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Biological Cybernetics Biol. Cybern. 73, 195-207 (1995) 9 Springer-Verlag 1995 Kinetic models of odor transduction implemented as artificial neural networks Simulations of complex response properties of honeybee olfactory neurons R. Malaka1, T. Ragg 1, M. Hammel~ I Institut fiir Logik, Komplexit~t und Deduktionssysteme, Universit~ Karlsruhe, D-76128 Karlsruhe, Germany 2 Institut f'tir Neurobiologie, Freie Universit/it Berlin, D-14195 Berlin, Germany Received: 4 August 1994 / Accepted in revised form: 14 March 1995 Abstract. We present a formal model of olfactory transduction corresponding to the biochemical reaction cascade found in chemosensory neurons. It assumes that odorants bind to receptor proteins which, in turn, activate transducer mechanisms corresponding to second messenger-mediated processes. The model is reformulated as a mathematically equivalent artificial neural network (ANN). To enable comparison of the computational power of our model, previously suggested models of chemosensory transduction are also presented in ANN versions. In ANNs, certain biological parameters, such as rate constants and affinities, are transformed into weights that can be fitted by training with a given experimental data set. After training, these weights do not necessarily equal the real biological parameters, but represent a set of values that is sufficient to simulate an experimental set of data. We used ANNs to simulate data recorded from bee subplacodes and compare the capacity of our model with ANN versions of other models. Receptor neurons of the nonpheromonal, general odor-processing subsystem of the honeybee are broadly tuned, have overlapping response spectra, and show highly nonlinear concentration dependencies and mixture interactions, i.e., synergistic and inhibitory effects. Our full model alone has the necessary complexity to simulate these complex response characteristics. To account for the complex response characteristics of honeybee receptor neurons, we suggest that several different receptor protein types and at least two second messenger systems are necessary that may interact at various levels of the transduction cascade and may eventually have opposing effects on receptor neuron excitability. 1. Introduction The senses for chemical signals like olfaction or taste are phylogenetically old. Both vertebrates and invertebrates have developed very similar solutions for the problem of recognizing volatile substances (Vogt et al. 1989). Odor molecules bind to receptor proteins (receptor sites) located in the membrane of the sensory cell. This interaction of odor molecules Correspondence to: R. Malaka or M. Hammer and receptor proteins activates G-protein mediated second messengers, such as cyclic adenosine monophosphate (cAMP) or IP3-inositol 1,4,5-triphosphate. Binding odorants to receptor sites induces a rapid rise in the concentrations of second messengers that, in turn, is thought to activate second messenger-gated ion channels (Breer et al. 1989; Firestein et al. 1991; Shepherd 1991; Reed 1992). This results in a conductivity change in the cell membrane and may eventually cause the generation of action potentials. So far, odor-induced second messenger processing involving IP3 and cAMP have been found in vertebrates and crustaceans. In various insect species, the response of receptor cells tuned to detect species-specific sex pheromones is mediated by IP 3 (Breer and Boekhoff 1992). Recent evidence implicated the nitric oxide NO/cGMP system as a third second messenger in olfactory signal transduction. Since the cGMP concentration in response to odor stimulation shows a delayed and lasting elevation, the (NO)/cGMP system has been suggested to be involved in stimulus adaption rather than in primary transduction (Breer and Boekhoff 1992; Breer and Shepherd 1993; Boekhoff et al. 1993). The question of whether one second messenger system is specific for certain odorants is still an unsolved issue. Evidence from a rat olfactory ciliae preparation demonstrated odor specificity of the IP3 and the cAMP system (Breer et al. 1990; Breer and Boeckhoff 1991; Boekhoff et al. 1991). However, single odorants can affect both systems in cultured rat olfactory cells (Ronnett et al. 1993). In the lobster, the same odorant can excite or inhibit different cells (Michel et al. 1991). Since in the lobster system IP3 and cAMP activate opposing ionic conductances that have excitatory or inhibitory effects, respectively (Fadool and Ache 1992; Michel and Ache 1992), the same odor ligand may be coupled to more than one second messenger pathway. If both the cAMP and the IP3 pathway co-occur in individual cells, a variety of odor ligand-second messenger interactions may be possible: cells may be tuned to be selectively excited or inhibited, thus serving as integration units (Ache 1993), or may respond to single odorants with opposing effects, for instance dependent on concentration or mixture interaction. 196 The existence of at least two second messenger systems allows considerable interaction at the level of individual cells dependent on stimulation with different odorants (Breer and Boekhoff 1992; Reed 1992; Ache 1993; Dionne 1994; Shepherd 1994). The cascade of reactions from odor molecules over receptor proteins and second messengers up to a changing of ion channel conductances and the generation of an action potential is shown in Fig. 1. In insects, olfactory sensory neurons are located in the antennae. They are grouped in pore plates or placodes (Schneider and Steinbrecht 1968). In honeybees, olfactory receptor neurons are broadly tuned and may respond to many different odorants in a broad range of different concentrations (Vareschi 1971; Akers and Getz 1992). Certain properties demonstrate that the chemical reactions performed by the receptor cell are complex. The response as a function of the odor concentration is highly nonlinear. The response to binary mixtures can be synergistic or inhibitory, according to the response to the components of the compound. A synergistic