 |
INTRODUCTION |
Variability has long attracted neurophysiologists as a tool to
investigate the biophysical mechanisms of sensory processing, the
integrative properties of nerve cells, and the encoding schemes used in
various parts of the nervous system (Baylor et al. 1979
; Hecht et al. 1942
; Shadlen et al. 1996
;
Softky and Koch 1993
). Until recently, most work has
focused on characterizing the response variability of nerve cells to
static stimuli, in part because simple measures such as the variance of
the number of spikes recorded in long time windows provide universal
and effective ways to quantify variability under such conditions
(Parker and Newsome 1998
).
Most biologically relevant stimuli, however, are not
static. Therefore, more recently, investigators have started to
characterize the trial-to-trial variability of responses to
time-varying, dynamic stimuli in vivo and in vitro (Bair and
Koch 1996
; Berry et al. 1997
; de Ruyter
van Steveninck et al. 1997
; Mainen and Sejnowski 1995
; Mechler et al. 1998
; Reich et al.
1997
; Stevens and Zador 1998
; Warzecha et
al. 1998
). When temporal variations are sufficiently strong to
induce locking of spikes to stimulus transients, measures such as the
standard deviation in the spike occurrence times following those
transients or the probability of spike occurrence within a given time
window from trial to trial may be used to provide a characterization of
variability (Bair and Koch 1996
; Mainen and
Sejnowski 1995
). However, these measures are not likely to carry over to more general stimulation conditions, when locking to
stimulus transients is absent or less pronounced. An alternative approach consists of extrapolating from the study of static stimuli and
to use the variance in the number of spikes observed in short time
windows as a measure of variability (referred to as the spike count variance) (Berry et al. 1997
; de
Ruyter van Steveninck et al. 1997
; Warzecha and Egelhaaf
1999
). Two goals of the present work are to clarify the limits
of the spike count variance as a measure of short term variability, and
to introduce a new measure of spike time jitter based on recent work by
Victor and Purpura (1996
, 1997
) that
should be applicable to a wide range of stimuli, independent of the
integrative properties of the investigated neurons.
Eigenmannia is a weakly electric gymnotiform fish of wave
type that discharges its electric organ at regular intervals 200-600 times per second. Two types of tuberous sensory afferent nerve fibers
convey information about the resulting electrical environment to the
brain (Scheich et al. 1973
). T-type afferent fibers
provide the first stage of a pathway specialized to process phase
information, called the timing pathway (Heiligenberg
1991
). They fire one spike per electric organ discharge (EOD)
cycle, each tightly phase locked to the zero crossings of the EOD and
thus signal phase modulations of the electric field. P-type afferents,
on the other hand, fire at most one spike per EOD cycle with loose
phase locking to the EOD and a probability that increases in direct
proportion to the mean amplitude of the field. They thus convey
information about amplitude changes of the electric field to higher
order neurons in the brain.
While it is well-known that the timing jitter of P-receptor afferent
spikes is greater than that of T-type afferents (Scheich et al.
1973
), variability in the amplitude pathway has received little
quantitative attention. In contrast, variability in the timing pathway
has been characterized in considerable detail, revealing the high
precision of neurons in encoding phase shifts of the EOD. T-type fibers
are able to fire spikes with a precision of approximately 30 µs
(Carr et al. 1986
). This precision increases at higher
stages of electrosensory processing because of the pooling and
averaging of T-type activity across the body surface (Rose and
Heiligenberg 1985
). Here we focus on the variability of P-type afferents and show that their firing is approximately 100 times less
precise. Nevertheless, our results demonstrate that the jitter in
P-receptor afferent spike trains lies within the appropriate range to
efficiently convey amplitude information to the electrosensory lateral
line lobe, the hindbrain nucleus that forms the first central stage of
the amplitude coding pathway.
Part of our results has been presented in abstract form (Kreiman
et al. 1998
).
 |
METHODS |
Preparation and electrophysiology
Eigenmannia specimens of 12-20 cm body
length were prepared for electrophysiological recordings as described
by Wessel et al. (1996)
and Metzner et al.
(1998)
. Briefly, the EOD frequency was measured, and
subsequently the animal was immobilized and its EOD amplitude
attenuated by an intramuscular injection of Flaxedil (gallamine
triethiodide, Sigma, St. Louis, MO; <5 µg/g body wt). Under local
anesthesia (2% lidocaine, Western Medical Supply, Arcadia, CA), the
posterior branch of the anterior lateral line nerve was exposed just
rostral to the operculum. Signals from P-receptor afferents were
recorded extracellularly from this nerve with glass micropipettes
filled with 3 M KCl (resistance: 40-60 M
), amplified with an
extra/intracellular electrometer (World Precision Instruments 767, Sarasota, FL), and stored on video tape using a PCM recording adapter
(Vetter 3000A, Rebersburg, PA; sampling rate: 40 kHz). They were
subsequently digitized using a commercial data analysis system
(Datawave, Denver, CO; sampling rate: 10 kHz/channel). A few recordings
were acquired and digitized using LabView (National Instruments,
Austin, TX). Data corresponding to one point in Fig. 13 (for the cutoff
frequency fc = 88 Hz, see Stimulation below for a complete description) were obtained
in a previous study (Wessel et al. 1996
).
Stimulation
P-receptor afferents were stimulated as described previously
(Metzner et al. 1998
; Wessel et al.
1996
). A sinusoidal carrier signal (Exact 519, Hillsboro, OR)
with a frequency matched to the EOD frequency
(fEOD) of the fish was
modulated in amplitude. The main difference with earlier work was that
electric field AMs were synthesized and stored digitally for playback
using commercial software (Signal Engineering Design, Belmont, MA;
sampling rate: 2 kHz), allowing for repeated presentations of identical
stimuli. The AM and the carrier signal were gated by the same trigger
signal and were therefore phase locked to each other. The stimuli were delivered via two carbon rod electrodes, one positioned either in front
of the animal or in its mouth, the other behind its tail. No
differences in the neuronal responses were observed between these two
configurations. The mean stimulus amplitude, measured at the side fin
perpendicular to the body axis, ranged from 1 to 5 mV/cm. To avoid
under-driving the afferents, it was adjusted individually for each
P-receptor afferent to stimulate it at 10-15 dB above threshold.
One set of stimuli consisted of random AMs (RAMs) with a flat power
spectrum (white noise) up to a fixed cutoff frequency (fc = 5, 10, 20, 40, and 60 Hz). These AMs were obtained using a modulation signal
s(t) that caused a doubling of the carrier signal
amplitude for s(t) = 1 V and a reduction to
zero for s(t) =
1 V (see Eq. 1 of Wessel et al. 1996
). The standard deviation,
, of
the stimulus s(t) (which can be thought of as the
stimulus contrast) was varied between 10 and 30% of the mean electric
field amplitude (
= 100, 150, 200, 250, 275, and 300 mV;
= 1 V corresponded to a 100% variation of the stimulus
amplitude). Consequently, amplitudes varied over a range of
20 to
10 dB of the mean stimulus amplitude. A single 15-s-long stimulus was
synthesized for each parameter pair
(fc,
) and was presented
10 times, drawn in pseudo-random order from a subset of all possible
(fc,
) combinations. We
usually started by presenting all fc
values at a fixed contrast (
= 250 mV) or all contrasts at two
cutoff frequencies (fc = 5, 60 Hz). Further (fc,
)
combinations were tested as time permitted.
The second set of stimuli consisted of sinusoidal AMs (SAMs) at a fixed
contrast (
= 250 mV) and at various temporal frequencies fs. The values used were
fs = 0.1, 0.5, 1, 5, 7, 10, 20, 50, 100, and 125 Hz. Each stimulus was 15 s long and was presented six times in pseudo-random order. These stimuli were presented interleaved with the RAMs protocol described above.
Characterization of spike train variability
Two methods were used to quantify inter-trial spike train
variability in response to repeated presentations of the same RAM stimulus. We first computed the spike count variance as a function of
the mean spike count in fixed time windows of length T
(see RESULTS and Fig. 5). The same RAM stimulus was
presented R = 10 times and the number of spikes,
ni, occurring in a fixed time window,
T, was determined for each trial i = 1, ... R. The average number of spikes occurring in that
window,
n
(mean spike count), and its variance,

