{"title": "The Sigmoid Nonlinearity in Prepyriform Cortex", "book": "Neural Information Processing Systems", "page_first": 242, "page_last": 248, "abstract": null, "full_text": "242 \n\nTHE SIGMOID NONLINEARITY IN PREPYRIFORM CORTEX \n\nFrank H.  Eeckman \n\nUniversity of California, Berkeley, CA 94720 \n\nABSlRACT \n\nWe report a study \u00b7on  the relationship  between EEG  amplitude  values  and  unit \nspike output in the prepyriform cortex of awake and motivated rats. This relationship \ntakes  the  form  of a  sigmoid  curve,  that  describes  normalized  pulse-output  for \nnormalized  wave  input.  The  curve  is  fitted  using  nonlinear  regression  and  is \ndescribed by its slope and maximum value. \n\nMeasurements were made for both excitatory and inhibitory neurons in the cortex. \nThese neurons  are known to form  a monosynaptic  negative  feedback  loop.  Both \nclasses of cells can be described by the same parameters. \n\nThe sigmoid curve is asymmetric in that the region of maximal slope is displaced \ntoward  the  excitatory  side.  The  data  are  compatible  with  Freeman's  model  of \nprepyriform  burst generation.  Other analogies  with  existing  neural  nets are being \ndiscussed,  and the implications for signal processing are reviewed. In particular the \nrelationship of sigmoid slope to efficiency of neural computation is examined. \n\nINTRODUCTION \n\nThe olfactory cortex of mammals generates repeated nearly sinusoidal bursts of \nelectrical activity  (EEG)  in  the  30 to  60 Hz.  range 1. These bursts ride on top  of a \nslower (  1 to  2 Hz.), high amplitude wave related to respiration.  Each burst begins \nshortly after inspiration and terminates during expiration. They are generated locally \nin the cortex.  Similar bursts  occur in the olfactory  bulb (OB)  and there is a  high \ndegree of correlation between the activity in the two structures!' \n\nThe two main cell types in the olfactory cortex are the superficial pyramidal cell \n(type A),  an excitatory neuron receiving direct input from the OB, and the cortical \ngranule  cell  (type  B),  an  inhibitory  interneuron.  These  cell  groups  are \nmonosynaptically connected in a negative feedback loop2. \n\nSuperficial  pyramidal  cells  are  mutually  excitatory3, 4,  5  as  well  as  being \nexcitatory to the granule cells. The granule cells are inhibitory to the pyramidal cells \nas well as to each other3, 4, 6. \n\nIn this paper we focus on the analysis of amplitude dependent properties: How is \nthe output of a cellmass (pulses) related to  the synaptic potentials (ie.  waves)? The \nconcurrent  recording  of  multi-unit  spikes  and  EEG  allows  us  to  study  these \nphenomena in the olfactory cortex. \n\nThe anatomy of the olfactory system has been extensively studied beginning with \nthe work of S. Ramon y Cajal 7. The regular geometry and the simple three-layered \narchitecture makes these structures ideally suitable for EEG recording 4,  8. The EEG \ngenerators in  the various olfactory regions have been identified and their synaptic \nconnectivities have been extensively studied9, 10,5,4, 11,6. \n\nThe EEG is the scalar sum of synaptic currents in the underlying cortex. It can \nbe recorded using low impedance \u00ab  .5  Mohm) cortical or depth electrodes.  Multi(cid:173)\nunit signals are recorded in the appropriate cell layers using high  impedance  (> .5 \nMohm)  electrodes and appropriate high pass filtering. \n\nHere we derive a function that relates waves (EEG)  to pulses in the olfactory \ncortex of the rat. This function has a sigmoidal shape. The derivative of this curve \n\n\u00a9 American Institute of Physics 1988 \n\n\f243 \n\ngives us  the gain curve for wave-to-pulse conversion.  