SERIAL DEPENDENCY IN NEURAL POINT PROCESSES
DUE TO CUMULATIVE AFTERHYPERPOLARIZATION
Charles E. Smith*# and Chao-lung Chen*
* Department of Statistics
# Biomathematics Graduate Program
North Carolina State University
Raleigh, North Carolina 27695-8203
Running head: Serial Dependency due to CAHP
Correspondence to:
Dr. Charles E. Smith
Department of Statistics
Biomathematics Graduate Program
Campus Box 8203
North Carolina State University
Raleigh, North Carolina 27695-8203
Acknowledgement: This research was supported by the Office of Naval
Research under contract N00014-85~K-0105.
ABSTRACT:
The effects of cumulative vs noncumulative afterhyperpolarization
(AHP) are examined through simulations of a stochastic neural model
(Smith and Goldberg, 1986). The afterhyperpolarization in the model is
due to a timevarying potassium conductance. Cumulative effects result
from summing the residual activity of the potassium conductance in the
preceding interspike interval. The variablity in the model is due to
random quantal transmitter release. The statistical properties of the
steady state discharge patterns that are independent of the serial
ordering of interspike intervals show only slight differences between
the two cases. Sever~forder dependent statistical measures are used
to show that a negative serial dependency results from cumulative AHP
at moderate to high discharge rates. The discussion considers the
robustness of the model and its relation to a generalized Poisson
process description of spike trains. Possible applications of the
results to neurons in the auditory and vestibular systems are also
examined.
1
1. INTRODUCTION.
A stochastic extension of Kernell's (1968,1972) spike
initiation model with cumulative afterhyperpolarization (CAHP) has
recently been developed to account for interspike-interval variability
in vestibular afferents (Smith and Goldberg, 1986). Simulations of the
model indicated that the postspike recovery of the sensory axons was
more important than synaptic noise in determining variations in
discharge regularity across the vestibular afferent population. The
model reproduced many features of the steady-state discharge of
peripheral vestibular afferents in response to natural and galvanic
stimulation, provided discharge rates were higher than 40 spikes/sec
/
(Goldberg, Smith and Fernandez, 1984). The model assumed that the AHP
was the result of an increased potassium conductance (gK) that was
time-dependent but not voltage dependent. It further assumed a
cumulati ve summation of AHP's, i. e., a definite proportion (p) of the
gK left over from the preceding activity was added to the gK triggered
by each spike. This is known as cumulative afterhyperpolarization. By
analogy with other repetitively discharging neurons, a value of p
=1
was used for the simulations. Some support for this choice of the
value of p came from the agreement of simulation and experimental
results on interposed short-shocks in regularly discharging afferents.
For irregularly discharging afferents, the interposed shock procedure
required many more stimulus repetitions than were practically feasible.
The present study examines the role of cumulative (p
= 1)
vs
noncumulative (p = 0) AHP in the above spike initiation model. The
goal is to distinguish between the two cases, p
=0
and p
= 1,
in
terms of first and second order statistics of the interspike
intervals. Furthermore, the procedure should be applicable to both
2
regularly and irregularly discharging neurons.
The problem can be stated in a different way as follows: If
the spike train is regarded as a stationary point process (Perkel
et al., 1967), then the p = 0 case produces a renewal process, the p =
1 case does not. The goal is then to characterize the serial
dependence between intervals that is characteristic of a cumulative
AHP.
The motivation for the study comes from several sources. AHP's
have been reported in many types of neurons (Crill and Schwindt, 1983)
and recently have been implicated in the biophysical basis of certain
types of associative learning (Alkon, 1984; Alkon et al., 1985). An
AHP can be viewed as a "recovery process" following an action
potential. Recent point process descriptions of discharge activity in
the peripheral (Johnson and Swami, 1983; Gaumond, Kim, Molnar, 1983;
Miller, 1985; Jones et al., 1985; Lutkenhoner and Smith, 1986) and
central (Tsuchitani and Johnson, 1985; Johnson et a!., 1986) aUditory
systems also use a "recovery process" in the specification of the
intensity of the neural point process. In at least some cases (e.g.,
Tsuchitani and Johnson, 1985; Johnson et al., 1986) the observed
serial dependency between intervals and its dependence on firing rate
parallel the results presented here for the stochastic AHP model.
Finally there was the original motivation of finding an alternative to
the interposed shock procedure for irregularly discharging vestibular
afferents.
The model and its assumptions are first reviewed, including a
brief tutorial illustration of the important parameters in the model.
A description of the methods specific to this stUdy then follow.
3
2. The Model and Methods.
2.1 Assumptions.
The main assumptions of the model are: (1) There is a single
spike initiation site or trigger zone. A spike or action potential
occurs whenever the site's transmembrane voltage crosses a fixed
threshold, VT• (2) The postsynaptic depolarization at the trigger site
is the result of excitatory synaptic inputs (in vestibular afferents
from the hair cell or cells). The discrete nature and random timing of
the quantal EPSP's lead to synaptic noise. Consequently the postspike
vol tage trajectories will fluctuate from one interspike interval to
the next. These fluctuations are assumed to be the only source of
interval variability.
(3)
The quantal EPSP's sum nonlinearly due to a
synaptic reversal potential, VS' (4) A spike at time t = 0 produces an
increase in the potassium conductance (gK) and a consequent AHP. This
conductance declines exponentially as a function of time with time
constant,
T
K"
(5) External (perilymphatic) currents influence
discharge by acting on the trigger site rather than elsewhere in the
stimulus transduction pathway, e.g., in the peripheral vestibular
system: the hair cells, endorgan mechanics or axon far distant from
the trigger site. (6) The membrane time constant is small compared to
the values of
T.
K
and of the duration of quantal events. Discussion of
the applicability of these assumptions in the peripheral vestibular
system has been presented elsewhere (Goldberg et al., 1984; Smith and
Goldberg, 1986). The model can be modified to account for
nonnegligible values of the membrane time constant and for different
external current effects, but they are not considered here.