effect occurs if the response of one sensory cell to a binary mixture of two odorants Al and A2, with concentrations [All and [A2], is larger than the response to component AI or A2, alone with the largest response at concentration [A1 + A2]. That is, synergism occurs if, for the same total number of odor molecules, mixtures elicit a higher response than any one component. An inhibitory effect occurs if the response to the mixture is smaller than the response to component A1 at concentration [All or Az at concentration [A2], respectively. That is, inhibition occurs if an added component decreases the response (for a similar, more elaborate definition see Akers and Getz 1993). In the bee, both effects occur at the level of subplacode units that most likely represent the response properties of single olfactory sensory neurons (Akers and Getz 1993; see Akers and Getz 1992 for a more detailed discussion on subplacode and single cell responses). Although considerable knowledge of the olfactory transduction mechanism has been accumulated, it is still not sufficient to explain even the basic problems of olfactory transduction, such as odor quality and quantity coding or mixture effects. We, therefore, developed a general kinetic model of the olfactory transduction process which is constructed in analogy to the biochemical reaction cascade in olfactory sensory neurons. This model is aimed at simulating the full range from specifically to broadly tuned receptor cells. It provides a minimal complexity to account for nonlinear effects, such as synergistic or inhibitory response properties. Furthermore, this model allows the integration of realistic responses of sensory neurons into larger models of sensory information processing at the central neural networks level. It is, however, not a realistic model implementing the exact values of biochemical and biophysical parameters. Rather, we used the input/output characteristics recorded from bee subplacodes for parameter fitting. We transformed our kinetic model into a mathematically equivalent artificial neural network (ANN). Note that we use a single ANN to model an individual sensory neuron. Thus, the neural structure of the ANN does not correspond to a biological neural structure, but rather to biochemical reaction cascades in a single neuron. ~" '~ [] ~ odormolecules # ~ j~l~176 secondmessenger s actionpotentials ionic influx Fig. 1. Reaction cascade in chemosensory neurons. Volatile odor molecules reach receptor proteins at the surface of the chemosensory neuron. Odoractivated receptor proteins in turn activate second messengers (e.g., Gproteins), which leads to second messenger-mediated conductivity changes of ion channels ANNs have been shown to be powerful tools for function approximation. Their parameters can be adapted by training input/output patterns. A single neuron-like element sj of an ANN receives input from elements of the previous layer sj and computes a weighted sum over the input stimuli. A bias 0 may be added to the sum. The neuron then computes its output with an activation function f . This is the most common model of a neuron-like element and can be varied by modifying the connectivity of the network and by selecting different activation functions (Rumelhart et al. 1986). Mathematically, the output value si of the ith neuron-like element can be described by where w~j is the weight from the neuron-like element j to element i. Weight adaption is performed with a learning rule or a learning algorithm. The most popular is the backpropagation rule (Rumelhart et al. 1986). In the following, we first present a formal description of a biochemical reaction cascade of a chemosensory neuron developed along previous models of chemosensory transduction (Renquist 1919; Lasareff 1922; Beidler 1962; Carr and Derby 1986; Ennis 1991). This assumes that odor molecules bind with different affinities to sets of receptor proteins, which in turn activate transducer mechanisms corresponding to second messenger mediated processes. Transducer activation, finally, determines the effect a certain stimulation has on the response of the sensory neuron. We then formulate an equivalent ANN which is used to simulate the data recorded from bee subplacodes (Akers and Getz 1993). Its capacity is compared with ANN versions of other models. We show that two receptor 197 protein types or a single transducer mechanism, respectively, are not sufficient to account for the response characteristics of bee subplacode units. Acceptable approximations are obtained for at least two transducer mechanisms and several receptor protein types. 2. Models for chemoreceptors Psychophysical and biochemical models of chemosensory neurons have many similarities in their mathematical structure. They can be used to model both biological sensory cells and industrial chemoreceptors. In the following we introduce some of the most common models and discuss their performance. For all models a vector of odorant concentrations [ A I ] , . . . , [An] serves as an input. [Ai] describes the total concentration of an odor ligand Ai and the full vector describes a mixture of odorants. The output E of the models corresponds to the total effect the stimulus input has on the sensory cell (e.g., graded receptor potential, spiking rate, etc.). The simplest kind of psychophysical model based on receptor properties is a linear one, where an odor ligand Ai at concentration [Ai] causes a response proportional to the concentration. Mixture interaction is obtained by adding the effect of the components weighted by odor-specific constant parameters ki: E = E ki[Ai] i (2) This linear relationship between concentrations and responses cannot describe any synergistic or inhibitory effects. The linear model is applied in a model of the honeybee's olfactory information processing by Getz and Chapman (1987) and Getz (1991). Obviously, it can be implemented by a single artificial neuron with a linear activation function. Because of the observed nonlinear logarithmic-like relationship between stimulus concentration and response