{"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}]}