242 
THE SIGMOID NONLINEARITY IN PREPYRIFORM CORTEX 
Frank H. Eeckman 
University of California, Berkeley, CA 94720 
ABSTRACT 
We report a study bn the relationship between EEG amplitude values and unit 
spike output in the prepyriform cortex of awake and motivated rats. This relationship 
takes the form of a sigmoid curve, that describes normalized pulse-output for 
normalized wave input. The curve is fitted using nonlinear regression and is 
described by its slope and maximum value. 
Measurements were made for both excitatory and inhibitory neurons in the cortex. 
These neurons are known to form a monosynaptic negative feedback loop. Both 
classes of cells can be described by the same parameters. 
The sigmoid curve is asymmetric in that the region of maximal slope is displaced 
toward the excitatory side. The data are compatible with Freeman's model of 
prepyriform burst generation. Other analogies with existing neural nets are being 
discussed, and the implications for signal processing are reviewed. In particular the 
relationship of sigmoid slope to efficiency of neural computation is examined. 
INTRODUCTION 
The olfactory cortex of mammals generates repeated nearly sinusoidal bursts of 
electrical activity (EEG) in the 30 to 60 Hz. range 1. These bursts ride on top of a 
slower ( 1 to 2 Hz.), high amplitude wave related to respiration. Each burst begins 
shortly after inspiration and terminates during expiration. They are generated locally 
in the cortex. Similar bursts occur in the olfactory bulb (OB) and there is a high 
degree of correlation between the activity in the two structures 1. 
The two main cell types in the olfactory cortex are the superficial pyramidal cell 
(type A), an excitatory neuron receiving direct input from the OB, and the cortical 
granule cell (type B), an inhibitory interneuron. These cell groups are 
monosynaptically connected in a negative feedback loop2. 
Superficial pyramidal cells are mutually excitatory3, 4, 5 as well as being 
excitatory to the granule cells. The granule cells are inhibitory to the pyramidal cells 
as well as to each other3, 4, 6. 
In this paper we focus on the analysis of amplitude dependent properties: How is 
the output of a cellmass (pulses) related to the synaptic potentials (ie. waves)? The 
concurrent recording of multi-unit spikes and EEG allows us to study these 
phenomena in the olfactory cortex. 
The anatomy of the olfactor system has been extensively studied beginning with 
the work of S. Ramon y Caj al 7. The regular geometry and the simple three-layered 
architecture makes these structures ideally suitable for EEG recording 4, 8. The EEG 
generators in the various olfactory regions have been identified and their synaptic 
connectivities have been extensively studied9, 10, 5, 4, 11, 6. 
The EEG is the scalar sum of synaptic currents in the underlying cortex. It can 
be recorded using low impedance (< .5 Mohm) cortical or depth electrodes. Multi- 
unit signals are recorded in the appropriate cell layers using high impedance (> .5 
Mohm) electrodes and appropriate high pass filtering. 
Here we derive a function that relates waves (EEG) to pulses in the olfactory 
cortex of the rat. This function has a sigmoidal shape. The derivative of this curve 
American Institute of Physics 1988 
243 
gives us the gain curve for wave-to-pulse conversion. This is the forward gain for 
neurons embedded in the cortical cellmass. The product of the forward gain values of 
both sets of neurons (excitatory and inhibitory) gives us the feedback gain values. 
These ultimately determine the dynamics of the system under study. 
MATERIALS AND METHODS 
A total of twenty-nine rats were entered in this study. In each rat a linear array of 
6 100 micron stainless steel electrodes was chronically implanted in the prepyriform 
(olfactory) cortex. The tips of the electrodes were electrolytically sharpened to 
produce a tip impedance on the order of .5 to 1 megaohm. The electrodes were 
implanted laterally in the midcortex, using stereotaxic coordinates. Their position was 
verified electrophysiologically using a stimulating electrode in the olfactory tract. 
This procedure has been described earlier by Freeman 12. At the end of the recording 
session a small iron deposit was made to help in histological verification. Every 
electrode position was verified in this manner. 
Each rat was recorded from over a two week period following implantation. All 
animals were awake and attentive. No stimulation (electrical or olfactory) was used. 
The background environment for recording was the animal's home cage placed in the 
same room during all sessions. 
For the present study two channels of data were recorded concurrently. Channel 
1 carried the EEG signal, filtered between 10 and 300 Hz. and digitized at 1 ms 
intervals. Channel 2 carried standard pulses 5 V, 1.2 ms wide, that were obtained by 
passing the multi-unit signal (filtered between 300 Hz. and 3kHz.) through a 
window discriminator. 
These two time-series were stored on disk for off-line processing using a Perkin- 
Elmer 3220 computer. All routines were written in FORTRAN. They were tested on 
data files containing standard sine-wave and pulse signals. 
