fromphysical dimensional arguments.By differentiating both sides of Eq.3 by η and thensetting η = 1, one obtains that ν (cid:48) ( c ) = βc ν ( c ) , (5)where β . = g (cid:48) (1) >
0, from which it follows that ν ( c ) ∝ c β . (6)Changing integration variable in Eq.2 by x (cid:55)→ y = c x ,then differentiating both sides by c , and finally makinguse of Eq.5, some calculus leads to (cid:90) + ∞ x dy f K (cid:16) yc (cid:17) ν ( y ) (cid:34) β + yc f (cid:48) K (cid:0) yc (cid:1) f K (cid:0) yc (cid:1) (cid:35) = 0 , ∀ c ∈ R + , (7)3here x = c (cid:98) K ( c ), which does not depend on c becauseof the third postulate. It follows that K f (cid:48) K ( K ) f K ( K ) = − − β, (8)from which f K ( K ) ∝ K − α , (9)where α . = 1 + β , which is a further testable prediction, asthe PDF of the sensitivity and that of the mean FR canbe separately estimated empirically.
4. Comparison with experimental data
To test the predicted dependence of mean FR on odor-ant concentration, experimental data was taken from Ta-ble S2 of Hallem and Carlson (2006), a study on the
Dro-sophila Melanogaster olfaction , which provides the FRsof 24 ORNs exposed to 10 odorants, each at 4 differentconcentrations, one combination at a time (dimensionlessdilution values: 10 − , 10 − , 10 − , 10 − ), measured fromcounting the number of spikes within 500ms after stimulusdelivery and averaging over 4-to-6 trials. The actual meanFRs are then plotted in Fig.1a, where each point-symbolrefers to one of the 10 odorants at the 4 different concen-trations, on a log–log scale ; the seemingly linear trend isalready suggestive of a power-law relationship. On the basis of a preliminary survey of the dataset,a probabilistic model is here proposed that takes into ac-count several sources of variability and allows for a morereliable analysis of the data and, hence, for extracting moreaccurate information from them. The statistical validity ofsuch model will be duly tested against the data. As it willappear more clearly in the following, one of the advantagesof building this model for the experimental data is that, atsome stage, several subsets of data-points can be pooledtogether, hence, in a way, providing a larger dataset forstatistical inference.The firing-rate Y of any ORN is modelled with thefunction Y = A e W c β + E , (10) Because of the ease of access and of the amenability to geneticmanipulation, electrophysiology and fluorescence imaging of fairlylarge populations of neurons, the
Drosophila ’s has become the bestknown and studied animal model for olfactory systems. The Table of Hallem and Carlson (2006) presents some negativevalues because the mean spontaneous activity was subtracted fromthe after-stimulus FRs, and some odorants had inhibitory effect onsome ORNs. As ORNs that do not fire at all, because inhibited orjust not excited enough by the odorant, obviously do not contributeto the actual stimulus-related mean FR of the population, which isthe object of this work, all negative values in the Table are here resetto zero. where: the positive random variable (RV) A is determinedby the choice of odorant; W is a Gaussian RV of meanzero, determined by the choice of the pair concentration-odorant; E is a zero-mean noise that originates from thefluctuations of spike-counts between trials. The RVs A , W and E are independent from each other. The RV A is constant across measurements that use the same odor-ant as stimulus and is, in a way, equivalent to a quenched noise, while W is constant across trials that use the sameodorant at the same concentration, and E is independentlyvariable between measurements with any odorant at anyconcentration. The assumption that the standard devia-tion (s.d.) of E be independent from concentration andodorant is consistent with data presented in figure n. 3 ofHallem and Carlson (2006), where the range of concentra-tion covers four orders of magnitude for 10 odorants and4 ORNs and where the s.d. also appears to be generallysmall in comparison to the FR.Both sides of Eq.10 are formally rescaled by a constantequal to 1 Hz, so that the quantities involved as well asthe experimental figures become all dimensionless, main-taining