Abstract
We investigated the variability of P-receptor afferent spike trains in the weakly electric fish, Eigenmannia, to repeated presentations of random electric field AMs (RAMs) and quantified its impact on the encoding of time-varying stimuli. A new measure of spike timing jitter was developed using the notion of spike train distances recently introduced by Victor and Purpura. This measure of variability is widely applicable to neuronal responses, irrespective of the type of stimuli used (deterministic vs. random) or the reliability of the recorded spike trains. In our data, the mean spike count and its variance measured in short time windows were poorly correlated with the reliability of P-receptor afferent spike trains, implying that such measures provide unreliable indices of trial-to-trial variability. P-receptor afferent spike trains were considerably less variable than those of Poisson model neurons. The average timing jitter of spikes lay within 1–2 cycles of the electric organ discharge (EOD). At low, but not at high firing rates, the timing jitter was dependent on the cutoff frequency of the stimulus and, to a lesser extent, on its contrast. When spikes were artificially manipulated to increase jitter, information conveyed by P-receptor afferents was degraded only for average jitters considerably larger than those observed experimentally. This suggests that the intrinsic variability of single spike trains lies outside of the range where it might degrade the information conveyed, yet still allows for improvement in coding by averaging across multiple afferent fibers. Our results were summarized in a phenomenological model of P-receptor afferents, incorporating both their linear transfer properties and the variability of their spike trains. This model complements an earlier one proposed by Nelson et al. for P-receptor afferents of Apteronotus. Because of their relatively high precision with respect to the EOD cycle frequency, P-receptor afferent spike trains possess the temporal resolution necessary to support coincidence detection operations at the next stage in the amplitude-coding pathway.
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 byVictor 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 MΩ), 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 f c = 88 Hz, seeStimulation 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 (f EOD) 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 (f c = 5, 10, 20, 40, and 60 Hz). These AMs were obtained using a modulation signals(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. 1of Wessel et al. 1996). The standard deviation, ς, 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 (ς = 100, 150, 200, 250, 275, and 300 mV; ς = 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 (f c, ς) and was presented 10 times, drawn in pseudo-random order from a subset of all possible (f c, ς) combinations. We usually started by presenting all f cvalues at a fixed contrast (ς = 250 mV) or all contrasts at two cutoff frequencies (f c = 5, 60 Hz). Further (f c, ς) combinations were tested as time permitted.
The second set of stimuli consisted of sinusoidal AMs (SAMs) at a fixed contrast (ς = 250 mV) and at various temporal frequenciesf s. The values used weref s = 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, ς〈n〉
2 (spike count variance), were estimated from
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 Δt ms is assigned a cost of q · ‖Δt‖ 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
andxj
. 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 inxj
: if the interval separating the spikes is larger than 2/q it is less “expensive” to transformxi
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 · ‖Δt‖; Fig. 1
B). It is therefore straightforward to computedij
(q) when q is large: let ni
andnj
be the number of spikes inxi
and xj
, respectively, and the integer cij
denote the number of coincident spikes in xi
andxj
(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 betweenxi
and xj
is obtained by first deleting (ni
−cij
) spikes inxi
and then adding (nj
− cij
) spikes in xj
. Thus
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 sinceq · ‖Δt 1‖ < 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, t
jitter, of the spike occurrence times was defined ast
jitter = 1/q
1/2 whereq
1/2 is the value of q such thatd
ij
n(q
1/2) = 1/2. This definition is motivated by the following arguments showing that t
jitter equals the average time interval, t̄
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 jittert
jitter is a generalization oft̄
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
α,n
β, andn
γ denote the number of spikes moved, deleted, and added when computing the distance betweenxi
and xj
. If we pool together all noncoincident spikes inxi
and xj
,ni
+ nj
− 2cij
, then each one of these spikes is either moved, deleted, or created when transformingxi
into xj
so that the following equation holds
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,
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,s
est(t), of the stimulus,s(t) (Fig.2
A). The waveformh(t) is chosen to optimize the match betweens
est(t) ands(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 1999for an introduction). For each spike trainxi
(t) (i = 1, … , 10) obtained on presentation of a RAMs(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 convolvinghi
(t) withxi
(t) (see Fig. 2
A). 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 ashi
(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, ε2, by cross-validation (Fukunaga 1990): each filterhi
(t) was convolved with a spike train xj
(t) different from the one used to computehi
(t) to avoid over fitting. This yielded an estimate ε̂ij
2
Quantification of stimulus encoding and of its robustness to spike time jittering. A: an estimate,s est(t), of the stimuluss(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 (ε2) 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 ςjitter. The modified spike trains are shown for increasing values of ςjitter (from top tobottom) on the left. From each distorted spike train, a new WK-filter and a new estimate,s est(t), of the stimulus were computed (right). Robustness was quantified by computing the rate at which the fraction of the stimulus encoded decreased with ςjitter (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 p add indicate the percentage of spikes added to the experimental spike train andp del 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 ςjitter (Fig. 2 B). We used ςjitter = 0, 1, 3, 5, 7, 10, 15, and 30 ms;p add = 0, 1, 5, 10, 20, and 30%;p del = 0, 1, 5, 10, 20, and 30%.
