{"title": "A Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytic Results", "book": "Advances in Neural Information Processing Systems", "page_first": 1022, "page_last": 1029, "abstract": null, "full_text": "A Formal Model of the Insect Olfactory \n\nMacroglomerulus: Simulations and \n\nAnalytical Results. \n\nChristiane Linster \n\nDavid Marsan \n\nESPCI, Laboratoire d'Electronique \n\n10, Rue Vauquelin \n75005 Paris, France \n\nMichel Kerszberg \n\nInstitut Pasteur \n\nCNRS (URA 1284) \n\nNeurobiologie Moleculaire \n\n25, Rue du Dr. Roux \n75015 Paris, France \n\nClaudine Masson \n\nLaboratoire de Neurobiologie Comparee des \n\nInvertebrees \n\nINRA/CNRS (URA 1190) \n\n91140 Bures sur Yvette, France \n\nESPCI, Laboratoire d'Electronique \n\nGerard Dreyfus \nLeon Personnaz \n\n10, Rue Vauquelin \n75005 Paris, France \n\nAbstract \n\nIt is known from biological data that the response patterns of \ninterneurons in the olfactory macroglomerulus (MGC) of insects are of \ncentral importance for the coding of the olfactory signal. We propose an \nanalytically tractable model of the MGC which allows us to relate the \ndistribution of response patterns to the architecture of the network. \n\n1. Introduction \n\nThe processing of pheromone odors in the antennallobe of several insect species relies on \na number of response patterns of the antennallobe neurons in reaction to stimulation with \npheromone components and blends. Antennallobe interneurons receive input from different \nreceptor types, and relay this input to antennal lobe projection neurons via excitatory as \nwell as inhibitory synapses. The diversity of the responses of the interneurons and \nprojection neurons as well the long response latencies of these neurons to pheromone \nstimulation or electrical stimulation of the antenna, suggest a polysynaptic pathway \n\n1022 \n\n\fA Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results \n\n1023 \n\nbetween the receptor neurons and these projection neurons (for a review see (Kaissling, \n1990; Masson and Mustaparta, 1990)). \n\nI. Pf-EROMONE ce..ERALlSTS \n\nA. Carnot Discrinilate Single Odors AN) Camot COde Ter1lXlI'aI Olanges \n\n,. Excited Type \n\nStml'S \n\nResooose \n\nBAL \nC1S \nBlend \n\n\u2022 \n\n2. Wtited Type \n\nStint'S \n\nResponse \n\nill' i. I 'I i \n: ... '-I:.!.! tU.i)! D II .. ua \n\n':IIIIIIIM~IIIH~11 l \n\nIII \n.~.~ _ . ..-._,,' .. \n\nI \n\nI \n\nI \n\nI \n\nI \n\nBAL \nC15 \nB1erd \n\n.111.J1IL._-_ '_'_'J1J~UlUlJ \n\n. . . . . \n\nn. PJ-EROMO\\E SPECtAUSTS \n\nA. Can Oiscrini1ate Singe Odors BUT Camot Code Terrporal Olanges \n\nStinhs \n\nResponse \n\n(1) 00 (2) \n\nBAl \nC15 \nBlend \n\n0 \n\u2022 \n\u2022 \nB. can Discriri1ate 5ilgIe Odors \n\n+ \n0 \n\u2022 \n\nI-\n\u2022 \u2022 \u2022 I \n\n,L \n\nf>K) Can Code T~a1 Olanges \n\nstmt\u00a7 \n\n8espoose \n\n(1) 00 (2) \n\n+ \n\n-/./-\n\n\u2022 \n-/./-\n\nBAL \nC15 \nBlend \n\nFigure 1: With courtesy of John Hildebrand, by permission from Oxford University \nPress, from: Christensen, Mustaparta and Hildebrand: Discrimination of sex \npheromone blends in the olfactory system of the moth, Chemical Senses, Vol 14, \nno 3, pp 463-477, 1989. \n\n\f1024 \n\nLinster, Marsan, Masson, Kerszberg, Dreyfus, and Personnaz \n\nIn the MOC of Manduca sexta, antennal lobe interneurons respond in various ways to \nantennal stimulation with single pheromone components or the blend: pheromone \ngeneralists respond by either excitation or inhibition to both components and the blend: \nthey cannot discriminate the components; pheromone specialists respond (i) to one \ncomponent but not to the other by either excitation or inhibition, (ii) with different \nresponse patterns to the presence of the single components or the blend, namely with \nexcitation to one component, with inhibition to the other component and with a mixed \nresponse to the blend. These neurons can also follow pulsed stimulation up to a cut-off \nfrequency (Figure 1). \nA model of the MOC (Linster et aI, 1993), based on biological data (anatomical and \nphysiological) has demonstrated that the full diversity of response patterns can be \nreproduced with a random architecture using very simple ingredients such as spiking \nneurons governed by a