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The Journal of Neuroscience, July 15, 2001, 21(14):5328-5343
Negative Interspike Interval Correlations Increase the Neuronal
Capacity for Encoding Time-Dependent Stimuli
Maurice J.
Chacron1, 2,
André
Longtin1, and
Leonard
Maler2
1 Department of Physics, University of Ottawa, Ottawa,
Ontario, Canada K1N-6N5, and 2 Department of Cellular and
Molecular Medecine, University of Ottawa, Ottawa, Ontario, Canada
K1H-8M5
 |
ABSTRACT |
Accurate detection of sensory input is essential for the survival
of a species. Weakly electric fish use amplitude modulations of their
self-generated electric field to probe their environment. P-type
electroreceptors convert these modulations into trains of action
potentials. Cumulative relative refractoriness in these afferents leads
to negatively correlated successive interspike intervals (ISIs). We use
simple and accurate models of P-unit firing to show that these
refractory effects lead to a substantial increase in the animal's
ability to detect sensory stimuli. This assessment is based on two
approaches, signal detection theory and information theory. The former
is appropriate for low-frequency stimuli, and the latter for
high-frequency stimuli. For low frequencies, we find that signal
detection is dependent on differences in mean firing rate and is
optimal for a counting time at which spike train variability is
minimal. Furthermore, we demonstrate that this minimum arises from the
presence of negative ISI correlations at short lags and of positive ISI
correlations that extend out to long lags. Although ISI correlations
might be expected to reduce information transfer, in fact we find that
they improve information transmission about time-varying stimuli. This
is attributable to the differential effect that these correlations have
on the noise and baseline entropies. Furthermore, the gain in
information transmission rate attributable to correlations exhibits a
resonance as a function of stimulus bandwidth; the maximum occurs when
the inverse of the cutoff frequency of the stimulus is of the order of
the decay time constant of refractory effects. Finally, we show that
the loss of potential information caused by a decrease in spike-timing
resolution is smaller for low stimulus cutoff frequencies than for high
ones. This suggests that a rate code is used for the encoding of
low-frequency stimuli, whereas spike timing is important for the
encoding of high-frequency stimuli.
Key words:
electrosensory afferent; electrolocation; interspike
intervals; spike train variability; weak signal detection; correlations; information theory; resonance
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INTRODUCTION |
Detecting external stimuli is
essential for an animal to survive in its environment. A variety of
methods have been used to quantify information transmission in sensory
systems, such as information theory (Borst and Theunissen,
1999 ; Bura as and Albright, 1999),
signal detection theory (Green and Swets, 1966;
Gabbiani and Koch, 1998), and stimulus reconstruction
(Rieke et al., 1997; Gabbiani and Koch,
1998). Information theory makes no assumptions on the nature of
the neural code and uses mutual information to quantify information
transfer in sensory systems. The mutual information I(X,Y) between two random variables X and
Y is equal to H(X) H(X|Y), where
H(X) is the entropy of X, and H(X|Y)
is the conditional entropy of X given Y. Because
entropy increases with variability, it is clear that I(X,Y)
will increase if the variability of X is high and the
variability of X given Y is low. In neural
systems, we can identify a time-varying stimulus with Y and
a spike train with X. Thus, information theory suggests that
high variability of the spike train in the absence of the stimulus,
combined with low variability in the presence of the stimulus, will
maximize information transmission. In particular, because correlations are known to reduce entropy (Shannon, 1948), one might
expect that the correlations in a spike train will degrade information transmission.
On the other hand, signal detection theory in the context of neural
spike trains aims to determine the presence versus the absence of a
stimulus that is based on differences between the number of spikes
counted in an interval of length T. The discriminability d between the two cases is given by the absolute value of
the difference between the mean of these spike counts with and
without stimulus, divided by the square root of the summed
variances of these spike counts (see Eq. 6). It is clear that
discriminability is enhanced by a low variability in the spike train,
because this will result in low variability of spike count (low
variance) over the counting window. In this paper, we show that
negative interspike interval (ISI) correlations, i.e., the
tendency for long ISIs to be followed by short ISIs (and vice versa),
reduce spike count variability, whereas positive ISI correlations
increase spike count variability. Together, these effects lead to an
optimal spike counting time at which discriminability is maximal.
Contrasting the information theoretic and signal detection approaches
points to an apparent contradiction with respect to spike train
variability: information theory requires high variability, whereas we
have just shown that signal detection requires low variability.
However, we will show, in the context of the electrosensory system,
that each approach is best suited to a particular stimulus frequency
range. Also, we will show that negative ISI correlations enhance both
the mutual information I and the discriminant measure d.
Gymnotiform weakly electric fish are particularly adept at detecting
prey (Nelson and MacIver, 1999) and each other
(Dulka et al., 1995; Bastian et al.,
2001). They emit a sinusoidal time-varying electric field
through their electric organ discharge (EOD; frequency, 600-1200 Hz).
P-type electroreceptors on their skin detect amplitude modulations
(AMs) of this field caused by nearby objects or conspecifics (for
review, see Bastian, 1981; Zakon, 1986).
These P-units fire action potentials when driven only by the EOD, i.e.,
without AMs. Therefore, detection of external stimuli is based
presumably on a change from this baseline activity (Ratnam and
Nelson, 2000). Furthermore, these P-units exhibit negative ISI
correlations at low lags (Longtin and Racicot, 1997;
Chacron et al., 2000; Ratnam and Nelson,
2000), resulting from cumulative relative refractoriness (Chacron et al., 2000).
For Apteronotus leptorhynchus (Brown ghost knife fish),
P-unit spike train variability decreases (as measured by the Fano factor; see Materials and Methods) by as much as two orders of magnitude for counting times varying between 40 and 1000 EOD cycles (Ratnam and Nelson, 2000) before increasing again. Our
computational study provides an explanation for this result and for the
fact that P-units can transmit information about both low- and
high-frequency stimuli. Our study uses simple and accurate
biophysically plausible models for P-unit activity to generate the
large data sets necessary for mutual information analysis and for
establishing the role of the ISI correlations in the enhancement of
stimulus coding.
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MATERIALS AND METHODS |
Interspike interval analysis
Let us denote by {Ij} the ISI
sequence with mean  I and variance VAR(I).
The coefficient of variation (CV) of the interspike interval
histogram (ISIH) provides a good measure of spike train variability on
time scales of the order of the mean ISI. If
{Ij} is stationary, then the coefficient of
variation is defined as:
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Serial correlation coefficients |
One common measure of memory effects in a time series of events
is the serial correlation coefficients (SCCs)
 j defined for lag j by:
The SCCs are a measure of linear ISI correlations in the spike train.
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Spectral density function |
The spectral density function (SDF) is the discrete Fourier
transform of the SCCs (Cox and Lewis, 1966) and is
defined for positive frequencies f as:
This formula can be inverted using the inverse Fourier transform
to yield an expression for j (Cox and
Lewis, 1966):
The spectral density function is always positive (Cox and Lewis,
1966). Moreover, specifying the SCC sequence allows us to uniquely
determine the SDF and vice versa. The two quantities are thus
completely equivalent (Cox and Lewis, 1966).
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Pulse number distributions and the Fano factor time curve |
The pulse number distribution (PND) (Barlow and Levick,
1969a,b;
Teich and Khanna, 1985) P(n,T) is defined as
the probability of observing n spikes during a counting time
T (the PND is sometimes referred to as the spike count
distribution). It is calculated by first dividing the spike train into
nonoverlapping time windows of length T and then counting
the number of action potentials in each window. A normalized histogram
of these numbers then yields the PND. The Fano factor (Fano,
1947) is defined as the variance to mean ratio of the PND:
and has units of spikes. The Fano factor curve F(T)
gives a measure of spike train variability on all time scales
T.
The Fano factor always approaches unity for small T
(Teich et al., 1997). For a Poisson process, we have
F(T) = 1 for any T (Cox and Lewis,
1966). Processes with F(T) < 1 are thus
considered less variable than Poisson, whereas those with
F(T) > 1 are more variable (Gabbiani and Koch,
1998). The asymptotic value F of the
Fano factor is related to the SCCs of the ISI sequence (Cox and
Lewis, 1966) according to:
|
(1)
|
where we have assumed that the series is convergent. Note that
positive and negative ISI correlations increase and decrease, respectively, the asymptotic value F.
| |
Signal detection theory |
The ideal observer paradigm is based on the optimal
discrimination between PNDs obtained in the presence and absence of a stimulus (Green and Swets, 1966) for examples of
applications to neural systems (see Nachimas, 1972;
Shofner and Dye, 1989; Gabbiani and Koch,
1998; Gabbiani and Metzner, 1999). Let
P0(n,T) be the PND obtained without stimulus and
let P1(n,T) be the one obtained in the presence
of stimulus. Then, we define the probability of false alarm
PFA, i.e., of reporting a signal when it is not there, as (Gabbiani and Koch, 1998):
and the probability of correct detection
PD as:
where m is some threshold. The overall performance of
the detector is characterized by varying m between zero and
infinity and plotting PD as a function of
PFA. This curve is called the receiver operating
characteristic (ROC) of the detector (Green and Swets,
1966; Gabbiani and Koch, 1998). The further this
curve lies above the diagonal PD = PFA (which corresponds to chance detection), the
better the performance of the detector for a counting time T
(Gabbiani and Koch, 1998).
| |
Information theory |
The entropy of a discrete random variable X with
probability density function P(X) is defined as
(Shannon, 1948; Cover and Thomas,
1991):
and is measured in bits.