n
2 (spike count
variance), were estimated from
Three window sizes were used (T = 10, 50, and 100 ms), and each time window was successively shifted by 5 ms to
cover the entire stimulus presentation interval. For highly variable
spike trains, such as those corresponding to independent
Poisson-distributed spike occurrence times, the spike count variance
equals the mean independent of the window T. Conversely, if
the R = 10 spike trains are exactly identical,

n
2 = 0 in each
window T. If, however, the spike trains are not exactly identical, the minimum nonzero variance may be computed by considering the discrete nature of spiking. With f lying in the interval
[0;1), we assume that a fraction (1
f) of
spike counts in a fixed interval of length T equals the
integer nT (where nT is
usually small) and the remaining fraction, f, contains one
additional spike, so that the spike counts equal
nT + 1. It then follows that the mean
spike count,
n
(a positive real number), is given by
and the minimal variance is
This last equation states that

n
2 is a quadratic
function of the fraction, f, of spike counts equaling nT + 1 in the interval T. As a
function of f, the minimal variance spans a parabola between
successive integer values of the mean spike count, taking its maximal
value (=1/4) at f = 1/2 and its minimal value
(=0) for integer spike counts (f = 0; see
RESULTS and Fig. 5). Similarly, if all spike counts in
T for all R repetitions are equal to
nT or nT + 1 except for
one spike count equal to nT
1 (or
nT + 2), then the variance still follows a
parabola, but translated by a factor 2/R along the vertical
axis: f(1
f) + 2/R.
Successive parabolas translated vertically are generated by an
analogous procedure (see RESULTS and Fig. 5).
A second measure of inter-trial variability that proved more sensitive
to changes in stimulus parameters (see RESULTS) was obtained by computing an average distance between spike trains obtained
in response to the same RAM. The distance measure employed was
introduced by Victor and Purpura (1996)
based on an
earlier one used to quantify the similarity of DNA sequences (Li
and Graur 1991
, chap. 3; Sellers 1974
).
Operationally, the distance between two spike trains is defined by the
following procedure: the first spike train is transformed into the
second one by a series of elementary steps. Each step is assigned a
"cost" and the distance is obtained by adding up the cost of all
elementary steps and finding the transformation sequence yielding the
minimal cost. This procedure is illustrated in Fig.
1: the two spike trains to be compared
are labeled 1 and 8, while spike trains 2-7 represent the sequence of
elementary steps in the transformation yielding the minimal cost. Only
three elementary steps are allowed: adding a spike (as in step 6 to 7),
deleting a spike (as in step 1 to 2) or moving a spike to a new
position (as in step 2 to 3). The first two elementary steps are
assigned an arbitrary cost of 1, whereas moving a spike by
t ms is assigned a cost of q · |
t| for q positive. Victor and
Purpura (1996
, 1997
) describe an algorithm for
determining the minimum cost transformation sequence and derive the
mathematical properties of the ensuing distance measure,
dij(q), between two spike
trains xi and
xj. The parameter q (measured in units of 1/time) characterizes the time interval for which the
occurrence of a spike in xi is considered
to be significantly different from the occurrence of a spike in
xj: if the interval separating the spikes
is larger than 2/q it is less "expensive" to transform
xi into xj by
first deleting the spike in xi and then
adding it in xj (at a cost of 2) than by
translating it to its new position (at a cost of q · |
t|; Fig. 1B). It is therefore straightforward to compute
dij(q) when q is
large: let ni and nj be the number of spikes in
xi and xj,
respectively, and the integer cij denote
the number of coincident spikes in xi and
xj (coincident within some discretization
interval). For large q's it is always less expensive to
delete and add spikes than to move them so that the distance between
xi and xj is
obtained by first deleting (ni
cij) spikes in
xi and then adding
(nj
cij)
spikes in xj. Thus
|
(1)
|
On the other hand, if the cost of moving a spike vanishes,
q = 0, each spike in xi
may be moved at zero cost to match the position of an arbitrary spike
in xj, and a cost of 1 is only endured for
each additional spike to be added or deleted in
xj. Therefore
|
(2)
|
measures the difference in the number of spikes between the two
spike trains. As q
0 increases,
dij(q) increases monotonically and reaches its maximum value (given by Eq. 1)
when 2/q is smaller than the minimal time interval between
two noncoincident spikes in xi and
xj. Note that if the two spike trains are
perfectly coincident
dij(q) = 0, independent of
q. The distance dij(q) was normalized by the total number of spikes in the two spike trains
|
(3)
|
so that dijn(0)
measures the difference in spike count normalized by the total spike
count and
dijn(q
) is the fraction of noncoincident spikes relative to the total
number of spikes.

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Fig. 1.
Computation of spike train distances. The distance between 2 spike
trains was obtained as the minimum cost to convert one spike train into
the 2nd one using 3 elementary steps. A: the minimum cost
path transforming spike train 1 into spike train 8 is illustrated (for
a fixed value of q). Each intermediate spike train 2-7
corresponds to one elementary step: moving (from 2 to 3), adding (from
6 to 7) or deleting (from 1 to 2) a single spike. The cost of each
elementary step is indicated on the right. Note that the
cost of moving a spike is proportional to the distance that it is moved
along the time axis. B: there are 2 alternatives to go from
spike train 2 to spike train 3 in A: i) delete
the last spike and add a new one or ii) move the last spike
to its new desired position. The latter alternative is less expensive
for the particular value of q illustrated here since
q · | t1| < 2 (the
dashed time interval of length 2/q corresponds to the
maximum displacement for which it is less expensive to move a spike).
|
|
The effective temporal jitter, tjitter, of
the spike occurrence times was defined as
tjitter = 1/q1/2 where
q1/2 is the value of q such
that
dijn(q1/2) = 1/2. This definition is motivated by the following arguments showing
that tjitter equals the average time
interval,
inter, by which
spikes are moved to transform one spike train into the other one if no
spikes have to be added or deleted (see Eq. 6). Thus the effective temporal jitter
tjitter is a generalization of
inter to situations where
spikes might also need to be added or deleted, as we now explain. For a
fixed value of q, let n
, n
, and
n
denote the number of spikes
moved, deleted, and added when computing the distance between
xi and xj. If
we pool together all noncoincident spikes in
xi and xj,
ni + nj
2cij, then each one of these spikes is
either moved, deleted, or created when transforming
xi into xj so
that the following equation holds
|
(4)
|
Using Eqs. 3 and 4 to express
dijn(q) directly in
terms of n
,
n
, and
n
we obtain
where |
ti| is the time
interval by which the ith spike (out of
n
) is moved. Therefore when
q = q1/2, rearranging
this last equation shows that the average time interval by which a spike is moved is given by
|
(5)
|
Let us assume from now on that the number of coincident spikes
is negligible, cij = 0 (see
RESULTS). If all spikes are moved to transform one spike
train into the other one (n
= n
= 0), Eq. 5 implies
that
|
(6)
|
and 1/q1/2 is the average time
interval,
inter, by which
spikes are moved. If n
0 and/or
n
0, then the distance by which
the remaining n
spikes are moved is
on average smaller to compensate for the extra cost imposed by spike
additions and deletions (see Eqs. 4 and 5; the
expression within the parentheses in Eq. 5 will be <1).
Note, however, that the total number of displaced spikes cannot be less
than half the average total number of spikes
since the right hand side of Eq. 5 has to be
positive. Thus tjitter provides an
appropriate measure of spike time jitter, which automatically takes
into account possible spike additions or deletions.
From the responses of a P-receptor afferent to 10 repetitions of a RAM
stimulus, we computed an estimate of the average normalized distance
between two spike trains as a function of q,
where npairs = 90 (npairs is obtained by considering all
possible pairs of trains among 10). Normalized distances were typically computed for q = 0, 0.05, 0.1, 0.25, 0.5, and 20 ms
1 (the last value
corresponds to the temporal resolution, 2/q = 0.1 ms,
at which spike occurrence times were digitized). According to
Eqs. 1 and 2, Dn(20)
measures the average fraction of noncoincident spikes, while
Dn(0) measures the average difference in
spike counts (normalized by the total spike count). The average
effective temporal jitter,
jitter = 1/
1/2, Dn(
1/2) = 1/2 measures the average jitter of the spike occurrence times under
repeated presentation of the same stimulus. The value of
1/2 was estimated to ±0.02 accuracy
[i.e.,
1/2 satisfied the
requirement: 0.48 < Dn(
1/2) < 0.52] by the bisection method (Press et al. 1992
,
chapt. 9). The percentage of spikes moved,
n
/(2n
+ n
+ n
), and the percentage of spikes
added or deleted, (n
+ n
)/(2n
+ n
+ n
), were computed over 3 s of
data and six stimulus repetitions (instead of the 15 s and 10 repetitions used to compute the distances) because this task was
computationally very intensive. We verified in a few cases that the
results were not altered significantly by this procedure. For this
latter task, a total of 15 units and 140 stimulus conditions were
analyzed. We checked that the distances Dn(
1/2)
computed over these reduced data sets lay between 0.45 and 0.55. This
was the case for 125 stimulus conditions; the other 15 conditions were
not considered further.
Stimulus estimation
The accuracy of single P-receptor afferent spike trains in
encoding RAMs was assessed by linearly estimating the stimulus from the
recorded spike trains. This technique essentially replaces each spike
in a spike train by a continuous waveform, h(t),
thus yielding an estimate,
sest(t), of the stimulus,
s(t) (Fig.
2A). The waveform
h(t) is chosen to optimize the match between
sest(t) and
s(t) and, at low firing rates, it closely
resembles the mean stimulus waveform preceding a spike (Gabbiani
and Koch 1996
; Wessel et al. 1996
). The
theoretical aspects of this signal processing technique and its
application to P-receptor afferent spike trains have been discussed in
detail elsewhere (Gabbiani and Koch 1998
; Wessel
et al. 1996
; see also Gabbiani and Metzner 1999
for an introduction). For each spike train
xi(t) (i = 1, ... , 10) obtained on presentation of a RAM
s(t), we subtracted the mean firing rate and
estimated the filter, hi(t),
that minimizes the mean square error between the stimulus and the
estimated stimulus obtained by convolving
hi(t) with
xi(t) (see Fig. 2A).
This filter is called a Wiener-Kolmogorov (WK) filter in the signal
processing literature (e.g., Poor 1994
) and plays a role
analogous to the impulse response used to estimate the instantaneous
firing rate of a neuron (see Fig. 1 of Gabbiani and Metzner
1999
). Each estimate of the WK filter depends on the recorded
spike train xi(t) from which it is computed and is therefore indexed accordingly as
hi(t). The WK filter was
computed using MATLAB M-files (The MathWorks, Natick, MA) available at
the following web address:
http://www.klab.caltech.edu/~gabbiani/signproc.html. We then
estimated the mean square estimation error,
2,
by cross-validation (Fukunaga 1990
): each filter
hi(t) was convolved with a
spike train xj(t) different
from the one used to compute hi(t) to avoid over fitting.
This yielded an estimate
ij2
|
|
where the brackets,
·
, denote time-averaging and *
denotes the time convolution operation (Gabbiani and Koch
1998
). An improved estimate was obtained by averaging over all
possible pairs
|
|
where npairs = 90. The fraction
of the stimulus encoded, or coding fraction, was evaluated as
|
|
where
is the standard deviation of the stimulus. In the
worst possible case, when the spike train is completely uncorrelated with the stimulus, the linear estimation algorithm predicts the stimulus mean value and the root mean square error equals the stimulus
standard deviation. The root mean square error is therefore always
smaller than the stimulus standard deviation (
) so that
the coding fraction,
, lies between 0 and 1. The coding fraction
represents the fraction of the stimulus, expressed in units of
,
that can be reconstructed by linear filtering of the spike train.