first order differential equation, and synapses modeled as simple \ndelay lines. In a model with uniform distributions of afferent, inhibitory and excitatory \nsynapses, the distribution of the response patterns depends on the following network \nparameters: the percentage of afferent, inhibitory and excitatory synapses, the ratio of the \naverage excitation of any interneuron to its spiking threshold, and the amount of feedback \nin the network. \nIn the present paper, we show that the behavior of such a model can be described by a \nstatistical approach, allowing us to search through parameter space and to make predictions \nabout the biological system without exhaustive simulations. We compare the results \nobtained with simulation of the network model to the results obtained analytically by the \nstatistical approach, and we show that the approximations made for the statistical \ndescriptions are valid. \n\n2. Simulations and comparison to biological data \n\nIn (Linster et aI, 1993), we have used a simple neuron model: all neurons are spiking \nneurons, governed by a first order differential equation, with a membrane time constant and \na probabilistic threshold 9. The time constant represents the decay time of the membrane \npotential of the neuron. The output of each neuron consists of an all-or-none action \npotential with unit amplitude that is generated when the membrane potential of the cell \ncrosses a threshold, whose cumulative distribution function is a continuous and bounded \nprobabilistic function of the membrane potential. All sources of delay and signal \ntransformation from the presynaptic neuron to its postsynaptic site are modeled by a \nsynaptic time delay. These delays are chosen in a random distribution (gaussian), with a \nlonger mean value for inhibitory synapses than for excitatory synapses. We model two \nmain populations of olfactory neurons: receptor neurons which are sensitive to the main \npheromone component (called A) or to the minor pheromone component (called B) project \nuniformly onto the network of interneurons; two types of interneurons (excitatory and \ninhibitory) exist: each interneuron is allowed to make one synapse with any other \ninterneuron. \n\nThe model exhibits several behaviors that agree with biological data, and it allows us to \nstate several predictive hypotheses about the processing of the pheromone blend. We \nobserve two broad classes of intemeurons: selective (to one odor component) and non(cid:173)\nselective neurons (in comparison to Figure 1). Selective neurons and non-selective neurons \nexhibit a variety of response patterns, which fall into three classes: inhibitory, excitatory \nand mixed (Figure 2). Such a classification has indeed been proposed for olfactory antennal \n\n\fA Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results \n\n1025 \n\nlobe neurons (local interneurons and projection neurons) in the specialist olfactory system \nin Manduca (Christensen and Hildebrand, 1987) and for the cockroach (Burrows et al, 1982; \nBoeckh and Ernst, 1987). \n\nAction \npotenUals \n\nMembrane \npotential \n\nStimulus A \n\nStimulus B \n\nInhibitory response \n\nI,!r\"'\"'' \",1\"\"\"\"\"\"\",,1', II II I.\"='.,_we.\" dlllI, \u2022 \u2022 ',IIIIIIII'''QII,'', \"III, I \n\nSimple mixed response \n\nExcitatory response \n\n,r-------., \n\n,,------\"\"\"', \n\n,,....-----\"\"\"'\\ \n\n'~----\"\"\"'\\ \n\nMixed responses \n\n....--\n\n......... -------.~ \n\nIbll \u2022\u2022\u2022 , ,111M \" \" IU! ,'d, \", \", \"b I, ,.\" \" .. , 11\" 111111 , ,\"\"I j 111,\" 1\".' I.h IgUIUIIi. I dil'\"'' I I rI \n\n........ \n\n500 ms \n\n,,-----------, \n\nOscillatory responses \n\n-\n\n,,---------.., \n,,---------.., \n\n4 \n\n~\"I II.\"I.! ,., I. II.\" II II\" , \u2022\u2022 I.h \u2022 \u2022 I.! I.,!\" '\" ,I,U'!, !., ,. ! .\".,\" ,,',,! II\" II \",! \n\nV\\fVrfl.]