The variability of the neural response to an ensemble of stimuli is
characterized by the total entropy Htotal
(Strong et al., 1998; Buraas and Albright
1999) and is estimated from the neural response to an
unrepeated Gaussian stimulus. The trial-to-trial variability of the
neural response to a repeated stimulus is characterized by the noise
entropy Hnoise (Strong et al.,
1998; Reinagel and Reid, 2000) defined as the
conditional entropy of the spike train given a stimulus (previously
referred to as H(X|Y) in the introductory remarks). Thus
Hnoise represents the variability ("noise")
in the spike train that cannot be accounted for by the stimulus.
A good stimulus encoder must have a neural response that varies highly
in response to different stimuli while having a very reliable response
to repeated presentations of the same stimulus. In the case of neurons
that are silent in the absence of sensory input, an estimate of the
mutual information about the stimulus is given by (Strong et
al., 1998):
This measure gives a lower bound for the mutual information, as
the estimate for Htotal will be lower than the
value obtained for all possible stimuli. For P-type electroreceptors,
the absence of stimulus corresponds to the fish's EOD that is
unmodulated by extrinsic signals. The resulting spike train defines the
baseline activity of the P-unit. P-units display a phase-locked random skipping pattern (CV = 0.44 on average) (Ratnam and Nelson,
2000) to this EOD (Xu et al., 1996;
Nelson et al., 1997; Chacron et al.,
2000). A decision about whether a stimulus is present or not
must thus presumably be made on the basis of a change from this
baseline activity (Ratnam and Nelson, 2000). Hence, it
is natural in our case to estimate mutual information I as
the difference between the entropy of the baseline activity and the
entropy estimated from the trial-to-trial variability of the neural
response to a repeated stimulus:
Note that we have I = 0 when no stimulus is present. We
divide information and entropy by stimulus duration measured in EOD cycles and express the results in bits per EOD cycle. To estimate these
quantities, we divide the spike train into bins of length   . If
n spikes occurred between i and
(i + 1) , then the value assigned to bin
i is n. The entropy of words comprising L bins is given by (Strong et al., 1998):
where w is a word of length L and
W(L,) is the set of all possible words of length
L. If the correlations have finite range, then we can expand
H(L,) as a Taylor series in powers of
L 1 (Strong et al.,
1998):
|
(2)
|
where H() is the entropy rate for infinite word
length and C1 and C2 are
constants. In some cases the L1 term is
sufficient (Strong et al., 1998; Reinagel and
Reid, 2000). However, this is not the case here; rather we
performed quadratic fits of the H(L,) versus
L1 data obtained to get the entropy
rates and infer the information rate. We used our models to generate
1,000 realizations, each containing 10,000 successive EOD cycles with
the same repeated stimulus (note that such amounts of data would have
to be obtained from recordings lasting in excess of 2-3 hr, which is
not currently feasible for electrosensory afferents). The baseline
entropies were estimated in the same way, except that no stimulus was present.
| |
Modeling |
Because we are interested in understanding how correlations of
P-unit ISIs affect information transfer, we use two models of P-type
electroreceptors. The Nelson model (Nelson et al., 1997) accounts for first-order ISI statistics (i.e., the ISIH) and for the
gain and phase response to sinusoidal AMs; however, it is memoryless in
the sense that the significant ISI correlations displayed by
experimental data are absent (see Fig. 1 and Chacron et al.,
2000). The other model uses the filtering property of the
Nelson model to extend the model proposed by Chacron et al. (2000). The latter model was shown to reproduce first- and
second-order statistics of experimental data, namely the ISIH, ISI
return map (Ij+1 as a function of
Ij), and the ISI correlations as measured by the SCCs. We start with a description of the Nelson model and then
describe ours. For convenience, a list of all symbols and acronyms used
in this paper is provided in appendix A, whereas appendix B summarizes
all equations used for both models.
The Nelson model
It is known from experiments that various filtering mechanisms
are at work inside a P-unit (Hopkins, 1976;
Bastian, 1981; Wessel et al., 1996;
Xu et al., 1996; Nelson et al., 1997).
Nelson et al. (1997) measured the gain and phase
response characteristics of P-units to sinusoidal AMs of frequencies in
the range 0.1-200 Hz, leading to the following set of differential
equations:
|
(3)
|
|
(4)
|
|
(5)
|
where the dot denotes differentiation with respect to time,
A(t) is the stimulus (i.e., the time-varying EOD amplitude
minus its baseline value), and X(t) is the filtered
stimulus. The G values are gains in units of spikes per
second per millivolt, and the values are time constants in units of
seconds. A baseline firing rate rbase is added
then to X(t), and the sum Z(t) is passed through
a clipping nonlinearity to account for saturation effects (Nelson et al., 1997):
The probability p(t) of firing per EOD cycle is thus
r(t)/fEOD, where fEOD is
the EOD frequency. At each maximum of the EOD, the P-unit has
probability p(t) of firing. If the unit fires, jitter is
added to the spike time in the form of Gaussian white noise of zero
mean and standard deviation 0.04 EOD cycles. Throughout this paper, we
take fEOD = 1000 Hz, hence an EOD cycle
corresponds to 1 msec. Furthermore, to reduce the coefficient of
variation of the ISIH, it is possible to implement m
independent random subprocesses, each with an event rate equal to the
spike rate r(t) (Nelson et al., 1997). Each
subprocess is simulated as described above, and output spikes are
generated at the time of occurrence of every mth subprocess
event. The model gives an ISIH similar to the data but does not display
the correlations seen experimentally (see Fig. 1 and Chacron et
al., 2000). The model was constructed to give the correct
responses to sinusoidal AMs (Nelson et al., 1997) and
was used to give the firing dynamics in response to changes in
transdermal potential caused by a prey (Nelson and MacIver,
1999).
A modified integrate-and-fire type model
Biophysical justification. We begin with a
description of P-type electroreceptors to biophysically justify our
model. A P-unit is composed of 25-40 receptor cells and a nerve fiber
making synaptic contact onto at least 16 active neurotransmitter
release sites per receptor cell (Bennett et al., 1989).
Although it is currently impossible to record intracellularly from
these cells, there is much indirect evidence that the EOD amplitude
changes individual receptor potentials that govern the rate of release
of neurotransmitter onto the afferent nerve. Fluctuations in this rate
are thus one expected source of variability in such systems
(Stein, 1965; Nelson et al., 1997).
Another possible source is the conductance fluctuations of the ionic
channels at the spike initiation zone in the axon.
As mentioned above, we expect relative refractoriness to be important
at such high firing rates (Stein, 1965). It leads to negative SCCs for the ISIs at low lags (Geisler and Goldberg, 1966; Chacron et al., 2000). Because there is
currently no biophysical characterization of the ionic conductances
inside the P-unit, we can only speculate as to the possible mechanisms
responsible for this experimentally observed relative refractoriness.
The physiological mechanisms responsible for these correlations could be presynaptic in origin; for example, long-term depression at the
synapses connecting the receptor cells to the afferent nerve (Bennett et al., 1989; Hausser and Roth,
1997) would lead to relative refractoriness. However, the
recovery time constant of the neurotransmitter at typical synapses is
usually in the thousands of milliseconds range (von Gernsdorff
et al., 1997), which is much too long for the phenomenon at
work here because serial correlations are significant only up to lag 2 (Chacron et al., 2000) (see Fig. 1). Thus, we are
looking at a time scale of about 10 EOD cycles (twice the mean ISI,
which is 5 EOD cycles long for the unit considered here). A likely
candidate would be a postsynaptic spike-activated potassium channel
that slowly deactivates and thus summates to produce a negative
adaptation current. The KV3.1 channel has the right activation and
deactivation kinetics (Wang et al., 1998). Members of
the KV3 family are richly expressed in the electrosensory system
(A. J. Rashid, personal communication), but it remains to be shown whether similar channels are present in P-units.
From the foregoing discussion, we feel that the best approach for
studying the effects of correlations on information transfer and signal
detection is to use a simple yet biophysically plausible model that
reproduces the essential features of P-unit baseline and evoked discharge.
Description. First, we will give an expression for the
synaptic current at the spike initiation zone in the axon and then describe the spiking mechanism. We use the simple model in
Chacron et al. (2000) that has been proposed to model
the baseline firing dynamics of P-type electroreceptors and extend it
to get the proper responses to time-varying AMs of the EOD amplitude.