4
2.2 The mode 1.
The model can be illustrated in terms of a simple lumped
electrical circuit (see Figure 1 in Smith and Goldberg, 1986)
consisting of three conductances, with an associated equilbrium
potential, in parallel with a membrane capacity (em) and an external
current source (I p ). The conductances are a leak conductance (GL)' an
excitatory synaptic conductance (G S )' and a time varying potassium
conductance (G K). GS is proportional to the input that the sensory
axon receives from hair cell or presynaptic neuron. GK is responsible
for an AHP. Membrane potentials are measured from the resting
potential and expressed as intracellular minus extracellular
potential. Equilibrium potentials were assumed to be VL
VK = -30 mV.
Normali~ing
= 0,
Vs
= 70,
all conductances to the leak conductance GL
and denoting them by the corresponding lower case symbols, the circuit
equation is
wi th
T
M = Cm/GL and VP = Ip/G L•
Firing occurs whenever Vet) reaches a fixed threshold VT• Since
gs is a random process, so is Vet). Firing times correspond to first
passage times for vet) to reach the absorbing barrier VT•
The conductance gs is assumed to be a homogeneous shot-noise
process composed of quantal events, rectangular in shape with
amp 11 tude II gs and duration lItS. This shot noise process wou 1 d resu 1 t
from passing a homogeneous Poisson impulse train with intensity A
(times of transmitter release) through a linear system with a finite
impu Ise response of the abo ve shape (opening of conductance channe Is).
5
The mean value, gs ' is
(2a)
The quantal EPSP size, (A), measured at rest, is
A=
(2b)
The time dependence of the potassium conductance following an
isolated spike (i.e., no preceding spike) is assumed to be
where t is the postspike time, gKO = gK(t = 0) and, K is a time
constant.
Cumu lati ve summation of AHP's
was represented as ~follows: a
fixed proportion (p) of the gK left over from the preceding activity
was added to the gK triggered by each spike. If the (i)th interspike
interval is t i , the activity until the (i+1)th spike has a gK given by
gK(t)
=
[gKO + P gK(ti)] exp(-t/, K)
= g*(t i ) exp(-tl
<3b)
'K)
that is, a higher effective gKO' namely g *• For p
~
0, the term g *(t i )
is a random variable due to its dependence on the random variable, t i •
However, in the times between interspike intervals, g*(t i ) is a
constant, i. e. a particular value of the random variable. Said
another way, the dependence of the initial value on the length of the
previous interval will make the firing times a 1-memory point process.
As in the earlier simulation study (Smith and Goldberg, 1986),
the membrane time constant, 'M' was set to zero in (1). The
expression for V(t) now becomes
6
( 4)
This simplification should correspond to neurons where
much shorter than the values of
evidence that
T
M
T
K and
~S'
T
M
has a value
In light of experimental
< 0.1 msec in vestibular afferents (Schessel, 1982)
and in other vertebrate axons (Tasaki, 1955), this may not be an
unreasonable assumption.
Equation (4) was simulated in VS Fortran on an IBM 3081. The
GGPOS subroutine of the International Mathematics and Statistical
Library (1980) was used to calculate the number of quantal events in
discrete time steps of 0.1 msec. VT was always set to 10 mV and
to
~ts
0.5 msec. The simulation always started with an isolated spike and
after discarding the next 10 interspike intervals, 500 or 2000
additional interspike intervals were simulated. If
T
M
is not set to
zero, then (1) would be the basis for the simulation and an order of
magnitude smaller time steps would be required.
2.3 Simulation parameters.
Four parameters determine the properties of the model neuron.
Three of these (gKO'
' p) specify the AHP. The fourth is the
K
quantal EPSP size (A). All four parameters remain fixed for a gi ven
T
model neuron. The discharge rate of the neuron is modulated by natural
and electrical stimulation, which were mimicked by varying gs (via
A)
and Vp ' respectively. The role of the AHP parameter, p, is the object
of the present study.
Table 1 gi ves the values of gKO'
T
K and A for our three model
neurons, or units, denoted as units 2, 3, and 5. For each unit the two
7
cases, p = 0 and p = 1, were examined. The p = 1 case corresponds to
the earlier simulation study (Smith and Goldberg, 1986). Units 2 and 3
represent regularly discharging neurons and unit 5 an irregularly
discharging neuron. We will mainly be concerned with units 2 and 5.
Unit 2, compared to unit 5, has smaller quantal EPSP's and a larger,
slower gK. The measure of discharge regularity was taken to be the
coefficient of variation
appropriate to a mean interval of 15 msec
and is denoted as CV* (Goldberg et al., 1984).
2.4 Illustration of the model.
A few pictorial examples will illustrate the working of the model
and the cumulative summation of AHP's. First (figure 1) consider what
happens if gs is not a shot-noise process but has a constant value and
for gK the parameter p equals 1. The three plots show corresponding
temporal events in unit 2 following an isolated spike for the membrane
voltage V(t) in (A), gs in (B), and gK in (C). After the first spike
(arbitrari ly drawn to be 0.5 msec), the initial value of gK is larger
than its previous initial value (horizontial reference line in (C)) by
an amount equal to the value of gK at the time of the first spike. The
result via equation (4) is a lower (more hyperpolarized) starting
value for the membrane voltage trajectory. Since it starts at a lower
value it takes longer for the membrane voltage to reach threshold and
the second interspike interval is slightly longer than the first one,
which had followed
an isolated spike. The progression continues for
the third, fourth and fifth spikes, but the summation of gK and change
in interspike interval are not nearly as dramatic as that following
the isolated spike.