strength, two models are proposed by Carr and Derby (1986) that use logarithmic stimulus dependencies. In the stimulus summation model, the response to the mixture of all Ai equals the response to the sum over all concentrations Instead of logarithmic functions one could also consider other negative accelerating functions, such as hyperbolic functions or root functions, if the response properties are modelled according to biochemical reaction cascades. In the model proposed by Renquist (1919) the reaction of a sensory cell is set proportional to the influx or adsorption of odor molecules into the cell. In our notation, we get E = d[AR]/dt, where [AR] is the concentration of activated receptor proteins at the cell membrane, which leads to: E = k[A] ~/n (5) where k and n are odorant-specific constants depending on adsorption and diffusion properties. An extension of this model for odor mixture effects can be achieved by changing the logarithmic functions in (3) and (4) into nth root functions. Lasareff (1922) proposed a model in which the effect of an odor ligand is proportional to the amount of activated receptor sites [AR] at equilibrium state. Therefore, he modelled the reaction kinetics of the reaction OLA A + R ~ AR (6) 3a with d[AR]/dt = aA[A][R] - flA[AR] (7) which at equilibrium (d[AR]/dt = 0) leads to: [AR] = k[A][R] (8) where [R] is the concentration of flee receptor proteins R and k = O~A/flA (k > 0) is a constant which describes the binding affinity of oder molecules A with receptor proteins R. If we introduce the constant total concentration of receptor proteins [~] = [AR] + [R], we get a nonlinear dependency of JAR] from [A]: [AR] = k[A][&]/(1 + k[A]) (9) If we identify the effect E of odorant A with the concentration of activated receptor proteins [AR] and introduce the hyperbolic function hyp(x) = x/(1 + x), (9) can be written as E = [&] hyp(k[A]) whereas in the response summation model, mixture interaction is achieved by adding the logarithms of the individual odor ligands: E =E log (kdAd) (4) i The stimulus summation model simply adds the concentrations of the ligands in a mixture and is thus not able to elicit synergisms or inhibitory effects. The response summation model can show synergistic, but not inhibitory, effects. The stimulus summation model and the response summation model can be expressed as a single artificial neuron with a logarithmic activation function or as a single artificial neuron with a linear activation function and logarithmic input transformations, respectively. (10) Thus, in Lasareff's formulation, the effect E is a hyperbolic, instead of a logarithmic, function of the concentration of an odorant. Beidler (1962) proposed two extensions to this model. First, he introduced different independent receptor proteins with different concentration dependencies to account for an overlay of different response characteristics dependent on various concentration levels. For example, with one type of receptor site specifically tuned to low concentrations and one to high concentrations, he was able to fit the responses of rat taste cells to ammonium chloride stimulation more effectively: E = E lj[~j] hyp(kj[A]) J (11) 198 This is actually a combination of the hyperbolic reaction function introduced by Lasareff, (10), and the response summation model shown in (4). Each different receptor protein type has its own binding characteristic defined by lj and The second extension describes the mixture interaction of different odor ligands with a single type of receptor protein. With [/~j] = ~i[AiR]+[R] and [AIR] = kdAi][R], the amount [AIR] of all activated receptor sites with odor ligand A~ is given by: [aiR] = ki[A,][R] / (l + ~--~ (12) Accordingly, the effect caused by a mixture can be interpreted as the sum over all receptor sites activated by the different odor ligands: This is actually a combination of the hyperbolic reaction function used in (10) and the stimulus summation model shown in (3). transducer mechanisms and a constant bias 0 that represents spontaneous activity. The reaction kinetics are an extension of (6) with multiple odor ligands Ai and receptor protein types Rj. The second level is the transducer reaction where odor-receptor complexes AiRj activate transducer mechanisms Tk: Ai + Rj ~ AiRj A~Rj + Tk ~ AiRjTk (14.1) (14.2) With the affinities kij and lijk describing the rate of reactions between odor ligands Ai and receptor proteins Rj or between odor-receptor complexes AiRy and transducers Tk, respectively, the amount of activated receptor proteins or transducers at equilibrium is given by: [A~Rj] = k~j[A~][Rj] [AiRjTk] = lijk[AiRj][Tk] (15.1) (15.2) A sensory neuron is defined by the total concentration (or amount) of receptor proteins [~] and transducers [~']. The total concentration of either type corresponds to the sum of the free and the activated sites: [/~j] = [Rj] + ~.[AiRj] + ~-~[AiRjTk] i (16.1) i,k [7'k] = [Tk] + ~[AiRjTk] 3. Receptor transducer models Ennis (1991) modelled the perception of two kinds of sugars, glucose and fructose. There is a strong synergistic effect in the perceived sweetness of mixtures in comparison with single components. Ennis showed that odor-receptor models such as those proposed by Beidler are not sufficient to describe these phenomena. Therefore, he suggested two extensions. Firstly, he proposed that receptor proteins may have multiple binding sites, e.g., different odor molecules must bind to one receptor protein in order to fully activate it. Secondly, he introduced the modelling of transducer mechanisms Ti, which represent a second step of reactions in olfactory transduction and may correspond to a G-protein/second messenger-mediated process. Ennis proposed two types of models involving transducer mechanisms, one in which each odor ligand activates a parallel pathway of independent receptor and transducer types, and one in which different ligands compete for activation of a common receptor transducer mechanism. For animals with a generalistic odor perception, such as the honeybee, which is able to detect hundreds or thousands