the same symbols. Taking the logarithm of bothsides of the equation, one obtainsln Y = β ln c + ln A + W + ln (cid:18) E c − β A e W (cid:19) . (11)The term with noise E in the RHS also involves a depen-dence on c , which implies that the variance of ln( Y ) withrespect to the joint PDF of all the random variables de-pends on the concentration. However, such dependenceresults to be statistically negligible, as also appears vi-sually from Fig.1a: for the lowest and the highest con-centrations, there is some boundary effect (accumulationtowards the extremes of activity, namely, quiescence andsaturation respectively), but the variances in the two mid-dle concentrations are statistically equivalent and similarin magnitude to the other two, despite the range of con-centrations covering four orders of magnitude. It followsthat the contribution of noise E is of minor relevance; infact, this should not come as a surprise, for the noise term E is, in a way, a remnant of fluctuations already averagedover (same odorant, same concentration) trials and thenORNs. Accordingly with this observation, the last termin Eq.11 will be ignored from this point onwards; the finalverification of the model will support such approximationas statistically legitimate.Each realisation of the RVs, after self-explanatory re-naming, is written as y ij = β x i + a j + W ij , where i and j run on the sets of C concentrations and D odorants, re-spectively. In the following, for convenience of notation,the unknown parameters ( a j ) j will be considered as thecomponents of the D -column a , the controlled parameters( x i ) i as the components of the C -column x , and the mea-sured values ( y ij ) ij as the components of the C × D -matrix y .4 -8 -6 -4 -2 (a) P opu l a t i on m ean f i r i ng -r a t e ( H z ) Concentration -4 -3 -2 -1 -8 -6 -4 -2 (b) P opu l a t i on sc a l ed m ean a c t i v i t y Concentration
Figure 1: Plots of relevance to the data analysis: (a) Raw data plot of population mean firing-rate (Hz) vs. odorants dilution (dimensionless),separately for each odorant (same symbol for the same odorant; log scale); (b) Firing-rates rescaled by the best estimate of the quenchednoise and best-fit straight line, the data from all odorants having being pooled (slope (cid:39) The values in y and x are provided by the experiments,while the parameters to estimate are β , a , and the variance σ of the Gaussian W . In order to lighten the notation,the controlled experimental variable x will be tacitly con-sidered as given without uncertainty. Therefore, given thedataset y , the probability density that the true values ofthe unknown parameters be σ , β and a is, by the Bayesformula, p ( σ , β, a | y ) = p ( y | σ , β, a ) p ( σ , β, a ) / p ( y ) , (12)where p ( σ , β, a ) is the prior for the unknowns. In order tofind the parameters values that are most likely to be thetrue ones given the empirical data, one can look for thevalues that maximise p ( σ , β, a | y ) or, more conveniently,its logarithm. If one ignores the prior p ( σ , β, a ), this ap-proach coincides with the standard Maximum LikelihoodMethod. Ignoring the prior is clearly equivalent to assum-ing arbitrarily that the prior is a positive constant in asufficiently wide range and zero outside of it, as if thatwas the best way to mathematically represent null bias.In general, when the number of independent data-pointsis large in comparison to the number of unknowns, thecontribution of the prior to the estimate is usually negligi-ble, as appears when taking the logarithm. In the presentcase, the number of parameters to estimate is D + 2 = 12and the number of data-points, once pooled together, is C · D = 48; therefore, being these figures of similar mag-nitude, it would be inappropriate to ignore the prior. Nullbias on the prior is realised here as a uniform distribu-tion over a (sub-)space of the PDFs rather than on thatof the parameters; this idea belongs with the theory ofInformation Geometry (Amari, 1985), built on the semi-nal works of ˇCencov (1972), Rao (1945), Hotelling (1930), Jeffreys (1946), and Fisher (1925). Using the parametersas coordinates on a probability manifold