\"'\" ''tMN\\ \",,,, -\\JWtJV'''' \"'J\"\" , \n\n~ \n\nI \n\n,,-------.., \n\n,,-------, \n,,-----\"\"\"', \n\nFigure 2: Response patterns of interneurons in the model presented, in response to \nstimulation with single components A and B, and with a blend with equal \ncomponent concentrations. Receptor neurons fIre at maximum frequency during the \nstimulations. The interneuron in the upper row is inhibited by stimulus A, excited \nby stimulus B, and has a mixed response (excitation followed by inhibition) to the \nblend: in reference to Figure 1, this is a pheromone specialist receiving mixed input \nfrom both types of receptor neurons. These types of simple and mixed responses can \nbe observed in the model at low connectivity, where the average excitation received \nby an interneuron is low compared to its spiking threshold. The neuron in the middle \nrow responds with similar mixed responses to stimuli A, Band A+B. The neuron in \nthe lower row responds to all stimuli with the same oscillatory response, here the \naverage excitation received by an interneuron approaches or exceeds the spiking \nthreshold of the neurons. Network parameters: 15 receptor neurons; 35 interneurons; \n40% excitatory interneurons; 60% inhibitory interneurons; afferent connectivity \n10%; membrane time constant 25 ms; mean inhibitory synaptic delays 100 ms; \nmean excitatory synaptic delays 25 ms, spiking threshold 4.0, synaptic weights + 1 \nand -1. \n\n\f1026 \n\nLinster, Marsan, Masson, Kerszberg, Dreyfus, and Personnaz \n\nIn our model, as well as in biological systems (Christensen and Hildebrand 1988, \nChristensen et ai., 1989) we observe a number of local interneurons that cannot follow \npulsed stimulation beyond a neuron-specific cut-off frequency. This frequency depends on \nthe neuron response pattern and on the duration of the interstimulus interval. \nTherefore, the type of response pattern is of central importance for the coding of the \nolfactory signal. Thus, in order to be able to relate the coding capabilities of a (model or \nbiological) network to its architecture, we have investigated the distribution of response \npatterns both analytically and by simulations. \n\n3. Analytical approach \n\nIn order to investigate these questions in a more rigorous way, some of us (C.L., D.M., \nG.D., L.P.) have designed a simplified, analytically tractable model. \nWe define two layers of interneurons: those which receive direct afferent input from the \nreceptor neurons (layer 1), and those which receive only input from other interneurons \n(layer 2). In order to predict the response pattern of any interneuron as a function of the \nnetwork parameters, we make the following assumptions: (i) statistically, all interneurons \nwithin a given layer receive the same synaptic input, (ii) the effect of feedback loops from \nlayer 2 can be neglected, (iii) the response patterns have the same distribution \nfor \nstimulations either by the blend or by pure components. Assumption (i) is correct because \nof the uniform distribution of synapses in the network of interneurons. Assumption (ii) is \nvalid at low connectivity: if the average amount of excitation received by an interneuron is \nlow as compared to its spiking threshold, its firing probability is low; therefore, the effect \nof the excitation from the receptors is vanishingly small beyond two interneurons: we thus \nneglect the effect of signals sent from layer 2. Thus, feedback is present within layer 1, and \nlayer 2 receives only feed forward connections. Assumption (iii) is plausible if we suppose \nthat the natural pheromone blend is more relevant for the system than the single \ncomponents of the blend. We further assume in the analytical approach (as in the \nsimulations) that the synaptic delays are longer on the average for inhibitory synapses than \nfor excitatory synapses . \nAn interneuron can thus respond with four types of patterns: non-response, which means \nthat it does not have a presynaptic neuron (this response pattern can only occur in layer 2, \nat low connectivity); excitation, meaning that an interneuron receives only afferent input \nfrom receptor neurons or from excitatory interneurons; inhibition, meaning that an \ninterneuron receives only input from inhibitory interneurons (this can occur in layer 2 \nonly); and mixed responses, covering all other combinations of presynaptic input. \nWe consider a network of N + Nr neurons, N (number of interneurons) and Nr (number of \nreceptor neurons) being random variables, N + Nr being fixed. We define the probability ni \nthat a neuron is