We write the transdermal potential on the fish's skin as
(A(t)+A0) sin(2  fEODt), where
A0 is the constant EOD amplitude corresponding
to baseline firing dynamics (this is similar to
rbase in Nelson's model) and A(t) is
the AM. The stimulus A(t) is filtered using Equations 3-5.
Because many receptors rectify a periodic input (French et al.,
1972; Gabbiani, 1996), we take the total
synaptic current to be:
where  and  are constants to make units match,
 1 and 2 are noise terms, and  is the
Heaviside function ((x) = 1 if x  0 and (x) = 0 if x < 0) to
account for rectification. Thus, the deterministic component of the
synaptic current is zero whenever sin(2fEODt) is negative or
X(t) + A0 is negative. This is to ensure that firings will always occur near the maxima of the EOD sine
wave, as experimentally observed (Scheich et al., 1973). Because our model is phenomenological, we take the synaptic current to
be dimensionless. Noise sources including conductance and synaptic fluctuations are modeled by two Ornstein-Uhlenbeck processes
1 and 2 given by:
where  1 and 2 are two independent
Gaussian random variables of zero mean and variance unity,
D1 and D2 are constants proportional to the
intensities of 1 and 2, and
1 and 2 are time constants. Figure
2a gives a time series for 1 and
2. It can be shown (Gardiner, 1985) that
1 and 2 are stationary Gaussian random
variables with zero mean and respective variances D1
1/2 and D2
2/2. However, unlike Gaussian white noise,
1 and 2 are correlated in time, and their
correlation functions decay exponentially with respective time
constants 1 and 2. We take
1 to be much less than an EOD cycle; hence,
1 can be thought of as "fast" compared to the model dynamics (it is almost white noise in fact) (see Fig. 2a).
It could thus model fluctuations that occur on time scales much faster than the EOD cycle (e.g., membrane noise caused by channel flicker low-pass filtered by the membrane capacitance) (Manwani and
Koch, 1999). In this case, D1 would be
related to the strength of the membrane noise.
In contrast, we take 2 to be much greater than the EOD
cycle. Hence, this noise term can be thought of as "slow" compared with the model dynamics (it is almost constant for time scales much
smaller than 2) (see Fig. 2a). It thus
models fluctuations on slow time scales (e.g., fluctuations in
vesicular release rate; see below). This term is needed to accurately
reproduce the Fano factor curve as shown in Results.
The spiking mechanism is a simple extension to the leaky
integrate-and-fire (LIF) model in which a spike is said to have
occurred when the membrane potential V reaches a constant
threshold  . Immediately afterward, the voltage is reset to its
resting value (usually taken to be 0). LIF models are memoryless in the
sense that consecutive ISIs are not correlated
(i = 0 for all i > 0). To include refractory effects, we make also a dynamical variable (Geisler and Goldberg, 1966). But instead of making it
random (Gestri et al., 1980; Gabbiani and Koch,
1996, 1998), we let
it carry the memory by the following firing rule (Chacron et
al., 2000): when voltage equals threshold, it is reset to zero
as in the LIF model, whereas threshold is incremented by a constant amount and kept constant for the duration of the absolute
refractory period Tr, after which it relaxes
exponentially toward its equilibrium value 0 until the
next spiking time. The equations for voltage and threshold between
times of occurrence of action potentials and after the
absolute refractory period are thus:
Like the synaptic current, v and are
dimensionless. A stretch of simulation showing v and is
shown in Figure 2b. The filter given by Equations 3-5 gives
the linear transfer properties of the afferent, whereas our spiking
mechanism gives the correct baseline dynamics. We thus combine the two
to get the correct responses to AMs. Kreiman et al.
(2000) had a similar approach to model the P-type
electroreceptors of another species of weakly electric fish
(Eigenmannia); they used a high-pass filter to give proper
AM response characteristics and fed the output to an LIF model.
However, because their LIF model had a random threshold, it did not
take into account the relative refractory effects that could exist in
this species.
For the remainder of this paper, we refer to our model as the leaky
integrate-and-fire with dynamic threshold (LIFDT) model. Note that our
dynamic threshold can model the aforementioned KV3.1 channel but could
also result from any current that leads to an increase in the effective
distance between voltage and threshold immediately after a spike. It is
thus very general and is used here to model cumulative refractory
effects and consequently endow the ISI sequence with proper
second-order statistics (Chacron et al., 2000) (Fig. 1).
The experimentally obtained SCC at lag 1 for the P-receptor data was
0.35 (Chacron et al., 2000), whereas it is 0.385 for
the model. Note that the three parameters for the spiking mechanism
(0, v, and
 ) can be adjusted to give the proper first- and
second-order statistics (ISIH, ISI return map, and SCCs) for other
P-units with different firing rates (data not shown). This spiking
model could thus be used to model other neural systems in which
negative ISI correlations at short lags have been observed (e.g., the
auditory system) (Lowen and Teich, 1992). The reason
explaining why the dynamic threshold gives rise to negative ISI
correlations can be understood as follows: suppose an ISI shorter than
I just occurred, then the threshold (having had less
time to decay) will typically be high after the spike and will thus
take a long time to decay. Consequently, the next ISI will (on average)
be longer than I. This gives rise to a negative SCC at
lag 1. This is also the case when an ISI longer than I
occurs; as a result, the threshold will now be lower, and the next ISI
will be shorter than I. However, because of the strong
noise 1, negative ISI correlations at longer lags will be washed out. An extended explanation of the role of the noise in
the model and why the dynamic threshold leads to negative ISI
correlations in the presence of noise can be found in Chacron et
al. (2001).
Also, our extended model gives the correct responses to sinusoidal AMs
(see Fig. 3 and the next section).
| |
Stimulation |
Baseline firing statistics were computed for both models. Their
ability to encode time-varying stimuli was tested using AMs of the EOD
amplitude. For the LIFDT model, the EOD amplitude minus its baseline
value was given by:
where s(t) is the stimulus and  stim is
the contrast. To calibrate the model, we first used sinusoidal AMs of
different frequencies and intensities to construct the phase and gain
response curves (see Fig. 3). We took the baseline transdermal
potential to have a root mean squared value of 0.566 mV, which is in
the physiological range (Xu et al., 1996; Nelson
et al., 1997). The gain and phase curves were constructed then
using the method outlined in Nelson et al. (1997). The
gain for sinusoidal AMs of frequency 1 Hz was 1060 spikes per second
per millivolt, which is in the experimentally observed range of values
(Nelson et al., 1997).
The stimulus c stims(t) was
presented then to the Nelson model, and the constant c was
adjusted so that both models gave identical gains and phases over the
frequency range of the sinusoidal AMs. For the purpose of quantifying
the amount of information transmitted, we then took s(t) to
be low-pass filtered Gaussian white noise of mean 0 and variance 1. A
Butterworth fourth-order filter was used with cutoff frequency
fc (Wessel et al., 1996). As
mentioned above, this type of stimulus has been used widely in
quantifying the ability of neurons to encode time-varying stimuli by
means of the stimulus reconstruction technique (Gabbiani,
1996; Gabbiani and Koch, 1996,
1998; Wessel et al.,
1996; Chacron et al., 2000; Kreiman et
al., 2000).
| |
RESULTS |
We first explain how negative ISI correlations (Fig. 1) lead to a
decrease in spike train variability and how positive ISI correlations
increase this variability. We then show how a combination of strong
negative ISI correlations at short lags and weak positive ISI
correlations induced by a weak correlated noise (Fig. 2) and extending
out to long lags gives rise to the minimum in the Fano factor curve
F(T) seen experimentally by Ratnam and Nelson
(2000). We consider the role of these ISI correlations for
signal detection using the ideal observer paradigm by considering two
models (see Materials and Methods) that have identical responses to
sinusoidal AMs (Fig. 3). We then study their contribution to the
ability of the receptor to encode time-varying stimuli in the form of low-pass filtered Gaussian white noise. This will be done by computing the mutual information rate as a function of stimulus contrast and
cutoff frequency. Finally, we show that signal detection is best suited
for low-frequency stimuli (e.g., electrolocation signals) (Nelson and MacIver, 1999) because spike train
variability is low on long time scales, and that information theory is
best suited for high-frequency stimuli (e.g., electrocommunication
signals) (Zupanc and Maler, 1993; Bastian et al.,
2001) because spike train variability is high on short time scales.
