Download Biol. Cybern. 73, 195-207 (1995) - Institut für Biologie Neurobiologie

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.
In a one-dimensional case, i.e., one receptor and transducer mechanism,
we get with a relation of [/~] : [T] of 100:l a relative error below 1%.
References
Ache BW (1993) Towards a common strategy for transducing olfactory
information. Semin. Cell Biol 5:55-63
Akers RP, Getz WM (1992) A test of identified response classes among
olfactory receptor neurons in the honeybee worker. Chem Senses 17:
191-209
Akers RP, Getz WM (1993) Response of olfactory receptor neurons in
honeybees to odorants and their binary mixtures. J. Comp. Physiol. 173:
169-185
Beidler LM (1962) Taste receptor stimulation. Prog. Biophys. Chem. 12:
107-151
Boekhoff I, Tareilus E, Strotmann J, Breer H (1991) Rapid activation of
alternative second messenger pathways in olfactory alia from rats by
different odorants. EMBOJ 9:2453-2458
Boekhoff I, Seifert E, GOggede S, Lindemann M, Kriiger B-W, Breer H
(1993) Pheromone-induced second messenger signaling in insect antennae. Insect Biochem. Mol. Biol. 23:757-762
Breer H, Boekhoff I (1991) Odorants of the same odor class activate different second messenger pathways. Chem Senses 16:19-29
Breer H, Boekhoff I (1992) Second messenger signalling in olfaction. Curr
Opin Neurobiol 2:439--443
Breer H, Shepherd GM (1993) Implications of the NO/cGMP system for
olfaction. Trends Neurosci 16:5-9
Breer H, Boekhoff I, Strotmann J, Raming K, Tareilus E (1989) Molecular
elements of olfactory signal transduction in insect antennae, in: Schild D
(ed.) Chemosensory information processing. Springer, Berlin Heidelberg
New York, p. 75-86
Breer H, Boekhoff I, Tareilus E (1990) Rapid kinetics of second messenger
formation in olfactory transduction. Nature 344:65--68
Carr WES, Derby CD (1986) Chemically stimulated feeding behavior in
marine animals, the importance of chemical mixtures and the involvement of mixture interactions. J. Chem. Ecol. 12:987-1009
Clapham DE (1994) Direct G-protein activation of ion channels? Annu Rev.
Neurosci. 17:441-464
Dionne VE (1992) Chemosensory responses in isolated olfactory receptor
neurons from Necturus maculosus. J. Gen. Physiol 99:415-433
Dionne VE (1994) Emerging complexity of odor transduction. Proc. Natl.
Acad. Sci. USA 91:6253-6254
Ennis DM (1991) Molecular mixture models based on competitive and
non-competitive agonism. Chem. Senses 16:1-17
Fadool DA, Ache BW (1992) Plasma membrane inositol 1.4.5-triphosphateactivated channels mediate signal transduction in lobster olfactory receptor neurons. Neuron 9:907-918
Firestein S, Zufall F, Shepard GM (1991) Single odorsensitive channels
in olfactory receptor neurons are also gated by cyclic nucleotides. J.
Neurosci 11, 3565-3572
Fujimura K, Yokohari F, Tateda H (1991) Classification of antennal olfactory receptors of the cockroach, Periplaneta americana. L. Zool. Sci. 8:
243-255
Getz WM (1991) A neural network for processing olfactory-like stimuli.
Bull. Math. Biol. 53:805-823
Getz WM, Chapman RF (1987) An odor discrimination model with application to kin recognition in social insects. Int. J. Neurosci. 32:963-978
Homberg U, Christensen TA, Hildebrandt JG (1989) Structure and function
of the deutocerebrum in insects Annu. Rev. Entomol. 34, 477-501
Lasareff P (1922) Untersuchungen fiber die lonentheorie der Reizung. II1.
Mitteilung, Ionentheorie der Geschmacksreizung. Arch. Ges. Physiol.,
194:293-297
Lucero MT, Horrigan FT, Gilly WF (1992) Electrical responses to chemical
stimulation of squid olfactory receptor cells. J. Exp. Biol 162, 231-249
Michel WL, Ache BW (1992) Cyclic nucleotides mediate an odor-evoked
potassium conductance in lobster olfactory receptor cells. J. Neurosci.
12, 3979-3984
Michel WL, McClintock TS, Ache BW (1991) Inhibition of lobster olfactory receptor cells by an odor activated potassium conductance J.
Neurophysiol 65, 446-453
Reed RR (1992) Signaling pathways in odorant detection. Neuron 8: 205209
Renquist Y (1919) 0bet den Geschmack. Scand. Arch. Physiol. 38:97-201
Riedmiller M, Braun H (1993) A direct adaptive method for faster backpropagation learning: the RProp algorithm. In: Ruspini H (ed) Proceedings of the IEEE International Conference on Neural Networks (ICNN),
San Francisco, pp 586-591
Ronnett GV, Cho H, Hester LD, Wood SF, Snyder SH (1993) Odorants
differentially enhance phosphoinositide turnover and adenylyl cylase in
olfactory receptor neuronal cultures. J. Neurosci. 13:1751-1758
Rumelhart DE, Hinton GE, Williams RJ (1986) Learning internal representations by error propagation. In: Rumelhart DE, McClelland JL (eds)
Parallel distributed processing, explorations in the microstructures of
cognition, Vol 1. MIT Press, Cambridge, Mass., p. 318-362
Schneider D, Steinbrecht RA (1968) Checklist of insect olfactory sensilla
Symp. Zool. Soc. Lond. 23:279-297
207
Shepherd GM (1991) Computational structure of the olfactory system. In:
Davis JL, Eichenbaum H (eds) Olfaction - a model system for computational neuroscience. MIT Press, Cambridge, Mass., p. 3--41
Shepherd GM (1994) Discrimination of molecular signals by the olfactory
receptor neuron. Neuron 13:771-790
Vareschi E (1971) Duftunterscheidung bei der Honigbiene - EinzelzellAbleitungen und Verhaltensreaktionen. Z Vergl. Physiol 75:143-173
Vogt RG, Rybczynski R, Lemer MR (1989) The biochemistry of odorant
reception and transduction. In: Schild D (ed) Chemosensory information
processing, Springer, Berlin Heidelberg New York, p. 33-76