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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:
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(6)
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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. |
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ABSTRACT
INTRODUCTION