of different odor ligands, a sensory cell with hundreds or thousands of different receptor protein types and second messengers is rather implausible. It is more likely that there are only about three interacting second messenger pathways that can be activated by many odorants (Breer and Boekhoff 1992). On the other hand, a system with only one receptor transducer mechanisms seems too simple for a very generalistic odor perception system. We therefore developed an extended version of Ennis' model in which neither receptor nor transducer proteins are odor-specific. Moreover, this model includes both excitatory and inhibitory transducer mechanisms. The effect of an odor stimulus is determined by the proportion of activated (16.2) i,j Ennis modelled the relative effect an odor has as the proportion of activated transducers to the total amount of transducers. For mixture interactions, this shows only additive but no inhibitory effects. We, therefore, divided the transducers into two types: inhibitory and excitatory transducers, i.e., transducers which either decrease or increase the probability of generation of action potentials, respectively. 6k = +l, -- l, if transducer if transducer Tk Tk is excitatory is inhibitory The effect an odor stimulus has can be set to the sum of all activated excitatory transducers minus the sum of all inhibitory transducers relative to the total amount of transducers. This model will always show a zero response if no odor is present. A additive constant 0 can be used to model spontaneous activity. Now the effect E of an odor in the extended receptor transducer model is For further mathematical analysis of the receptor transducer model, we formulate it in a closed form as an ANN. 3.1. Receptor transducer models using ANNs With (15.2) the effect E defined in (17) can be reformulated to E= _.]~k[~k------~] l~+~ii~31~Ak[~j]Sk[Tk] +0 (18) 199 With the hyperbolic function hyp this gives [Ai] E - Ek[~k] k We now define netk as the weighted sum over all [AiRj] which yields with (15.1) netk = ~ lijk[AiRj] = ~ lijkkij[Ai][Rj] i,3 (20) ij Using the simplification [~/j] = [R/] + ~i[AiRj] instead of (16.1), which is sufficient for [/~/j] >> [Tk] (Ennis 1991), we can express netk as dependent on the constant [J~] instead of the variable [_R]: netk = ~ ~-~il~jkkij[Ai] [j~j] J 1 + ~-~ k~j[Ai] (21) The error is evaluated in the Appendix and is less than [:F]/[/~] for the one-dimensional case. Two special cases can be considered concerning the type of odor ligand-receptor-transducer interaction. Case 1 (ljk). lijk = Irnjk, the affinity of an odor-receptor complex to activate a transducer does not depend on the odor ligand it is activated with. Under this assumption, (21) becomes: j - A chemosensory system with exactly two transducer mechanisms - one inhibitory and one excitatory - can be modelled as a subcase of case 1, where T1 and T2 mimic the transducer mechanisms with 61 = 1 and 62 = - 1 , respectively. Receptor proteins with affinities to only one of the two transducers are either inhibitory or excitatory. Those with affinities to both transducer mechanisms represent either proteins that can be coupled with the inhibitory and excitatory transducer pathway or two distinct receptor proteins with the same affinities to odors that either activate the inhibitory transducer or the excitatory transducer, respectively. Case 2 (AiRiTk). There is exactly one type of receptor site for each odor ligand. Under this assumption, (21) becomes netk = ~ l~ik[/~] hyp(k~ [A~]) (23) i This case is a generalization of Ennis" model with parallel odor ligand-receptor-transducer pathways. As in the cases of the simple models without transducer mechanisms, receptor-transducer models can be expressed as ANNs. In case 1, the corresponding network is a 4-layer ANN with the concentrations of the odor ligands [AO as input layer, two hidden layers corresponding to receptor proteins and transducers, and one output element in layer 4 which represents the effect caused by the input. The weight between the ith element of the input layer to the jth element of the first hidden layer is kij and from there Fig. 2. ANN equivalent to the full receptor-transducer model (i.e., case 1, see text). The input layer corresponds to the concentration of odor ligands [Ai], the first hidden layer corresponds to activated receptor protein types, the second to activated transducer mechanisms. The output neuron computes the effect E of the sensory cell. The weights between the input layer and the first hidden layer are kij, and ljk[~j] between hidden layer one and hidden layer two. The weights from hidden layer two to the output element to the kth element of the second hidden layer,/jk[/~j] [see (22)]. The weight from element k of hidden layer 2 to the output element is 6k[Tk]/~k[Tk] [see (19)]. The adaptive elements of the hidden layers have hyperbolic activation functions hyp. The network structure is shown in Fig. 2. In case 2, the model differs in the connectivity between the input layer and the first hidden layer. Only connections between input i and element i in the first hidden layer are possible. Ennis' receptor-transducer model, in which kij and lijk are nonzero only for / = j = k, is expressible as an ANN with only one hidden layer and weights (1 + [J~i]liii)kii between input element i and element i in the hidden layer and weights [Ti]hyp([Ri]li~)/~[T~] between the /th element of the hidden layer and the output unit. With its linear output element, this model is not able to simulate inhibitory mixture interactions. Due to the hyperbolic input transformation, synergistic effects are possible. Mathematically, the model is equivalent to the response summation version of Beidler's model described in (11). 