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Fig. 2.
Quantification of stimulus encoding and of its robustness to spike time
jittering. A: an estimate,
sest(t), of the stimulus
s(t) was obtained from the spike train by
convolving it with a Wiener-Kolmogorov (WK) filter (see main text for
details). The accuracy of stimulus encoding by the spike train was
assessed by computing the mean square error ( 2) between
the stimulus and the estimate. The brackets, · , denote
averaging over time. B: temporal jitter was introduced
by adding to each spike time a random variable taken from a zero-mean
gaussian distribution with standard deviation jitter.
The modified spike trains are shown for increasing values of
jitter (from top to
bottom) on the left. From each distorted
spike train, a new WK-filter and a new estimate,
sest(t), of the stimulus were
computed (right). Robustness was quantified by computing
the rate at which the fraction of the stimulus encoded decreased with
jitter (see inset to Fig. 13). A similar
procedure was used when spikes were randomly added or deleted from the
spike trains.
|
|
Robustness of RAM encoding to spike time jitter, and random spike
additions or deletions
To investigate the effect of spike time jitter, spike failures
and the occurrence of spikes unrelated to the stimulus on the encoding
of RAMs by P-receptor afferents, we created synthetic spike trains from
the experimental ones by randomly adding, deleting or jittering spikes
(Bialek et al. 1991
). The stimulus was then estimated
from these synthetic spike trains, and the coding fraction was
monitored as a function of the parameters determining the amount of
jitter and the number of spikes added or deleted. Each one of these
three types of modifications was introduced separately. In all cases, a
minimum separation of 2 ms was imposed between two spikes of the
modified spike trains to take into account the refractory period of the
afferent fibers.
Let padd indicate the percentage of
spikes added to the experimental spike train and
pdel the percentage of spikes randomly deleted. For spike time jittering, the spikes were moved from their
actual occurrence times by a random distance taken from a zero-mean
gaussian distribution with various standard deviations
jitter (Fig. 2B). We used
jitter = 0, 1, 3, 5, 7, 10, 15, and 30 ms;
padd = 0, 1, 5, 10, 20, and 30%;
pdel = 0, 1, 5, 10, 20, and 30%.
Let
(padd),
(pdel), and
(
jitter) denote the coding fractions for a
given value of padd,
pdel, and
jitter,
respectively. The robustness of RAM encoding by P-receptor afferent
spike trains was evaluated by plotting the normalized coding fraction
n(x) =
(x)/
(0) where x = padd,
pdel, or
jitter as a function of x (Fig. 13,
inset). In most cases, the normalized coding fraction was
linearly related to the distortion parameter x (see
RESULTS). We therefore performed linear fits of
n as a function of x = padd,
pdel, or
jitter
where
add,
del, and
jitter are
the slopes of the regression lines. The robustness was defined as the
amount of distortion required to cause a 50% drop in coding fraction
The values of padd50,
pdel50, and
jitter50 were obtained by
linear interpolation between adjacent values of the normalized coding fraction plotted as a function of the perturbation or by extrapolation at low stimulus cutoff frequencies (see the point
fc = 5 Hz in Fig. 13).
Modeling of P-receptor afferent spike trains
Modeling of P-receptor afferent spike trains was performed in
three steps. In the first step, the variability of P-receptor afferent
spike trains during RAM stimulation was compared with that of standard
nonleaky integrate-and-fire models with a random voltage threshold
(Fig. 3A) (Gabbiani and
Koch 1998
; Reich et al. 1997
). The properties of
the model random threshold determine the variability of the resulting
spike trains. The random threshold was taken from a gamma distribution
with parameters n and Vth
where
Larger values of n correspond to more reliable spike
trains (see RESULTS and Fig. 7) (see also Gabbiani
and Koch 1998
, Fig. 9.3), and the mean voltage threshold
Vth determines the mean firing rate of
the model. An absolute refractory period of 2 ms was inserted after
each spike occurrence. The order of the gamma distribution was varied
between 1 (corresponding to an exponential distribution leading to
Poisson distributed spike times), 3, 5, 10, and 100 (effectively
implementing the limit n
, which corresponds to a
perfect integrator). The mean voltage threshold value,
Vth, was fixed so as to match the mean
firing rate of the model to the one of each P-receptor afferent. Ten
repetitions of the same RAM used to stimulate P-receptor afferents were
fed to each model, and the distances between spike trains were computed
as explained above.

