A Formal Model of the Insect Olfactory 
Macroglomerulus: Simulations and 
Analytical Results. 
Christiane Linster 
David Marsan 
ESPCI, Laboratoire d'Electronique 
10, Rue Vauquelin 
75005 Paris, France 
Michel Kerszberg 
Institut Pasteur 
CNRS (URA 1284) 
Neurobiologie Mo16culaire 
25, Rue du Dr. Roux 
75015 Pads, France 
Claudine Masson 
Laboratoire de Neurobiologie Compar6e des 
Invertbr6es 
INRA/CNRS (URA 1190) 
91140 Bures sur Yvette, France 
G6rard Dreyfus 
L6on Personnaz 
ESPCI, Laboratoire d'Electronique 
10, Rue Vauquelin 
75005 Paris, France 
Abstract 
It is known from biological data that the response patterns of 
interneurons in the olfactory macroglomerulus (MGC) of insects are of 
central importance for the coding of the olfactory signal. We propose an 
analytically tractable model of the MGC which allows us to relate the 
distribution of response patterns to the architecture of the network. 
1. Introduction 
The processing of pheromone odors in the antennal lobe of several insect species relies on 
a number of response patterns of the antennal lobe neurons in reaction to stimulation with 
pheromone components and blends. Antennal lobe interneurons receive input from different 
receptor types, and relay this input to antennal lobe projection neurons via excitatory as 
well as inhibitory synapses. The diversity of the responses of the interneurons and 
projection neurons as well the long response latencies of these neurons to pheromone 
stimulation or electrical stimulation of the antenna, suggest a polysynaptic pathway 
1022 
A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results 1023 
between the receptor neurons and these projection neurons (for a review see (Kaissling, 
1990; Masson and Mustaparta, 1990)). 
PI-E'ROMONE ALISTS 
A. Camot Discriminate Single Odors 
1. Excited Type 
BAL * 
C15 * 
Blend * 
2. Inl'ted Type 
BAL - 
C15 - 
Blend - 
ANO Cannot Code Temporal Changes 
!1. PI-ER SPECIALISTS 
A. Can Discriminat Single Odors 
(1) OR (2) 
BAt_ +  
C15 0  
B. Can Discrirniate Single Odors 
Carnot Code Tempal Changes 
i !  
Can Code Temporal Changes 
BAL 
C15 
(1) OR (2) 
-/o/- -/o/- 
I I I ! ! 
Figure 1: With courtesy of John Hildebrand, by permission from Oxford University 
Press, from: Christensen, Mustaparta and Hildebrand: Discrimination of sex 
pheromone blends in the olfactory system of the moth, Chemical Senses, Vol 14, 
no 3, pp 463-477, 1989. 
1024 Linster, Marsan, Masson, Kerszberg, Dreyfus, and Personnaz 
In the MGC of Manduca sexta, antennal lobe interneurons respond in various ways to 
antennal stimulation with single pheromone components or the blend: pheromone 
generalists respond by either excitation or inhibition to both components and the blend: 
they cannot discriminate the components; pheromone specialists respond (i) to one 
component but not to the other by either excitation or inhibition, (ii) with different 
response patterns to the presence of the single components or the blend, namely with 
excitation to one component, with inhibition to the other component and with a mixed 
response to the blend. These neurons can also follow pulsed stimulation up to a cut-off 
frequency (Figure 1). 
A model of the MGC (Linster et al, 1993), based on biological data (anatomical and 
physiological) has demonstrated that the full diversity of response patterns can be 
reproduced with a random architecture using very simple ingredients such as spiking 
neurons governed by a first order differential equation, and synapses modeled as simple 
delay lines. In a model with uniform distributions of afferent, inhibitory and excitatory 
synapses, the distribution of the response patterns depends on the following network 
parameters: the percentage of afferent, inhibitory and excitatory synapses, the ratio of the 
average excitation of any interneuron to its spiking threshold, and the amount of feedback 
in the network. 
In the present paper, we show that the behavior of such a model can be described by a 
statistical approach, allowing us to search through parameter space and to make predictions 
about the biological system without exhaustive simulations. We compare the results 
obtained with simulation of the network model to the results obtained analytically by the 
statistical approach, and we show that the approximations made for the statistical 
descriptions are valid. 
