{"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. Freeman, Mass Action in the Nervous System. (Academic Press, N.Y., \n1975), Chapter 3. \n5 L.B. Haberly and G.M. Shepherd, Neurophys.~: 789 (1973). \n6 L.B. 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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}]}