and consideringthe submanifold spanned by the parameters ranges, givingequal weight to all volume elements of the submanifoldtranslates generally into a non-uniform weighing of vol-ume elements of the parameters space, the weight beinggiven by √ det g , where g is the Riemannian metric on themanifold. It should be stressed that this approach is notentirely exempt from arbitrariness, for only a submanifoldis taken into consideration rather than the whole spaceof PDFs, but is nevertheless much more representative ofnull-bias than the assumption of uniform PDF on the spaceof parameters. It was proved by ˇCencov (1972) and later,differently and more generally, by Campbell (1986) that,under what are essentially just consistency requirements,the only metric suitable for a probability manifold mustbe, but for an inconsequential arbitrary scaling constant,equal to g uv = (cid:90) dy p ( y | σ , β, a ) · (13) · (cid:2) ∂ u ln p ( y | σ , β, a ) (cid:3) · (cid:2) ∂ v ln p ( y | σ , β, a ) (cid:3) , known as the Rao-Fisher metric, where u and v are anytwo of the parameters, which in this case are σ , β and allthe components of a . Because the outcomes of the RV W are independent across concentration-odorant pairs, onehas that p ( y | σ , β, a ) = (cid:89) i,j √ πσ exp (cid:40) − ( y ij − βx i − a j ) σ (cid:41) , (14)where i and j run, as in previous expressions, over the C concentrations and the D odorants respectively. From thisequation, one can calculate the components of the metric g and hence its determinant, resulting in √ det g ∝ σ − ( D +3) ,5here, again, the proportionality coefficient is an inconse-quential positive constant and D is the number of odor-ants, that is, D +3 is the number of parameters to estimateplus one. The values of the parameters that maximise thelogarithm of the LHS of Eq.12 are given by: (cid:98) β = xy − x yx − x (cid:98) a j = 1 C (cid:88) i y ij − (cid:98) β x (cid:99) σ = 1 CD + D + 3 (cid:88) i,j (cid:16) y ij − (cid:98) β x i − (cid:98) a j (cid:17) (15)where the caps indicate the estimates and the over-bar in-dicates the arithmetic mean over all the C · D data-points(e.g., y = (cid:80) i,j y ij / ( CD )). The estimates of β and of a result to be unbiased. The estimate of σ , instead, is bi-ased, which is not uncommon in maximisation estimations;in order to become unbiased, it only has to be multipliedby the factor ( CD + D + 3) / ( CD − D − (cid:98) β (cid:39) . (cid:99) σ (cid:39) . a , it is possible now to derive from the original dataset anodorant-independent (pseudo-)sample that pools togetherall of its C · D = 40 data-points by rescaling the FRs,that is, { y ij − (cid:98) a j , ∀ i, j } . The relative plot vs. the odorantconcentration is shown in Fig.1b together with the best-fitstraight line. The empirical cumulative distribution func-tion (eCDF) of the residuals is (cid:98) F N ( t ) . = 1 N N (cid:88) n =1 T ( r n ≤ t ) , ∀ t ∈ R , (16)where T is the logical truth function, which is equal to 1if the argument is true and to 0 if it is not, N . = CD ,and ( r n ) n are the residual values. The Glivenko-Cantellitheorem guarantees that the eCDF converges uniformly tothe theoretical CDF almost surely in the limit of infinitei.i.d. data-points. The eCDF is plotted against the best-fitGaussian CDF in Fig.2; the apparent goodness of the fitwill be quantified by means of a distribution-independenttest. The plausibility of the model is evaluated using theeCDF; the first, immediate advantage of this approach isthat it does not require ‘binning’, whose intrinsic arbi-trariness affects many other statistical tests. To quantifyhow well the model obtained in the previous Sections fitsthe data, use is made here of the goodness-of-fit (GOF) Smirnov-Cram´er-von Mises (SCvM) test: ω N . = N (cid:90) dF ( t ) (cid:104) (cid:98) F N ( t ) − F ( t ) (cid:105) . (17)The SCvM test belongs in the distribution-independentSmirnov subfamily (Smirnov, 1937) of the Cram´er-von Misestests (Cram´er, 1928; von Mises, 1931). That the test isdistribution-independent is made even more evident by thealgebraic explicitation: ω N = ( z n ) n ↑ (cid:88) n (cid:20) F ( z n ) − n − N (cid:21) + 112 N , (18)where