Fano factor
The Fano factor curve obtained for the LIFDT model with
D2 = 0 is plotted in Figure 4
(triangles). We see that the electroreceptor is more regular
at all time scales than a Poisson process because F(T) < 1 (Cox and Lewis, 1966). F(T) decreases
for T in the 1-5000 EOD cycle range and has an asymptotic
value of 0.00685 (n = 5 line). If we take only the SCCs
to be nonzero up to lag 5, then we get F = 0.00681 from Equation 1, which is very close to the observed
asymptotic value of 0.00685. For comparison, the Fano factor time curve
obtained by random shuffle of the ISI sequence is also plotted in
Figure 4 (diamonds). Because all ISI correlations have been
eliminated by this operation, we now have a renewal process (Cox
and Lewis, 1966) for which F(T) tends toward
CV2 from Equation 1 (CV2 line in
Fig. 4). Note that because CV2  0.0436 < 1 in our case, hence we have F < 1
even in the absence of ISI correlations. The two curves
(triangles and diamonds) are almost on top of one
another for short counting times (<10 EOD cycles), implying that
correlations do not play a significant role over this range from the
Fano factor curve perspective; however, as we will see below, this is
not the case from an information theoretic perspective. However, they
become different for longer times; the Fano factor curve without
correlations tends toward CV2, whereas the one with
correlations has a lower asymptotic value. We plot the Fano factor
curve that was obtained with the Nelson model in Figure 4
(squares). We see that it matches the one obtained for
randomly shuffled ISI sequences from the LIFDT model. This match is not
surprising, because the two models have for all practical purposes
identical ISI distributions and thus identical CVs (Fig. 1). Furthermore, it demonstrates that
there are no significant ISI correlations in the Nelson model. Because
the SCCs are effectively negligible beyond lag 5 in the model, the Fano
factor tends toward a constant for long counting times given by
Equation 1; this is not what is observed experimentally.
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Figure 1.
Comparison of the two models used in our study.
a, ISIH obtained from the analysis of 10,000 consecutive
ISIs from the LIFDT model [I = 4.9912 EOD cycles,
VAR(I) = 1.1449 EOD cycles2,
CV = 0.2143]. b, SCCs obtained with the model.
c, ISIH obtained from 10,000 consecutive ISIs with Nelson's
model [I = 4.9982 EOD cycles,
VAR(I) = 1.1003 EOD cycles, CV = 0.2098].
d, SCCs obtained. Note that both models have the same
distribution of ISIs but that the Nelson model does not exhibit any
significant ISI correlations; the small negative SCC at lag 1 is not
significant because the model has the same Fano factor curve as our
model with shuffled ISIs (Fig. 4).
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We now plot the Fano factor curve obtained with
D2 = 9 × 106
(EOD cycles)2 in the LIFDT model. This curve is on top of
the other ones for short counting times. In particular, the noise
2 is weak (Fig. 2a)
and has negligible effect on the ISIH and the SCCs at low lags (see
below). However, the Fano factor curve differs from the others by
increasing in a power law fashion for long counting times before
saturating. The behavior can be understood from a plot of the mean and
variance of the PND (Fig. 5). The mean
increases linearly with counting time; this is because the mean number
of spikes that is expected in a time window of length T is
equal to the length of that window multiplied by the mean firing rate. However, the variance is almost constant for short counting times; hence, F(T) decreases. The variance then increases at a
greater rate than the mean; hence, F(T) increases. Finally,
the variance and mean both increase with the same rate, and the Fano
factor is constant.
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Figure 2.
a, Noise terms 1
(top curve) and 2 (bottom curve)
as a function of time. 2 varies more slowly than
1 and is four orders of magnitude smaller. b,
Voltage (bottom curve) and threshold (top curve)
trace obtained with the LIFDT model for baseline activity
(no AMs) showing the firing rule. When voltage equals threshold, a
spike is said to have occurred, and voltage is reset to zero, whereas
threshold is incremented by a constant . The threshold is kept
constant to simulate the absolute refractory period
Tr (equal to one EOD cycle) and then decays
exponentially with time constant to its equilibrium
value 0. Parameter values used are given in Appendix
A.
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Figure 3.
Gain and phase response curves obtained with both
models for sinusoidal AMs of various frequencies. The root mean squared
baseline transdermal potential is
A0/ = 0.566 mV, which is
in the physiological range (Xu et al., 1996;
Nelson et al., 1997; Nelson and MacIver,
1999). As in Nelson et al. (1997), we say that a
sinusoidal AM has 0 dB intensity when it produces a 1 mV change (RMS)
in transdermal potential. SAMs of various frequencies were presented to
the model with the same intensities used in Nelson et al.
(1997) to construct the phase and gain curves. The gains have
been normalized by the value 1060 spikes per second per millivolt (this
value is in the physiological range) obtained for
fstim = 1 Hz.
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Figure 4.
Fano factor curve obtained with the models. The
Nelson model has no significant serial correlations amongst ISIs;
hence, the Fano factor tends toward the coefficient of variation
squared. Random shuffling of the ISI sequence obtained with the LIFDT
model removes ISI correlations and gives the same results as the Nelson
model. The negative ISI correlations decrease the Fano factor,
resulting in a lower asymptotic value. However, adding a weak noise
with a long correlation time leads to an increase in the Fano factor at
higher counting times (see text and Appendix C for an
explanation).
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Figure 5.
Mean and variance of the PND as a function of
counting time T for the LIFDT model with slow noise
intensity D2 = 9 × 106 (EOD cycles)2. The mean
increases linearly with counting time. The variance is at first almost
constant, which leads to a decrease in F(T); it then
increases faster than the mean [F(T) increases]. At long
counting times, both increase at the same rate; hence F(T)
is constant.
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We now show that the increase in the Fano factor at long counting times
is caused by the presence of weak positive SCCs that extend out to long
lags. These positive correlations are extremely small and cannot be
seen from a plot of the SCCs j as a function of j. However, their presence is revealed by the increase of
the spectral density function at low frequencies (Cox and Lewis,
1966) (Fig. 6). This can be seen
on the following simple example in which the following form for the
SCCs at lags >0 is assumed:
|
(6)
|
where  ij is the Kronecker delta function
(ij = 1 if i = j and
ij = 0 if i  j). The
corresponding spectral density function is plotted in Figure 6. We have
only retained the negative SCC at lag 1 because it is dominant. We see
that the spectral density function corresponding to the SCCs given by
Equation 6 is very similar to the one obtained with our model for
D2 = 9 × 106
(EOD cycles)2. Because the SDF and the SCC sequence are
completely equivalent (Cox and Lewis, 1966) (see also
"Signal detection theory"), this justifies our assumptions. These
positive correlations at lags >1 sum up according to Equation 1 to
give the increase in the Fano factor. Thus, adding a slow additive
noise to the membrane voltage increases spike train variability at long
counting times. ISI sequences obtained from other neural systems have
been shown in some cases to display negative ISI correlations at short
lags and positive ISI correlations at long lags (Lowen and
Teich, 1992). Furthermore, an increase in the Fano factor curve
has been observed in many preparations (Teich, 1992;
Lowen and Teich, 1992, 1996; Teich et al., 1996,
1997; Turcott and Teich,
1996) and has been modeled by driving the rate of a Poisson
spike generator with colored noise (Teich, 1992;
Teich et al., 1996, 1997). This is a feature of our biophysically plausible
simple model of the P-unit. An intuitive explanation of this
interesting phenomenon is given in appendix C while a full explanation
is beyond the scope of this paper and will be presented elsewhere.
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Figure 6.
Spectral density function obtained with the LIFDT
model with D2 = 0 and
D2 = 9 × 106
(EOD cycles)2. Power law behavior is observed with
D2 = 9 × 106
(EOD cycles)2 for low frequencies. The spectral density
function obtained by assuming a form for the SCCs (see text) is also
shown (solid line) and matches the curve obtained for
D2 = 9 × 106
(EOD cycles)2.
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It is known experimentally that the Fano factor curves obtained for
different P-units have the same qualitative shape, although their
minimum occurs between 40 and 1000 EOD cycles (Ratnam and Nelson, 2000). Our model produces Fano factor curves
quantitatively similar to those obtained experimentally. Furthermore,
the location of the minimum is found to be dependent mainly on
D2; in fact, by a suitable choice of
D2, we can obtain Fano factor curves matching the full experimental range (data not shown). For example, if we take
D2 = 104 (EOD
cycles)2, then the minimum of the Fano factor curve is at
T = 40 EOD cycles. On the other hand, if we take
D2 = 106 (EOD
cycles)2, then the minimum is at T = 1000
EOD cycles.
These results imply that the remarkable regularity of P-unit spike
trains at counting times of ~250 EOD cycles (within the experimentally observed range) (Ratnam and Nelson, 2000)
can be entirely explained by negative ISI correlations that are present experimentally and in the LIFDT model at low lags. These arise from
cumulative relative refractoriness exhibited by our dynamic threshold.
We have shown here that negative SCCs contribute to the reduction of
the Fano factor and that positive SCCs increase the Fano factor
according to Equation 1. Ratnam and Nelson (2000) have
shown that modeling the ISI sequence by a first-order Markov chain gave
the correct SCC at lag 1. However, their SCCs at longer lags had higher
absolute values than observed experimentally and alternated in sign. As
a consequence, the SCC sum for their model was greater (less negative)
than the one calculated from the experimental data. Thus, the Fano
factor calculated with their Markov chain model did not decrease as
much as the Fano factor from their experimental data. In contrast, our
model reproduces the descending part of the Fano factor curve seen
experimentally [compare our Fig. 4 with Ratnam and Nelson
(2000), their Fig. 11F].