In figure 2 (A,B,C), the corresponding three plots are shown for
8
unit 2 with p = 1 when gs now is a shot noise process with a mean
value equal to the constant value in figure (1B). As expected this
produces some noisiness in the voltage trajectories and hence some
variation (here small) in the interspike times and initial values of
gK. In figure 2 (D,E,F) the corresponding plots are shown with p = O.
gK now resets to the same value following each spike. Even though the
gs process is identical to that in figure (2B), some differences in
the voltage trajectory can be seen, particularly right after each
spike. The higher initial value of the voltage results in shorter
interspike intervals. The value of
A
required to produce this value of
gs was 560/msec. If we continue the progression of spikes unti I there
are say 2010 intervals, and average the last 2000 spikes (since the
process started from an isolated spike) a mean interval of around 10.4
msec resu Its for the p
= 1 case
and around 8.4 msec for the p
=0
case.
For unit 5, a somewhat different picture develops for the
conditions corresponding to a mean interval of around 10 msec as shown
in
figure 3. As in the previous figure (A,B,C) are for p
= 1 and
(D,E,F) are for p = 0, and the shot noise process sample paths in (B)
and (E) are identical. First note that the mean value of gs is smaller
but the process has a larger variance, recall it has a larger quantal
EPSP size. The required release rate
A
is also much smaller 14.6/msec
as compared with 560/msec for unit 2. The gK process is smaller in
magnitude and decays faster due to a smaller gKO and TKo The voltage
trajectory "recovers faster" if you like. The asymptotic value of the
mean voltage trajectory is below the voltage threshold. The only way
the neuron can fire is from fluctuations about its mean, while in unit
9
2 the mean trajectory crossed the threshold and the shot-noise induced
variation was a small perturbation about the mean behaviour. These two
situations will be termed nondeterministic and deterministic crossings
respectively. The other point to note from figure 3 is that the
differences between the p = 0 and p = 1 cases are now much smaller due
to the smaller value of gK (for 2000 intervals the corresponding mean
intervals differ by less than 1%). At higher values of A some
differences do occur as will be seen later.
In summary, at least at a mean interval of around 10 msec, the
chief difference between units 2 and 5 was the deterministic vs
nondeterministic firing. The effect of cumulative vs noncumulative AHP
seemed IOOre pronounced in the regular unit, unit 2. A.quantification
of these differences in terms of statistics of .the first (marginal)
and second (bivariate with consecutive intervals) order distributions
of the interspike intervals is presented in the results section.
2.5 Measures of serial dependence.
The methods specific to this paper, not used earlier (Smith and
Goldberg, 1986), primarily concern statistical measures of serial
dependence of the interspike intervals, which were examined in units 2
and 5. The first three serial correlation coefficients, denoted as
rho1, rho2 and rho3 were calculated using the BEICOR subroutine of
IMSL (1980).
The conditional mean and conditional median plots were calculated
from a first order joint interval histogram of 2000 simulated
intervals. For unit 5, the conditioning intervals were binned as
successive 5 percentile points (to within ties) of their marginal
distributions. This binning method was also used in unit 2 at moderate
10
firing rates (firing rate is taken to be the reciprocal of the mean
interspike interval). At higher rates, a fixed binwidth of 0.1 or 0.2
msec was used. Least-squares linear fits for the conditional mean and
median plots were calculated using the GLM procedure of SASe
Conditional median lines were unweighted. Conditional mean lines were
weighted by the reciprocal of the standard deviation of the
conditioned intervals for the percentile binning cases and by the
number of conditioned intervals for the fixed binwidth cases.
11
3. RESULTS
Two types of statistical measures were computed for the simulated
intervals: those that don't depend on the serial order of the
intervals, denoted as order-independent measures and computed from the
first order (marginal) distributions, and those that do depend on the
serial order, denoted as order-dependent measures and computed from
bivariate distributions of successive intervals (Perkel et al., 1967).
For each type of measure we wi 11 see if the cases p
= 1 and
p
=0
can be distinguished for an irregular neuron, unit 5, and a regular
neuron, unit 2. Unit 3, also a regular unit, is used to further
illustrate some of the order-independent results. We begin by looking
at some order independent measures of the spike train that parallel
the figures in the initial study (Smith and Goldberg, 1986).
3.1 Order-independent measures.
In figure 4, the coefficient of variation,
cv,
is plotted against
the mean interspike interval, MI, in msec on a log-log scale for units
2, 3 and 5. The solid lines correspond to p = 1 and the open symbols
to p = 0 with the unit number coded on the graph. The lower set
(connected curve and symbols) is the most regular unit, unit 2; unit 3
is next; and uni t 5 is the upper set. Points for uni ts 2 and 5
correspond to 2000 simulated intervals and 500 intervals for unit 3.
For a given unit, different points reflect different values of the
release rate A and hence SS. Higher values of A produce smaller mean
intervals. This type of plot would result from applying a stimulus at
a series of amplitudes or strengths.
The effect of changing p from 1 to 0 is to increase the CV at a
given mean interval up until the mean interval is around 2.5 to 3
12
• Since T decreases as the unit number increases, the effect
K
K
diminishes as the unit becomes more irregular. This small but
T
consistent effect is somewhat expected since the p
=1
case produces a
higher effective gKO and can be predicted from a sensitivity analysis
of equation (4). However, given the uncertainty about the true values
of the other parameters, this plot can hardly be used as a diagnostic
for determining the value of p. For unit 3, simulations were also run
for p = 0.3 and 0.6 with resu lts falling between the p = 0 and 1
curves as expected. Recall that our measure of discharge regularity,
CV*, was the CV at a mean interval of 15 msec. The figure also
illustrates that the value of p had a negligible effect on CV*.
Some indication of the effect of p on the shape of the marginal
distributions can be seen in figure 5. For unit 5 (p
=0
as
0 and p
= 1 as X) a measure of skewness, Pearson's square root of (31' is
plotted against CV for the same marginal distributions used for unit 5
in figure 4. The three solid lines on the graph represent the curves
expected from several commonly used two-parameter skewed
distributions: lognormal (upper curve), inverse gaussian (next curve)
and gamma (lower curve) distributions. The simulation data is somewhat
U-shaped and crosses all three curves. A plot of fourth vs third
moment (Pearson's method) also shows that none of the above two
parameter distributions adequately represent all of the simulation
data. Further confirmation came from a goodness of fit test developed
by Dennis Boos (1981) which uses an Anderson-Darling distance to
measure the similarity between the empirical and theoretical
distributions.
