Stationarity of Synaptic Coupling Strength Between 
Neurons with Nonstationary Discharge Properties 
Mark R. Sydorenko and Eric D. Young 
Dept. of Biomedical Engineering & Center for Hearing Sciences 
The Johns Hopkins School of Medicine 
720 Rutland Avenue 
Baltimore, Maryland 21205 
Abstract 
Based on a general non-stationary point process model, we computed estimates of 
the synaptic coupling strength (efficacy) as a function of time after stimulus onset 
between an inhibitory interneuron and its target postsynaptic cell in the feline dorsal 
cochlear nucleus. The data consist of spike trains from pairs of neurons responding 
to brief tone bursts recorded in vivo. Our results suggest that the synaptic efficacy is 
non-stationary. Further, synaptic efficacy is shown to be inversely and 
approximately linearly related to average presynaptic spike rate. A second-order 
analysis suggests that the latter result is not due to non-linear interactions. Synaptic 
efficacy is less strongly correlated with postsynaptic rate and the correlation is not 
consistent across neural pairs. 
1 
INTRODUCTION 
The aim of this study was to investigate the dynamic properties of the inhibitory effect of 
type II neurons on type IV neurons in the cat dorsal cochlear nucleus (DCN). Type IV cells 
are the principal (output) cells of the DCN and type II cells are inhibitory interneurons (Voigt 
& Young 1990). In particular, we examined the stationarity of the efficacy of inhibition of 
neural activity in a type IV neuron by individual action potentials (APs) in a type II neuron. 
Synaptic efficacy, or effectiveness, is defined as the average number of postsynaptic (type IV) 
APs eliminated per presynaptic (type II) AP. 
This study was motivated by the observation that post-stimulus time histograms of type IV 
neurons often show gradual recovery ("buildup") from inhibition (Rhode et al. 1983; Young 
& Brownell 1976) which could arise through a weakening of inhibitory input over time. 
11 
12 Sydorenko and Young 
Correlograms of pairs of DCN units using long duration stimuli are reported to display 
inhibitory features (Voigt & Young 1980; Voigt & Young 1990) whereas correlograms using 
short stimuli are reported to show excitatory features (Gochin et al. 1989). This difference 
might result from nonstationarity of synaptic coupling. Finally, pharmacological results 
(Caspary et al. 1984) and current source-density analysis of DCN responses to electrical 
stimulation (Manis & Brownell 1983) suggest that this synapse may fatigue with activity. 
Synapfic efficacy was investigated by analyzing the statistical relationship of spike trains 
recorded simultaneously from pairs of neurons in vivo. We adopt a first order (linear) non- 
stationary point process model that does not impose a priori restrictions on the presynaptic 
process's distribution. Using this model, estimators of the postsynaptic impulse response to a 
presynaptic spike were derived using martingale theory and a method of moments approach. 
To study stationarity of synaptic efficacy, independent estimates of the impulse response 
were derived over a series of brief time windows spanning the stimulus duration. Average 
pre- and postsynaptic rate were computed for each window, as well. In this report, we 
summarize the results of analyzing the dependence of synaptic efficacy (derived from the 
impulse response estimates) on post-stimulus onset time, presynaptic average rate, 
postsynaptic average rate, and presaptic interspike interval. 
2 METHODS 
2.1 DATA COLLECTION 
Data were collected from unanesthetized cats that had been decerebrated at the level of the 
superior colliculus. We used a posterior approach to expose the DCN that did not require 
aspiration of brain tissue nor disruption of the local blood supply. Recordings were made 
using two platinum-iridium electrodes. 
The electrodes were advanced independently until a type II unit was isolated on one electrode 
and a type IV unit was isolated on the other electrode. Only pairs of units with best 
frequencies (BFs) within 20% were studied. The data consist of responses of the two units to 
500-4000 repetitions of a 100-1500 millisecond tone. The frequency of the tone was at the 
type II BF and the tone level was high enough to elicit activity in the type II unit for the 
duration of the presentation, but low enough not to inhibit the activity of the type IV unit 
(usually 5-10 dB above the type II threshold). Driven discharge rates of the two units ranged 
from 15 to 350 spikes per second. A silent recovery period at least four times longer than the 
tone burst duration followed each stimulus presentation. 
