Effects of Firing Synchrony on Signal Propagation in Layered Networks 141 
Effects of Firing 
Propagation in 
Synchrony on Signal 
Layered Networks 
G. T. Kenyon,  E. E. Fetz, 2 R. D. Puff x 
Department of Physics FM-15, 2Department of Physiology and Biophysics SJ-40 
University of Washington, Seattle, Wa. 98195 
ABSTRACT 
Spiking neurons which integrate to threshold and fire were used 
to study the transmission of frequency modulated (FM) signals 
through layered networks. Firing correlations between cells in the 
input layer were found to modulate the transmission of FM sig- 
nals under certain dynamical conditions. A tonic level of activity 
was maintained by providing each cell with a source of Poisson- 
distributed synaptic input. When the average membrane depo- 
larization produced by the synaptic input was sufficiently below 
threshold, the firing correlations between cells in the input layer 
could greatly amplify the signal present in subsequent layers. When 
the depolarization was sufficiently close to threshold, however, the 
firing synchrony between cells in the initial layers could no longer 
effect the propagation of FM signals. In this latter case, integrate- 
and-fire neurons could be effectively modeled by simpler analog 
elements governed by a linear input-output relation. 
1 Introduction 
Physiologists have long recognized that neurons may code information in their in- 
stantaneous firing rates. Analog neuron models have been proposed which assume 
that a single function (usually identified with the firing rate) is sufficient to char- 
acterize the output state of a cell. We investigate whether biological neurons may 
use firing correlations as an additional method of coding information. Specifically, 
we use computer simulations of integrate-and-fire neurons to examine how various 
levels of synchronous firing activity affect the transmission of frequency-modulated 
142 Kenyon, tz and Puff 
(FM) signals through layered networks. Our principal observation is that for certain 
dynamical modes of activity, a sufficient level of firing synchrony can considerably 
amplify the conduction of FM signals. This work is partly motivated by recent 
experimental results obtained from primary visual cortex [1, 2] which report the 
existence of synchronized stimulus-evoked oscillations (SEO's) between populations 
of cells whose receptive fields share some attribute. 
2 Description of Simulation 
For these simulations we used integrate-and-fire neurons as a reasonable compro- 
mise between biological accuracy and mathematical convenience. The subthreshold 
membrane potential of each cell is governed by an over-damped second-order dif- 
ferential equation with source terms to account for synaptic input: 
where b is the membrane potential of cell k, N is the number of cells, Tj is the 
synaptic weight from cell j to cell k, tj are the firing times for the jth cell, Tp is the 
synaptic weight of the Poisson-distributed input source, p are the firing times of 
Poisson-distributed input, and r, and rd are the rise and decay times of the EPSP. 
The Poisson-distributed input represents the synaptic drive from a large presynaptic 
population of neurons. 
Equation 1 is augmented by a threshold firing condition 
= o - q) 
if 4(t! > 0 
then { (2) 
+ 0+) = 
where O(t- t) is the threshold of the k th cell, and r, is the absolute refractory 
period. If the conditions (2) do not hold then b continues to be governed by 
equation 1. 
The threshold is oo during the absolute refractory period and decays exponentially 
during the relative refractory period: 
{ o, if t - t[ < r.; 
O(t - t) = Ope-('-g.)/*, + 0o, otherwise, (3) 
where, O0 is the resting threshold value, 0p is the maximum increase of 0 during the 
relative refractory period, and r is the time constant characterizing the relative 
refractory period. 
2.1 Simulation Parameters 
and rd are set to 0.2 msec and 1 msec, respectively. Tp and Tij are always 
(1/100)0o. This strength was chosen as typical of synapses in the CNS. To sustain 
Effects of Firing Synchrony on Signal Propagation in Layered Networks 143 
IO 
o 
o 
Mode I b) Mode lI 
<v.>  85 msec -a <v.>  .023 _ 
-, [  I ,  t I - 
20 40 0 20 40 
(meec) (msec) 
Figure 1: Example membrane potential trajectories for two different modes of 
activity. EPSP's arrive at mean frequency, v,,, that is higher for mode I (a) than for 
mode II (b). Dotted line below threshold indicates asymptotic membrane potential. 
activity, during each interval rt, a cell must receive  (Oo/T,) = 100 Poisson- 
distributed inputs. Resting potential is set to 0.0 mV and 00 to 10 mV. ;fi, and 
, are set to 0.0 mV and -1.0 mV/msec, which simulates a small hyperpolarization 
after firing. r and rp were each set to 1 msec, and 0p to 1.0 reV. 
3 Response Properties of Single Cells 
Figure 1 illustrates membrane potential trajectories for two modes of activity. In 
mode I (fig. la), synaptic input drives the membrane potential to an asymptotic 
value (dotted line) within one standard deviation of 0. In mode II (fig. lb), the 
asymptotic membrane potential is more than one standard deviation below 0. 