4. Simulation results All models described in the previous sections can be interpreted as ANNs. Applying learning algorithms like backpropagation or RProp, it is possible to find parameter settings for optimal (or local optimal) simulations of chemosensory cells with given response characteristics. The activation functions of the network neurons are set to hyp for the hidden neurons and to a linear function for the output neuron of the network. Since affinities and concentrations are positive values, all weights from the input layer to the first hidden layer (k~j) and between the hidden layers (/jk[J~j]) must be positive. These constraints are enforced during learning. Otherwise, in the case of negative weights, the singularity of 200 the hyperbolic activation function hyp(x) at x = - 1 would cause severe problems. In our simulations, the best training results were achieved by using the fast learning algorithm RProp, which is an improved version of backpropagation (Riedmiller and Braun 1993). To train the different ANN versions, we used recordings from olfactory cells of the honeybee made by Akers and Getz (1993). They recorded extracellularly from single placodes of worker honeybee antennae, applying different odorants and their binary mixtures. With mathematical methods, they sorted the overlaid responses of multiple receptors into responses of subplacode units that most likely represent single sensory neurons (Akers and Getz 1992). The data set for training the ANNs consists of the responses of 54 subplacodes to the four odorants, geraniol, citral, limonene, linalool, their binary mixtures, and the mixture of all four odorants. Each odor was applied at two concentration levels. Together with a blank stimulus, we were thus able to use a data set of responses to 23 different odor stimuli for each subplacode. The mean response over all subplacode responses is 18.15 spikes (standard deviation 9.8). Akers and Getz achieved different odor concentrations by adding a certain amount of the odorant to a fixed amount of mineral oil in a syringe. An air stream delivered through the syringe then moved odor molecules from the solution to the antennae. As a measure for odorant concentration we always refer to the amount of odor in the syringe. The response of a subplacode is measured as the number of spikes it generated when the antenna was stimulated with an odor. We identify the effect E with this number. relative error (per sample point) 0.3 0.25 0.2 0.15 0.i 0.05 1 0 receptor protein types transducer mechanisms Fig.3. Relative error of all responses for the full model with different network sizes (mean absolute error divided by mean response). Network sizes differ in the number of receptor protein types and the number of transducer mechanisms. Each network was trained ten times with 50000 learning steps. Each error value shown is the best of ten networks Models with more than 6 receptor protein types and more than 5 transducer mechanisms result in errors of less than 2%. To avoid overtraining effects that may occur if a model size is large and the training set is restricted, we used smaller models from then on. The large relative error for models with R < 2 and a single T indicates that these numbers are not sufficient to elicit the observed responses of honeybee sensory neurons. Small numbers may, however, be sufficient for a non-generalistic, highly specific olfactory system, such as the sex pheromone system in various insect species. 4.2. Comparison of the models 4.1. Size of the full model In a series of simulation runs with varying numbers of receptor protein types and transducer mechanisms the full model described in (19) and (22) was trained to fit the data set. Depending on the network size, the model was able to simulate the responses of the 54 subplacodes. The size of the first hidden layer corresponds to the number of receptor protein types (R) in the model, the size of the second hidden layer corresponds to the number of transducing mechanisms (T). Figure 3 shows the relative error per output neuron for all possible combinations of one to six receptor types and one to six transducer mechanisms and for combinations with 10, 20, and 50 receptor protein types. All models with only one receptor protein type (R = 1) or only one transducer mechanism (T = 1) have relative errors of more than 21%. Models with R = 2 still yield high errors of more than 15%. Good simulation results are obtained with models, where R + T > 8, i.e., for models with R = 4 and T > 4, the error is less than 8%; for models with R = 3, T > 5 and for models with R = 5, T >_ 3, the error lies below 11%. Models with only two transducer mechanisms need at least R = 9 to simulate the data with a relative error of less than 10%. Here we investigated the general case of T = 2 in which the nature of the transducer mechanisms, inhibitory or excitatory, was not restricted before training. To compare the computational capabilities of the different models of chemosensory neurons, we investigated the simulation results of the full model, the particular case of the full model with two transducer mechanisms, (one excitatory, one inhibitory) the Ennis model, and the stimulus and response summation models. Each model was trained ten times. Since initialization of the network before training allows its convergence to different solutions, these ten runs lead to slightly different results. As shown in Table 1, the resulting networks of the different runs do not vary much (see column deviation). It can thus be concluded that for each model, the minima of the error function are very similar. The full model with six receptor protein types and four transducer mechanisms yields an error of 4%, while the smaller (R = 4, T = 4) model has a relative error of 8%. The full model for the particular case of two transducer mechanisms (one excitatory and one inhibitory) with R = 10 or R = 20 results in errors of 10% and 7%, respectively. Both the Ennis and the response summation models lead to similar high relative errors of more than 26%, while the stimulus summation model has the highest relative error (63%). Thus, only the full receptor transducer model is able to simulate the complex response characteristics of the given data set. To investigate the simulation capabilities of these models in more detail, we investigated a single sensory cell that exhibits significantly different response characteristics 201 Table 1. Model comparision of the full model with R = 6 and T = 4, R = 4 and T = 4, the full model with two transducer mechanisms (one inhibitory and one excitatory) with R = 10 and R = 20, the Ennis model, the response summation model (Resp.Sum.), and the stimulus summation model (Stim.Sum.). For each model ten corresponding ANNs were trained, and the error (-4-SD) of the best ANN (best) and the average error (4- SD) are displayed. Absolute errors are given in spikes per output and stimulus within one second. The SD of the errors and the deviations over the ten runs are given in the column Deviation R = 6, T = 4 R = 4, T = 4 R = 10, T = 2 R = 20, T = 2 Ennis Resp.Sum. Stim.Sum. Absolute error Best Average Deviation 0.75 4.0.89 1.52 4.1"50 1.85 4.1"87 0.82 4.0.98 1.58 4.1'77 1.85 +-1"85 0.07 4.