This is  the forward  gain for \nneurons embedded in the cortical cellmass. The product of the forward gain values of \nboth sets of neurons (excitatory and inhibitory) gives us  the feedback  gain values. \nThese ultimately determine the dynamics of the system under study. \n\nMATERIALS AND METI-IODS \n\nA total of twenty-nine rats were entered in this study. In each rat a linear array of \n6 100 micron stainless steel electrodes was chronically implanted in the prepyriform \n(olfactory)  cortex.  The  tips  of the  electrodes  were  electrolytically  sharpened to \nproduce  a  tip  impedance on  the order of .5  to  1 megaohm.  The electrodes  were \nimplanted laterally in the midcortex, using stereotaxic coordinates. Their position was \nverified electrophysiologically using a stimulating electrode in the olfactory tract. \nThis procedure has been described earlier by Freeman 12. At the end of the recording \nsession  a  small  iron deposit  was  made to  help  in  histological  verification.  Every \nelectrode position was verified in this manner. \n\nEach rat was recorded from over a two week period following implantation. All \nanimals were awake and attentive. No stimulation (electrical or olfactory) was used. \nThe background environment for recording was the animal's home cage placed in the \nsame room during all sessions. \n\nFor the present study two channels of data were recorded concurrently. Channel \n1 carried the  EEG  signal,  filtered  between  10  and 300 Hz.  and digitized  at  1 ms \nintervals. Channel 2 carried standard pulses 5 V,  1.2 ms wide, that were obtained by \npassing  the  multi-unit  signal  (filtered  between  300  Hz.  and  3kHz.)  through  a \nwindow discriminator. \n\nThese two time-series were stored on disk for off-line processing using a Perkin(cid:173)\n\nElmer 3220 computer. All routines were written in FORTRAN. They were tested on \ndata files containing standard sine-wave and pulse signals. \n\nDATA PROCESSING \n\nThe procedures for obtaining a two-dimensional conditional pulse probability \n\ntable have been described earlier 4. This table gives us the probability of occurrence \nof a spike conditional on both time and normalized EEG amplitude value. \n\nBy counting  the  number  of pulses  at  a  fixed  time-delay,  where  the  EEG  is \nmaximal in amplitude, and plotting them versus the normalized EEG amplitudes, one \nobtains a sigmoidal function:  The Pulse probability Sigmoid Curve (PSC)  13,  14. \nThis function is normalized by dividing it by the average pulse level in the record. It \nis  smoothed  by  passing  it  through  a  digital  1: 1: 1 filter  and  fitted  by  nonlinear \nregression. \n\nThe equations are: \n\nQ = Qmax ( 1- exp [ - ( ev - 1) I Qmax ])  for v> - uO \nfor v < - uO \nQ = -1 \n\n(1  ) \n\nwhere uO is the steady state voltage, and Q = (p-PO)/pO. \nand Qmax =(Pmax-PO)/pO. \nPO  is the background pulse count, Pmax is the maximal pulse count. \nThese equations rely on one parameter only. The derivation and justification for \n\nthese equations were discussed in an earlier paper by Freeman 13. \n\n\f244 \n\nRESULTS \n\nData were obtained from all  animals. They express normalized pulse counts, a \ndimensionless value as a function of normalized EEG values, expressed as a Z-score \n(ie.  ranging from - 3 sd. to + 3 sd., with mean of 0.0).  The true mean for the EEG \nafter filtering is very close to 0.0 m V and the distribution of amplitude values is very \nnearly Gaussian. \n\nThe recording convention was such that high EEG-values (ie. > 0.0 to + 3.0 sd.) \ncorresponded to  surface-negative waves.  