the up-arrow specifies that the terms in the sequenceof data-points ( z n ) n are in ascending order; indeed, assum-ing that the data-points are from independent trials, thePDF of F is uniform as long as F is a continuous CDF.Therefore, the PDF of the test ω N itself only depends onthe number of data-points. In the numerical evaluation ofthe test variable, F is a Gaussian CDF with mean zeroand variance (cid:99) σ . Although the asymptotic distribution ofthe SCvM test is known to be the integral of the square ofthe canonical Brownian Bridge stochastic process and canbe expressed explicitly as a series (Smirnov, 1937; Doob,1949; Kac, 1949), the small size of the dataset makes re-lying on such analytical derivation incautious. Therefore,an evaluation of the distribution of the test-variable wasmade through a Monte Carlo simulation with one millionrealisations over a set of the appropriate size ( N = 40; thepseudo-random number generation algorithm was adaptedfrom routine ran2.c of Press et al. (2007), with double-precision arithmetic) . The Monte Carlo results show thatthe probability of the SCvM test-variable taking any valuebelow 0.02739 (critical value) by mere chance is 1.5%,while the critical value for 1.0% is 0.02522; the value ob-tained using Eq.18 is ω (cid:39) . p (cid:39) . p < .
5. Conclusions
Biological olfactory systems are generally capable ofmaintaining the same odorant identification across a widerange of its concentration. This work was based on thefollowing general hypotheses: (1) The olfactory systemscompensate for changes of concentrations by normalisa-tion; (2) The sensitivity K of any ORN and the concen-tration c of the presented odorant only enter the equations In order to eliminate the need for sorting routines, a less compactbut equivalent algebraic explicit expression for ω N can be easily de-rived from Eq.17 without assuming that the data values are ordered.Alternatively, use can be made of the ziggurat algorithm of Marsagliaand Marsaglia (2004), which directly generates ordered sets of uni-form variates. One could also easily modify the formula to keep intoaccount possible multiplicity of any data value, though with N = 40and 64-bit precision arithmetic this possibility can be safely ignored. C u m u l a t i v e d i s t r i bu t i on Residual value
Figure 2: Empirical (crosses) and model (dashed line) cumulative distributions of residuals (all residuals pooled together; p < . of the response dynamics through the product Kc ; (3) Thesensitivity is statistically broadly distributed over ORNsand odorants and the statistical dependence between sen-sitivities is of minor relevance when considering statisticsof large sets of ORNs and odorants.Hypothesis (1) was translated into a mathematical re-lationship stating that the scaling of the concentration ofthe odorant leads to a corresponding scaling of the meanFR of the population of ORNs. Together with hypotheses(2) and (3), this lead mathematically to three predictions:(a) The population mean FR is a power-law function ofthe odorant concentration; (b) The PDF of the sensitivityacross the population obeys a power-law; (c) The expo-nent of the power-law of the sensitivity is equal to that ofthe power-law of the concentration dependence plus one.All hypotheses and predictions have to be intended to bevalid over a wide range of the relative variables but notover the whole range, as boundary and saturation effectsmay intervene.Prediction (a) was tested against publicly available ex-perimental data. In order to accurately analyse the data,a probabilistic model was introduced which takes into ac-count separately three possible sources of randomness: odor-ant-dependent finite-size effect, multiplicative noise, andtrial-by-trial spike-count variability. The values of the pa-rameters were estimated by means of info-geometric Bayes-ian maximisation followed by prefactor adjustment for un-biasedness, and the GOF was quantified by means of afinite-size distribution-independent SCvM test. The esti-mated model results to have p < . (cid:98) β (cid:39) .
23, which leads to the prediction for the exponent ofthe sensitivity power-law (cid:98) α (cid:39) .