DATA PROCESSING 
The procedures for obtaining a two-dimensional conditional pulse probability 
table have been described earlier 4. This table gives us the probability of occurrence 
of a spike conditional on both time and normalized EEG amplitude value. 
By counting the number of pulses at a fixed time-delay, where the EEG is 
maximal in amplitude, and plotting them versus the normalized EEG amplitudes, one 
obtains a sigmoidal function: The Pulse probability Sigmoid Curve (PSC) 13, 14. 
This function is normalized by dividing it by the average pulse level in the record. It 
is smoothed by passing it through a digital 1:1:1 filter and fitted by nonlinear 
regression. 
The equations are: 
Q = Qmax ( 1- exp [ - ( ev - 1) / Qmax ] ) for v > - u0 
Q = -1 for v < - u0 
(1) 
where u0 is the steady state voltage, and Q = (p-p0)/p0. 
and Qmax =(pmax-p0)/p0. 
p0 is the background pulse count, Pmax is the maximal pulse count. 
These equations rely on one parameter only. The derivation and justification for 
these equations were discussed in an earlier paper by Freeman 13. 
244 
RESULTS 
Data were obtained from all animals. They express normalized pulse counts, a 
dimensionless value as a function of normalized EEG values, expressed as a Z-score 
(ie. ranging from - 3 sd. to + 3 sd., with mean of 0.0). The true mean for the EEG 
after filtering is very close to 0.0 mV and the distribution of amplitude values is very 
nearly Gaussian. 
The recording convention was such that high EEG-values ( ie. > 0.0 to + 3.0 sd.) 
corresponded to surface-negative waves. These in turn occur with activity at the 
apical dendrites of the cells of interest. Low EEG values (ie. from - 3.0 sd. to < 0.0) 
corresponded to surface-positive voltage values, representing inhibition of the cells. 
The data were smoothed and fitted with equation (1). This yielded a Qmax value 
for every data file. There were on average 5 data files per animal. Of these 5, an 
average of 3.7 per animal could be fitted succesfully with our technique. In 25 % of 
the traces, each representing a different electrode pair, no 6orrelations between spikes 
and the EEG were found. 
Besides Qmax we also calculated Q' the maximum derivative of the PSC, 
representing the maximal gain. 
There were 108 traces in all. In the first 61 cases the Qmax value described the 
wave-to-pulse conversion for a class of cells whose maximum firing probability is in 
phase with the EEG. These cells were labelled type A cells 2. These traces 
correspond to the excitatory pyramidal cells. The mean for Qmax in that group was 
14.6, with a standard deviation of 1.84. The range was 10.5 to 17.8. 
In the remaining 47 traces the Qmax described the wave-to-pulse conversion for 
class B cells. Class B is a label for those cells whose maximal firing probability lags 
the EEG maximum by approximately 1/4 cycle. The mean for Qmax in that group 
was 14.3, with a standard deviation of 2.05. The range in this group was 11.0 to 
18.8. 
The overall mean for Qmax was 14.4 with a standard deviation of 1.94. There is 
no difference in Qmax between both groups as measured by the Student t-test. The 
nonparametric Wilcoxon rank-sum test also found no difference between the groups 
( p = 0.558 for the t-test; p = 0.729 for the Wilcoxon). 
Assuming that the two groups have Qmax values that are normally distributed ( in 
group A, mean = 14.6, median = 14.6; in group B, mean = 14.3, median = 14.1), 
and that they have equal variances ( st. deviation group A is 1.84; st. deviation group 
B is 2.05) but different means, we estimated the power of the t-test to detect that 
difference in means. 
A difference of 3 points between the Qmax'S of the respective groups was 
considered to be physiologically significant. Given these assumptions the power of 
the t-test to detect a 3 point difference was greater than .999 at the alpha .05 level for 
a two sided test. We thus feel reasonably confident that there is no difference 
between the Qmax values of both groups. 
The first derivative of the PSC gives us the gain for w,e-to-pulse conversion4. 
The maximum value for this first derivative was labelled Q. The location at which 
the maximum Q' occurs was labelled Vmax. Vmax is expressed in units of standard 
deviation of EEG amplitudes. 
The mean for Q' in group A was 5.7, with a standard deviation of .67, in group B 
it was 5.6 with standard deviation of .73. Since Q' depends on Qmax, the same 
statistics apply to both: there was no significant difference between the two groups 
for slope maxima. 
245 
Figure 1. Distribution of Qmax values 
group A group B 
14 
' lO 
8 
6 
4 
2 
0 
1011121314151617181920 
qmax values 
14 
12 
6 
4 
2 
1011121314151617181920 
Qmx values 
The mean for Vmax was at 2.15 sd. +/- .307. In every case Vmax was on the 
excitatory side from 0.00, ie. at a positive value of EEG Z-scores. All values were 
greater than 1.00. A similar phenomenon has been reported in the olfactory bulb 4, 
14, 15. 
Figure 2. Examples of sigmoid fits. 