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Fig. 3.
Comparison of P-receptor afferent spike trains to integrate-and-fire
models. A: the variability of experimental spike trains
was compared with the variability of perfect integrate-and-fire (I&F)
neurons with a random threshold. In this model, the sum of the stimulus
and a constant bias term (corresponding to the spontaneous activity) is
integrated, and a spike is emitted each time that the threshold
(Vthresh) is reached. After each spike, a
refractory period of 2 ms is imposed and a new threshold value is
chosen from a gamma probability distribution. B: to
model the linear transfer properties of P-receptor afferent spike
trains, the AM was first linearly filtered, with a high-pass filter
fitted from the responses of P-receptor afferent to sinusoidal AMs
(SAMs; see Fig. 14) and then delayed. The output
z(t) was clipped and injected into a
perfect I&F neuron with random threshold and refractory period equal to
2 ms.
|
|
In the second step, the linear transfer properties of P-receptor
afferents were characterized using a model based on an earlier one
proposed by Nelson et al. (1997)
for P-receptor
afferents of Apteronotus leptorhynchus (see Fig.
3B). An alternative biophysical model proposed by
Kashimori et al. (1996)
was not considered here, as our
goal was to obtain the simplest possible description of P-receptor
afferent spike trains taking into account their linear transfer
function and the variability of their spike trains. The stimulus was
passed through a first-order high-pass filter with transfer function
H(s)
|
(7)
|
simulating the linear transfer properties of P-receptor
afferents. In this equation, Ga and
Gc are gain and offset terms, respectively,
a is the time constant of the
filter and s = i
= 2
if
is the complex circular frequency of the input signal. The parameters
Ga, Gc, and
a were obtained by fitting the gain G(f) = |H(2
if)| and the phase
(f) = tan
1[ImH(2
if)/ReH(2
if)]
of the model to experimentally measured gains and phases obtained from
responses to SAMs. For each SAM stimulus, the mean instantaneous firing
rate was computed over the full stimulus cycle and fitted to the
function
|
(8)
|
(see RESULTS and Fig. 14). The fit parameters
Gfs and
fs are the experimental gain
and phase at the frequencies fs used
in the SAMs protocols, respectively (see Stimulation above). The constant c represents an offset between stimulus and response.
In the third and last step, the variability characterized in the first
step and the linear filtering properties obtained in the second step
were combined to obtain a complete model reproducing both the
variability of P-receptor afferents and their linear filtering
properties. The high-pass filtered signal was delayed by 2.5 ms
(corresponding to the synaptic delay between tuberous receptors and
afferent fibers), and a mean spontaneous activity was added
(Nelson et al. 1997
) (see Fig. 3B). The
resulting signal, z(t), was then passed through a
clipping nonlinearity, effectively half-wave rectifying it, and
imposing a maximal firing rate of 1 spike per EOD cycle
The output, r(t) (see Fig. 3B),
was fed as input to a perfect integrator with gamma-distributed
threshold, as described above, to determine when a spike was fired. The
order n of the gamma distribution for the threshold was
selected to match the spike train variability in response to SAMs, as
assessed by computing interspike interval distributions and distances
between spike trains (see above and Gabbiani and Koch
1998
). The responses to RAM stimuli, when available, were then
compared with the model predictions (see RESULTS). In some
cases the mean firing rate of the model was adjusted to take into
account changes in the experimental firing rate during a recording session.
 |
RESULTS |
This study is based on recordings and analysis from 69 P-receptor
afferent fibers obtained in 34 different animals.
Responses of P-receptor afferents to repeated presentations of
identical RAMs
To investigate the variability of P-receptor afferent spike trains
and its relation to the encoding of electric field AMs, we recorded
their responses to repeated presentations of identical RAMs of a
sinusoidal electric field. The mean firing rates of afferent fibers
were widely distributed, ranging from 25 spike/s to 374 spike/s
(117 ± 69 spike/s, mean ± SD). The coefficient of variation
of the interspike interval (ISI) distribution (CV = mean/SD)
ranged from 0.16 to 1.7 (0.59 ± 0.36, mean ± SD). These values were similar to those observed in spontaneously active units
(range: 0.12-1.12) (Wessel et al. 1996
, Fig.
2B2), although several units analyzed here had higher CVs
under RAM stimulation than those observed spontaneously.
Figure 4 illustrates the range of
responses to repeated RAMs recorded under a variety of stimulus
conditions and mean firing rates. In a few cases, the responses of
P-receptor afferents were highly reproducible from trial to trial (see
in particular Fig. 4C and, to a lesser extent, Fig.
4D) as has sometimes been observed in other preparations
(Bair and Koch 1996
; Berry and Meister
1998
; Mainen and Sejnowski 1995
). A
clear locking of the responses to the stimulus was usually observed at
high contrasts (
> 200 mV) and cutoff frequencies
(fc > 40 Hz; see Fig. 4,
C and D). Furthermore, the mean firing rate of
the afferent fibers had to be low (<125 spike/s; compare Fig. 4,
C and G). Decreasing the cutoff frequency or the
stimulus contrast tended to decrease the reproducibility of the spike
occurrence times (Fig. 4, A and B). At high
firing rates (>125 spike/s), P-receptor afferent responses did not
show clear trends of changes in reproducibility with stimulus
parameters (Fig. 4, E-H). These preliminary observations
suggested that the variability across trials of P-receptor afferent
spike trains depended on stimulus parameters as well as on the mean
firing rate of the units.

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Fig. 4.
P-receptor afferent responses to RAMs exhibit a broad range of
variability. A portion of the stimulus presented to each P-receptor
afferent is shown on top. Each raster of spikes (9 per
panel, 500 ms long) illustrates the response of the same P-receptor
afferent to a single presentation of the stimulus. The left
column (A, C, E, and G)
illustrates responses for fixed stimulus contrast ( = 250 mV)
of a neuron with low mean firing rate (A and
C: mfr = 65 ± 2 spikes/s) and a different
neuron with high firing rate (E and G:
mfr = 137 ± 1 spikes/s) to stimuli with low and high cutoff
frequencies (A and E:
fc = 5 Hz; C and
G: fc = 60 Hz). The
right column (B, D, F, and
H) illustrates the responses for a fixed cutoff
frequency (fc = 60 Hz) of a
neuron with low firing rate (B and D:
mfr = 62 ± 1 spikes/s) and a different neuron with high
firing rate (F and H: mfr = 151 ± 1 spikes/s) to stimuli with low and high contrasts (B
and F: = 100 mV; D and
H: = 275 mV).
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Quantification of response variability
The spike count variance over short time windows has often been
considered as an indicator of spike train variability across repeated
trials of the same stimulus (Berry et al. 1997
;
de Ruyter van Steveninck et al. 1997
). As a
first step in quantifying P-receptor afferent spike train variability,
we therefore plotted the spike count variance versus mean spike count
across trials in windows of various sizes (10, 50, and 100 ms) as
illustrated in Fig. 5. At low firing
rates (Fig. 5, top row) the observed mean spike count in a
given window was typically low (<10 spikes per window), and the
variance across trials as a function of the mean had a scalloped
appearance, reproducing almost perfectly a series of parabolas stacked
onto each other along the vertical axis. Similar observations have been
made in other preparations [in ganglion cells of the salamander retina
(Berry and Meister 1998
); in a wide-field visual
tangential neuron of the fly lobula plate (de Ruyter van
Steveninck et al. 1997
)]. The lowest series of parabolas corresponded to the minimal possible variance that is achieved when the
spike count is either equal to n or n + 1 (where
n is an integer) in a given window (see
METHODS). Higher parabolas corresponded successively to all
spike counts equal to n or n + 1, except for one
equal to n
1 (or n + 2), etc. ... per
window. This result indicated that the number of spikes per window was reliable (either n, n + 1, or n
1) and
well below that expected for a Poisson process (mean equals variance;
dashed line in Fig. 5). However, since the scalloping was observed
independently of the stimulus cutoff frequency, it did not correlate
with the reliability of spike occurrence times, as observed in spike
rasters (see Fig. 4, A-D). At higher firing rates, the mean
spike count reached up to 25 spikes or more per window (Fig. 5,
bottom row), and the variance increased considerably,
ranging from the theoretical minimum up to the mean equals variance
line. On average, the variance was still below that of a Poisson
process.