We now discuss the power law increase of the Fano factor curve in more
detail. We first note that because we are using Ornstein-Uhlenbeck noise, which has a finite correlation time 2, the
Fano factor will eventually saturate to a finite value. This value is
equal to 0.36188 (Fig. 4, n = 4000 line) and is given
approximately by taking the SCCs up to lag 4000 in Equation 1 (hence
j = 0 for j > 4000).
This implies that the 4000th ISI is still correlated to the first one.
Ratnam and Nelson (2000) found that, for many P-units,
the ISI sequence cannot be modeled by a Markov process of order <10.
Our results suggest that a Markov chain of ISIs of order at least 4000 would be required to correctly reproduce this feature of the Fano
factor curve. In contrast, our simple dynamical model accurately
accounts for the full behavior of the Fano factor curve. One plausible
origin of the slow Ornstein-Uhlenbeck noise might be fluctuations in
synaptic neurotransmitter secretion rates that exhibit long-term
correlations (Lowen et al., 1997). However, it could
also be caused by slow drifts in EOD amplitude or frequency
(Moortgat et al., 1998).
Pulse number distributions and ROC curves
We show in Figure 7 the PNDs that
were obtained in the presence and absence of ISI correlations for four
different counting times for the LIFDT model. We see that ISI
correlations have minimal effects at short counting times such as 20 EOD cycles (Fig. 7a). The effect of negative ISI
correlations increases with counting time (Fig. 7b). It is
very important around 255 EOD cycles (Fig. 7c), at which the
variance of the PND is reduced while the mean is left unchanged. This
effect diminishes for longer counting times (Fig. 7d) at
which positive ISI correlations contribute to the broadening of the
PND.
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Figure 7.
PNDs obtained for both models for various counting
times: (a) 20, (b) 90, (c) 255, (d) 3000 EOD cycles. ISI correlations reduce the variance of
the PND while keeping the mean unchanged. This effect is maximal at
counting times in which the Fano factor is minimal.
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This has implications for weak signal detection using the ideal
observer paradigm (Green and Swets, 1966). Intuitively,
if P0(n,T) and P1(n,T) do
not overlap much, then we have a very good detector (Gabbiani
and Koch, 1998). As mentioned previously, a good measure for
this is the discriminant measure d (Green and Swets,
1966; Nachimas, 1972; Snippe and
Koenderink, 1992) defined by:
|
(7)
|
where the vertical bars denote the absolute value and
 and µi are
the respective variance and mean of Pi(n,T). We
have assumed in Equation 7 that the Pi(n,T) are
Gaussian (i.e., we have neglected their third and higher moments). This
is not too restrictive as they approach Gaussian distributions for high
T by the central limit theorem. Furthermore, the PNDs
obtained with the models are bell shaped (Fig. 7), and the Gaussian
approximation is reasonable. Optimal discrimination, and hence
detection, requires d to be higher than some threshold
dcrit. Let f0 be the
baseline steady-state firing rate of the electroreceptor and suppose
that the stimulus varies slowly with time and leads to a new
steady-state firing rate f1. Here, we do not
consider transients in P-unit firing rate that can occur after a change
in EOD amplitude (Xu et al., 1996; Nelson et al.,
1997), but rather only the new steady-state firing rate. Furthermore, suppose that the signal is weak, and that as a
consequence, the variances of the PNDs are approximately equal. Hence,
µ1(T) = (f1/f0) µ0(T)
and  (T)   (T)
and using Equation 7, the inequality d dcrit becomes:
|
(8)
|
because F(T) = (T)/µ0(T). Furthermore,
0(T) does not vary much if we consider low counting times (Fig. 5), hence it can be considered constant to a first
approximation. From Equation 8, the lower the Fano factor, the better
the detector. A good value for dcrit that gives
almost no overlap between the P0(n,T) and
P1(n,T) distributions is three. Using this
value, one can find a lower bound for
|(f1/f0) 1| from Equation 8. From the data for T = 255
EOD cycles, 0(255) = 0.78, F(255) = 0.012, hence a single P-unit can near perfectly discriminate
steady-state response to slow amplitude modulations as low as 6.5%
from baseline firing. The negative ISI correlations at low lags make
the PNDs narrower and lead to a significant improvement in the ROC
curve (Fig. 8). These fish are very good
at detecting prey using their electrosensory system (Nelson and
MacIver, 1999). Because there are relatively few numbers of
false strikes (Nelson and MacIver, 1999),
PFA must be low (Ratnam and Nelson,
2000). Our results show that the improvement in the detection
probability PD attributable to ISI correlations
is in fact greatest for low PFA. Thus, the
animal significantly improves its chances at detecting prey by having
negatively correlated ISIs.
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Figure 8.
ROC curve obtained for both models. Correlations
improve the ROC curve by decreasing the variance of the PNDs, leading
to a better discriminability between the distributions. The stimulus
was a 4% step increase in EOD amplitude (the new amplitude value is
equal to 1.04 times the old value). Including the transients leads to a
further increase of the mean of the PND with stimulus, which further
increases discriminability and, hence, improves further the ROC
curve.
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In the above analysis, we did not consider the effect on signal
detection of transients in firing rates that are associated with
changes in EOD amplitude (Xu et al., 1996; Nelson
et al., 1997). Transients will help the animal in detecting
weak signals by increasing or decreasing the firing rate, thus shifting
the mean of the PND with stimulus away from the baseline PND. This is
confirmed in Figure 8 in which the transients resulting from a 4% step
increase in EOD amplitude lead to a near perfect detection of the
signal. These transients are present in both models because both
incorporate the filter given by Equations 3-5. A full exploration of
their effects will be done elsewhere.
Thus, there are two phenomena at work here: transients resulting from a
change in EOD amplitude will shift the mean of the PND, whereas
negative correlations will reduce the variance. The combined effect is
better discriminability between the PNDs, hence a better ROC curve and
a lower stimulus contrast threshold for signal detection.
Usually, fluctuations caused by noise average out over time, and the
ROC curve improves (Gabbiani and Koch, 1998). In our case, the ROC curve becomes worse at longer counting times as the
positive ISI correlations increase variability (Fig. 7d). Thus, there is a time window in which signal detectability is optimal
for the animal, and it corresponds to the counting time at which the
Fano factor is minimal. This optimal counting time has been shown to
vary between 40 and 1000 EOD cycles (Ratnam and Nelson,
2000) within the P-unit population.
Entropy
All the previous analysis assumed a rate code, and thus spike
timing was considered unimportant. A recent study on the
electroreceptors of a very similar electric fish (Kreiman et
al., 2000) showed that significant jitter could be added to the
spike train without affecting the quality of encoding by using the
stimulus reconstruction technique (Rieke et al., 1997;
Gabbiani and Koch, 1998) when the stimulus cutoff
frequency was low (<20 Hz). This suggests that a rate code might be
more relevant for the encoding of low-frequency stimuli.
However, electrocommunication signals contain much higher frequency
components (Metzner and Heiligenberg, 1991;
Zupanc and Maler, 1993; Dulka et al.,
1995). In fact, the animal can detect AM frequencies >200 Hz
(Bastian et al., 2001). We use information theoretic
measures to assess the quality of encoding time, varying stimuli by
P-units at such frequencies. In particular, we will assess the role of
ISI correlations by comparing the information rates obtained with and
without their presence.
To assess the ability of the P-unit to encode different frequency
stimuli, we used low-pass filtered Gaussian white noise with a variable
cutoff frequency fc (Wessel et al.,
1996). The same stimulus was presented to both models, and the
resulting baseline and noise entropies for word lengths up to 16 were
calculated. Because the unit can fire at most once per EOD cycle, it is
natural to make the bin size equal to one EOD cycle. We
have also plotted results obtained with a binomial process (each bin
has probability p of being assigned the value 1 and
probability 1 p of being assigned the value 0, p being the probability of firing per EOD cycle equal to 0.2 in our case). For such a process, the entropy of words of length
L as a function of L1 is
constant (Shannon, 1948; Cover and Thomas,
1991); this permits us to verify the accuracy of our algorithm.
The calculated entropy for the binomial process was significantly lower
than the true value (Fig. 9) for words of
length >6. This is attributable to the finite length of the spike
train that is considered, which leads to undersampling for words of
longer lengths (Strong et al., 1998). We thus only
considered words up to length 6 (i.e., 1/L goes from 1 to
1/6). We plot the baseline entropies obtained for each model in Figure
9. Because a clear deviation from linearity can be seen in each case,
we performed a quadratic fit according to Equation 2. Note that the
baseline entropy values for our model are always lower than for
Nelson's as correlations reduce entropy (Shannon, 1948;
Cover and Thomas, 1991). However, noise entropies will
also be lower for our model for the same reason. It is thus not clear a
priori what effect correlations will have on information transfer.
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Figure 9.