2.3 DATA ANALYSIS 
The stimulus duration is divided into 3 to 9 overlapping or non-overlapping time windows ('a' 
thru 'k' in figure 1). A separate impulse response estimate, presynapfic rate, and postsynaptic 
rate computation is made using only those type II and type IV spikes that fall within each 
window. The effectiveness of synaptic coupling during each window is calculated from the 
area bounded by the impulse response feature and the abscissa (shaded area in figure 1). The 
effectiveness measure has units of number of spikes. 
The synapfic impulse response is estimated using a non-stationary method of moments 
algorithm. The estimation algorithm is based on the model depicted in figure 2. The thick 
gray line encircles elements belonging to the postsynapfic (type IV) cell. The neural network 
surrounding the postsynaptic cell is modelled as a J-dimensional multivariate counting 
process. Each element of the J-dimensional counting process is an input to the postsynaptic 
Stationarity of Synaptic Coupling Strength Between Neurons 13 
cell. One of these input elements is the presynapfic (type II) cell under observation. The 
input processes modulate the postsynaptic cell's instantaneous rate function, ,j(t). Roughly 
speaking, (t) is the conditional firing probability of neuron j given the history of the input 
events up to time t. 
HISTOGRAM 
0 
SR 
a b 
Post-Stimulus Time 
0 
a b 
K:(t) 
TYPE IV PST 
HISTOGRAM 
I 
k. Post-StimulusTime 
Figure 1: Analysis of Non-stationary Synaptic Coupling 
Figure 2 
N.1 
The transformation K describes how the 
input processes influence ,j(t). We model 
this transformation as a linear sum of an 
intrinsic rate component and the contribution 
of all the presynaptic processes: 
Xj(t) = Koj(t)+k=, K,,(t,u) dNk(u) (1) 
where K0 describes the intrinsic rate and the 
K1 describe the impulse response of the 
postsynaptic cell in response to an input 
event. The output of the postsynaptic neuron 
is modeled as the integral of this rate 
function plus a mean-zero noise process, the 
innovation martingale (Bremaud 1981): 
Nj(t) = Xj(u)du + Mj(t). 
0 
(2) 
An algorithm for estimating the first order 
kernel, K1, was derived without assuming 
14 Sydorenko and Young 
anything about the distribution of the presynaptic process and without assuming stationary 
first or second order product densities (i.e., without assuming stationary rate or stationary 
auto-correlation). One or more such assumptions have been made in previous method of 
moments based algorithms for estimating neural interactions (Chomoboy et al. 1988 describe 
a maximum likelihood approach that does not require these assumptions). 
Since K1 is assumed to be stationary during the windowed interval (figure 1) while the 
process product densities are non-stationary (see PSTHs in figure 1), K1 is an average of 
separate estimates of K 1 computed at each point in time during the windowed interval: 
y. e) 
na ha-t=ta; teI (3) 
where K1 inside the summation is an estimate of the impulse response of neuron i at time t 
to a spike from neuron j at time t? (times are relative to stimulus onset); the digitization bin 
width A (= 0.3 msec in our case) determines the location of the discrete time points as well 
as the number of separate kernel estimates, n A, within the windowed interval, I. The time 
dependent kernel, KI(','), is computed by deconvolving the effects of the presynaptic process 
distribution, described by rii below, from the estimate of the cross-cumulant density, qij: 
where: 
r.ij tu ,v / = 9 '- 1 , 
uA, v 
3j(t?) -- #{spike in neuronj during [t-, t+-)}/(#{trials} A), 
(4) 
(5) 
(6) 
(7) 
(8) 
pij (t,t?) #{spike in i during tt-, tlx+-) and spike in j during tt-, 
{fls} a 2 (9) 
where b(.) is e dirac delta function; fd 1 me the D d inverse D, respectively; 
d {.} is e number of members in e set described iide e braces.  the presynaptic 
process is Poisson sbuted, expression (4) simplffies to: 
pj(tj) (10) 
Under mild (physiologically justffiable) conditions, e estimator given by (3) converges in 
quaatic me d yiel  ymptoficflly bi estimate of the e impulse response 
hncfion (in e generfl, (4), d Poisson presapfic process, (10), cases). 