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Fig. 5.
Scalloping of the variance vs. mean spike count relation is not a
predictor of spike timing variability. Plots of spike count variance
vs. mean spike count in windows T of 10, 50, and 100 ms.
A-D were obtained in a neuron firing at low rate
(mfr = 65 ± 2 spikes/s) for fixed contrast ( = 250 mV) and various cutoff frequencies fc (as
indicated on the top of each panel; A is
the same experiment as in Fig. 4A). E-H
were obtained in a different neuron with high firing rate (mfr = 151 ± 1 spikes/s) for the same contrast and cutoff frequency
values. Note that the variance vs. mean spike count curves follow the
theoretical minimum curves in A-D in spite of the fact
that reliable spike timing was only observed at high
fcs (see Fig. 4, A-D). At
higher firing rates (E-H), scalloping is still observed
in some cases but is masked by a general increase in spike count
variability. The 3 clusters evident in G and
H, and to a lesser extent in F,
correspond to the 3 window sizes (if T varies
continuously between 10 and 100 ms, no clusters are observed). In all
panels, mean equal to variance is indicated by a straight dashed
line.
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|
Thus according to the experimental results plotted in Fig. 5,
scalloping did not appear to be directly related to the precision of
spike timing across trials. To confirm this point, we artificially modified the spike trains obtained in response to repeated
presentations of identical RAMs to alter the precision in spike timing
without changing the statistical properties of the spike trains. We
took the 10 rasters of units exhibiting scalloping of the spike count variance versus mean spike count relation and firing with varying degrees of reliability in response to RAMs (such as the rasters for the
unit illustrated in Figs. 4, A and C, and 5,
A-D) and successively shifted the spikes with a
fixed delay tshift. In other words, if
x1(t), ... ,
x10(t) represent the
original spike trains, new spike trains were defined as
1(t) = x1(t),
2(t) = x2(t + tshift), ... ,
10(t) = x10(t + 9 · tshift). The parameter tshift took three values: 1, 5, and 10 ms. We then computed the variance versus mean relations exactly as in
Fig. 5. In all cases (5 units, 14 conditions) and irrespective of
whether the timing of spikes was reliable or not, the scalloping
remained present, independently of the value of
tshift. In some cases the number of
vertical rows of parabolas increased with
tshift. These points are illustrated
in Fig. 6, A and B.
Similar results were obtained in integrate-and-fire neuron models as
illustrated in Fig. 6, C and D. Thus in the worst
case, tshift = 10 ms, the timing of spikes drifted by 90 ms between the first spike train
1(t) and the last
spike train
10(t)
without affecting the scalloping in windows of 10, 50, and 100 ms.
Since it was possible to largely eliminate any precision in the spike
occurrence times from trial to trial without altering the scalloping of
the spike count variance, this analysis confirmed that scalloping in
these time windows was not related to the reliability of spike
occurrence times.

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Fig. 6.
Scalloping of the variance vs. mean spike count relation measured
across trials is preserved even after large shifts in the timing of
individual spike trains. A: the top 10 rasters represent
the response of a P-receptor afferent (mfr = 65 ± 2 spikes/s; same experiment as in Figs. 4A and
5A) to repeated presentations of a random electric field
AM (RAM) stimulus ( = 250 mV,
fc = 10 Hz). The corresponding spike
count variance vs. mean spike count plot is scalloped as illustrated
below. B: the spike trains were successively shifted by
10 ms as illustrated on top (see main text), and the
variance vs. mean spike count relations was recomputed. Note that the
scalloping remained present, although the variance increased as
compared with A. C and D:
same stimulation and analysis procedure as in A and
B for an I&F neuron model with gamma order 10 (mfr = 81 spikes/s; see main text and Fig. 7A for a more
detailed description of the model).
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Because the spike count variance as a function of mean spike count did
not offer a reliable indication of spike train variability under our
experimental conditions, we turned to a second measure based on the
calculation of distances between spike trains obtained under repeated
RAM stimulation. This measure,
Dn(q), depends on a parameter
q (in units of 1/time), which determines the temporal precision at which the distance between two spike trains is computed (higher values of q correspond to higher temporal
precisions, see METHODS). For two identical spike trains
Dn(q) = 0 independent of
q. The maximum,
Dn(q) = 1, is obtained for
large q's only if no spikes in the two spike trains
occurred exactly at the same time. The value at which
Dn(q) = 1/2, called
1/2, may be used to summarize
spike train variability: if we set
jitter = 1/
1/2, then
jitter measures the
average time by which spikes have to be moved to transform one spike
train into the second one, or equivalently, the average jitter in spike
timing. By definition, this jitter also takes into account differences
in spike numbers between the two spike trains (i.e., the need to create
or delete spikes to transform one spike train into the other; see
METHODS and Fig. 1).
We computed the average distance between all pairs of spike trains
obtained in response to the same RAM stimulus for our sample of 69 P-receptor afferents. The spike train distances
Dn(q) were compared with those
obtained from a family of gamma neuron models indexed by a parameter
n controlling spike train variability (see METHODS). A value of n = 1 (gamma-1
neuron) corresponds to Poisson-distributed spike occurrence times in
response to the stimulus while for large n
(n > 100) the gamma model is identical to an
integrate-and-fire neuron. Figure
7A illustrates in one example
how the variability observed in P-receptor afferents compared with the
model variability. The top 10 rasters labeled "P-unit" correspond
to the response of a P-receptor afferent, while the next 10 rasters
were obtained by simulating a Poisson (gamma-1) model. The P-receptor
afferent spike trains are considerably more regular than those of a
Poisson neuron and match quite well those of the gamma-10 model
illustrated at the bottom of Fig. 7A.
Accordingly, the average distance between two spike trains of this
P-receptor afferent followed closely those of the gamma-10 neuron (see
Fig. 7B,
and
) and were always smaller than those of
a Poisson model (Fig. 7B,
). The value Dn(0) in Fig. 7B yields the
average difference in spike number between two spike trains normalized
by the total spike count. The small distance value,
Dn(0) = 0.02, indicates that the
number of spikes was very reproducible from one trial to the next with an average variability of 2%. On the other hand,
Dn(20) is the fraction of noncoincident
spikes in two spike trains at 0.1-ms resolution. The value
Dn(20) = 0.98 in this experiment
indicates that <2% of spikes occurred at the same time (±0.05 ms)
and thus the spike trains were clearly not reproducible at a 0.1-ms
resolution. The average temporal jitter in spike occurrence times,
jitter, was in this
case equal to 2.9 ms (with 86% of spikes moved and 14% of spikes
added or deleted), corresponding to 1.3 EOD cycles (fEOD = 438 Hz). Furthermore,
the largest deviation between
Dn(q) in the gamma-10 model (or
the P-receptor afferent) and the Poisson model was observed for
q values lying in the interval (0.05-0.25 ms
1; Fig. 7C).
A value of q = 0.25 ms
1 was used to
illustrate our results in subsequent figures.

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Fig. 7.
Spike train distances of P-receptor afferents match those of gamma
integrate-and-fire neurons with order n in the 3-10
range. A: the spike trains of a P-receptor afferent (top
10 rasters, labeled P-unit; same experiment as in Fig.
5B) recorded in response to a RAM stimulus
[s(t), shown on top] are
illustrated together with those elicited by the same stimulus in 2 I&F
models with random threshold (labeled n = 1 and 10;
see Fig. 3A). The n = 1 model
corresponds to Poisson spike occurrence times and matches poorly the
observed variability, while the n = 10 model
matches it quite well. B: plot of the mean distance
Dn(q) between 2 spike
trains in response to s(t) for the
P-receptor afferent and I&F models shown in A. The close
match between P-unit and n = 10 I&F distances
confirms quantitatively the visual observation in A
(standard errors are too small to be visible,
npairs = 90). C: plot of
the difference in mean distances between n = 1 and
n = 10 models (mean ± SE,
npairs = 90) as a function of
q.
Dn(q) = Dn(q)Poisson Dn(q)gamma order
10, where
Dn(q)Poisson
corresponds to the filled circles in B and
Dn(q)gamma
order 10 to the squares. Note that the largest difference in
distances is observed in the range of q values between
0.05 and 0.25 ms 1. In B and
C the value q = 20 ms 1 was not plotted because it would lie
off-scale (see main text).
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Similar results were obtained in all 69 P-receptor afferents analyzed.
The relative difference in spike count,
Dn(0), ranged from 0.01 to 0.1 (0.03 ± 0.04, mean ± SD), while the fraction of noncoincident spikes,
Dn(20), ranged from 0.87 to 1.0 (0.97 ± 0.04). The distribution of average temporal jitters is plotted in
Fig. 8A for 69 P-receptor
afferents. The range of values was between 0.6 and 23.2 ms (3.5 ± 3.9) with 77 ± 7% of spikes moved (range: 62-87%) and 23 ± 7% of spikes added or deleted (range: 13-38%). Figure
8B replots the average temporal jitter in units of the EOD
cycle (1/fEOD) as measured for each
fish prior to the experiment. The temporal jitter ranged from a
fraction of the EOD cycle (0.29) up to several cycles (8.7; 1.4 ± 1.5). The temporal jitter was dependent on the firing rate of the
afferent fibers. High firing rate afferents (arbitrarily defined as
those with mean firing rate above >125 spike/s) had a mean jitter,
jitter, of 1.7 ± 0.3 ms (range: 0.6-2.45 ms) corresponding to 0.8 ± 0.3 EOD
cycles (range: 0.3-1.2). The mean jitter of low firing rate afferents
(<125 spike/s) was typically higher, 6.3 ± 6.0 ms (range: 1.7-23.2 ms) corresponding to 2.4 ± 2.3 EOD cycles (range:
0.7-8.7).