~ 0.06 4.{1"14 0.00 +-0.02 1.324.1"4~ 1.35+-143 0.034.0.08 4.75 4.3.96 4.80 +-3.99 0.05 +-0.05 6.59+-5.34 11.60+-9.93 6.62+-5.37 11.60+-9.93 0.02+-~176 0.00 + ~ 1 7 6 Relative error Best Average 0.044.0.05 0.044.0.05 0.08+-0.08 0.09+~176 Deviation 0.10 +-0'10 0.104"0'10 0.00384.0"0077 0 .0031 +0'0tr/7 0.0001+-0'0010 0.074.0.08 0.26+-0.22 0.36+-0.29 0.63+-0.54 0.074.0.08 0.26+-0.22 0.36+-0.29 0.63+-0.54 0 .0018+~ 0.0028+0'oo25 0-0011+~176176176 0.0001+o.tx~02 dependent on stimulus interaction in two different binary mixtures. It responds to geraniol and citral as well as to their binary mixtures with similar spike rates, whereas the interaction of limonene and linalool exhibits strong synergistic effects, i.e., the response to mixtures of both odorants is much higher than the responses to the single odorants. The full model (with R = 4 and T = 4) is able to simulate this behavior, as shown in Fig. 4a and b. The Ennis model provides an acceptable interpolation for the interaction of geraniol and citral (Fig. 4c) but fails to simulate the synergistic effects of the limonene-linalool interaction (Fig. 4d). The mean error for this sensory cell is higher than four spikes. This is comparable to the full model with one receptor protein type and T < 5, but worse than all simulations with the full model with R _> 2 and T > 2. As shown in the previous section, the Ennis model can also be interpreted as a response summation model with a hyperbolic function instead of a logarithmic function. Thus, the response summation model has similar computational properties and similar problems in fitting the data. For this reason, an interpolation is not shown. The particular case of the full model with two transduction mechanisms (one excitatory, one inhibitory) is shown in Fig. 4e and f. The mean error depends on the number of receptor protein types. It varies for 4 to 20 types between 3.1 and 1.3 spikes. The interpolation was computed by a network with 20 receptor protein types. The data are fitted well, except for the strong synergistic effects between limonene and linalool. Figure 4g and h shows the results of the stimulus summation model with a logarithmic activation function. The error is in the same range as those of the Ennis and the response summation models. Thus, the Ennis model, the response summation model, and the stimulus summation model fail to simultaneously simulate two different types of interactions between different odor ligands of a single sensory cell. They show good simulation results for the nonsynergistic interaction between geraniol and citral (Fig. 4c,e,g) but fail to simulate the highly synergistic interaction between linalool and limonene (Fig. 4d,f,h). Moreover, since these three models cannot account for inhibitory mixture interactions, they do not have the computational power to simulate the data recorded by Akers and Getz (1993). In the next section we show that the full model is not only powerful enough to express highly synergistic mixture interaction effects, but also inhibitory effects. 4.3. Simulation capabilities of the full model The capacity of the full model to simulate synergistic and inhibitory mixture interactions as well as specificity to a single odor ligand is shown in further simulation results with a model size of R = 4 and T = 4 (Figs. 5 and 6). Since this capacity does not critically depend on the binding affinities between odor molecules and receptor sites, the weights between the first and second layer of the ANN, i.e., the parameters kij, are the same for all modelled sensory cells. Most of the sensory ceils recorded by Akers and Getz (1993) showed generalistic response profiles, that is, spiking activity was elicited by many odorants. Some cells, however, were odor specialists responding to a single odor ligand only. Figure 5 shows simulations for a cell that is specific to the odor ligand linalool. It responds to linalool in a concentration-dependent way, independent of the presence or absence of any other odorant. The response characteristic of the sensory cell shown in Fig. 6 is a good example of inhibitory mixture interactions as well as the strong nonlinear response profiles found in olfactory sensory neurons. In the case of mixture interactions between geraniol and limonene, Fig. 6 shows clear inhibitory effects. Reactions to geraniol are highly nonlinear, nonmonotonous concentration-dependent. At a specific concentration, the response reaches a significant maximum, and higher concentrations elicit smaller responses. 5. Discussion In this study we developed a formal model of olfactory transduction corresponding to the biochemical reaction cascade found in olfaction, i.e., the competitive receptor transducer model. This model was primarily aimed at accounting for the complex response characteristics of honeybee olfactory sensory cells (Akers and Getz 1992, 1993). In contrast to the highly specific sex pheromone subsystems of vari- 202 spikes spikes 40 35 35 30 30 25 25 20 20 15 15 10 5 64 D 64 92 16 8 4 16 32 64 2 4 u.zJ ~.~ 0.06 ~ ~ 0.25 32 16 8 4 2 16 32 64 8 8 [citral] u.~J 0 . 0~. ~ / 0.06 0 ~ 0.25 0.06 [linalool] b) spikes spikes 40 40 - 35 35 " 30 30 " 25 25 9 20 20 15 15 9 io5 . 5 N 64 64 32 16 8 4 2 4 u.~ 0 . 0 _ ~ / ~ 0.06 0.25 B 16 32 64 16 8 4 2 [citral] 8 u.