3 RESULTS 
Figure 3 displays estimates of synaptic impulse response functions computed using traditional 
cross-correlation analysis and compares them to estimates computed using the method of 
moments algorithms descried above. (We use the definition of cross-correlation given by 
Voigt & Young 1990; equivalent to the function given by dividing expression (10) by 
Stationarity of Synaptic Coupling Strength Between Neurons 15 
expression (9) after averaging across all tj.) Figure 3A compares estimates computed from 
the responses of a real type II and type IV unit during the first 15 milliseconds of stimulation 
(where nonstationarity is greatest). Note that the cross-correlation estimate is distorted due to 
the nonstationarity of the underlying processes. This distortion leads to an overestimation of 
the effectiveness measure (shaded area) as compared to that yielded by the method of 
moments algorithm below. Figure 3B compares estimates computed using a simulated data 
set where the presynaptic neuron had regular (non-Poisson) discharge properties. Note the 
characteristic ringing pattern in the cross-correlation estimate as well as the larger feature 
amplitude in the non-Poisson method of moments estimate. 
(A) Cross-correlogram 
60. 
-100 
-10 
20. 
-20, 
-60, 
-5 0 5 10 
milliseconds 
(B) Cross-correlogram 
30 
0: 
-15" 
-30: 
-50 -25 0 25 50 
milliseconds 
60. 
 
-lO0 
-10 
Method of Moments 
20. 
-20. 
-60, 
-5 0 5 10 
milliseconds 
Method of Moments 
30 
15-' 
-15-' 
-30' 
-50 -25 0 25 50 
milliseconds 
Figure 3 
Results from one analysis of eight different type II / type IV pairs are shown in figure 4. For 
each pair, the effectiveness and the presynaptic (type ID average rate during each window are 
plotted and fit with a least squares line. Similar analyses were performed for effectiveness 
versus postsynaptic rate and for effectiveness versus post-stimulus-onset time. The number of 
pairs showing a positive or negative correlation of effectiveness with each parameter are 
tallied of table 1. The last column shows the average correlation coefficient of the lines fit to 
the eight sets of data. Note that: Synaptic efficacy tends to increase with time; there is no 
consistent relationship between synaptic efficacy and postsynaptic rate; there is a strong 
inverse and linear correlation between synaptic efficacy and presynaptic rate in 7 out of 8 
pairs. 
If the data appearing in figure 4 had been plotted as effectiveness versus average interspike 
interval (reciprocal of average rate) of the presynaptic neuron, the result would suggest that 
synaptic efficacy increases with average inter-spike interval. This result would be consistent 
with the interpretation that the effectiveness of an input event is suppressed by the occurrence 
of an input event immediately before it. The linear model initially used to analyze these data 
neglects the possibility of such second order effects. 
16 Sydorenko and Young 
Table 1: Summary of Results 
GRAPH 
NUMBER OF 
PAIRS WITH 
POSITIVE 
SLOPE 
NUMBER OF 
PAIRS WITH 
NEGATIVE 
SLOPE 
AVERAGE LINEAR 
REGRESSION 
CORRELATION 
COEFFICIENT 
Effectiveness 
-vs- 7/8 1/8 
Post Stimulus Onset Time 
Effectiveness 
-vs- 5/8 3 /8 
Average Postsynaptic Rate 
Effectiveness 
-vs- 1/8 7/8 
Average Presynaptic Rate 
0.83 
0.72 
0.89 
0.2 0.2 
0.15  
0.05  
0 
50 100 150 200 
Type II Rate (spikes/sec) 
250 
0.15  
 
0.05  
, : 
0 5 10 15 20 
Type II Inter-spike Interval (millisec) 
Figure 4 Figure 5 
We used a modification of the analysis described in the methods to investigate second order 
effects. Rather than window small segments of the stimulus duration as in figure 1, the entire 
duration was used in this analysis. Impulse response estimates were constructed conditional 
Stationarity of Synaptic Coupling Strength Between Neurons 17 
on presynaptic interspike interval. For example, the first estimate was constructed using 
presynaptic events occurring after a 1 ms interspike interval, the second estimate was based 
on events after a 2 ms interval, and so on. 