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Fig. 8.
Distribution of mean spike time jitter on 69 P-receptor afferents
(corresponding to 508 different RAM stimulations). A:
distribution of the average temporal jitter (bin size: 0.4 ms). For
display purposes, the probability distribution is shown only up to 10 ms; 8.2% of the cumulative distribution was between 10 ms and the
maximal value observed (23.2 ms). B: same distribution
of jitter in units of electric organ
discharge (EOD) cycles (2 bins per EOD cycle). In each panel, the
arrows indicate the means of the distributions.
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Dependence of temporal jitter on stimulus cutoff frequency
Next, we investigated the dependence of spike timing jitter on
stimulus parameters and P-receptor afferent firing rates. Figure 9A illustrates the change in
temporal jitter as a function of cutoff frequency for a low firing rate
unit (mfr = 65 spike/s). When the stimulus cutoff frequency was
increased from 5 to 40 Hz, the timing jitter decreased 1.4-fold from
4.8 to 3.5 ms. This increase in spike timing precision was quantified
by the slope of linear regression lines fitted to the data (see Fig.
9A, - - -). As illustrated in Fig. 9B, an
increase in temporal precision was observed mainly for units firing at
low rates. The left panel shows the distribution of slopes
for units with a mean firing rate below 125 spike/s, and the
right panel shows the distribution of slopes for units with
mean firing rates above 125 spike/s. The slopes calculated for low
firing rate units were negative on average (
0.052 ± 0.066 ms/Hz, mean ± SD) and significantly different from 0 (P < 0.05, 2-tailed t-test) while they were
not significantly different from zero at high firing rates (0.01 ± 0.02 ms/Hz; P > 0.4). Correspondingly, correlation
coefficients between fc and
jitter were negative
at low rates (
0.59 ± 0.35) but not at high firing rates
(0.29 ± 0.62).

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Fig. 9.
The timing jitter decreases with stimulus cutoff frequency at low but
not at high firing rates. A: plot of the mean jitter in
spike occurrence times as a function of stimulus frequency for a neuron
firing at low rate (mfr = 65 spikes/s, =250 mV,
fEOD = 438 Hz; percentage of spikes
moved: 85 ± 3%). B: the slope of timing jitter
vs. cutoff frequency plots (see A) is negative at low
firing rates but not at high firing rates. The 2 distributions are
significantly different (Wilcoxon rank-sum test, P < 0.0001).
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Similar results were obtained for the distance measure
Dn(q) over a broad range of the
spike distance parameter q, as illustrated in Fig.
10. At fixed, intermediate values of
q, the average distance decreased as a function of stimulus
cutoff frequency for low-firing rate units (Fig. 10, A-C).
At low temporal resolution [i.e., when q = 0 ms
1 and
Dn(q) measures differences in
spike counts] the slopes and correlation coefficients of
Dn(q) versus
fc regression lines were not
significantly different from 0 (P > 0.2 at
q = 0 ms
1, 2-tailed
t-test). That is, no trend in spike count variability versus
stimulus bandwidth could be observed. The same result was true at very
high temporal resolution (P > 0.2 at q = 20 ms
1). At
intermediate temporal resolutions, units firing at high rates did not
show slopes or regression coefficients significantly different from
zero (q in the range of 0.05-0.75
ms
1; P > 0.05) while low firing rate units yielded a significant decrease in
variability with stimulus bandwidth (P < 0.01 over the
same range of values). The strongest tendencies were observed for
values of q between 0.25 and 0.5 ms
1 (Fig. 10,
B and C).

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Fig. 10.
Increase in timing precision with stimulus cutoff frequency at low but
not at high firing rate is observed across a broad range of spike
moving costs. A: mean distance between 2 spike trains as
a function of cutoff frequency for a value of q = 0.25 ms 1 (1/q = 4 ms) in
a low firing rate neuron (mfr = 65 spike/s; = 250 mV).
This represents a particularly clear example. B: average
slopes (mean ± SE) of distance
[Dn(q)] vs. cutoff
frequency (fc) relations (computed as in
A) at low ( , average of
n = 21 neurons) and high ( , average
over n = 12 neurons) firing rates plotted as a
function of q. C: average correlation
coefficient (mean ± SE) of distance vs. cutoff frequency as a
function of q (computed as in A).
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Variability and stimulus contrast
The dependence of spike time jitter on stimulus contrast was very
similar to the one found for stimulus cutoff frequency. Figure
11A illustrates an example
of a low firing rate P-receptor afferent for which spike timing jitter
decreased twofold as the stimulus contrast was changed from 10 to 30%.
The effect of stimulus contrast on spike timing jitter is summarized in
Fig. 11B, which reports the slopes of linear regression
lines for
jitter
versus
in P-units firing at low and high rates (left and
right panels, respectively). Increasing stimulus contrast
was generally less effective than increasing cutoff frequency at
reducing spike time jitter as may be seen from the larger fraction of
units with slopes close to zero, even at low firing rates.

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Fig. 11.
The timing jitter decreases with increasing stimulus contrast at low
but not at high firing rates. A: plot of the mean jitter
as a function of stimulus contrast for a neuron at low firing rate
(mfr = 52 spikes/s, fc = 5 Hz,
fEOD = 438 Hz; percentage of spikes
moved: 79 ± 4%). B: the slope of the timing
jitter vs. stimulus contrast relation (see A) is
negative at low firing rates (left, 0.030 ± 0.041) but not at high firing rates (right, 0). The 2 distributions are significantly different (Wilcoxon rank-sum test,
P < 0.001).
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Figure 12 reports the same
results directly in terms of spike train distances at all values of
q used. At low firing rates and for intermediate values of
the temporal resolution parameter, the average distance between two
spike trains decreased as a function of stimulus contrast (Fig.
12A). Accordingly, the slopes of linear regression lines and
their correlation coefficients were significantly different from zero
for low firing rate units (P < 0.01, 2-tailed t-test) but not for high firing rate units
(P > 0.05) at those values of q (Fig. 12,
B and C). At very low or very high temporal resolution (q = 0 or 20 ms
1) changes with
stimulus contrast were not statistically significant (P > 0.05).