an inhibitory interneuron, and the probability ne that it is an excitatory \ninterneuron. Any interneuron has a probability c to make one synapse (with synaptic \nweight +1 or -1) with any other interneuron and a probability (1 - c) not to make a synapse \nwith this interneuron; Cr is the afferent connectivity: any receptor neuron has a probability \nCr to connect once to any interneuron, and a probability (1 - cr) not to connect to this \ninterneuron. Then na = 1 - (1 - cr)Nr is the probability that an interneuron belongs to layer \n1, and the number of interneurons in layer I obeys a binomial distribution with expectation \nvalue N nQ and variance N na (1 - na). In the following, the fixed number of interneurons in \nlayer 1 will be taken equal to its expectation value. Similarly, the number of interneurons \nin layer 2 is taken to be N (1 - na). \n\n\fA Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results \n\n1027 \n\nlayer 2, we have \n\nto consider two cases: (i) at \n\nBecause of the assumptions made above, in both layers, we take into account for each \ninterneuron the N na c synapses from presynaptic neurons of layerl. In layer 1, these \nneurons respond with excitatory or mixed responses. P 1 = nena N C is the probability that \nan interneuron in layer 1 responds with an excitation, and p~= 1 - neflaN c is the \nprobability that an interneuron in layer 1 receives mixed synaptic input. \nlow connectivity, if \nIn \nN c na < 1, P6 = 1 - N c na is the probability that an interneuron of layer 2 does not \nreceive a synapse, thus does not respond to stimulation, P; = N c nane is the probability \nthat a neuron in layer 2 responds with excitation, p? = N c nam is the probability that an \ninterneuron responds with inhibition; (ii) at higher connectivity, N c na > 1, P6 = 0, \nP; = ne naN c and pl = m naN c. In both cases (i) and (ii), the probability that an \ninterneuron in layer 2 has a mixed response pattern is P; = 1 - P6 -Pe - Pl. \nThus, an interneuron in the model responds with excitation with probability \nP e = na P; + (1 - na) P;, with inhibition with probability Pi = na p/ + (1 - na) p? and has \na mixed response with probability Pm =na P ~ + (1 - na) p;. \n\nP \n0 .80 \n\n0 .60 \n\n0 .40 \n\n0 .20 \n\nP \n0.80 \n\n0 .60 \n0 .40 \n\n0.20 \n\n0 .80 \n0.60 \n\n0.40 \n\n0.20 \n\nLayer 1 \n\n0 . 10 \n\n0.20 \n\n0 .30 \n\n0 .40 \n\n0.50 \n\n0 .60 \n\n0 .70 C \n\nLayer 2 \n\n0.20 \n\n0.30 \n\n0.40 \n\n0 .50 \n\n0 .60 \n\n0 .70 C \n\nLayers 1 & 2 \n\n0.10 \n\n0.20 \n\n0 .30 \n\n0 .40 \n\n0 .50 \n\n0.60 \n\n0 .70 C \n\nFigure 4: Analytically derived distribution of the response patterns in a typical \nnetwork (35 interneurons, 15 receptor neurons, 40% excitation, 60% inhibition, \nspiking threshold 4.0); the curves show the percentage of interneurons in the model \nwhich respond with a given pattern, as a function of the connectivity c. In this case, \nthe average excitation an interneuron receives from other interneurons is 3.15 at \nc=O.3. \n\nFigure 4 shows the distribution of the response patterns computed analytically for a typical \nset of parameters. In order to perform comparisons between computed pattern distributions \nand pattern distributions obtained from simulations with the model, we designed an \nautomatic classifier for the response patterns, based on the perceptron learning rule and the \npocket algorithm (Gallant 1986). The classifier is trained to classify the responses of \n\n\f1028 \n\nLinster, Marsan, Masson, Kerszberg, Dreyfus, and Personnaz \n\nindividual interneurons, based on their membrane potential, into 5 typical response classes: \nnon-response, pure excitation, pure inhibition, simple mixed response and oscillatory \nresponses. Figure 5 shows the simulation results for the same set of parameters as for \nFigure 4. The agreement between the two curves shows that the approximations which we \nhave made in order to describe the analytical model are valid. \nFigure 6 shows how the mixed responses in the simulations divide into simple mixed and \noscillatory responses. When the validity limit of the approximations made in the analytical \napproach is reached, all neurons fire at maximum frequency and the network oscillates. \nTherefore, the analytical model describes satisfactorily the whole range of connectivity in \nwhich the pattern distribution does not reduce to oscillations. The oscillation frequency is \ndetermined by the mean synaptic delays and by the membrane time constants; more detailed \nresults on the oscillatory behavior will be published in a future paper. \n\n~ ................ . \n\n../'4- Mixed \n\n..-* \n\nLayer 1 \n\n0.3 \n\n0.4 \n\no~ \n\nc \n\nLayer 2 \n\nLayers 1 & 2 \n\nP \n80 \n60 \n40 \n\n20 \n\nP \n80 \n60 \n40 \n\n20 \n\n80 \n60 \n40 \n\n20 \n\n0.4 \n\nFigure 5: Distribution of the response patterns obtained from simulations of the \nmodel with the set of parameters described above. The curves show the percentages \nof interneurons that respond with a given pattern, as a function of connectivity c. \nFor each value of c, 100 simulation runs with three different stimulation inputs have \nbeen averaged. \n\npr-------------------7-=-=--------==~-----------------=-------\n80 \n60 \n40 \n\nLayers 1 & 2 \n\n20 \n\no.~ \n\no.b \n\no. \n\nc \n\nFigure 6: Distribution of simple mixed and oscillatory responses in the simulation \nmodel. With the set of parameters chosen, condition ne c :::: e is satisfied for c::::O.3. \n\n\fA Formal Model of the Insect Olfactory Macroglomerulus: Simulations and Analytical Results \n\n1029 \n\n4. Conclusion \n\nIn the olfactory system of insects and mammals, a number of response patterns are \nobserved, which are of central importance for the coding of the olfactory signal. In the \npresent paper, we show that, under some constraints, an analytical model can predict the \nexistence and the distribution of these response patterns. We further show that the \ntransition between non-oscillatory and oscillatory regimes is governed by a single \nparameter (ne c / E\u00bb. It is thus possible, to explore the parameter space without exhaustive \nsimulations, and to relate the coding capabilities of a model or biological network to its \narchitecture. \n\nAcknowledgements \nThis work was supported in part by a grant from Ministere de la Recherche et de la \nTechnologie (Sciences de la Cognition). C. Linster has been supported by a research grant \n(BFR91/051) from the Ministere des Affaires Culturelles, Grand-Duche de Luxembourg. \n\nReferences \nBoeckh, J. and Ernst, K.D. (1987). Contribution of single unit analysis in insects to an \n\nunderstanding of olfactory function. 1. Compo Physiolo. AI61:549-565. \n\nBurrows, M., Boeckh, J., Esslen, J. (1982). Physiological and Morphological Properties \nof Interneurons in the Deutocerebrum of Male Cockroaches which respond to \nFemale Pheromone. 1. Compo Physiolo. 145:447-457. \n\nChristensen, T.A., Hildebrand, J.G. (1987). Functions, Organization, and Physiology of \nthe Olfactory Pathways in the Lepidoteran Brain. In Arthropod Brain: its Evolution, \nDevelopment, Structure and Functions, A.P. GuPta, (ed), John Wiley & Sons. \n\nChristensen, T.A., Hildebrand, J.G. (1988). Frequency coding by central olfactory neurons \n\nin the spinx moth Manduca sexta. Chemical Senses 13 (1): 123-130. \n\nChristensen, T.A., Mustaparta, H., Hildebrand, J.G. (1989). Discrimination of sex \npheromone blends in the olfactory system of the moth. Chemical Senses 14 \n(3):463-477. \n\nKaissling, K-E., Kramer, E. (1990). Sensory basis of pheromone-mediated orientation in \n\nmoths. Verh. Dtsch. Zoolo. Ges. 83:109-131. \n\nLinster, C., Masson, C., Kerszberg, M., Personnaz, L., Dreyfus, G. (1993). \nComputational Diversity in a Formal Model of the Insect Olfactory \nMacroglomerulus. Neural Computation 5:239-252. \n\nMasson, C., Mustaparta, H. (1990). Chemical Information Processing in the Olfactory \n\nSystem of Insects. Physiol. Reviews 70 (1): 199-245. \n\n\f", "award": [], "sourceid": 677, "authors": [{"given_name": "Christiane", "family_name": "Linster", "institution": null}, {"given_name": "David", "family_name": "Marsan", "institution": null}, {"given_name": "Claudine", "family_name": "Masson", "institution": null}, {"given_name": "Michel", "family_name": "Kerszberg", "institution": null}, {"given_name": "G\u00e9rard", "family_name": "Dreyfus", "institution": null}, {"given_name": "L\u00e9on", "family_name": "Personnaz", "institution": null}]}