In summary the first order distributions, at least as indicated
13
by their first four moments, do not distinguish between the p
=0
and
p = 1 cases in a practically useful way.
The input-output curves for these pairs (figure 6), at first
glance, appear more promising for determining the value of p. In
figure 6A, the firing rate in spikes/sec is plotted against the mean
synaptic input, !s. The three unit pairs are coded as in the first
figure, with the p
=1
cases being solid lines and the p
being symbols. The vertical line at gs
= 1/6
=0
cases
demarcates the
deterministic (to the right) from nondeterministic (to the left)
firings. Unit 5 is fairly linear for both the p
= 1 and
The regular units show a somewhat different behaviour.
= 0 cases.
The p = 1 cases
p
are fairly linear for the deterministic firings, but the p
=0
cases
have a supra linear relationship as has been noticed in deterministic
AHP models, e.g. MacGregor and Oli ver, 1974. Figure 68 shows that a
supra linear relation also occurs at higher firing rates in response to
galvanic currents as illustrated by unit 3. The vertical line
corresponds to a resting (no electrical stimulation) discharge rate of
100 spikes/sec. Cathodal currents, to the right, increase the firing
rate above the resting level, while anodal currents decrease the
firing rate.
The problem with using the nonlinearity in (6A) to distinguish
between p
=1
and p
=0
is that the mean synaptic input gs is not
measured directly. There is some evidence (Hudspeth and Corey, 1977)
that the input-output curve is sigmoidal in the preceding stage (the
hair cell) of the transduction scheme for vestibular afferents. This
would serve to linearize to some degree the nonlinear input-output
relation seen above. Including an absolute refractory period (dead
time) in the model would also diminish the nonlinear firing rate
14
dependence.
The problem with using the electrical stimulation plot is that
the currents must be shown to act solely on the afferent terminal or
axon and in the additive manner indicated by equation (4). There is
some evidence for this site of action of the applied currents for
vestibular afferents (Goldberg et al., 1984). However, even for the
vestibular afferents there are problems. Since the differences seem
restricted to "deterministic firings", they are of no use for
determining p in irregular afferents. Other factors, such as the
geometry of stimulating electrode, saturation of firing rate at high
stimulus levels, may also limit the practical utility of this result.
In any case the order-dependent measures of the next section appear to
be applicable to both regularly and irregularly firing
neurons.
3.2 Order -dependent measures.
For vestibular afferents, a negative serial dependence was
suggested by the lengthening effects of an interposed shock on the
subsequent interval in regular afferents (Goldberg et al., 1984,
figures 10, 11). Using unit 2 with p
= 1,
the model matched this
lengthening effect reasonably well, as well as its dependence on
firing rate (Smith and Goldberg, 1986, figure 58). As mentioned above,
the corresponding procedure is not practical with irregularly
discharging vestibular afferents and use of the procedure in other
systems requires establishing the site of action of applied currents.
A negative serial dependence can be examined in several other ways.
One common method to measure order dependent effects is the
serial correlation between the (i)th and (i+k)th intervals. The next
15
set of figures (figure 7) show the first three serial correlation
coefficients (rho1, rho2, rho3) for the unit 5 pairs with p
as X and p
=0
= 1 coded
denoted as *. The abscissa is the mean interspike
interval. As above, each point represents 2000 intervals. The 5%
significance level for testing rho
=0
is 0.044 for a single point.
Beginning with rho1 (figure 7A), the p = 0 case, which should
behave like a renewal process, is bouncing around a value of zero,
with only one point of the 27 lying outside the 5% level for a single
point. The p
=1
case shows a systematic trend, being quite
significantly negative up until a mean interval of around 7 msec
(about 3
T
K). Then the two curves appear to be in phase. This is
expected since after several time constants, gK is neglible and the
same random number seeds were used for both p
=0
and p
= 1.
The second and third order serial correlations (figure 7B,C) are
a different story. Both plots produce nonsignificant values and
without trends for both values of p.
Figure 7 shows that cumulative afterhyperpolarization can
produce an appreciable negative first order serial correlation
provided the mean interval is less than several
T
K'. However rho1
measures the linear association between adjacent intervals. Is the
relation predominantly linear? The answer apppears to be yes as can be
seen from conditional plots of the joint intervals.
The next two figures (8A, p = 0, 8B, P = 1) are scatterplots of
the (i-nth vs (i)th intervals for unit 5 at the highest value of
synaptic input
(A
= 27/msec).
These joint interval plots were the
basis for the first set of points on our rho 1 graph (fig. 7A). While
there is some asymmetry in the p = 1 scatterplot, a better way to
characterize these 3-D plots is needed. What was done was to condition
16
the (i-1)th interval to a given bin width and then compute the mean
and median of the following interval (c.f. Tsuchitani and Johnson,
1985). This is illustrated for these two scatterplots in the next pair
of figures (9A,conditional mean, 9B, conditional median). The symbol
coding is X for p = 1 and
* for
p = O. The lower tick marks show the
bin widths used in calculating the mean and median, and represent (to
within ties) the 5 percentile points of the one dimensional
distribution. The p
= 1 case
shows a systematic linear relation. The
solid line is the least squares fit weighted by the standard deviation
of the intervals in each bin. The p
=0
case shows a positive but
nonsignificant trend. The mean interval of the 2000 points is shown as
a diamond on the fitted line for both cases. The conditional median
plot (fig. 9B) produces quite similar results, here however the least
squares line was unweighted.
The next pair of figures (10A, conditional mean, lOB, conditional
median) illustrate the
~orresponding plots
value of synaptic input used
(A
for the smallest paired
= 12/msec, c.f. last set of points in
figure 7A). The mean interval is now around 18 msec (about 7
the p
=1
and p
=0
'i)
and
cases are almost identical as suggested from the
serial correlation plots. The slopes of the least squares lines were
plotted for all 27 pairs and are shown in the pair of figures (llA,
means, 11B, medians). The two figures are very similar and the values
correspond closely to those found in the rho1 graph.