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Fig. 12.
Increase in timing precision as a function of stimulus contrast is
observed at low but not at high firing rates across a broad range of
spike moving costs. A: mean distance between 2 spike
trains as a function of stimulus contrast for a value of
q = 0.25 ms 1 in a low
firing rate neuron (mfr = 52 spikes/s,
fc = 5 Hz,
fEOD = 438 Hz). This represents a
particularly clear example. B: average slopes (mean ± SE) of distance
[Dn(q)] vs. stimulus
contrast ( ) relations (computed as in
A) at low ( , average over
n = 23 neurons) and high ( , average
over n = 8 neurons) firing rates as a function of
q. C: average correlation coefficient
(mean ± SE) of distance vs. cutoff frequency as a function of
q (computed as in A).
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In summary, the study of spike train distances demonstrated that the
timing precision of P-receptor afferents increased with stimulus cutoff
frequency and, to a lesser extent, with stimulus contrast. Low firing
rate units appear to be less variable than high firing rate units.
Robustness of stimulus encoding
To assess the impact of alterations in spike timing on the
accuracy of RAMs encoding, we modified experimental spike trains by
randomly adding, deleting, or moving spikes. The stimuli were then
estimated from the modified spike trains (see Fig. 2B and METHODS), and the change in coding fraction was monitored.
The inset of Fig. 13 reports
in one example the fraction of the stimulus encoded as a function of
spike time jitter, normalized by its baseline value, the coding
fraction of the original spike train. In most cases the addition and
the deletion of spikes or the addition of spike time jitter resulted in
a linear decrease of the normalized coding fraction as the perturbation
parameter was increased. Correlation coefficients ranged from 0.80 to
0.97 for 96% of the data. In those cases, the robustness of encoding
was characterized by the perturbation value required to cause a 50%
drop in coding fraction (see Fig. 13, inset for the
definition of
jitter50). P-receptor
afferent spike trains were in general quite robust to such
perturbations. As illustrated in Fig. 13, at low cutoff frequencies,
spike time jittering as high as 125 ms was required to cause a 50%
drop in
. The robustness to spike time jitter decreased as the
stimulus cutoff frequency increased, reaching a value of 6 ms for fast
changing stimuli (fc = 88 Hz). The robustness to spike additions or deletions did not show a
dependency on stimulus bandwidth for
fc > 5 Hz (see Table
1). For those stimuli, a drop of 50% in
the coding fraction was obtained after 36% random spike deletions and
41% additions. Robustness was not significantly dependent on stimulus
contrast (data not shown).

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Fig. 13.
Robustness of RAM encoding decreases with stimulus bandwidth. Plot of
the timing jitter (mean ± SE) causing a 50% reduction in the
coding fraction as a function of stimulus bandwidth. Averages were
computed on n = 58, 38, 21, 22, 38, and 9 stimulus
conditions from low to high fc, respectively
(the large error at fc = 5 Hz is due to
extrapolation from shallow slopes, see METHODS and
inset for the computation of
jitter50).
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Modeling of P-receptor afferent variability and linear transfer
properties
The results reported above were summarized by building a model of
P-receptor afferent spike trains able to account for the encoding of
RAMs and the spike train variability observed experimentally across
trials (see Fig. 3 and METHODS). We used an approach
similar to the one adopted by Nelson et al. (1997)
in
modeling P-receptor afferents of Apteronotus leptorhynchus.
The transfer functions of P-receptor afferents have been described as
high pass in the species of weakly electric fish investigated so far
(Bastian 1981
; Nelson et al. 1997
). We
confirmed this and characterized quantitatively the transfer function
in Eigenmannia by recording responses to SAMs. Gains and
phases were extracted from linear fits to sinusoids (Fig.
14A and Eq. 8 in
METHODS) at various frequencies
fs. The experimental gains and phases
were then fitted by maximum likelihood to a first-order high-pass
filter (Fig. 14B and Eq. 7). The resulting fits
had
2-values divided by the number of degrees
of freedoms (
2/DOF) (Press
et al. 1992
, chapt. 15) for the fits between 0.7 and 8.0 (for
15 afferent fibers), except for two outliers
(
2/DOF = 22.8 and 24.2, respectively). The mean values of the filter parameters were
Ga = 120 ± 82 spikes/s (range:
16-300 spikes/s), Gc = 40 ± 26 spikes/s (range: 7-99 spikes/s), and
a = 4 ± 5 ms (range:0.2-17.5 ms). In contrast to the results of
Nelson et al. (1997)
in Apteronotus, fitting
the data with a second-order filter improved only slightly the
2/DOF-values of the fits (range:
0.4-6.2). Since the additional parameters were not well constrained,
this approach was not pursued further. The static nonlinearity
illustrated in Fig. 3B was needed in the model to prevent
z(t) from becoming negative, leading to firing
rates lower than those observed experimentally. The variability of
P-receptor afferent spike trains was estimated from repeated presentations of SAM stimuli and was in the same range as the one
observed for RAMs.

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Fig. 14.
Fit of linear transfer function properties of a P-receptor afferent by
a 1st-order high-pass filter. A: plot of the mean
instantaneous firing rate as a function of the normalized period
fraction pn
(pn = t
fs/2 , bin size: 1/20 of the period
cycle) for 3 different sinusoidal AMs
(fs = 1, 20, and 100 Hz, respectively).
The solid line represents the fit with Eq. 8 (see
METHODS; r2 is the correlation
coefficient between the data and fit). B: fits of the
mean gain and phase (±SD) obtained from A (see
Eq. 7) with a 1st-order high-pass filter (same neuron as
in A; fit parameters: Ga = 147 ± 9 spikes/s, Gc = 20 ± 2 spikes/s, a = 1.2 ± 0.8 ms,
2/degress of freedom = 3.2).
|
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The ability of the model to predict responses to RAMs was tested in 10 P-receptor afferents by computing coding fractions and spike train
distances as a function of stimulus contrast and cutoff frequency.
Figure 15 illustrates two examples for
a P-receptor afferent firing at low rate (A-D) and a second
P-receptor afferent at high firing rate (E-H). The model
successfully reproduced both the dependence of coding fraction and
spike train distances observed experimentally on
fc and
.