The results of the second order analysis are shown in figure 5. Note that there is no 
systematic relationship between conditioning interspike interval and effectiveness. In fact, 
lines fitted to these points tend to be horizontal, suggesting that there are no significant 
second order effects under these experimental conditions. 
Our results suggest that synaptic efficacy is inversely and roughly linearly related to average 
presynaptic rate. We have attempted to understand the mechanism of the observed decrease 
in efficacy in terms of a model that assumes stationary synaptic coupling mechanisms. The 
model was designed to address the following hypothesis: Could the decrease in synaptic 
efficacy at high input rates be due to an increase in the likelihood of driving the stochastic 
intensity below zero, and, hence decreasing the apparent efficacy of the input due to clipping? 
The answer was pursued by attempting to reproduce the data collected for the 3 best type II / 
type IV pairs in our data set. Real data recorded from the presynaptic unit are used as input 
to these models. The parameters of the models were adjusted so that the first moment of the 
output process had the same quantitative trajectory as that seen in the real postsynaptic unit. 
The simulated data were analyzed by the same algorithms used to analyze the real data. Our 
goal was to compare the simulated results with the real results. If the simulated data showed 
the same inverse relationship between presynaptic rate and synaptic efficacy as the real data, 
it would suggest that the phenomenon is due to non-linear clipping by the postsynaptic unit. 
The simulation algorithm was based on the model described in figure 2 and equation (1) but 
with the following modifications: 
 The experimentally determined type IV PST profile was substituted for K0 (this term 
represents the average combined influence of all extrinsic inputs to the type IV cell plus 
the intrinsic spontaneous rate). 
 An impulse response function estimated from the data was substituted for K 1 (this kernel 
is stationary in the simulation model). 
 The convolution of the experimentally determined type II spikes with the first-order 
kernel was used to perturb the output cell's stochastic intensity: 
dS2 (ui) =8 
where: dN2(t) = Real type II cell spike record, and 
Pl(0 = PST profile of real type IV cell. 
 The output process was simulated as a non-homogeneous Poisson process with )l(t) as 
its parameter. This process was modified by a 0.5 msec absolute dead time. 
 The simulated data were analyzed in the same manner as the real data. 
The dependence of synaptic efficacy on presynaptic rate in the simulated data was compared 
to the corresponding real data. In 1 out of the 3 cases, we observed an inverse relationship 
between input rate and efficacy despite the use of a stationary first order kernel in the 
simulation. The similarity between the real and simulated results for this one case suggests 
that the mechanism may be purely statistical rather than physiological (e.g., not presynaptic 
depletion or postsynaptic desensitization). The other 2 simulations did not yield a strong 
dependence of effectiveness on input rate and, hence, failed to mimic the experimental 
results. In these two cases, the results suggest that the mechanism is not due solely to 
clipping, but involves some additional, possibly physiological, mechanisms. 
18 
Sydorenko and Young 
4 CONCLUSIONS 
1) 
(2) 
(3) 
(4) 
(5) 
The mount of inhibition imparted to type IV units by individual presynaptic type II unit 
action potentials (expressed as the expected number of type IV spikes eliminated per type 
II spike) is inversely and roughly linearly related to the average rate of the type II unit. 
There is no evidence for second order synaptic effects at the type II spike rates tested. In 
other words, the inhibitory effect of two successive type II spikes is simply the linear 
sum of the inhibition imparted by each individual spike. 
There is no consistent relationship between type II / type IV synaptic efficacy and 
postsynapfic (type IV) rate. 
Simulations, in some cases, suggest that the inverse relationship between presynaptic rate 
and effectiveness may be reproduced using a simple statistical model of neural 
interaction. 
We found no evidence that would explain the discrepancy between Voigt and Young's 
results and Gochin's results in the DCN. Gochin observed correlogram features 
consistent with monosynaptic excitatory connections within the DCN when short tone 
bursts were used as stimuli. We did not observe excitatory features between any unit 
pairs using short tone bursts. 
Acknowledgements 
Dr. Alan Karr assisted in developing Eqns. 1-10. 
Research supported by NIH grant DCOO115. 
E. Nelken provided helpful comments. 
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