23. Experimental data arealso in agreement with predictions (b) and (c), but theirsupport is weaker than for prediction (a) because of thelarge uncertainties in the datasets and relative extrapola-tions. Specifically, Si et al. (2019) extrapolated from theirdata sensitivity values that they found to follow a power-law PDF with exponent equal to about 1.42, but the rela-tive statistical uncertainty is quite large and, additionally,they experimented on the Drosophila larva instead of theadult insect.
The physical interpretation of the random variable A is that of a finite-size effect: the number of ORs is quitesmall and it is unsurprising that the mean FR depends onthe choice of odorant, while the population model predictsthis dependence to become negligible for large numbers ofORs and ORNs.The multiplicative noise e W has a log-normal PDF andmay be interpreted physically as an approximation of the The authors adopted an unusual p -value, defined as the prob-ability of obtaining a fictitious ‘empirical’ dataset whose fitness beby mere chance further away from their best-fit model than the ac-tual dataset, such probability resulting to be equal to about 0.17.The roots of this significant uncertainty appear to largely reside inthe uncertainties of the best-fit interpolations of response curves. Inparticular, the majority of the interpolated datasets do not cover theneuronal response range up to saturation, therefore complicating theanalysis and introducing even larger uncertainties. The authors alsomade use of a statistical method that eventually led them to excludeabout half of their sensitivity data-points. W was chosen to be zerobecause the most likely outcome of the multiplicative noiseis expected to be such to not affect the true value of themeasured quantity; however, including a non-zero meanwould not affect the results even if it depended on thechoice of odorant, for it would be equivalent to an incon-sequential scaling of A .The exponent of the dependence of mean FR on odor-ant concentration is significantly smaller than the unit( (cid:98) β (cid:39) .
23) with high statistical confidence, which mayseem at odds with the hypothesis of divisive normalisa-tion. However, while the results of Olsen et al. (2010)indicate that the inter-glomerular inhibition is an approx-imately linear function of the ORNs mean FR, they alsoindicate that such inhibition enters the response functionof the relay neurons non-linearly. Therefore, the fact thatthe ORNs mean FR scales sub-linearly with the odorantconcentration does not preclude normalisation as a plau-sible means to achieve concentration invariance.The present model is statistical and is therefore likelymore accurate for the mammalian olfactory system. In-deed, usually mammals have a much larger repertoire of re-ceptor types as well as a much larger number of ORNs thatexpress them than insects. These larger numbers shouldindeed make finite-size effects smaller, but they may pos-sibly make the assumptions of this model be more closelyfulfilled also because of a further reason: the larger num-bers of receptor types and ORNs may have left the systemless affected by evolutionary optimisation bias (includingthe incidence of inhibitory responses) towards the odorantsmost important for the survival of the species.It seems worth noting that, if the subjective perceptionof the intensity (concentration) of an odorant is propor-tional to the mean FR of the population of ORNs, the pre-dicted power-law dependence of mean population FR onodorant concentration is in agreement with the renownedpsychophysical Stevens Law (Stevens, 1957):
Equal stimu-lus ratios produce equal subjective ratios , that is, in math-ematical terms, the perceived intensity of a stimulus isproportional to a power of the actual physical intensity, atleast over a relatively large range of the latter.All in all, the approach adopted in this work shows howa wide variety of empirical evidence, ranging from molec-ular to psychophysical, may in fact be modelled startingfrom very few principles. The need for further experi-mental scrutiny of these principles might in itself suggestfruitful lines of study.