Let γ(p
add), γ(p
del), and γ(ςjitter) denote the coding fractions for a given value of p
add, pdel, and ςjitter, respectively. The robustness of RAM encoding by P-receptor afferent spike trains was evaluated by plotting the normalized coding fraction γn(x) = γ(x)/γ(0) where x =p
add,p
del, or ςjitter as a function of x (Fig. 13,inset). In most cases, the normalized coding fraction was linearly related to the distortion parameter x (seeresults). We therefore performed linear fits of γn as a function of x =p
add,p
del, or ςjitter
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. 3
A) (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 V
th
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 (V thresh) 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 outputz(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.3
B). An alternative biophysical model proposed byKashimori 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 functionH(s)
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. 3
B). 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
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.2 B2), 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. 4 C and, to a lesser extent, Fig.4 D) 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 (ς > 200 mV) and cutoff frequencies (f c > 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.
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 (ς = 250 mV) of a neuron with low mean firing rate (A andC: 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:f c = 5 Hz; C andG: f c = 60 Hz). Theright column (B, D, F, andH) illustrates the responses for a fixed cutoff frequency (f c = 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 (Band F: ς = 100 mV; D andH: ς = 275 mV).
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 (wheren is an integer) in a given window (seemethods). 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.
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 (ς = 250 mV) and various cutoff frequencies f c (as indicated on the top of each panel; A is the same experiment as in Fig. 4 A). E–Hwere 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 highf cs (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 andH, 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.
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 t shift. In other words, ifx 1(t), … ,x 10(t) represent the original spike trains, new spike trains were defined asx̃ 1(t) =x 1(t),x̃ 2(t) =x 2(t +t shift), … ,x̃ 10(t) =x 10(t + 9 ·t shift). The parametert shift 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 oft shift. In some cases the number of vertical rows of parabolas increased witht shift. 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, t shift = 10 ms, the timing of spikes drifted by 90 ms between the first spike trainx̃ 1(t) and the last spike train x̃ 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.
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. 4 A and 5A) to repeated presentations of a random electric field AM (RAM) stimulus (ς = 250 mV,f c = 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 andB for an I&F neuron model with gamma order 10 (mfr = 81 spikes/s; see main text and Fig. 7 A for a more detailed description of the model).
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 parameterq (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 trainsDn (q) = 0 independent ofq. 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 whichDn (q) = 1/2, calledq̄ 1/2, may be used to summarize spike train variability: if we sett̄ jitter = 1/q̄ 1/2, thent̄ 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; seemethods 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 distancesDn (q) were compared with those obtained from a family of gamma neuron models indexed by a parametern controlling spike train variability (seemethods). 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. Figure7 A 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. 7 A. Accordingly, the average distance between two spike trains of this P-receptor afferent followed closely those of the gamma-10 neuron (see Fig. 7 B, ▴ and ■) and were always smaller than those of a Poisson model (Fig. 7 B, ●). The valueDn (0) in Fig. 7 B 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 valueDn (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,t̄ 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 (f EOD = 438 Hz). Furthermore, the largest deviation betweenDn (q) in the gamma-10 model (or the P-receptor afferent) and the Poisson model was observed forq values lying in the interval (0.05–0.25 ms− 1; Fig. 7 C). A value of q = 0.25 ms− 1 was used to illustrate our results in subsequent figures.
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.5 B) 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. 3 A). 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 distanceD n(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,n pairs = 90). C: plot of the difference in mean distances between n = 1 andn = 10 models (mean ± SE,n pairs = 90) as a function ofq.ΔD n(q) =D n(q)Poisson− D n(q)gamma order 10, whereD n(q)Poissoncorresponds to the filled circles in B andD n(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 andC the value q = 20 ms− 1 was not plotted because it would lie off-scale (see main text).
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. 8 A 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%). Figure8 B replots the average temporal jitter in units of the EOD cycle (1/f EOD) 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,t̄ 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).
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 t̄ 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.
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. Figure9 A 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.9 A, – – –). As illustrated in Fig. 9 B, 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 theright 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 f c andt̄ jitter were negative at low rates (−0.59 ± 0.35) but not at high firing rates (0.29 ± 0.62).
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, ς =250 mV,f EOD = 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).
Similar results were obtained for the distance measureDn (q) over a broad range of the spike distance parameter q, as illustrated in Fig.10. At fixed, intermediate values ofq, 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 andDn (q) measures differences in spike counts] the slopes and correlation coefficients ofDn (q) versusf c regression lines were not significantly different from 0 (P > 0.2 atq = 0 ms− 1, 2-tailedt-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).