Baseline entropies of words of length n
calculated from 1000 realizations of duration 10,000 EOD cycles each
for both models. Comparison with a binomial process for which these
entropies should be constant is shown to verify the accuracy of the
results. Finite sampling errors occur for words of length >7 and lead
to results that are lower than the true value (Strong et al.,
1998). We thus take only the results for words up to
n = 6 for the fits. Clear deviation from linearity is
still seen for the first few points in both cases. Entropies are lower
for the LIFDT model due to correlations. Also shown are the best fits
obtained for the data.
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We thus calculated the information rates at different stimulus
contrasts stim for both models. The baseline and noise
entropy rates for different contrasts were calculated at a fixed cutoff frequency fc = 100 Hz to assess the
capacity of the P-unit to encode high-frequency stimuli (as discussed
above). We plot in Figure 10 the
information rate for both models. As expected, it increases with
stimulus contrast for both models; however, the information for the
LIFDT model is always higher than the one obtained for Nelson's model.
For example, for stimulus contrasts between 0.04 and 0.05 mV, the gain
in information rate is approximately 0.04 bits per EOD cycle, which
corresponds to 40 bits per second. For both models, the correlation
between successive spikes in the presence of a stimulus increases with
stim, resulting in a decrease in noise entropy
rate. This leads to an increase in the information rate. However, this
correlation is higher for the LIFDT model because of the dynamic
threshold, which reduces the noise entropy even more and leads to a
higher information rate.
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Figure 10.
Information rate as a function of stimulus
contrast for both models. The stimulus is low-pass filtered Gaussian
white noise with cutoff frequency 100 Hz and has mean of zero and SD
stim. Information rate increases with contrast for both
models, but it is higher in the LIFDT model than in the Nelson
model.
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We now look at the dependence of information rate on stimulus cutoff
frequency fc. We thus construct an
"information tuning curve" for the two models (Fig.
11a), i.e., the dependence
of information rate on stimulus cutoff frequency. For both models,
information rates are small at low cutoff frequencies and increase for
higher cutoff frequencies, such as those near the mean baseline firing rate of the neuron (200 Hz in our case). The general shape is attributable to the fact that the noise entropy decreases as a function
of fc (over the range of interest). This
decrease may be attributable in part to the high-pass filter
characteristics of both models (Eqs. 3-5) over the range of
frequencies considered. A more complete analysis of this result will be
presented elsewhere.
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Figure 11.
a, Mutual information rate as a
function of stimulus cutoff frequency. The stimulus contrast was
stim = 0.03 mV. The information rate increases with
frequency for both models. However, information rate is higher for the
LIFDT model than for the Nelson model. b, Difference between
the information rate for the LIFDT model and the Nelson model
I as a function of fc. This
quantity displays a resonance for frequencies on the order of the
inverse of the decay time constant of the dynamic threshold
.
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Information rates were higher for the LIFDT model, compared with the
Nelson model over the full range considered. A surprising result is the
fact that the gain in information caused by ISI correlations exhibits a
maximum at frequencies ~100-150 Hz (Fig. 11b). Hence, our
dynamic threshold enhances the information tuning curve, and this
effect is maximal for stimuli with a cutoff frequency fc of the order of the inverse of the decay time
constant of the dynamic threshold . This effect can
be understood intuitively as follows: cumulative relative
refractoriness leads to a less variable neural response to repeated
stimuli; if the unit fires on the rising phase of the stimulus, then it
has time to recover and fire on the next rise of the stimulus, which
occurs on average at least after one correlation time
f . One thus expects this effect to be
maximal when f is on the order of the
time constant of the cumulative relative refractoriness (modeled here
by our dynamic threshold). If fc is too low,
then the dynamic threshold does not enhance information transmission,
because the stimulus dynamics occur on a much longer time scale. Hence
the information rates obtained with both models are approximately the
same (Fig. 11a, less than fc = 50 Hz). For fc high, the dynamic threshold
cannot follow the fast stimulus variations, and the two information
rates are again the same. There is thus a resonance in the gain in
information rate at ~100 Hz because of cumulative relative
refractoriness. The maximum gain is ~0.03 bits per EOD cycle, which
corresponds to 30 bits per second.
Behavioral experiments have demonstrated that the animal can reliably
detect AMs with 100-200 Hz frequencies and that this might be relevant
for courtship behavior (Bastian et al., 2001). We
hypothesize that the resonance in the P-unit information tuning curve
caused by our dynamic threshold may be responsible for this observed sensitivity.
Note that if the stimulus correlation time
f is high, then one might expect that
the information rate over an EOD cycle would be lower than the one
obtained if f was low. This is
attributable to oversampling. Note also that for a stimulus cutoff
frequency fc, the electrosensory system might
integrate the input using a time window proportional to f. These possible effects can be
eliminated by dividing the information rate by fc. This results in an estimate
(Ic) of the average information transmitted during a time window equal to the correlation time f of the stimulus. This quantity is
shown to increase as a function of fc for both models (Fig. 12a) but
saturates for high fc. Thus, more information is
transmitted about high-frequency stimuli as compared to low ones, even
when the integration times are normalized. This occurs because the
increase in Ic is limited by the sampling rate
allowed by the baseline firing rate of the P-unit (Nyquist theorem).
However, Ic is still higher for the LIFDT model.
The gain also has a resonance with the maximum ~100 Hz (Fig.
12b).
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Figure 12.
Average information transmitted
Ic during a time window whose length is equal to
the stimulus correlation time f as a
function of cutoff frequency fc. This quantity
increases with fc (see text for
explanation).
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Effect of blurring on information rate
Finally, we show that the importance of spike timing increases
with stimulus cutoff frequency. We introduce "blurring" in the
spike train by making the bin width greater and studying the
incurred loss in potential information for different cutoff frequencies
of the stimulus. We see that this loss is greater for higher cutoff
frequencies (Fig. 13). This suggests
that spike timing has minimal contributions in encoding information
about slowly varying stimuli. Furthermore, this finding agrees with the
fact that significant jitter does not affect the quality of the
reconstructed stimulus for low stimulus cutoff frequencies (Kreiman et al., 2000). However, spike timing is
important for high-frequency time-varying stimuli, and our results show
in fact that electroreceptors can encode stimuli that vary on time
scales at least as fast as 5-10 EOD cycles. This agrees with the
results from a previous study that found that spike timing jitter in
electroreceptors was on the order of one to two EOD cycles
(Kreiman et al., 2000). This also justifies our use of
PNDs for low-frequency stimuli.
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Figure 13.
Information loss caused by blurring the spike
train. We counted the number of spikes that occurred in successive time
intervals of six EOD cycles in length and computed the information rate
for that spike train. We can see that fractional loss of information
increases with fc.
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Contrasting signal detection and information theory
As the fish swims by a prey, it will experience a small change in
transdermal potential in a time window of ~200 EOD cycles (Nelson and MacIver, 1999). However, information rates
are almost zero for such low cutoff frequencies (this corresponds to
frequencies of ~5 Hz, the information rate is almost zero for
fc < 50 Hz for the particular P-unit cell
modeled here) (Fig. 11a). The fact that the animal can
readily detect these signals (Nelson and MacIver, 1999)
suggests that the mutual information rate that was calculated using
baseline entropy is poorly suited for coding of these low-frequency stimuli. This is also the case after the correlation time of the stimulus has been taken into account (Fig. 12a). However, we
have shown that measures based on signal detection theory were adept at
quantifying the ability of the electroreceptor to transmit information
about this type of stimuli (Figs. 7, 8). It is thus more natural to
analyse slow time-varying stimuli by looking at the trial-to-trial
variability of the PND.
We use a weak, slow time-varying stimulus (stim = 0.01 mV, fc = 1.96 Hz) and look at the PND
in a time window of 255 EOD cycles. The portion of stimulus used is
shown in Figure 14a. The PNDs obtained with both models are shown in Figure 14b. Note
that for each model, the PNDs with and without stimulus have almost the
same variances. Thus, the main factor for discriminating between the
two distributions is the difference in their means. This is not
captured by entropy measures because they do not depend on the mean of
the distribution used (Shannon, 1948).
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Figure 14.
a, A realization of low-pass filtered
Gaussian white noise with cutoff frequency fc = 1.96 Hz and contrast stim = 0.01 mV. The
counting time window is between the horizontal lines (from 500 to 755 EOD cycles). b, PND obtained with this portion of stimulus
as compared with the one obtained from baseline dynamics. The variances
of the PNDs are nearly identical and low, whereas the means are very
different, which leads to a good discriminability d. The
difference between the means is not captured by entropy measures,
because they do not depend on the mean and also assume infinite spike
trains. Because there is no averaging over the stimulus in this case,
signal detection yields better results than information theory.
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However, as shown previously (Fig. 7), the variance of the PNDs that
were obtained with the LIFDT model are smaller than those obtained with
the Nelson model. Note that the means are separated by as few as three
spikes in both cases (as low as 6% difference). The lesser overlap for
the LIFDT model results in a dramatic improvement in discriminability.