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Fig. 15.
Comparison of spike train distances and stimulus encoding properties of
P-receptor afferents and model. A and E:
coding fraction (mean ± SE) as a function of stimulus cutoff
frequency for 2 different neurons with low and high firing rates,
respectively ( ) and models ( ; = 250 mV). C and G: coding fraction as a
function of stimulus contrast for the same 2 neurons
(fc = 5 Hz). B
and F: average spike train distances for the same
stimuli as in A and E, respectively.
D and H: average spike train distances
for the same stimuli as in C and G,
respectively. Model parameters were set as follows.
A-D: Ga = 17 spikes/s,
Gc = 7 spikes/s, a = 6 ms, fEOD = 375 Hz,
rbase = 5 spike/s,
n = 3, Vth = 80 mV.
E-H: Ga = 165 Hz,
Gc = 34 Hz, a = 2 ms, fEOD = 575 Hz,
rbase = 65 spikes/s,
n = 3, Vth = 130 mV.
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 |
DISCUSSION |
We characterized the variability of P-receptor afferent responses
to RAMs under a variety of stimulus conditions using a new measure of
distance between spike trains. Our results provide insight into the
relationship between the variance in the number of spikes and the mean
spike count as a measure of variability across repeated trials. They
also shed light on the impact of variability on the processing of
electric field AMs by the electrosensory system in weakly electric fish.
Quantification of spike train variability
Spike train variability has often been quantified by computing the
spike count variance as a function of the mean spike count in fixed
windows of length T (for a review, see Teich et al.
1996
). The benchmark stochastic process to which these values
are compared is the Poisson process for which the generation of
independent spikes yields a variance equal to the mean. The spike count
variance provides an appropriate measure of neural noise in tasks where the mean spike count (averaged over T) is used to assess a
neuron's ability to discriminate between two alternatives (for
reviews, see Gabbiani and Koch 1998
; Parker and
Newsome 1998
). For long time intervals (T
1 s) variances larger than mean spike counts are often observed,
indicative of positive long-term correlations in the spike trains
(Teich et al. 1996
). Such time windows are, however,
inadequate to assess the ability of neurons to convey information about
time-varying stimuli by rapid changes in instantaneous firing rate.
Recently, the spike count variance has also been used as a measure of
variability at short time scales (T
300 ms)
(Berry et al. 1997
; de Ruyter van Steveninck et
al. 1997
; Warzecha and Egelhaaf 1999
). In our
data, minimal nonzero values for the spike count variance were observed
in windows smaller than 100 ms, as has been reported in these studies.
However, they were not correlated with the reliability of spike
occurrence times assessed from raster plots (such as stimulus-dependent
phase locking to the sinusoidal carrier signal) or with objective
measures of the information encoded in the time-varying firing rate
like the coding fraction (Figs. 4-6). Therefore reliable spike timing
is not a necessary prerequisite for minimum nonzero variance curves:
they may be observed independently of whether spike timing is
reproducible at the millisecond level from trial to trial or not. Such
curves should therefore be interpreted with caution (see also
Barberini et al. 2000
; Warzecha and
Egelhaaf 1999
, footnote 21). One effect leading to variances
smaller than the mean over short time windows is the presence of a
refractory period that introduces negative correlations between spike
occurrence times. The addition of a refractory period to a Poisson
stochastic process has recently been shown to be sufficient to account
for the variability observed in retinal ganglion cells under dynamic
stimulation (Berry and Meister 1998
). Similar
observations were made in other preparations (for a review, see
Johnson 1996
). Figure 7B shows that a simple Poisson process with a 2-ms refractory period driven by the stimulus did not reproduce the spike train variability of P-receptor afferents. A comparison of variance versus mean spike count with theoretical results (Vannucci and Teich 1981
) suggests that the
regularizing effect of the refractory period is not sufficient to
account entirely for the low variability observed in our data. In
addition to the refractory period, the generation of spikes in
P-receptor afferents appears to be governed by biophysical mechanisms
that exhibit intermediate levels of variability lying between those of
Poisson and perfect integrate-and-fire models and corresponding to the factors n = 3-10 of our gamma models.
Because of their mathematical definition and properties, the
distances Dn(q) and the average
timing jitter
jitter
are well suited to characterize the reproducibility of spike occurrence times from one trial to the next. These measures are equally effective with deterministic or random stimuli and are applicable in cases, such
as here, where simpler measures like the timing precision or
reliability cannot be used (see Fig. 4) (Bair and Koch
1996
; Berry et al. 1997
). By definition, the
average jitter
jitter is a measure that automatically incorporates possible differences in
spike number between two spike trains. For example, since on average
23% of the spikes had to be added or deleted to transform one spike
train to a second one in our data set (see RESULTS, Quantification of response variability, last paragraph), the
average time interval by which the remaining spikes were moved was
actually smaller by 15% than that reported in Fig. 8. This may be seen from Eqs. 3 and 4: if
n
and/or
n
are different from zero, the
parenthesis on the right hand side of Eq. 4 will be smaller
than one (0.85 in the present case), implying that
jitter = 1/
1/2 is larger than the
average time interval given by the left hand side of Eq. 4.
The additional 15% increase in
jitter converts the
added or deleted spikes into an effective time jitter equivalent.
Our use of spike train distances is different from the one originally
introduced by Victor and Purpura (1996
,
1997
). These authors employed spike train distances to
assess the information conveyed by stimulus-dependent clustering of
spike trains from neurons of the monkey visual cortex. In the present
study, spike distances were used only to assess the variability across
identical trials; the performance at conveying stimulus-dependent
information was monitored with a second, independent measure, the
coding fraction.
Variability under various stimulus conditions
The results illustrated in Figs. 9-12 show that the timing
precision of P-receptor afferent spikes increases with the cutoff frequency of the stimulus and, to a lesser extent, with the contrast of
the RAMs. These results are consistent with observations made in other
preparations reporting that fast transients are likely to increase the
precision of spike occurrence times (Berry et al. 1997
;
Mechler et al. 1998
; Warzecha et al.
1998
). Similarly, our findings that spike trains can be more
reproducible at low than at high firing rates (see Fig. 4) is
consistent with earlier observations (Berry et al. 1997
;
de Ruyter van Steveninck et al. 1997
; Warzecha
and Egelhaaf 1999
). In contrast, no significant differences in
reliability were observed for RAM or SAM stimuli. Under the assumption
that the RAMs employed here are closer to natural stimuli than SAMs
(de Ruyter van Steveninck et al. 1997
), our results do
not support the idea that spike timing is more reliable under natural
stimulation (Warzecha and Egelhaaf 1999
). Behavioral
experiments show that Eigenmannia is able to perform remarkably precise jamming avoidance behaviors under artificial stimulation (for a review, see Kawasaki 1997
).
Accordingly, the reliability of spike timing recorded in the
time-coding pathway is very precise under such conditions. In contrast
to the amplitude-coding pathway, high reliability in the time-coding
pathway is necessary for the jamming avoidance response.
Variability and robustness of encoding
Our results show that the average jitter in the timing of
P-receptor afferent spikes
jitter, is in most
cases below 4 ms. On the other hand, the robustness of encoding to
spike time jitter yields values of
jitter50 well above 4 ms for most
of the behaviorally relevant range of stimulus cutoff frequencies (see
Table 1). Even at high stimulus cutoff frequencies (e.g.,
fc = 60 Hz), a jitter of 4 ms leads to
a relative decrease in coding fraction of at most 18% (see Table 1;
jitter50 = 11 ms implies that
n (4 ms) = 0.82). A similar observation is valid for spike additions and deletions. Therefore the jitter observed in P-receptor afferents is in a temporal range that does not
significantly affect the information transmitted by single spike trains
for most units and stimulus conditions. On the other hand, a small
amount of spike time jitter is beneficial to improve the stimulus
estimate obtained from several independent spike trains by averaging.
We verified this by computing estimates of the stimulus from
n = 2-10 spike trains (recorded successively from one
neuron) simultaneously (Kreiman et al. 1998
). The coding fraction increased when additional spike trains were included and
started to saturate for n = 6-7 spike trains. Thus our
results suggest that the spike timing jitter of P-receptor afferents
lies in a range for which the information transmitted by single units (when assessed by linear estimation) is not degraded (for the range of
behaviorally relevant stimulus cutoff frequencies considered here) but
which still allows for improvement by averaging over a small number of
afferents. Additional experiments recording simultaneously from several
P-receptor afferents under repeated presentations of the same RAMs are
needed to confirm this result.
Variability and the processing of AM in the electrosensory lateral
line lobe (ELL)
At the next stage of the amplitude coding pathway, the information
carried by P-receptor afferent spike trains is processed by pyramidal
cells of the ELL. These neurons represent the output elements of the
amplitude pathway and project to various higher order brain structures
specialized in the processing of electrosensory information. There are
two types of pyramidal cells, E- and I-type, which receive direct
excitatory input and indirect input via inhibitory interneurons,
respectively. At least two transformations have been identified in the
representation of AMs between the afferent input and the pyramidal cell
output to the ELL: 1) the detection threshold of pyramidal
cells for AMs appears considerably lower than the one of P-receptor
afferents (Bastian 1981
) and 2) E- and I-type
pyramidal cells appear less sensitive to the detailed time course of
AMs than P-receptor afferrents and seem to extract the occurrence of
upstrokes and downstrokes in AMs, respectively (Metzner et al.
1998
). Both these transformations are likely to play a role in
the generation of electrolocation and electrocommunication behaviors.
In particular, understanding how the increased sensitivity of pyramidal
cells arises might contribute to explain jamming avoidance responses to
extremely weak AMs and the detection of small preys using the electric
sense (Kawasaki 1997
; Nelson and MacIver
1999
).
Some sort of averaging operation across several afferent fibers
converging onto a pyramidal cell is likely to contribute to this
increased sensitivity (Bastian 1981
), as
discussed in the previous section. One biophysical mechanism
specifically proposed to enhance the sensitivity of E-type pyramidal
cells to upstrokes in the AM waveform is coincidence detection
(Berman and Maler 1999
): in slices of the ELL of
Apteronotus, the stimulation of afferents in the deep fiber
layer produces compound postsynaptic potentials consisting of an
initial, fast-rising excitatory postsynaptic potential followed by an
inhibitory postsynaptic potential that limits the time window of
integration to <10 ms and could therefore act as a high-pass filter
for coincident spikes occurring within 1-2 EOD cycles of each other
(Berman and Maler 1998
; see Softky 1995
for a similar theoretical result). Our experimental results show that
under repeated RAM stimulation, more than 77% of P-receptor afferent
spikes will on average occur within 1-2 EOD cycles of each other in
response to the same AM (Fig. 8B). Thus within the range of
stimulus parameters investigated in this study, spike trains of
P-receptor afferents appear indeed able to provide the information
necessary for such coincidence detection operations.
The authors thank Dr. M. Stopfer for carefully reading the
manuscript and suggesting many improvements. Experiments were carried out by R. Krahe and W. Metzner. Data analysis was carried out by G. Kreiman and F. Gabbiani. F. Gabbiani and G. Kreiman co-wrote the paper.
The original experiments were designed by F. Gabbiani, C. Koch, and W. Metzner.
This work was supported by a National Science Foundation grant to C. Koch and W. Metzner, by a National Institute of Mental Health grant to
C. Koch, and by the Sloan Center for Theoretical Neuroscience.
Address for reprint requests: F. Gabbiani (E-mail:
gabbiani{at}klab.caltech.edu).
The costs of publication of this article were defrayed in part
by the payment of page charges. The article must therefore be hereby
marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.