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; ς = 250 mV). This represents a particularly clear example. B: average slopes (mean ± SE) of distance [D n(q)] vs. cutoff frequency (f c) relations (computed as inA) at low (●, average ofn = 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).
Variability and stimulus contrast
The dependence of spike time jitter on stimulus contrast was very similar to the one found for stimulus cutoff frequency. Figure11 A 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. 11 B, which reports the slopes of linear regression lines for t̄ jitterversus ς in P-units firing at low and high rates (left andright 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.
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, f c = 5 Hz,f EOD = 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).
Figure 12 reports the same results directly in terms of spike train distances at all values ofq 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.12 A). 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-tailedt-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).
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 ofq = 0.25 ms− 1 in a low firing rate neuron (mfr = 52 spikes/s,f c = 5 Hz,f EOD = 438 Hz). This represents a particularly clear example. B: average slopes (mean ± SE) of distance [D n(q)] vs. stimulus contrast (ς) relations (computed as inA) at low (●, average overn = 23 neurons) and high (■, average over n = 8 neurons) firing rates as a function ofq. C: average correlation coefficient (mean ± SE) of distance vs. cutoff frequency as a function ofq (computed as in A).
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. 2 B andmethods), 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 ςjitter 50). 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 γ. The robustness to spike time jitter decreased as the stimulus cutoff frequency increased, reaching a value of 6 ms for fast changing stimuli (f c = 88 Hz). The robustness to spike additions or deletions did not show a dependency on stimulus bandwidth forf c > 5 Hz (see Table1). 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).
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 f c, respectively (the large error at f c = 5 Hz is due to extrapolation from shallow slopes, see methods andinset for the computation of ςjitter 50).
Robustness to spike time jittering, and random spike additions or deletions
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.14 A and Eq. 8 inmethods) at various frequenciesf s. The experimental gains and phases were then fitted by maximum likelihood to a first-order high-pass filter (Fig. 14 B and Eq. 7). The resulting fits had χ2-values divided by the number of degrees of freedoms (χ2/DOF) (Press et al. 1992, chapt. 15) for the fits between 0.7 and 8.0 (for 15 afferent fibers), except for two outliers (χ2/DOF = 22.8 and 24.2, respectively). The mean values of the filter parameters wereG a = 120 ± 82 spikes/s (range: 16–300 spikes/s), G c = 40 ± 26 spikes/s (range: 7–99 spikes/s), and τa = 4 ± 5 ms (range:0.2–17.5 ms). In contrast to the results ofNelson et al. (1997) in Apteronotus, fitting the data with a second-order filter improved only slightly the χ2/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. 3 B was needed in the model to preventz(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.
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 p n(p n = t f s/2π, bin size: 1/20 of the period cycle) for 3 different sinusoidal AMs (f s = 1, 20, and 100 Hz, respectively). The solid line represents the fit with Eq. 8 (seemethods; r 2 is the correlation coefficient between the data and fit). B: fits of the mean gain and phase (±SD) obtained from A (seeEq. 7 ) with a 1st-order high-pass filter (same neuron as in A; fit parameters: G a= 147 ± 9 spikes/s, G c = 20 ± 2 spikes/s, τa = 1.2 ± 0.8 ms, χ2/degress of freedom = 3.2).
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 onf c and ς.
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 (■; ς = 250 mV). C and G: coding fraction as a function of stimulus contrast for the same 2 neurons (f c = 5 Hz). Band 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: G a = 17 spikes/s,G c = 7 spikes/s, τa= 6 ms, f EOD = 375 Hz,r base = 5 spike/s,n = 3, V th = 80 mV.E–H: G a = 165 Hz,G c = 34 Hz, τa = 2 ms, f EOD = 575 Hz,r base = 65 spikes/s,n = 3, V th = 130 mV.
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 alsoBarberini 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, seeJohnson 1996). Figure 7 B 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 t̄ jitterare 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 t̄ jitteris 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 : ifn α and/orn β 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 thatt̄ jitter = 1/q̄ 1/2 is larger than the average time interval given by the left hand side of Eq. 4.The additional 15% increase int̄ 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 spikest̄ jitter, is in most cases below 4 ms. On the other hand, the robustness of encoding to spike time jitter yields values of ςjitter 50 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.,f c = 60 Hz), a jitter of 4 ms leads to a relative decrease in coding fraction of at most 18% (see Table 1; ςjitter 50 = 11 ms implies that γn (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 fromn = 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 ofApteronotus, 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 1995for 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. 8 B). 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.
Acknowledgments
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.
Footnotes
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F. Gabbiani (E-mail:gabbiani{at}klab.caltech.edu).
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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.
- Copyright © 2000 The American Physiological Society
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