The source of variability of W seems likely to lie inthe experimental settings and procedures, probably in thecontrol on the delivery of the odorant, whose concentrationmay vary because of uncertainties inherent in the appara-tus. However, it cannot be entirely ruled out that there be at least a partial contribution from the studied system it-self. In any case, the results suggest that such randomnessarise from multiplicative sources. It would be of coursemore interesting if these fluctuations resulted not to beentirely an artifact, as physical processes responsible forthe multiplicative randomness would have to be soughtwithin the neural system.It is notable that, were the sensitivity power-law coef-ficient α equal to 1, as it would be if the neuronal inputfollowed the MAL of the simplistic ligand-receptor model L + R (cid:29) LR , the ORN population mean FR would beindependent of the concentration ( β = 0) but possibly forpseudo-random finite-size effects. Therefore, it is plausi-ble that natural evolution has led to α being significantlydifferent from 1 so that the population mean FR carry re-liable information on concentration. However, while thisobservation may be of some interest in itself, it does notexplain why, apart from being different from 1, the value of α is what it is. It seems likely that such value resulted fromsome kind of optimisation, but it seems equally likely thatmore understanding in this direction will require takinginto account also further stages of the processing streamand their intrinsic interactions.While the already available data seem sufficient to con-firm the power-law dependence of ORNs mean FR on odor-ant concentration, at least in some animals, they are notenough to test with high accuracy the predictions on thesensitivity power-law and on the relationship between thepower-laws respective exponents. More data are certainlyneeded, especially from experiments in which the sensitiv-ities and the FRs are measured on the same animal model.This holds also true for a tighter verification of the postu-late according to which the statistical dependence betweenaffinities is small enough to be treated, to a good approx-imation on a large scale, as negligible “short-range” cor-relation. As already mentioned, the present modelling ap-proach should be expected to be better suited to the mam-malian olfactory system especially because of its muchlarger number of receptor types. Competing interests
No competing interests.
Acknowledgments
All computer codes were compiled, using
GCC ( GNUCompiler Collection , Free Software Foundation, Inc.), andrun on the open-source Linux-kernel
Debian
OS.
References
Amari, S. (1985).
Differential-Geometrical Methods in Statistics ,Vol. 28 of
Lecture Notes in Statistics , Springer-Verlag.Amit, D. J., Gutfreund, H. and Sompolinsky, H. (1985). Spin-glassmodels of neural networks,
Phys. Rev. A : 1007-1018. uck, L. and Axel, R. (1991). A novel multigene family may encodeodorant receptors: A molecular basis for odor recognition, Cell : 175-187.Campbell, L. L. (1986). An extended ˇCencov characterization of theinformation metric, Proc. Am. Math. Soc. : 135-141.ˇCencov, N. N. (1972). Statistical Decision Rules and Optimal Infer-ence , Nauka, Moskow (in Russian). English transl.:
Transl. Math.Monographs , Am. Math. Soc., 1981.Cram´er, H. (1928). On the composition of elementary errors, Skand.Akt. : 13-74, 141-180.Doob, J. L. (1949). Heuristic approach to the Kolmogorov-Smirnovtheorems, Ann. Math. Stat. : 393-403.Duchamp-Viret, P., Chaput, M. A. and Duchamp, A. (1999). Odorresponse properties of rat olfactory receptor neurons, Science : 2171-2174.Firestein, S., Shepherd, G. M. and Werblin, F. S. (1990). Timecourse of the membrane current underlying sensory transductionin salamander olfactory receptor neurones,
J. Physiol. (London) : 135-158.Fisher, R. A. (1925). Theory of statistical estimation,