For example, the probability of obtaining 53 spikes for the LIFDT model
is 0.3 with stimulus and 0.05 without stimulus, corresponding to a
ratio of 6. For the Nelson model, this ratio is 2.5. Our results thus
show that a single P-unit could discriminate as few as two extra spikes
in a time window of 255 EOD cycles as already suggested by
Ratnam and Nelson (2000). Their first-order Markov
process was not able to reproduce the experimentally observed
probabilities of correct detection. This is because the Fano factor for
their model was higher than the one for the experimental data. The key
factor in this remarkable sensitivity is the negative ISI correlations
that were observed experimentally and that result from cumulative
relative refractoriness exhibited by our simple model with dynamic threshold.
| |
DISCUSSION |
Using two models, one with baseline ISI correlations and one
without, we have shown that negative ISI correlations that are present
experimentally play an important role in the animal's ability to
detect both slowly and rapidly time-varying stimuli. A first analysis
based on signal detection theory, which assumes rate coding, revealed
that the ISI correlations dramatically enhanced the detectability of
low-frequency weak signals. We then used information theory to quantify
the ability of the afferents to encode time-varying stimuli with
various cutoff frequencies. We found that ISI correlations also helped
in this case and that the effect was maximal for cutoff frequencies on
the order of the inverse of the decay time constant of our dynamic
threshold that was used to include cumulative relative refractoriness.
By comparing both approaches, our study suggests that a rate code can
be assumed for low-frequency stimuli, whereas spike timing is important
for high-frequency signals.
Comparison of models
Our previous simple model (Chacron et al., 2000)
reproduced baseline first- and second-order ISI statistics that were
seen experimentally, such as the ISI correlations. Here, this model was
extended to include experimentally measured linear response properties
to sinusoidal AMs by Nelson et al. (1997). To quantify the effects of correlations, we used a second model proposed by Nelson et al. (1997) that gave identical first-order
statistics and responses to sinusoidal AMs, except that there were no
ISI correlations.
Spike train variability and signal detection
Spike train variability as measured by the Fano factor was
computed for both models at various time scales. Although able to
reproduce first-order ISI statistics as well as responses to AMs, the
Nelson model did not reproduce experimentally observed spike train
variability as measured by the Fano factor at counting times >10 EOD
cycles (Fig. 4) (Ratnam and Nelson, 2000). The
asymptotic value of the Fano factor for the Nelson model was shown to
equal the square of the coefficient of variation of the ISIH as
expected from Equation 1 in the absence of ISI correlations. The Fano
factor obtained from our LIFDT model also tends toward this value when the ISI sequence is randomly shuffled to eliminate ISI correlations, thus proving that the Nelson model did not display any significant ISI correlations.
The negative ISI correlations obtained with our simple LIFDT model
bring the Fano factor down (Cox and Lewis, 1966) (Eq. 1) to experimentally determined values for longer counting times; a
discrepancy was, however, observed for counting times >150 EOD cycles
as the Fano factor decreased monotonically instead of increasing. To
get this increase in variability at long counting times, it was
necessary to add a weak noise with a long correlation time to the model
dynamics. This noise had no effect at low counting times, as explained
in "Fano factor" and in Appendix C. However, it led to positive ISI
correlations extending to long lags as seen by the increase of the
spectral density function at low frequencies (Fig. 6). The positive ISI
correlations were weak, decayed exponentially up to long lags, and
summed up according to Equation 1 to give the increase in the Fano
factor. This increase has been observed in many neurons and may be of
synaptic origin or attributable to drifts in EOD amplitude or
frequency. These ISI correlations make modeling of electroreceptors by
Markov chains of ISIs difficult because ISI correlations can extend out
to lags in the thousands. However, the addition of an extra noise term in our simple model gave quantitatively accurate results and could be
used to incorporate these effects into other neuron models.
We used the ideal observer paradigm (Green and Swets,
1966) to optimally discriminate between pulse number
distributions with and without stimulus (Gabbiani and Koch,
1998). It was shown that negative ISI correlations
reduced the variance of these pulse number distributions without
changing their means significantly, hence increasing their
discriminability. This led to an improvement in the receiver operating
characteristic curve that was maximal at counting times at which the
Fano factor was minimal. A criterion for near perfect discrimination
was derived, and it was shown that signals leading to a 6.5% increase
in firing rate would be discriminated in this manner on the basis of
steady-state dynamics alone. Transients caused by the filter further
lowered this detection threshold by increasing or decreasing the firing
rate computed over the duration of the stimulus, thus leading to a
further increase in discriminability. Signal detection analysis is not
appropriate for zero mean high-frequency signals (e.g., beat
frequencies generated by fish with high EOD frequency differences)
because spike train variability is high at low counting times
(Softky and Koch, 1993; Borst and
Theunissen, 1999). It is, however, appropriate for
low-frequency signals such as those caused by prey.
Using information theory to quantify the coding of
time-varying stimuli
The trial-to-trial variability of the neural response to a
repeated stimulus was characterized by the noise entropy rate. The
entropy rate of the baseline spike train was estimated, and the
difference between baseline and noise entropy rates was used to
quantify the information transfer rate about the stimulus. This
definition is natural in our case because we have baseline firing
activity and signal detection must be based on a change from this
baseline activity (Ratnam and Nelson, 2000). We compared the information rates obtained with our model and Nelson's. It was
shown that the information rate increased in both cases with stimulus
contrast. This agrees with the fact that the electroreceptor is better
at encoding stronger stimuli (Wessel et al., 1996). However, negative ISI correlations could help the electroreceptor in
the coding of fast time-varying stimuli because information rates
computed with our model were higher. This is caused by the fact that
correlations reduce the noise entropy even more than the baseline
entropy. We note that our results are consistent with previous studies
that have demonstrated that a refractory period can improve the linear
correlation between the stimulus and the instantaneous firing rate
(Chialvo et al., 1997) as well as the neural precision
(and hence the mutual information) (Berry and Meister,
1998). However, the models did not incorporate ISI correlations. Furthermore, in Berry and Meister (1998),
the increase in refractory period led to a decrease in mean firing
rate, and the causes of the increase in mutual information were not
clear. In our analysis, we examined the effect of ISI correlations on mutual information and signal detection without concomitant changes in
the mean firing rate.
Information rates as a function of stimulus cutoff frequency were also
computed. Our results show that the information rate increased with
cutoff frequency as expected from the high-pass characteristics of
P-afferents (Xu et al., 1996). Information rates were
higher for the LIFDT model, and the gain attributable to correlations
exhibited a resonance with a maximum of ~100 Hz. This frequency
corresponds to the inverse of the decay time constant of our dynamic
threshold. Remarkably, there is evidence that these fish respond
preferentially to electrocommunication signals with a frequency content
of ~100 Hz (Bastian et al., 2001).
Comparing signal detection and information theory
Finally, we have shown that the loss of information about the
stimulus incurred by decreasing the spike-timing resolution increases
with cutoff frequency (Fig. 13). This suggests that spike timing is not
as important for low-frequency stimuli as it is for high-frequency
stimuli. However, signal detection theory assumes a rate code and can
be applied in cases in which spike timing is not important. Our results
show that an optimal detector receiving a single P-unit spike train
with baseline negative ISI correlations can detect the presence of
stimuli that would give rise to as few as two extra spikes over a
counting time of ~250 EOD cycles, as suggested by Ratnam and
Nelson (2000). For low-frequency stimuli, signal detection
theory is appropriate because the change in mean firing rate (computed
over an appropriate time window) without a concomitant change in firing
rate variance can signal the presence or absence of the stimulus.
Without a change in variance, there will be no change in entropy
(because the entropy of a random variable does not depend on its mean)
(Shannon, 1948) and thus almost no mutual information as
per our measure. For zero-mean high-frequency stimuli, there may be
almost no change in mean firing rate computed over a long time window;
hence, signal detection theory is not appropriate. However, because
baseline entropy is high (this is caused by the high variability at
short counting times) and because ISI correlations reduce the noise
entropy, the mutual information will be high. The ideas presented above are compatible with a recent analysis by Salinas and Sejnowksi (2000) in which neurons can be driven either by the mean
excitatory level (mean firing rate as assessed by the PND in our case)
or by fluctuations around this mean (as assessed by mutual information in our case). Hence, the high variability of P-afferent spike trains
observed for short counting times gives a high mutual information when
looking at high-frequency stimuli, whereas the low variability at
longer counting times caused by negative ISI correlations at short lags
improves signal detectability at low frequencies.
Conclusion and outlook
We have shown that negative ISI correlations that are seen
experimentally can improve the ability of a neuron to code both slow
and fast time-varying stimuli. The P-units we have studied are known to
converge onto basilar pyramidal cells of the electrosensory lateral
line lobe (ELL) (Bastian, 1981). Population averaging is
thus expected and might explain the extreme behavioral sensitivity to
AMs, down to 0.1% of baseline EOD (Knudsen,
1974; Nelson and MacIver, 1999). Also, as
mentioned before, the position of the minimum of the Fano factor is
highly variable (40-1000 EOD cycles). Moreover, it is possible that
different P-units (probability of firing per EOD cycle ranges from 0.1 to 0.5) (Nelson et al., 1997) will have different
cumulative relative refractoriness decay time constants. It will thus
be very interesting to study ELL decoding of slowly versus rapidly
time-varying input processed by this heterogeneous P-unit population.
| |
FOOTNOTES |
Received Jan. 16, 2001; revised April 25, 2001; accepted May 1, 2001.