~J ~ . / 0.06~0.06 0.25 16 32 64 [linalool] c) spikes spikes 40 4O 35 35 30 30 25 25 20 20 15 10 I0 10 5 9 64 5 n 4 31 8 64 6491 16 16 32 8 4 . 16 32 64 2 i 2 u.z~ 0.06 ~ 0.06 spikes / ~ " ~ ' - 0.25 0 906 . . 4 8 [linalool ] . . . . 2 4 spikes 40 35 35 30 30 25 25 20 20 15 15 5 64 649216 2 64 8 4 16 32 64 2 1 8 [geraniol] " 0.06 ~) ~ u.z~ ~ 0.06 / ~ 0.25 [linalool ] 0.06 l) Fig.4. Simulation results using ANNs of the full model (a,b), the Ennis model (e,d), the subcase of the full model with exactly two transducer mechanisms (e,f), and stimulus summation (g,h). The responses of simulated sensory cells are given in number of spikes per stimulus within one second. Left column (a,e,e,g) represents receptor neuron responses to mixtures of geraniol and citral, right column (b,d,f,h) represents sensory cell responses to binary mixtures of limonene and linalool. The concentrations of the odorants are depicted on a logarithmic scale from 2 - 5 to 26 /~g (0.03 to 64/zg). See text for definition of odor concentration. Crosses indicate deviations of simulated from experimental data. All other responses on the surface of the 3D-plots are simulated data 203 spikes spikes 40r 40 35}- 35 30}- 30 25~" 25 20 F 20 15 i0 5 5 64~ 64 64 ~2 8 4 2 4 1 "~ " 8 64 16 32 2 U..~D 0.25 0.06 ~ [citral ] 0.06 spikes spikes 40 40 35 35 30 3O 25 25 20 2O 15 15 I0 5 64 32 16 8 4 2 4 1 " " U.Z~ ~ 0.06 0.25 ~ 8 16 32 64 643~ 2 [linalo01] 0.06 ~ spikes spikes 40 40 35 35 30 30 25 25 20 20 15 15 2 64 i0 5 64 6, 16 32 64 ~2 16 8 4 2 - - 2 u.zo 0.25 0.06 ~ 4 ~i 16 8 8 [linalool] 0.06 4 16 32 64 2 [l i m ~ 1 2 nu . z~z 0.25 0.06 ~ 4 8 I''[ l i n a l"o 0 1"" ] 0.06 Fig. 5. Response in spikes per stimulus within 1 s of a single simulated sensory cell to different binary mixtures of odorants at different concentrations. The six single diagrams show the simulated responses of the sensory cell to all binary mixtures of geraniol, citral, limonene, and linalool. The simulated cell is an expert for linalool. There is little mixture interaction in the responses of the cell ous insect species, receptor neurons of the nonpheromonal, general odor processing subsystem are broadly tuned and respond to one or more classes of odorants, both in bees (Vareschi 1971; Akers and Getz 1992, 1993) and cockroaches (Fujimura et al. 1991). The generally overlapping response spectra of different cells vary in their concentration dependencies, specificities, and mixture integration, e.g., synergistic or inhibitory. We expressed our competitive receptor transducer model as well as previously proposed models in the form of ANNs in which single elements and weights of the net correspond to certain molecules (such as receptor proteins and second messengers) and rate constants, respectively, thus mimick- ing the biochemical reactions assumed to underlie olfactory transduction. Since in A N N s parameters can be fitted by training the nets with a given set of experimental data, this approach provides an elegant way of comparing the computational and simulation capacities of different model versions. In general, learning algorithms used for parameter fitting, e.g., backpropagation, optimize parameters so that differences between experimental data and simulation results are minimized. Since there may be several solutions for this task, it cannot be concluded that any set of found parameters corresponds to real physiological entities, such as affinities between molecules. However, if the learning algorithm is 204 spikes spikes 35 30 25 20 4O 35 30 25 20 15 ,o~ T 2 16 8 4 16 32 64 2 4 8 u.~J _ _ _ ~ 0.06 ...~ ~""'~ 0.06 0.25 ~2 16 8 4 2 [citral] 8 16 32 64 .... ~ . . . . . . . . . spikes 35 3O 25 2O 402 15 iR ~ [ ~ 2 64 [limonene ] ~ 64 0 3 4 2 [cit spikes 4 .... 8 16 32 ~ - / 0.25 [limonene ] 0.06 ~ " " " ~ 0.06 ............... spikes 409 35 25 2O 15 10 5 61 40r 35 25 20 15 640~ 9 16 0.25 spikes 25 20 15 I0 5 64~ 64 8 16 32 64 4 u.~ ~ 0.06 / ~"'-"'Y 0.25 o . o6 [linalool ] - ~ 8 4 2 8 .... 0 . 0~_ ~ .-~ 6 0.25 16 32 64 [linalool ] Fig. 6. Response in spikes per stimulus within 1 s of a single simulated sensory cell to different binary mixtures of odorants at different concentrations. The simulated cell shows inhibitory effects and a highly nonlinear response characteristic not able to fit the experimental data, it can be inferred that the model under investigation is not sufficient. Thus, the smallest model which is able to simulate a given data set covers the minimal necessary complexity. Only the full receptor transducer model has the necessary complexity to simulate the nonlinear response characteristics of honeybee chemosensory cells. Models, such as the response summation or the Ennis' model, in which the response of a sensory cell is determined only via the interaction of odor ligands with receptor proteins or by assuming a single common transducer mechanism or parallel independent receptor transducer pathways, respectively, are not sufficient. For instance, they cannot account for inhibitory mixture interactions and synergistic effects. Simulation of both highly synergistic and inhibitory mixture interactions requires receptor proteins with broadly tuned binding affinities and multiple transducer mechanisms that may exhibit excitatory or inhibitory effects. To account for the complex response properties of honeybee subplacodes, at least two transducer mechanisms with several different receptor protein types are necessary. Dependent on the number of receptor protein types and transducer mechanisms, our competitive receptor transducer model can also account for less complex effects, such as the odor specificity of individual receptor neurons. So far, there are no experimental data available for ttie number and specificity of receptor protein types expressed in single olfactory sensilla in insects. Moreover, signal trans- 205 duction seems to be mediated by a single second messenger, i.e., IP3, in the sex pheromone subsystem (Breer et al. 1990). Pheromones, utilized in intraspecies communication, seem to be processed via highly specific receptor neurons converging onto specialized