A cell B cell 
14 14 
12 12 
10 10 
4 4 
2 2 
o m o 
-4 -4  
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 
ormalized EES amplitmte 
Qm = 14.0 Qm = 13.4 
246 
COMPARISON WITH DATA FROM THE OB 
Previously we derived Qmax values for the mitml cell population in the olfactory 
bulb14. The mitral cells are the output neurons of the bulb and their axons form the 
lateral olfactory tract (LOT). The LOT is the main input to the pyramidal cells (type 
A) in the cortex. 
For awake and motivated rats (N = 10) the mean Qmax value was 6.34 and the 
standard deviation was 1.46. The range was 4.41- 9.53. For anesthetized animals 
(N= 8) the mean was 2.36 and the standard deviation was 0.89. The range was 1.15- 
3.62. There was a significant difference between anesthetized and awake animals. 
Furthermore there is a significant difference between the Qmax value for cortical cells 
and the Qmaxvalue for bulbar cells ( non - overlapping distributions). 
DISCUSSION 
An important characteristic of a feedback loop is its feedback gain. There is ample 
evidence for the existence of feedback at all levels in the nervous system. Moreover 
specific feedback loops between populations of neurons have been described and 
analyzed in the olfactory bulb and the prepyriform cortex 3, 9, 4. 
A monosynaptic negative feedback loop has been shown to exist in the PPC, 
between the pyramidal cells and inhibitory cells, called granule cells 3, 2, 6, 16. 
Time series analysis of concurrent pulse and EEG recordings agrees with this idea. 
The pyramidal cells are in the forward limb of the loop: they excite the granule 
cells. They are also mutually excitatory 2,4,16. The granule cells are in the feedback 
limb: they inhibit the pyramidal cells. Evidence for mutual inhibition (granule to 
granule) in the PPC also exists 17, 6. 
The analysis of cell firings versus EEG amplitude at selected time-lags allows one 
to derive a function (the PSC) that relates synaptic potentials to output in a neural 
feedback system. The first derivative of this curve gives an estimate of the forward 
gain at that stage of the loop. The procedure has been applied to various structures in 
the olfactory system 4, 13, 15, 14. The olfactory system lends itself well to this type 
of analysis due to its geometry, topology and well known anatomy. 
Examination of the experimental gain curves shows that the maximal gain is 
displaced to the excitatory side. This means that not only will the cells become 
activated by excitatory input, but their mutual interaction strength will increase. The 
result is an oscillatory burst of high frequency ( 30- 60 Hz.) activity. This is the 
mechanism behind bursting in the olfactory EEG 4, 13. 
In comparison with the data from the olfactory bulb one notices that there is a 
significant difference in the slope and the maximum of the PSC. In cortex the values 
are substantially higher, however the Vmax is similar. C. Gray 15 found a mean 
value of 2.14 +/- 0.41 for Vmax in the olfactory bulb of the rabbit (N= 6). Our value 
in the present study is 2.15 +/- .31. The difference is not statistically significant. 
There are important aspects of nonlinear coupling of the sigmoid type that are of 
interest in cortical functioning. A sigmoid interaction between groups of elements 
("neurons") is a prominent feature in many artificial neural nets. S. Grossberg has 
extensively studied the many desirable properties of sigmoids in these networks. 
Sigmoids can be used to contrast-enhance certain features in the stimulus. Together 
with a thresholding operation a sigmoid rule can effectively quench noise. Sigmoids 
can also provide for a built in gain control mechanism 18, 19. 
247 
Changing sigmoid slopes have been investigated by J. Hopfield. In his network 
changing the slope of the sigmoid interaction between the elements affects the 
number of attractors that the system can go to 20. We have previously remarked 
upon the similarities between this and the change in sigmoid slope between waking 
and anesthetized animals 14. Here we present a system with a steep slope (the PPC) 
in series with a system with a shallow slope (the OB). 
Present investigations into similarities between the olfactory bulb and Hopfield 
networks have been reported 21, 22. Similarities between the cortex and Hopfield- 
like networks have also been proposed 23. 
Spatial amplitude panems of EEG that correlate with significant odors exist in the 
bulb 24. A transmission of "wave-packets" from the bulb to the cortex is known to 
occur 25. It has been shown through cofrequency and phase analysis that the bulb 
can drive the cortex 25, 26. It thus seeems likely that spatial patterns may also exist 
in the cortex. A steeper sigmoid, if the analogy with neural networks is correct, 
would allow the cortex to further classify input patterns coming from the olfactory 
bulb. 
In this view the bulb could form an initial classifier as well as a scratch-pad 
memory for olfactory events. The cortex could then be the second classifier, as well 
as the more permanent memory. 
These are at present speculations that may turn out to be premature. They 
nevertheless are important in guiding experiments as well as in modelling. 
Theoretical studies will have to inform us of the likelihood of this kind of processing. 
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