These in  turn  occur with  activity  at  the \napical dendrites of the cells of interest. Low EEG values (ie. from - 3.0 sd. to < 0.0) \ncorresponded to surface-positive voltage values, representing inhibition of the cells. \n\nThe data were smoothed and fitted with equation (1). This yielded a Qrnax value \nfor every data file.  There were on average  5 data files  per animal.  Of these  5, an \naverage of 3.7 per animal could be fitted succesfully with our technique. In 25 % of \nthe traces, each representing a different electrode pair, no correlations between spikes \nand the EEG were found. \n\nBesides Qmax  we  also calculated  Q'  the  maximum derivative of the  PSC, \n\nrepresenting the maximal gain. \n\nThere were 108  traces in all.  In the first 61  cases  the Qrnax value described the \nwave-to-pulse conversion for a class of cells whose  maximum firing probability is in \nphase  with  the  EEG.  These  cells  were  labelled  type  A  cells  2.  These  traces \ncorrespond to the excitatory pyramidal cells. The mean for Qmax in that group was \n14.6, with a standard deviation of 1.84. The range was 10.5 to 17.8. \n\nIn the remaining 47 traces the Qmax described the wave-to-pulse conversion for \nclass B cells. Class B is a label for those cells whose maximal firing probability lags \nthe EEG maximum by approximately  1/4 cycle. The mean for Qrnax in that group \nwas  14.3, with a standard deviation  of 2.05.  The range in  this group  was  11.0 to \n18.8. \n\nThe overall mean for Qmax was  14.4 with a standard deviation of 1.94. There is \nno difference in Qmax between both groups as measured by the Student t-test.  The \nnonparametric Wilcoxon rank-sum test also found no difference between the groups \n( p = 0.558 for the t-test; p = 0.729 for the Wilcoxon). \nAssuming that the two groups have Qmax values that are normally distributed (in \ngroup  A,  mean = 14.6,  median =  14.6; in group  B, mean = 14.3, median = 14.1), \nand that they have equal variances ( st. deviation group A is 1.84; st. deviation group \nB is 2.05) but different means,  we  estimated the power of the t-test to  detect that \ndifference in means. \n\nA  difference of 3  points  between the  Qmax's  of the  respective  groups  was \nconsidered to be physiologically  significant. Given these assumptions the power of \nthe t-test to detect a 3 point difference was greater than .999 at the alpha .05 level for \na  two  sided  test.  We  thus  feel  reasonably  confident  that  there  is  no  difference \nbetween the Qmax values of both groups. \n\nThe first derivative of the PSC gives us the gain for  wave-to-pulse conversion4. \nThe maximum value for this first derivative was labelled Q'.  The location at which \nthe maximum Q' occurs was labelled Vmax. Vmax is expressed in units of standard \ndeviation of EEG amplitudes. \n\nThe mean for Q' in group A was 5.7, with a standard deviation of .67, in group B \nit  was 5.6 with  standard deviation  of .73.  Since  Q'  depends  on  Qmax,  the  same \nstatistics apply to both:  there was no significant difference between the two groups \nfor slope maxima. \n\n\f245 \n\nFigure 1. Distribution of Qmax values \n\ngroup A \n\ngroup B \n\n-\n\n14 \n\nH  12 \nCII \n.Q  10 \n\n~  8 \n\n-\n\nI-\n\n-\n-\n\n~ \n\nI-\n\nI-\n\n~  p: \n, \n, \n, \n\n.-\n\n; \n\n,~ \n\n\"  ~\" \n,  , \n, \n, \n, \n\"  ~ \n, \n, \n,  ;: \n, \n, \n, \n, \n\" \n,... \n, \n, \n, \n, \n, \n, \n\" \n, \n, \n, \n, \n, \n\" \n, \n, \n, \n, \n, \n\"!l~. \n, \n, \n, \n' \n, \n\" \n, \n, \n,  .f71. \n\n14 \n\nH  12 \nCII \n.Q  10 \n~  8 \n\n6 \n\n4 \n\n2 \no \n\n; \n\n\" , \n, \n, \n~ \n\" r- ' \n, \n> r-,  , \n' \n, \n. , \n, \n\" \n' \n, \n' \n\" \n, \n,  1', \n, \n': 1', \n, \n' \n\" \n, \n,  1'; \n, \n\" \n, \n\n; \n; \n\n\" \n\n; \n\nr\"': \n; \n, \n; \n, \n, \n, \n, \n\n; \n\n; \n\n~ \n,~ \n\n, \n, \n, \n\" \n\" \n\" \n.' \n\n\"  \" \n\nv, \nv, \n\nI-\n\n, \nr \n, \nI- ~ \n, \n, \n, \n, \n, \n, \n, \n\n.