2. Simulations and comparison to biological data 
In (Linster et al, 1993), we have used a simple neuron model: all neurons are spiking 
neurons, governed by a first order differential equation, with a membrane time constant and 
a probabilistic threshold O. The time constant represents the decay time of the membrane 
potential of the neuron. The output of each neuron consists of an all-or-none action 
potential with unit amplitude that is generated when the membrane potential of the cell 
crosses a threshold, whose cumulative distribution function is a continuous and bounded 
probabilistic function of the membrane potential. All sources of delay and signal 
transformation from the presynaptic neuron to its postsynaptic site are modeled by a 
synaptic time delay. These delays are chosen in a random distribution (gaussian), with a 
longer mean value for inhibitory synapses than for excitatory synapses. We model two 
main populations of olfactory neurons: receptor neurons which are sensitive to the main 
pheromone component (called A) or to the minor pheromone component (called B) project 
uniformly onto the network of interneurons; two types of interneurons (excitatory and 
inhibitory) exist: each interneuron is allowed to make one synapse with any other 
interneuron. 
The model exhibits several behaviors that agree with biological data, and it allows us to 
state several predictive hypotheses about the processing of the pheromone blend. We 
observe two broad classes of interneurons: selective (to one odor component) and non- 
selective neurons (in comparison to Figure 1). Selective neurons and non-selective neurons 
exhibit a variety of response patterns, which fall into three classes: inhibitory, excitatory 
and mixed (Figure 2). Such a classification has indeed been proposed for olfactory antennal 
A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results 1025 
lobe neurons (local interneurons and projection neurons) in the specialist olfactory system 
in Manduca (Christensen and Hildebrand, 1987) and for the cockroach (Burrows et al, 1982; 
Boeckh and Ernst, 1987). 
Action 
potentials 
Membrane 
potential 
Inhibitory response Excitatory response Simple mixed response 
I1' i ll U,h,,, I i,,,,,,, i, l,,  ,li,, l 1, I 111111111111lilalfilami#lil i I I ,I d III '" III tlllllllltllill i  
Stimulus A 
Stimulus B 
! \ 
! \ 
Mixed responses 
,lilt t Illit ,Ill Ill IIldU Ill, ih th ttlh t ilh  ]Ill, t llil illIll , 
500 ms r 
Oscillatory responses r 
 - 
......... a, a,, ,a , a, ,a, 
/ 
Figure 2: Response patterns of interneurons in the model presented, in response to 
stimulation with single components A and B, and with a blend with equal 
component concentrations. Receptor neurons fire at maximum frequency during the 
stimulations. The interneuron in the upper row is inhibited by stimulus A, excited 
by stimulus B, and has a mixed response (excitation followed by inhibition) to the 
blend: in reference to Figure 1, this is a pheromone specialist receiving mixed input 
from both types of receptor neurons. These types of simple and mixed responses can 
be observed in the model at low connectivity, where the average excitation received 
by an interneuron is low compared to its spiking threshold. The neuron in the middle 
row responds with similar mixed responses to stimuli A, B and A+B. The neuron in 
the lower row responds to all stimuli with the same oscillatory response, here the 
average excitation received by an interneuron approaches or exceeds the spiking 
threshold of the neurons. Network parameters: 15 receptor neurons; 35 interneurons; 
40% excitatory interneurons; 60% inhibitory interneurons; afferent connectivity 
10%; membrane time constant 25 ms; mean inhibitory synaptic delays 100 ms; 
mean excitatory synaptic delays 25 ms, spiking threshold 4.0, synaptic weights +1 
and -1. 
1026 Linster, Marsan, Masson, Kerszberg, Dreyfus, and Personnaz 
In our model, as well as in biological systems (Christensen and Hildebrand 1988, 
Christensen et al., 1989) we observe a number of local interneurons that cannot follow 
pulsed stimulation beyond a neuron-specific cut-off frequency. This frequency depends on 
the neuron response pattern and on the duration of the interstimulus interval. 