Math. Proc.Camb. Phil. Soc. : 700-725.Grabe, V. and Sachse, S. (2018). Fundamental principles of olfactorycode [Review], BioSystems : 94-101.Hallem, E. A. and Carlson, J. R. (2006). Coding of odors by areceptor repertoire,
Cell : 143-160.Hendin, O., Horn, D. and Tsodyks, M. V. (1998). Associative mem-ory and segmentation in an oscillatory neural model of the olfac-tory bulb,
J. Comput. Neurosci. : 157-169.Hildebrand, J. G. and Shepherd, G. M. (1997). Mechanisms of olfac-tory discrimination: converging evidence for common principlesacross phyla [Review], Ann. Rev. Neurosci. : 595-631.Hill, A. V. (1910). The possible effects of the aggregation of themolecules of haemoglobin on its dissociation curves, J. Physiol.(London) : iv-vii.Hopfield, J. J. (1982). Neural networks and physical systems withemergent collective computational abilities, Proc. Natl. Acad. Sci.USA : 2554-2558.Hotelling, H. (1930). Spaces of statistical parameters, Bull. Am.Math. Soc. : 191 (abstract only).Jeffreys, H. (1946). An invariant form for the prior probability inestimation problems, Proc. Roy. Soc. London Ser. A : 453-461.Kac, M. (1949). On deviations between theoretical and empiricaldistributions,
Proc. Natl. Acad. Sci. USA : 252-257.Kurahashi, T. and Shibuya, T. (1990). Calcium-dependent adaptiveproperties in the solitary olfactory receptor cells of the newt, BrainRes. : 261-268.Little, W. A. (1974). The existence of persistent states in the brain,
Math. Biosci. : 101-120.Ma, M. and Shepherd, G. M. (2000). Functional mosaic organizationof mouse olfactory receptor neurons, Proc. Natl. Acad. Sci. U.S.A. : 12869-12874.Malnic, B., Hirono, J., Sato, T. and Buck, L. B. (1999). Combina-torial receptor codes for odors, Cell : 713-723.Marsaglia, G. and Marsaglia, J. (2004). J. Stat. Soft. .Monod, J., Wyman, J. and Changeux, J.-P. (1965). On the nature ofallosteric transitions: a plausible model, J. Mol. Biol. : 88-118.Olsen, S. R., Bhandawat, V. and Wilson, R. I. (2010). Divisivenormalization in olfactory population codes, Neuron : 287-299.Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P.(2007). Numerical Recipes: The Art of Scientific Computing ,Cambridge University Press, Cambridge, U.K.Prinz, H. (2010). Hill coefficients, dose-response curves and allostericmechanisms,
J. Chem. Biol. : 37-44.Rao, C. R. (1945). Information and the accuracy attainable in theestimation of statistical parameters, Bull. Calcutta Math. Soc. : 81-91.Rubin, B. D. and Katz, L. C. (1999). Optical imaging of odorantrepresentations in the mammalian olfactory bulb, Neuron : 499-511.Si, G., Kanwal, J. K., Hu, Y., Tabone, C. J., Baron, J., Berck, M., Vignoud, G. and Samuel, A. D. T. (2019). Structured odor-ant response patterns across a complete olfactory receptor neuronpopulation, Neuron : 950-962.Smirnov, N. V. (1937). On the distribution of the ω criterion of vonMises, Rec. Math. (NS) : 973-993. In a footnote at pg. 974, creditis given to V. I. Glivenko for the idea of modifying the Cram´er-vonMises formula so to obtain a distribution-independent test.Stevens, S. S. (1957). On the psychophysical law, Psychol. Rev. : 153-181.Stopfer, M., Jayaraman, V. and Laurent, G. (2003). Intensity versusidentity coding in an olfactory system, Neuron : 991-1004.Su, C.-Y., Menuz, K. and Carlson, J. R. (2009). Olfactory percep-tion: Receptors, cells, and circuits [Review], Cell : 45-59.Treves, A. (1990). Threshold-linear formal neuron in auto-associativenets,
J. Phys. A: Math. Gen. : 2631-2650.von der Malsburg, C. and Schneider, W. (1986). A neural cocktailparty processor, Biol. Cybern. : 29-40.von Mises, R. E. (1931). Wahrscheinlichkeitsrechnung und Ihre An-wendung in der Statistik und Theoretischen Physik , Deuticke, Vi-enna.Wachowiak, M., Cohen, L. B. and Zochowski, M. R. (2002). Dis-tributed and concentration-invariant spatial representations ofodorants by receptor neuron input to the turtle olfactory bulb,
J. Neurophysiol. : 1035-1045.Zufall, F. and Leinders-Zufall, T. (2000). The cellular and molecularbasis of odor adaptation [Review], Chem. Senses : 473-481.: 473-481.