-\n\n6 \n\n4 \n\n2 \no \n\n1011121314151617181920 \n\n1011121314151617181920 \n\nQmax  values \n\nQmax  values \n\nThe mean for Vmax  was  at 2.15  sd.  +/- .307.  In every case Vmax  was on the \nexcitatory side from 0.00, ie.  at a positive value of EEG  Z-scores. All values were \ngreater than  1.00.  A similar phenomenon has been reported in the olfactory bulb 4, \n14,  15. \n\nFigure  2.  Examples  of  sigmoid  fits. \n\nA cell \n\nB cell \n\n14 \n12 \n10 \n~ \n~  8 \n\u00b7rot \n~  6 \nCII \nog \n4 \nCII \n11\\ \n~  2 \nPo  0 \n-2 \n-4 \n\n-3 \n\n14 \n12 \n10 \n\n8 \n6 \n\n4 \n2 \n0 \n\n-2 \n-4 \n\n-3 \n\n-2 \n\n-1 \n\n0 \n\n1 \n\n2 \n\n3 \n\n-2 \n\n-1 \n\n0 \n\n1 \n\n2 \n\n3 \n\nnormalized  EEG  amplitude \n\nQm = 14.0 \n\nQm = 13.4 \n\n\f246 \n\nCOMPARISON WITH DATA FROM TIIE OB \n\nPreviously we derived  Qrnax values for the mitral cell population in the olfactory \nbulb14. The mitral cells are the output neurons of the bulb and their axons form the \nlateral olfactory tract (LOT). The LOT is the main input to the pyramidal cells (type \nA) in the cortex. \n\nFor awake and motivated rats (N =  10)  the mean Qmax value was 6.34 and the \nstandard deviation was  1.46.  The range was 4.41- 9.53.  For anesthetized animals \n(N= 8) the mean was 2.36 and the standard deviation was 0.89. The range was 1.15-\n3.62. There was a significant difference between anesthetized and awake animals. \nFurthermore there is a significant difference between the Qmax value for cortical cells \nand the Qmaxvalue for bulbar cells (non - overlapping distributions). \n\nDISCUSSION \n\nAn important characteristic of a feedback loop is its feedback gain. There is ample \nevidence for the existence of feedback at all levels in the nervous system. Moreover \nspecific feedback loops  between populations of neurons have been described and \nanalyzed in the olfactory bulb and the prepyriform cortex 3, 9, 4. \n\nA  monosynaptic negative feedback loop  has  been  shown to exist in  the  PPC, \nbetween  the  pyramidal cells  and inhibitory cells,  called granule cells  3, 2, 6,  16. \nTime series analysis of concurrent pulse and EEG recordings agrees with this idea. \n\nThe pyramidal cells are in the forward limb of the loop:  they excite the granule \ncells. They are also mutually excitatory 2,4,16. The granule cells are in the feedback \nlimb:  they inhibit the pyramidal cells.  Evidence for  mutual  inhibition (granule to \ngranule) in the PPC also exists 17, 6. \n\nThe analysis of cell firings versus EEG amplitude at selected time-lags allows one \nto derive a  function  (the PSC)  that relates synaptic potentials  to output in a neural \nfeedback  system. The first derivative of this curve gives an estimate of the forward \ngain at that stage of the loop. The procedure has been applied to various structures in \nthe olfactory system 4,  13,  15,  14. The olfactory system lends itself well to this type \nof analysis due to its geometry, topology and well known anatomy. \n\nExamination of the experimental  gain curves  shows that the maximal gain is \ndisplaced  to the  excitatory  side.  This  means  that  not  only  will  the  cells  become \nactivated by excitatory input, but their mutual interaction strength will increase. The \nresult is an oscillatory  burst of high  frequency  (  30- 60 Hz.)  activity.  