Therefore, the type of response pattern is of central importance for the coding of the 
olfactory signal. Thus, in order to be able to relate the coding capabilities of a (model or 
biological) network to its architecture, we have investigated the distribution of response 
patterns both analytically and by simulations. 
3. Analytical approach 
In order to investigate these questions in a more rigorous way, some of us (C.L., D.M., 
G.D., L.P.) have designed a simplified, analytically tractable model. 
We define two layers of intemeurons: those which receive direct afferent input from the 
receptor neurons (layer 1), and those which receive only input from other interneurons 
(layer 2). In order to predict the response pattern of any interneuron as a function of the 
network parameters, we make the following assumptions: (i) statistically, all interneurons 
within a given layer receive the same synaptic input, (ii) the effect of feedback loops from 
layer 2 can be neglected, (iii) the response patterns have the same distribution for 
stimulations either by the blend or by pure components. Assumption (i) is correct because 
of the uniform distribution of synapses in the network of interneurons. Assumption (ii) is 
valid at low connectivity: if the average amount of excitation received by an interneuron is 
low as compared to its spiking threshold, its firing probability is low; therefore, the effect 
of the excitation from the receptors is vanishingly small beyond two interneurons: we thus 
neglect the effect of signals sent from layer 2. Thus, feedback is present within layer 1, and 
layer 2 receives only feedforward connections. Assumption (iii) is plausible if we suppose 
that the natural pheromone blend is more relevant for the system than the single 
components of the blend. We further assume in the analytical approach (as in the 
simulations) that the synaptic delays are longer on the average for inhibitory synapses than 
for excitatory synapses. 
An interneuron can thus respond with four types of patterns: non-response, which means 
that it does not have a presynaptic neuron (this response pattern can only occur in layer 2, 
at low connectivity); excitation, meaning that an interneuron receives only afferent input 
from receptor neurons or from excitatory interneurons; inhibition, meaning that an 
interneuron receives only input from inhibitory interneurons (this can occur in layer 2 
only); and mixed responses, covering all other combinations of presynaptic input. 
We consider a network of N + N r neurons, N (number of interneurons) and N r (number of 
receptor neurons) being random variables, N + N r being fixed. We define the probability n i 
that a neuron is an inhibitory interneuron, and the probability ns that it is an excitatory 
interneuron. Any interneuron has a probability c to make one synapse (with synaptic 
weight +1 or -1) with any other interneuron and a probability (1 - c) not to make a synapse 
with this interneuron; cr is the afferent connectivity: any receptor neuron has a probability 
c to connect once to any interneuron, and a probability (1 - cr) not to connect to this 
interneuron. Then n, = 1 - (1 - cr)N is the probability that an interneuron belongs to layer 
1, and the number of interneurons in layer 1 obeys a binomial distribution with expectation 
value N n and variance N n, (1 - n,). In the following, the fixed number of interneurons in 
layer 1 will be taken equal to its expectation value. Similarly, the number of interneurons 
in layer 2 is taken to be N (1 - n,). 
A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results 1027 
Because of the assumptions made above, in both layers, we take into account for each 
interneuron the N n,, c synapses from presynaptic neurons of layerl. In layer 1, these 
neurons respond with excitatory or mixed responses. P N c is the probability that 
e '- rtetla 
an interneuron in layer 1 responds with an excitation, and  
Prn = 1 - tlena N c is the 
probability that an interneuron in layer 1 receives mixed synaptic input. 
In layer 2, we have to consider two cases: (i) at low connectivity, if 
N c n, < 1, P02 = 1 - N c n,, is the probability that an interneuron of layer 2 does not 
receive a synapse, thus does not respond to stimulation, p2 = N c n,,n is the probability 
that a neuron in layer 2 responds with excitation, Pi 2 = N c nani is the probability that an 
interneuron responds with inhibition; (ii) at higher connectivity, N c n,, > 1, P02 = 0, 
p2 = n n,N c and p/2 = ni n,,Nc. In both cases (i) and (ii), the probability that an 
interneuron in layer 2 has a mixed response pattern is Pm 2 = 1 - P02 - pZ_ p/2. 