Provided the larger value of 'K (6.50 msec for unit 2 as
opposed to 2.36 msec for unit 5) is taken into account, the resu Its
for the unit 2 pairs closely parallel those of unit 5. To reiterate
the main points with unit 2: The first order serial correlation
17
coefficient (figure 12A) is negative for high firing rates and its
value decreases to nonsignificant levels as the mean interval
approaches around 3
• The second and third order correlations
K
(figures 12B,C) are small and show no trends. The conditional mean and
T
median plots (figure 13A,B) parallel the rh01 plots in shape and
values, both estimating the linear serial order dependence at each
mean interval.
One technical note is in order for the unit 2 pairs. The
simulations were done with a time step of 0.1 msec. The regularity of
unit 2 at high firing rates (standard deviations of a few time steps)
produced many ties when trying to use equal percentile points to bin
the condi tiona 1 p lots. So fixed bin widths of 0.1 or 0.2 msec were
used instead. The result was that some bins had only a few intervals
in them. For the conditional means, a weighting corresponding to the
number of intervals in each bin was used for the least squares fits.
The conditional median fits were again unweighted. An artifact of this
procedure is that the slopes from the conditional mean (median) vs
conditioning interval plots can drastically change depending on the
value of the first few points. This can be seen from the next set of
figures (Figure 14A, p
median; 14C, p
= 1,
= 0,
conditional mean; 14B, p
conditional mean; 14D, p
= 1,
= 0,
conditional
conditional median).
The conditional means and medians are shown as connected points for
all the fixed binwidth cases in the unit 2 plots. For unit 5, the
percentile method was used throughout. The erratic behaviour in figure
14 due to the tails of each conditional distribution can be removed by
truncating the lower and upper 2.5% of each distribution, reiterating
that the effect is the result of a few points in each case. A smoother
curve for p
=1
in Figure 13B also is obtained, as well as reducing
18
4. DISCUSSION.
Our main conclusion is that the presence of cumulative
afterhyperpolarization can be seen in moderate sample sizes using
conventional measures of serial order dependence for point processes
provided the mean interval is less in value than 2 to 3
T
K• There
are however several points that need to be discussed: how robust is
the model; what other factors might either cause similar serial
dependence or mask the negative serial dependence due to a CAHP; and
where might these results be applicable.
Some feeling for the robustness of the present model comes from
looking at other models. In simulations of an Ornstein-Uhlenbeck type
model with a decaying threshold and summating AHP's, Geisler and
Goldberg (1966) found negative first order serial correlations at high
firing rates that decreased as the mean interval rose to several time
constants of the AHP buildup (their figure 9B). They also found that
the values of rho1 were not very sensitive to the level of the
synaptic noise and that higher order serial correlations were not
significantly different from zero. In summary their findings also
suggested a 1-memory point process description of the spike trains.
The order-independent effects on mean interval due to p
=1
can be
seen in deterministic AHP models (see Jack et al., 1975, pp 311-315
for a review). For example, the linearization of the input-output
curves due to AHP accumulation (Figure 6) is also illustrated in
deterministic versions of the model (MacGregor and Oliver, 1974,
Figure 4A). The main difference among the deterministic models is the
functional form chosen for the potassium conductance underlying the
AHP (e.g., Baldissera et al., 1976). These findings suggest that the
qualitative features of the present model are somewhat robust with
20
respect to the time dependence of the potassium conductance and the
specific method used to achieve summation of AHP's.
Several factors could either mimic or mask the negative serial
dependence due to a CAHP. A serious concern is the assumption of
stationarity. In some cases, such as static tilt stimulation of
otoliths or the steady state portion of tone bursts for central
auditory neurons, this may be a reasonable assumption (however, c.f.
Correia and Landolt, 1977) and can be tested in any case. The
importance of assuring that the spike train is stationary comes from
the effects of firing rate trends on the first order serial
correlation. A trend in firing rate, say due to adaptation, can
produce positive first order serial correlations which would mask the
negative ones produced by cumulative AHP. Other mechanisms such as
facilitation at the input synapse, changing synaptic efficacy, or
serial correlation in the input spike train are all examples of
confounding factors that need to be examined.
With regard to the use of the results, the peripheral vestibular
system was the motivation for the original modelling study. Another
set of neurons where serial dependency, similar to that produced by
our model, has been observed is the lateral superior olive of the
auditory brainstem. The single unit studies of Goldberg et ale (1964)
and Tsuchitani and Johnson (1985) found first order serial
correlations and conditional mean dependencies on mean interval that
parallel the model results presented here. Provided that these reflect
an AHP (see discussion pp 738-740, Goldberg et al., 1964) and not
serial correlation of the input (see discussion pp 1494-1495,
Tsuchitani and Johnson, 1985) and that the other model assumptions are
21
applicable, then these plots allow a rough estimate of the time course
of the AHP in these neurons. This estimate would be useful as an aid
to the technically more difficult intracellular studies required to
guarentee an AHP in these neurons.
Our approach has been to construct a physiologically based model of
events occurring at the spike initiation site and then examine the
statistical properties of the resultant spike train. A second approach
is to begin with the spike train and consider it as a generalized
Poisson process, whose "intensity" can depend both on time and on the
past history of the point process (Snyder, 1975). As mentioned in the
Introduction, this approach has been extensively used recently in
modelling the discharges of the auditory nerve afferents and lateral
superior oli ve (LSO) neurons.