This is  the \nmechanism behind bursting in the olfactory EEG 4,  13. \n\nIn comparison with the data from the olfactory bulb one notices that there is a \nsignificant difference in the slope and the maximum of the PSC. In cortex the values \nare  substantially  higher,  however  the  Vmax is  similar.  C.  Gray  15  found  a  mean \nvalue of 2.14 +/- 0.41 for V max in the olfactory bulb of the rabbit (N = 6).  Our value \nin the present study is 2.15 +/- .31. The difference is not statistically significant. \n\nThere are important aspects of nonlinear coupling of the sigmoid type that are of \ninterest in cortical functioning.  A  sigmoid interaction between groups of elements \n(\"neurons\") is a prominent feature in many artificial neural nets.  S.  Grossberg has \nextensively  studied the many desirable properties of sigmoids in  these  networks. \nSigmoids can be used to contrast-enhance certain features in the stimulus. Together \nwith a thresholding operation  a sigmoid rule can effectively quench noise. Sigmoids \ncan also provide for a built in gain control mechanism 18, 19. \n\n\f247 \n\nChanging sigmoid slopes have been investigated by J.  Hopfield. In his network \nchanging  the  slope of the  sigmoid  interaction  between  the  elements  affects  the \nnumber of attractors that the system can go  to 20.  We  have  previously  remarked \nupon the similarities between this and the change in sigmoid slope between waking \nand anesthetized animals 14. Here we present a system with a steep slope (the PPC) \nin series with a system with a shallow slope (the DB). \n\nPresent investigations into similarities between the olfactory bulb and Hopfield \n\nnetworks have been reported 21, 22.  Similarities between the cortex and Hopfield(cid:173)\nlike networks have also been proposed 23. \n\nSpatial amplitude patterns of EEG that correlate with significant odors exist in the \nbulb 24.  A transmission of \"wave-packets\" from the bulb to the cortex is known to \noccur 25.  It has been shown through cofrequency and phase analysis that the bulb \ncan drive the cortex 25, 26.  It thus seeems likely that spatial patterns may also exist \nin  the  cortex.  A  steeper sigmoid,  if the  analogy  with  neural  networks  is correct, \nwould allow  the cortex to further classify input patterns coming from the olfactory \nbulb. \n\nIn  this  view  the  bulb  could form  an  initial  classifier as  well  as  a  scratch-pad \nmemory for olfactory events. The cortex could then be the second classifier, as well \nas the more permanent memory. \n\nThese  are  at  present  speculations  that  may  turn  out  to  be  premature.  They \nnevertheless  are  important  in  guiding  experiments  as  well  as  in  modelling. \nTheoretical studies will have to inform us of the likelihood of this kind of processing. \n\nREFERENCES \n\n1 S.L. Bressler and W.J. Freeman, Electroencephalogr. Clin. Neurophysiol. ~: 19 \n(1980). \n. \n2 W.J.  Freeman, J.  Neurophysiol. ll: 1 (1968). \n3 W.J. Freeman, Exptl.  Neurol. .lO.:  525  (1964). \n4 W.J. 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Grossberg, Studies in Applied Mathematics, Vol LII, 3 (MIT Press,  1973) \np  213. \n19  S.  Grossberg, SIAM-AMS  Proc. U:  107  (1981). \n20 J.J Hopfield,  Proc. Natl.  Acad.  Sci. USA  8.1:  3088 (1984). \n21  W.A.  Baird, Physica 2.m:  150  (1986). \n22 W.A.  Baird, AlP Proceedings ill: 29 (1986). \n23  M.  Wilson  and J.  Bower,  Neurosci.  Abstr.  387,10  (1987). \n\n\f248 \n\n24 K.A. Grajski and W.J. Freeman, AlP Proc.lS.l:  188  (1986). \n25 S.L.  Bressler, Brain Res.  ~: 285  (1986). \n26 S.L.  Bressler, Brain Res.~: 294 (1986). \n\n\f", "award": [], "sourceid": 8, "authors": [{"given_name": "Frank", "family_name": "Eeckman", "institution": null}]}