Thus, an interneuron in the model responds with excitation with probability 
P = n,, P + (1 - 2 ........ 
na) P, with nhlbton with probability Pi -' na Pi 1 + (1 - ha) Pi 2 and has 
 .. 1 2 
a mixed response with probablhty Pm "'tla Pm+ (1 - rta) Pro. 
0.804 
0.80-' 
o .4o  
o.2o-' 
o1o 
o .20 
0-401 t Inhibition 
Excitaaon ' o   o ' o :,o ' 
o .8o.: Mixed 
0.60! 
0.40' 
o. ! o o .20 
Layer 1 
o3o oko oso oeo ' o7o c 
Layer 2 
0 30 ] 0 
o.,o .so oleo 0.70 c 
Layers 1 & 2 
0.30 0,40 0.50 0.e0 0.70 C 
Figure 4: Analytically derived distribution of the response patterns in a typical 
network (35 interneurons, 15 receptor neurons, 40% excitation, 60% inhibition, 
spiking threshold 4.0); the curves show the percentage of interneurons in the model 
which respond with a given pattern, as a function of the connectivity c. In this case, 
the average excitation an interneuron receives from other interneurons is 3.15 at 
c=0.3. 
Figure 4 shows the distribution of the response patterns computed analytically for a typical 
set of parameters. In order to perform comparisons between computed pattern distributions 
and pattern distributions obtained from simulations with the model, we designed an 
automatic classifier for the response patterns, based on the perceptton learning rule and the 
pocket algorithm (Gallant 1986). The classifier is trained to classify the responses of 
1028 Linster, Marsan, Masson, Kerszberg, Dreyfus, and Personnaz 
individual interneurons, based on their membrane potential, into 5 typical response classes: 
non-response, pure excitation, pure inhibition, simple mixed response and oscillatory 
responses. Figure 5 shows the simulation results for the same set of parameters as for 
Figure 4. The agreement between the two curves shows that the approximations which we 
have made in order to describe the analytical model are valid. 
Figure 6 shows how the mixed responses in the simulations divide into simple mixed and 
oscillatory responses. When the validity limit of the approximations made in the analytical 
approach is reached, all neurons fire at maximum frequency and the network oscillates. 
Therefore, the analytical model describes satisfactorily the whole range of connectivity in 
which the pattern distribution does not reduce to oscillations. The oscillation frequency is 
determined by the mean synaptic delays and by the membrane time constants; more detailed 
results on the oscillatory behavior will be published in a future paper. 
Layer 1 
80 ]l None 
l\ ' ...... .- Mixed 
: ............ 
- _  Inhibition 
Excit,ibl ' 6!1 ...... 
0'.3 Oh 0. b.t 05- c 
Layer 2 
o.ls ..... 
Excitation 
.''"-'_,Inh Layers 1 & 2 
ibition 
;'o3 ...... 6.5' ..... 05 ..... 6!i ....... o'. ..... . ..... 0.5' ....  ......... 
Figure 5: Distribution of the response patterns obtained from simulations of the 
model with the set of parameters described above. The curves show the percentages 
of interneurons that respond with a given pattern, as a function of connectivity c. 
For each value of c, 100 simulation runs with three different stimulation inputs have 
been averaged. 
fions Layers 1 & 2 
6!i ..... ;!i o!3 ..... ! ..... . ..... oi[ b.[ ....  ..... 
Figure 6: Distribution of simple mixed and oscillatory responses in the simulation 
model. With the set of parameters chosen, condition n e c = O is satisfied for c--0.3. 
A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results 1029 
4. Conclusion 
In the olfactory system of insects and mammals, a number of response patterns are 
observed, which are of central importance for the coding of the olfactory signal. In the 
present paper, we show that, under some constraints, an analytical model can predict the 
existence and the distribution of these response patterns. We further show that the 
transition between non-oscillatory and oscillatory regimes is governed by a single 
parameter (n e c / I). It is thus possible, to explore the parameter space without exhaustive 
simulations, and to relate the coding capabilities of a model or biological network to its 
architecture. 
Acknowledgements 
This work was supported in part by a grant from Ministre de la Recherche et de la 
Technologie (Sciences de la Cognition). C. Linster has been supported by a research grant 
(BFR91/051) from the Ministre des Affaires Culturelies, Grand-Duch6 de Luxembourg. 
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