This research was supported by the Natural Sciences and Engineering
Research Council of Canada (M.J.C. and A.L.) and Canadian Institutes of
Health Research (L.M.) Canada. We thank B. Doiron, D. Bueti, and J. Bastian for useful discussions. We give special thanks to P. Lánský, J. Lewis, and A. M. Oswald for their careful reading of this manuscript as well as useful discussions. Finally, we
thank two anonymous reviewers for their helpful comments.
Correspondence should be addressed to Maurice J. Chacron, Physics
Department, 150 Louis-Pasteur, Ottawa, Ontario, Canada K1N-6N5. E-mail: mchacron{at}physics.uottawa.ca.
| |
APPENDIX A |
In this appendix, we give a list of all the symbols and acronyms
used in this paper. When a symbol is constant throughout the paper
(e.g., model parameter), its value is given. Units are only given for
symbols.
| Symbol/acronym |
Description |
Value/units
|
| |
Constant |
1
(spikes/EOD cycle)1 |
| |
Bin width |
1 EOD
cycle |
| |
Constant by which threshold is
incremented |
0.05 |
| |
Constant |
0.3266 (mV)1
|
| 1 |
Uncorrelated Gaussian random
variable |
Dimensionless |
| 2 |
Uncorrelated
Gaussian random variable |
Dimensionless
|
| 1 |
Ornstein-Uhlenbeck process |
Dimensionless
|
| 2 |
Ornstein-Uhlenbeck process |
Dimensionless
|
| µ |
Mean of the spike count |
Spikes
|
| j |
Serial correlation coefficient at lag
j |
Dimensionless |
| |
SD of the spike count |
Spikes
|
| stim |
Stimulus contrast |
mV |
| |
Threshold
for the LIFDT model |
Dimensionless (see text)
|
| 0 |
Equilibrium value of the threshold |
0.03
|
| 1 |
Time constant of 1 |
0.025 EOD
cycles |
| 2 |
Time constant of
2 |
50,000 EOD cycles
|
| a |
Time constant in Equation 3 |
0.0026 sec |
| b |
Time constant in
Equation 4 |
0.21 sec |
| v |
Voltage decay
time constant |
1 EOD cycle |
| |
Threshold decay
time constant |
7.75 EOD cycles |
| |
Heaviside
function |
Dimensionless |
| A(t) |
Amplitude modulation
of the EOD |
mV |
| A0 |
Baseline EOD
amplitude |
0.8 mV |
| AM |
Amplitude modulation (of the EOD)
|
| c |
Constant |
0.84
|
| C1 |
Fitting constant |
Dimensionless
|
| C2 |
Fitting constant |
Dimensionless
|
| CV |
Coefficient of variation |
Dimensionless
|
| d |
Discriminability |
Dimensionless
|
| dcrit |
Discriminability threshold |
3
|
| D1 |
Intensity of 1 |
8
(EOD cycles)2 |
| D2 |
Intensity of
2 |
0 or 9 × 106
(EOD |
|
|
cycles)2
|
| dt |
Integration time step |
0.0025 EOD cycles
|
| EOD |
Electric organ discharge |
| F |
Fano
factor |
Spikes |
| f |
Frequency |
Hz
|
| fc |
Stimulus cutoff frequency |
Hz
|
| fEOD |
EOD frequency |
1000 Hz
|
| Ga |
Gain term in Equation 3 |
14,100
spikes/sec/mV |
| Gb |
Gain term in Equation 4 |
470 spikes/sec/mV |
| Gc |
Gain term in
Equation 5 |
670 spikes/sec/mV |
| H |
Entropy or entropy
rate |
Bits or bits/EOD cycle |
| I |
Mutual
information or mutual information rate |
Bits or bits/EOD cycle
|
| Ic |
Average information transmitted during
the correlation |
Bits |
|
time of the
stimulus |
|
| Ii |
ith ISI |
EOD cycles
|
| Isyn |
Synaptic current in LIFDT
model |
Dimensionless (see text) |
| I |
Mean of the
ISI distribution |
EOD cycles |
| ISI |
Interspike interval
|
| ISIH |
Interspike interval histogram |
| L |
Word
length |
Dimensionless |
| LIF |
Leaky integrate-and-fire
|
| LIFDT |
Leaky integrate-and-fire with dynamic threshold
|
| m |
Number of subprocesses used in Nelson's
model |
18 |
| p |
Probability of firing per EOD
cycle |
Dimensionless |
| PD |
Probability of
correct detection |
Dimensionless
|
| PFA |
Probability of false
alarm |
Dimensionless |
| PND |
Pulse number distribution
|
| r |
Time-dependent firing rate in Nelson's model |
Hz
|
| rbase |
Baseline firing rate in Nelson's
model |
200 Hz |
| ROC |
Receiver operating characteristic
|
| s |
Time-varying stimulus |
Dimensionless
|
| SCC |
Serial correlation coefficient |
| SDF |
Spectral density
function |
Dimensionless |
| T |
Counting time window
length |
EOD cycles |
| Tr |
Absolute
refractory period duration in LIFDT |
1 EOD cycle
|
| v |
Membrane voltage in the LIFDT
model |
Dimensionless (see text) |
| VAR |
Variance of the
ISI distribution |
(EOD cycles)2
|
| X |
Filtered stimulus |
Spikes/EOD cycle
|
| Xa |
Filter variable in Equation 3 |
Spikes/EOD cycle |
| Xb |
Filter variable
in Equation 4 |
Spikes/EOD cycle |
| |
APPENDIX B |
In this appendix, we summarize the equations used for both models.
Nelson model
The AM A(t) is filtered using the following set of
equations:
A baseline firing rate rbase is
added to X(t) and Z(t) = X(t) + rbase is rectified according to:
The probability of firing per EOD cycle is p(t) = r(t)/fEOD. Thus, at each maximum of the EOD, the
P-unit has probability p(t) of firing. If there is a firing,
jitter is added to the spike time in the form of Gaussian white noise
of zero mean and SD 0.04 EOD cycles. The ISIH can be made more regular
by simulating m copies of the subprocess and generating
action potentials at the time of occurrence of the mth event
(Nelson et al., 1997).
LIFDT model
In our model, the AM A(t) is filtered using the same
set of equations as for the Nelson model:
The total nondimensionalised synaptic current arriving at the
spike initiation zone is given by:
where and are constants to ensure units are matched,
A0 is the baseline EOD amplitude,
fEOD is the EOD frequency, and is the
Heaviside function used for rectification [(x) = 1
if x 0 and (x) = 0 if
x < 0]. The noise sources 1 and
2 are Ornstein-Uhlenbeck processes given by [see text
for a brief introduction and Gardiner (1985) for further
information]:
The noise sources 1 and 2 are shown
in Figure 2a. For example, the synaptic current for baseline
dynamics is just a rectified sine-wave (i.e., the negative part is set
to zero) perturbed by noise. In the time window between action
potentials and after the absolute refractory period, the voltage
v and the threshold are given by:
A spike is said to have occurred when v = .
Immediately afterward, v is reset to zero, and is
incremented by a constant . The threshold is then maintained
constant for the duration of the absolute refractory period
Tr before decaying exponentially until the next
spike time. A stretch of simulation showing voltage and threshold is
shown in Figure 2.
| |
APPENDIX C |
In this appendix, we give an intuitive explanation of why adding a
slow noise to our model will give rise to positive ISI correlations.
We first recall that the Ornstein-Uhlenbeck process 2
varies on a time scale much greater than either the voltage or
threshold variables in the LIFDT model because 2 = 50,000 EOD cycles is much greater than an EOD cycle. Thus, we can treat
2 as a quasistatic variable with respect to voltage and
threshold because it is almost constant when considering time scales
much smaller than 2. Because the mean ISI is five EOD
cycles long, many ISIs (10,000 on average) will have occurred during a
time window of length 2. Let us imagine that
2 has some value; then 2 will have that
value (approximately) for a long time. If the value is positive, then
the synaptic current Isyn is bigger than it
would be if 2 = 0 and we thus expect a long
sequence of ISIs of shorter duration (but not much shorter because
2 is weak) than on average (when 2 = 0). When 2 has a negative value, one can expect
sequences of ISIs longer than average by the same argument. Thus, we
will get long sequences of ISIs that are shorter than average and long
sequences of ISIs that are longer than average. It is these long
sequences that will lead to positive SCCs (the SCC at a given lag is
positive only when two ISIs separated by this lag are both shorter or
longer than average). Because those sequences can be very long, the ISI correlations extend to long lags. As mentioned in the text, a full
mathematical explanation will be presented elsewhere.
| |
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ABSTRACT
INTRODUCTION