odor-processing subsystems of the brain, such as the macroglomerulus (Homberg et al. 1989). They may not provide a model for the general odor-processing system. Recent evidence suggests that, in the lobster, different olfactory receptor neurons may express different receptor proteins that, in turn, may excite or inhibit the cells in response to the same odor mediated via IP3 or cAMP (Michel et al. 1991; Fadool and Ache 1992; Michel and Ache 1992). In other species, excitatory or inhibitory electrophysiological effects in response to stimulation with the different odors of single receptor neurons were demonstrated (Dionne 1992; Lucero et al. 1992). A feature that appears to be implied by our model investigation is that individual receptor protein types may affect more than one second messenger system. Based on biochemical and electrophysiological results, it has been suggested that receptor proteins are specific for only one of the two second messenger systems (IP3 or cAMP) (see, e.g., Reed 1992; Ache 1993). Our simulation of this particular case (see Fig. 4e,f) revealed reasonably good fits for a high number of receptor protein types expressed in a single olfactory receptor neuron. Since one of the two transducer mechanisms is inhibitory, models based on this assumption can account for inhibitory mixture interactions. Further improvement of the simulations, however, can be achieved when interactions between receptor protein types and different transduction mechanisms are possible. These may occur on several levels. For instance, receptor-activated G-proteins exert their effects by a GTP-driven dissociation into Gc~ and GZ.~ subunits that may affect different targets (Clapham 1994). Moreover, the inositol-lipid pathway diverges onto several different second messenger systems, such as IP3, Ca 2§ and diacylglycerol (DAG). Other potential sites of interaction between second messenger systems are second messenger-activated protein kinases or proteins involved in second messenger turnover. Thus, olfaction may involve complexity at various levels of the transduction cascade: (a) A single ligand may bind to different receptor proteins, e.g., with different affinities; (b) olfactory receptor neurons can be equipped with two or more odor-driven second messengers, one of which may be inhibitory; (c) second messenger systems may interact, and a single receptor protein type may activate different second messenger systems. Determining the mechanisms underlying olfactory transduction will require the combined approach of psychophysics, electrophysiology, biochemistry, molecular biology, and modelling. Based on our model, we suggest that olfactory sensilla of the general processing subsystem in insects may also be equipped with several receptor protein types and at least two interactive second messenger systems, one of which has inhibitory effects on excitability. Independent of the physiological realization, the complex nonlinear response characteristics of honeybee chemosensory neurons require special processing machinery in the nervous system in order to extract and classify olfactory information. Attempts to model this computational task rely on realistic input signals. Our ANN version of the olfactory transduction mechanism provides a convenient method to generate such signals. Models of the honeybee olfactory system using an ANN version of the competitive receptor transducer model to generate responses of sensory neurons are currently under investigation. Acknowledgements. We want to thank Pat Akers and Wayne Getz for giving us subplacode response data to train the ANNs. We also want to thank Wayne Getz for fruitful discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG), SPP Physiologic und Theofie neuronaler Netze. Appendix Relative error of the receptor transducer models We give an upper bound for the relative error which is caused by the simplification of [/~j]. We restrict this to the case that all (5k are 1. The more general case with inhibitory transducers can be handled analogously by computing two upper bounds, one for the error with excitatory transducers and one for that with inhibitory transducers, which results in the same upper boundary for the relative error. To assess the relative error of the model, two inequations are used: (1) Let a, b >_ O, c > O, then a/b>(a+c)/(b+c) r a>_b (2) Let bi, b > O, al/bi <_ a/b, i = 1 , . . . , n then The proof is simple and is omitted here. With (16.1) and the simplification [/~j] = [Rj] + ~ i [ A i R j ] the concentration of free receptor proteins R j evaluates to [Rj] in the simplified case and to [R~] in the correct version: l The simplified effect E and the correct E p can be calculated with [Rj] and [R~] in 08). Since E ' _< E because of [R~] _< [Rj] the relative error e becomes: Proof: e = (E- E')/E' = E/E' - 1 ( k k i,j ] / i,j / t Because the concentration of free Tk depends on the concentration of free R j , we also distinguish between [T~] and [Tk]. 206 with the second inequation and ajk e<-max((~-~[AiRjTk]l / (~-~[AIR;Tk]I - 1 \ =max i,j / [Tk] ~-~j ajk[R d] =max Edadk[Rd] (1+ - \ := ~-~i lidkklj[Ai]: i,j / 1 ~-~dajk[R~]) --1 k ~jajktR;](l+~-~jajk[Rd]) = -1 max with inequation (1) and [R~] < [Rj]: ~~,d(adk[Rj] + ajk[Rj]adk[R~]) e < max k ~j(adk[R~] + -1 adk[Rd]adk[R~] Reapplication of inequation (2) leads to [Rd]ajk + [Rd]ajk[R~]ajk k,d [R~]adk + [Rd]adk[R~]ajk e < max = max [Rj.__~] 1 + [RIj]adk k,j [R~] 1 + [Rd]ajk = max 1/thai + a~k k,~ l / J R d] + adk =max k,j - 1 1 1 1 + Y~i kij[Ai] + ~ i k lijkkij[Ad[T~] + ajk[t~j] ' 1 + ~-~i klj[Ai] + ajk[l~j] - I ~ i , k lijkkij [Ai][T~] = max k,j 1 + Y~i kij[Ai] + Y~i lijkkij[Ai][Rj] Let n be the number of elementary odors we get: [Ai], then with inequation (2) ~-~k lidkkij [Ai][T~] e< max Lj,k 1/n + klj[Ai] + lijkkij[Ai][t~j] tijktTs = max ~ i,j,k k T' ~ k l i J k [-~ k] i,j,k 1 + lijk[Rj] = max n k~ tAd 1 + (1 + lijk[Rj])[Ai] (1 - 1/(1 +(1 n klj +lijk[Rj])[Ai]nkij)) Because of lijk, [/~j], [Ai], n, kid _> 0, the assertion of the theorem is proven. 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