Of particular relevance for our study are the results of
Tsuchitani and Johnson (1985) and Johnson et ale (1986) for cat LSO
units. For stationary portions of their data, a serial dependence
solely on the duration of the previous interval was found. The
conditional mean plots for these units are quite similar to those
illustrated here in figure 9. Except for units with bimodal marginal
distributions, the first order serial correlation coefficient was
significantly negative while higher serial dependence seemed
negligible. There was also some evidence for the source of the serial
dependency being in the LSO neurons rather than in their inputs
(Johnson et al., 1986, pp 156-157). The connection between the two
approaches is that a possible physiological mechanism, namely a
cumulative AHP, is suggested for their generalized point process
description.
In this study only stationary spike trains were considered.
22
Following a characterization of the intensity in stationary segments
of the data, Johnson et ale (1986) used this description to examine
the non-stationary responses via simulation. The corresponding study
in the stochastic AHP model could be done but would require much more
computation time. A more limited objective would be to examine the
adaptation that results from a rapid change in synaptic input rate. If
a cumulative AHP mechanism is present, a rapid change in the synaptic
input rate would produce an initially high firing rate that would
dec line to a steady state leve 1. The degree of "ear ly adaptation" has
been studied in deterministic
AHP models (see Jack et a1., 1975, pp
352-367 for a review), but not in stochastic CAHP models. Such a study
could be relevant to the mammalian aUditory nerve where "rapid
adaptation" has been studied both experimentally and by the general
point process approach (Lutkenhoner and Smith, 1986). There is some
evidence from intracelluar records of auditory afferent terminals
(Siegel and Dallos, 1986) that suggest a change in our model's
parameters would be needed. The quantal EPSP's appear even larger than
that for our unit 5. The time course of the "recovery process" appears
to be comparable to unit 5, or even shorter, and also depends on the
length of the previous interval (Lutkenhoner et al., 1980; Gaumond et
a1., 1982, 1983).
The combining of the general point process approach and the spike
initiation mechanism approach should provide a hierarchial description
of discharge activity with testable predictions. Several types of
neurons in the auditory and vestibular systems provide a "ripe"
testing ground for this combined approach.
23
REFERENCES
Alkon, D. L. (1984)."Persistent calcium-mediated changes of identified
membrane currents as a cause of associati ve learning", in Primary
Neural Substrates of Learning and Behavioral Change, edited by D. L.
Alkon and J. Farley (Cambridge Uni v. Press, Cambridge, MA),pp 291-324.
Alkon, D. L., Sakakibara, M., Forman, R., Harrigan, J., Lederhendler,
I., Far ley, J. (1985)."Reduction of two vol tage-dependent K+ currents
mediates retention of a learned association", Behavioral and Neural
Biology, ~4, 278-300.
Baldissera, F., Gustafsson, B., Parmiggiani, F. (1976)."'A model for
refractoriness accumulation and secondary range firing in spinal
motoneurons", BioI. Cybernetics 2~, 61-65.
Boos, D. D. (1981). "Minimum distance estimators for location and
goodness of fit", J. Amer. Stat. Assoc. 76, 663-670.
Correia, M. J. and Landolt, J. P. (1977). "A point process analysis
of the spontaneous activity of anterior semicircular canal units in
the anesthetized pigeon", BioI. Cybernetics 27, 199-213.
Crill, W. E. and Schwindt, P. C. (1983)."Active currents in mammalian
central neurons", Trends Neurosci. 6, 236-240.
Gaumond, R. P., Kim, D.O., Molnar, C. E. (1983). "Response of
cochlear nerve fibers to brief acoustic stimuli: Role of dischargehistory effects", J. Acoust. Soc. Amer. 7~, 1392-1398.
Gaumond, R. P., Molnar, C. E., Kim, D. O. (1982). "Stimulus and
recovery dependence of cat cochlear nerve spike discharge
probability", J. Neurophysiology 48, 856-873.
Geisler, C. D., and Goldberg, J. M. (1966)."A stochastic model for the
repetiti ve acti vity of neurons", Biophys. J. 6, 53-69.
Goldberg, J. M., Adrian, H. 0., Smith, F. D. (1964)."Responses of
neurons of the superior olivary complex of the cat to acoustic stimuli
of long duration", J. Neurophysiol. 27, 706-749.
Goldberg, J. M., Ferncfndez, C., Smith, C. E. (1982)."Responses of
vestibular-nerve afferents in the squirrel monkey to externally
applied galvanic currents", Brain Research, 252, 156-160.
Goldberg, J. M., Smith, C. E., Fernandez, C. (1984)."The relation
between discharge regularity and responses to externally applied
galvanic currents in vestibular nerve afferents of the squirrel
monkey", J. Neurophysio1., 51, 1236-1256.
Hudspeth, A. J. and Corey, D. P. (1977). "Sensitivity, polarity, and
conductance change in the response of vertebrate hair cells to
controlled mechanical stimuli", Proc. Natl. Acad. Sci. 7~, 2407-2411.
24
International Mathematics and Statistical Library Reference Manual,
8th edition, vol 2, (1980). International Mathematics and Statistical
Library, Inc., Houston, Texas.
Jack, J. J. B., Nobel, B., Tsien, R. W. (1975). Electric Current Flow
in Excitable Cells. (Claredon, Oxford Univ. Press, UK).
Johnson, D. H. and Swami, A. (1983).''The transmission of signals by
auditory-nerve fiber discharge patterns", J. Acoust. Soc. Am. 74, 493501.
Johnson, D. H., Tsuchitani, C., Linebarger, D. A., Johnson, M. J.
(1986). "Application of a point process model to responses of cat
lateral superior olive units to ipsilateral tones", Hearing Res. 21,
135-159.
Jones, K., TUbis, A., Burns, E. M. (1985l."On the extraction of the
signal-excitation function from a non-Poisson cochlear neural spike
train", J. Acoust. Soc. Am. 78, 90-94.
Kernell, D. (1968). ''The repetitive impulse discharge of a simple
neurone model compared to that of spinal motoneurones", Brain Res.
11, 685-687.
Kernell, D. (1972). ''The early phase of adaptation in repetitive
impulse discharges of cat spinal motoneurones", Brain Res. 41, 184186.
