Fig. 2. Concentration – response curve with logistic black line and arc tangent grey line fittings. Same example as in Fig. 1.
Response properties maximum firing frequency F
M
, threshold C
t
, saturation C
s
, and dynamic range DC are indicated for the logistic curve.
zero frequency; in the case of logistic the same was done for all concentrations up to and includ-
ing C . As in a previous modelling work e.g.
Rospars et al., 1996, these calculated parameters were not used directly but converted to more
meaningful characteristics, the maximum response F
M
horizontal asymptote, the logconcentrations at threshold C
t
and close to saturation C
s
, and the dynamic range DC=C
s
− C
t
giving the ratio in log units of the extreme concentrations see Fig.
2.
3. Results
3
.
1
. Global characteristics of responses and cur6es Pooling all odorants and concentrations to-
gether, the firing frequency F is found to lay in the range 2.9 – 26.3 with a mean of 9.2 spikes.
This shows that even the strongest responses are relatively weak. Two-way analysis of variance
shows that the response frequency cannot be pre- dicted from the knowledge of the odorant type
alone, which means that all odorants act on fre- quency in the same way. The knowledge of the
neuron alone is more informative and still more that of both the neuron and the odorant.
The weakest stimulus C eliciting a response is −
11, the strongest −
4.9, and the median strength being − 6.7. These values depend on the
odorant. CAM appears as the most effective odorant, ISO the least efficient one, with LIM and
ANI of similar intermediate efficiencies. Interest- ingly, this ordering is consistent with that based
on the proportion of significant responses with respect to the total number of stimulations, CAM
giving the highest proportion 81 and ISO the lowest 65.
Plots of the response frequency F as a function of log concentration C showed that most of the
plots were monotonically increasing. However, some curves in at most four neurons were an
exception to this rule. They showed with some odorants a long plateau at relatively low firing
frequency ca. 7 spikes starting at low concen- tration followed in two cases with a terminal
increase culminating at ca. 20 spikes. In these
2
.
3
. Cur6e fitting The C – F curves were fitted to 3-parameter
logistic and arc tangent functions Fig. 2. The logistic function is
FC = a
1 + exp − bC − d .
1 where a \ 0, b \ 0 and d are the parameters esti-
mated from the data. It possesses two asymptotes, the lower one is zero when C decreases and the
upper one is a when C increases to infinity. The arc tangent function is
FC = 2a
p a tanbC − g,
for C \ g. 2
where a \ 0, b \ 0 and g are the estimated parameters. It possesses one asymptote alpha
when C tends to infinity. Parameters a and a characterize the maximum response of the neu-
ron. In most cases 96 of logistic and 86 of arc tangent the three parameters of the fitted curves
were obtained using a nonlinear least-square al- gorithm. A generalization of 1 introducing an
exponent Hill coefficient greater than one has not been used because determining a fourth
parameter was usually not feasible due to the number of points available. In the case of arc
tangent fits, the first deliverable concentration C
smaller than that eliciting the first significant re- sponse was considered as giving a ‘response’ of
three cases only the part of the curve corresponding to the initial plateau was kept for fitting.
3
.
2
. Maximum firing frequency Fig.
3
The asymptotic frequency a predicted by the logistic fit 13.5 9 8.1 spikes, median 11.5 is
slightly smaller than that a predicted by the arc tangent fit 15.5 9 10.1 spikes, median 13. The
predictions are conservative; only in rare cases they suggest that the true maximum might be above the
maximum observed experimentally. The median values of a, a and the maximum observed frequency
are all in the range 11 – 13 spikes. The distributions of these parameters are asymmetrical Fig. 3 and
the hypothesis of a normal distribution is rejected for both a and a. This global behavior masks
differences
between odorants.
Whatever the
parameter a or a considered, CAM produced the strongest maximum responses median 14 – 19
spikes and LIM the weakest median 7 – 8 spikes, whereas ANI and ISO are barely distinguishable
median 11 – 15 spikes, see Fig. 4a.
3
.
3
. Concentration at threshold Fig.
5
For all neurons and odorants pooled together, the logconcentration C
0.05
at which the logistic curves reach 5 of their maximum varies between
− 11.5 and − 5.5, with a median at − 7.8. This is
very similar to the range of concentrations C − 11.2 to − 5.6 at which the arc tangent curves
rise from 0. For both curves the histograms are remarkably uniform Fig. 5. Part of the variability
between curves comes from differences between odorants Fig. 4b, the curve corresponding to
CAM being shifted to the left median threshold is at C
0.05
C −
8.6 with respect to the other odorants − 6.7 ISO, − 7.0 ANI, − 7.7 LIM.
Analysis of variance shows that ISO and CAM are significantly different.
3
.
4
. Concentration close to maximum response Fig.
6
The approach to the asymptote is quantified by the logconcentration C
0.95
and C
0.80
giving 95 logistic and 80 arc tangent of the maximum
response. These values vary between − 9.5 and − 3 with a median at − 7.0 for the logistic curves and
between − 9.1 and − 3.3 with a median at − 6.9 for the arc tangent curves. These estimates are in
good agreement. The histograms are asymmetric with modes greater than the medians Fig. 6. Here
again CAM curves are shifted to the left with respect to ANI, ISO and LIM Fig. 4c. The median
saturation moves approximately from − 7.0 for CAM and − 6.2 for LIM, to − 5.3 for ANI and
ISO. The odorants do not form an homogeneous group and CAM is significantly different from ANI
for both logistic and arc tangent.
3
.
5
. Dynamic range Fig.
7
The range of concentrations DC over which the response frequency increases from 5 to 95 in
Fig. 3. Maximum frequency in spikes. Histograms for logis- tic and arc tangent curves. Comparison of logistic and arc
tangent values outliers removed.
Fig. 4. Odors have different response characteristics. Comparison of cumulative frequencies for F
m
, C
t
, C
s
and DC see Fig. 2 for CAM, ANI, ISO and LIM.
logistic curves varies from 0.2 to 3.9 rejecting four outliers log units depending on neurons and
odorants. The
distribution of
DC is very asymmetric with a median at 1.0 log unit Fig. 7.
With arc tangent curves, and a response interval going from 0 to 80, the dynamic range lays
principally in the interval 0.1 to 4 log units rejecting two outliers with median 1.0. It has
been seen above that CAM curves are in general shifted to the left. This shift is uniform because it
has the same dynamic range as the other three odorants. Whatever the odorant the median
dynamic range is about 0.3 – 1.2 logistic and 0.7 – 1.4 log units arc tangent. Therefore the
estimates of Delta C do not depend on the model curve which is expected and are also remarkably
independent of the odorant considered Fig. 4d.
3
.
6
. Correlation between characteristics Fig.
8
The correlations between C
t
and F
M
, and C
t
and DC are low, which means that a high sensiv- ity response curve can have a low or a high
maximum frequency and a narrow or wide dy- namic range.
4. Discussion