Lutkenhoner, B., Hoke, M., Bappert, E. (1980)."Influence of refractory
properties on the response of single auditory nerve fibres to
sinusoida 1 stimu 11 ", Hearing Res. 2, 565-572.
Lutkenhoner, B. and Smith, R. L. (1986). "The role of discharge
history effects in rapid adaptation of aUditory-nerve fibers", J.
Acoust. Soc. Am. Supple 1, 79, 33.
MacGregor, R. J. and Oli ver, R. M. (1974)."A model for repetitive
firing in neurons", Kybernetik 16, 53-64.
Miller, M. I. (1985)."Algorithms for removing recovery-related
distortion from aUditory-nerve discharge patterns", J. Acoust. Soc.
Am. 77, 1452-1464.
Perkel, D. H., Gerstein, G. L., Moore, G. P. (1967l."Neurona 1 spike
trains and stochastic point processes I. The single spike train",
Biohys. J. 7, 391-418.
Schessel, D. A. (1982). "Chemica 1 synaptic transmission between type I
vestibular hair cells and the primary afferent chalice: An
intracellular study using horseradish peroxidase", Ph.D.
Dissertation), Albert Einstein College of Medicine, Bronx, NY.
Siegel, J. H. and Dallos, P. (1986). "Spike acti vity recorded from the
organ of Corti", Hearing Res. 22, 245-248.
25
Smith, C. E. and Goldberg, J. M. (1986)."A stochastic afterhyperpolarization mode 1 of repetiti ve acti vity in vestibu lar afferents",
Biol. Cybernetics 54, 41-51.
Snyder, D. L. (1975). Random Point Processes (Wiley, New York).
Tasaki, I. (1955). "New measurements of the capacity and resistance of
the myelin sheath and the nodal membrane of the isolated frog nerve
fiber", Am. J. Physiol. 181, 639-650.
Tsuchitani, C. and Johnson, D. H. (1985).''The effects of ipsilateral
tone burst stimulus level on the discharge patterns of cat lateral
superior olive", J. Acoust. Soc. Am., 77, 1484-1496.
26
TABLE 1.
Parameters used for model neurons.
Unit
Parameters
N
gKO
A
'K
(msec)
(mV)
2
2000
2.15
6.50
0.136
3
500
1.32
5.50
0.265
5
2000
0.50
2.36
1.000
N, number of intervals, each simulation run. Other symbols
are explained in text. Unit numbering corresponds to that
in Smith and Goldberg, 1986.
27
FIGURE LEGENDS
Figure 1Illustration of the model with constant synaptic input.
Five consecu-
tive spikes following an isolated spike are shown in (A) for model unit 2
with p = 1.
The voltage trajectories in (A) are related to the synaptic
input gs (B) and the potassium conductance gK (C) via equation (4).
Spikes in (A) are shown as 0.5 msec pulses.
The lower horizontal reference
line is the resting potential, the upper line is the threshold for firing
and has a value of 10 mY.
The value of gs in (8) is 0.5444, corresponding
to the mean synaptic input due to a release rate A of 560/msec in model
unit 2.
The horizontial references line in (C) is the initial value of
gKO following an isolated spike.
Figure 2.
Illustration of the model with shot noise synaptic input resulting from
Poisson transmitter release.
(D,E,F) with p = O.
(A,B,C) represent unit 2 with p = 1 and
The synaptic input gs is identical in (8) and (E);
and has a mean value equal to the constant value of gs in figure 1.
The
voltage trajectories (A) and (D) show slight differences that reflect the
differences in the potassium conductances either summating (C, p
= 1)
residual activity or reseting to a fixed value (F, p = 0).
Figure 3.
Illustration of the model for unit 5.
corresponds to p
=1
and (D,E,F) to p
= O.
As in the previous figure, (A,B,C)
The release rate A is l4.6/msec.
The other parameters for unit 5 are in table 1.
28
The upper reference line
for the potassium conductance has the same value as in figure 2; the lower
reference line is the initial value of gKO for an isolated spike in unit 5
(c.f. table 1).
Figure 4.
Relations between coefficient of variation (CV) and mean interval (MI)
for p
=0
2(~),
3(0) and 5(0).
log-log.
(points) and p
=1
(connected lines) versions of model units
The unit number is displayed.
Note the scale is
For units 2 and 5, the number of simulated interspike intervals,
N, represented by each plotted point is 2000; for unit 3, N
= 500.
Figure 5.
The relation between skew and CV for model unit 5.
Pearson's square root of ale
X denotes p
Skew is measured as
= 1, and 0 denotes p = o. The
three reference curves are the corresponding relations for the following
two parameter distributions: lognormal (LN), inverse gaussian (IG) and
gamma (GA).
Figure 6.
Simulated input-output curves for the model.
spikes/sec vs mean synaptic input gS.
figure 4.
(A) Firing rate in
Coding for unit number is as in
The vertical line' demarcates deterministic (to the right) vs
nondeterministic (to the left) firings. (B) Response in spikes/sec vs
level of externally applied galvanic polarization (V p ) in mV for unit 3.
Open triangles represent p
=
0 and connected points p
represents no external polarization
=
1.
Vertical line
with a corresponding mean interval
of 10 msec.
29
Figure 7.
Serial correlation coefficients vs mean interval for unit 5.
interval (MI) is in msec.
The symbols * and x denote p
=0
Mean
and 1 respectively.
First (A), second (B) and third (C) order serial correlation coefficients
are denoted as rhol, rho2 and rho3 respectively.
line is at rho
Horizontal reference
= O.
Figure 8.
Three dimensional display of first order joint interval histogram for
Unit 5, A
= 27/msec.
(A) p
= 0,
(B) P
= 1.
The joint distribution between
th
th
.
successive intervals, l.e. (n-l)
vs (n) , is shown for a binwidth of
0.2 msec.
The x-y axes are labelled in terms of bin number.
case corresponds to the first pair of points in figure 7.
The A = 27
The corresponding
mean interval of the 2000 intervals was 4.10 msec for p = 1 and 3.62 msec
for p
= o.
Figure 9.
Conditional mean and conditional median plots corresponding to figure 8.
The mean (A) and median (B) of the conditional intervals ((n)th lSI) that
follow the conditioning interval ((n_l)th lSI) is plotted as a function
of the conditioning interval.
respectively.
upper; p
=
The symbols x and * denote p = 1 and 0
The two horizontal lines just above the abscissa (p
=
1,
0, lower) display the binwidths used in calculating the
conditional mean (A) and median (B).
Except for ties, they represent
consecutive 5 percentile points of the conditioning interval distribution.
The other pair of lines in each plot represent least squares fits, weighted
by the reciprocal of the standard deviation for the conditional mean plot
30
and unweighted for the conditional median plots.
In (A), diamonds denote
lower range of data, mean of the 2000 intervals, and upper range of data
respectively in terms of values on the abscissa.
denotes the median of the 2000 intervals.
In (8), the middle diamond
lSI is interspike interval.
Figure 10.
Conditional mean (A) and median (8) plots for unit 5 with A = l2/msec.
Display symbols are as in figure 9.
A
=
12 corresponds to the last pair
of points in figure 7, with a corresponding mean interval for the 2000
intervals of 17.92 msec (p
= 1)
and 17.89 msec (p
= 0).
Figure 11.
Slopes of conditional mean (A) and of conditional median (B) vs the mean
interval of the 2000 simulated intervals for unit 5.
denote p = 1 and p = 0 respectively.
The symbols x and *
The slope of the upper least squares
fit line in figure 9A gives the first point for p
=1
-0.147 at a mean interval of 4.10 msec; likewise for p
with a value of
= 0,
the slope of
the lower least-squares line in figure 9A has a value of 0.026 at a mean
interval of 3.62 msec.
The corresponding slopes in figure 10 give the right
most pair of points in (A) and (B) at a mean interval of 17.92 msec (p • 1)
and 17.89 msec (p
=
0).
Figure 12.
Serial correlation coefficients vs mean interval for unit 2. Symbols and
legends are as in figure 7.
31
Figure 13.
Slopes of conditional mean (A) and median (B) vs the mean interval of the
2000 simulated intervals for unit 2.
figure 11.
Symbols and legends are as in
The 5 percentile method of binning was used, as for unit 5, for
calculating the value of the rightmost 12 points in (A) and in (B) for both
p = 0 and p = 1.
Fixed binwidths were used for the remaining cases.
For
p = I, a 0.1 msec binwidth was used for the first 16 points, and 0.2 msec for
the next 8 points.
For p
= 0,
a 0.1 msec binwidth was used for the first
6 points, and 0.2 msec for the next 8 points.
The conditional median least
squares lines were unweighted as with unit 5.
The fitted lines for the
conditional mean were weighted by the reciprocal of the standard deviation
or by the number of intervals for the percentile and fixed binwidth conditions
respectively.
Figure 14.
Conditional mean (A,C) and median (B,D) plots for unit 2 that used fixed
binwidths.
The data points are drawn as connected lines rather than as
individual points (c.f. figures 9 and 10).
14 curves are shown for
p = 0 (A,B) corresponding to A values, starting with lowest curve, of 850,
800, 750, 700, 650, 600, 550, 500, 450, 400, 375, 350, 325 and 300/msec
respectively.
For p = 1 (C,D), 24 curves are shown with corresponding A values,
starting with lowest curve of
200~
1750, 1500, 1400, 1300, 1200, 1100, 1000,
950, 900, 850, 800, 750, 700, 650, 600, 550, 500, 450, 400, 375, 350, 325
and 300/msec respectively.
The curves for the last 8 values of A in each
plot (A, B, C, D) have a binwidth of 0.2 msec, the others have a binwidth of
0.1 msec.
32
UNIT 2
15
v
p
s
0
T
E
N
T
J:
A.
L
M
V
-6
P=l
A
e
UNIT 2 p=!
15
A
v
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o
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E
N
T
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A
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Ii
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a
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20
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60
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UNIT 5 P=!
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16
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0
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1.00000
5
0.31620
c
V
0.10000
0.03162
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31.620
10.000
HI (MSEC)
Figure 4
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SECURITy' ClASSiHCAT10N OF THIS PAGE
I
REPORT DOCUMENTATION PAGE
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Cldsslf/c~t;on'
SERIAL DEPENDENCY IN NEURAL POINT PROCESSES DUE TO CUMULATIVE AFTERHYPERPOLARIZATION
1.2. PERSONAL AUTHOR(S)
Charles E. Smith and Chao-lung Chen
13a. TYPE QF REPORT
~chnical
10.
11~!> Ti"~E CO'JERED
FROM _
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114 DATE OF REPORT (YI.',Ir. Month, OAyl
August 1986
TO
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50
SUPFLEMENTARY NOTATION
17.
FIELD
COSATl CODES
GROUP
SUB·GROUP
18. SUBJECT TERMS (Continue on
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necessary and
19. ABSTRACT (Continue on reverse if "ec~1Jary .nd identify by blo</c num~r)
;d~ntlfy by blo<lc num~r'
.
The effects of cumulative vs noncumulative afterhyperpolarization (AHP) are examined
through simulations of a stochastic neural model (Smith and Goldberg, 1986). The afterhyperpolarization in the model is due to a timevarying potassium conductance. Cumulative
effects result from summing the residual activity of the potassium conductance in the
preceding interspike interval. The variability in the model is due to random quantal
transmitter release. The statistical properties of the steady state discharge patterns
that are independent of the serial ordering of interspike intervals show only slight
differences between the two cases. Several order dependent statistical measures are used
to show that a negative serial dependency results from cumulative AHP at moderate to high
discharge rates. The discussion considers the robustness of the model and its relation to
a generalized Poisson process descripti9n of spike trains. Possible applications of the
results to neurons in the auditory and vestibular systems are also examined.
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