Figure 2 illustrates the change in average firing rate produced by an EPSP, as 
measured by a cross-correlation histogram (CCH) between the Poisson source and 
the target cell. In mode I (fig. 2a), the CCH is characterized by a primary peak 
followed by a period of reduced activity. The derivative of the EPSP, when mea- 
sured in units of 0, approximates the peak magnitude of the CCH. In mode II 
(fig. 2b), the CCH peak is not followed by a period of reduced activity. The EPSP 
itself, measured in units of 0o and divided by 'd, predicts the peak magnitude of 
the CCH. The transform between the EPSP and the resulting change in firing rate 
has been discussed by several authors [3, 4]. Figures 2c and 2d show the cumula- 
tive area (CUSUM) between the CCH and the baseline firing rate. The CUSUM 
asymptotes to a finite value, A, which can be interpreted as the average number of 
additional firings produced by the EPSP. 
/x increases with EPSP amplitude in a manner which depends on the mode of 
activity (fig. 2e). In mode II, the response is amplified for large inputs (concave 
up). In mode I, the response curve is concave down. The amplified response to large 
inputs during mode II activity is understandable in terms of the threshold crossing 
mechanism. Populations of such cells should respond preferentially to synchronous 
synaptic input [5]. 
144 Kenyon, Fetz and Puff 
o 6 o 6 
b) 
.O75 
.050 
.026 
CCH d) CUSUM 
_ e/dt 
.0! 
',l,,,, {,H 
o 5 o 5 
(mec) 
mode 1 
Figure 2: Response to EPSP for two different modes of activity. 
Cross-correlogram with Poisson input source. 
c) and d) CUSUM computed from a) and b). 
modes of activity. 
0 , .1 
EPSP Amplitude in unJ of o 
a) 
and b) 
Mode I and mode II respectively. 
e) A vs. EPSP amplitude for both 
4 Analog Neuron Models 
The histograms shown in Figures 2a,b may be used to compute the impulse response 
kernel, U, for a cell in either of the two modes of activity, simply by subtracting the 
baseline firing rate and normalizing to a unit impulse strength. If the cell behaves 
as a linear system in response to a small impulse, U may be used to compute the 
response of the cell to any time-varying input. In terms of U, the change in firing 
rate, 6F, produced by an external source of Poisson-distributed impulses arriving 
with an instantaneous frequency F(t) is given by 
6F(t) = U(t- t')F,(t')Tedt' 
oo 
(4) 
where, Te is the amplitude of the incoming EPSP's. For the layered network used in 
our simulations, equation 4 may be generalized to yield an iterative relation giving 
the signal in one layer in terms of the signal in the previous layer. 
= :v v(t- (s) 
--013 
Effects of Firing Synchrony on Signal Propagation in Layered Networks 145 
.25 
.20 
.25 
.2 
.20 
0 4 0 4 0 4 
(msec) 
-4 0 4 -4 0 4 -4 0 4 
(msec) 
Figure 3: Signal propagation in mode I network. a) Response in first three layers 
due to a single impulse delivered simultaneously to all cells in the first layer. Ratio 
of common to independent input given by percentages at top of figure. First row 
corresponds to input layer. Firing synchrony does not effect signal propagation 
through mode I cells. Prediction of analog neuron model (solid line) gives a good 
description of signal propagation at all synchrony levels tested. b) Synchrony be- 
tween cells in the same layer measured by MCH. Firing synchrony within a layer 
increases with layer depth for all initial values of the synchrony in the first layer. 
where, tiFi is the change in instantaneous firing rate for cells in the ita layer, T/+x, 
is the synaptic weight between layer i and i + 1, and N is the number of cells per 
layer. Equation 5 follows from an equivalent analog neuron model with a linear 
input-output relation. This convolution method has been proposed previously [6]. 
5 Effects of Firing Synchrony on Signal Propagation 
A layered network was designed such that the cells in the first layer receive impulses 
from both common and independent sources. The ratio of the two inputs was 
adjusted to control the degree of firing synchrony between cells in the initial layer. 
Each cell in a given layer projects to all the cells in the succeeding layer with equal 
strength,  All simulations use $0 cells per layer. 
Figure 3a shows the response of cells in the mode I state to a single impulse of 
strength  delivered simultaneously to all the cells in the first layer. In this and 
all subsequent figures, successive layers are shown from top to bottom and synchrony 
(defined as the fraction of common input for cells in the first layer) increases from 
146 Kenyon, Fetz and Puff 
a I 20 40 
1 20 40 
-404 -404 -404 
Figure 4: Signal propagation in mode II network. Same organization as fig. 3. 
a) At initial levels of synchrony above , 30%, signal propagation is amplified sig- 
nificantly. The propagation of relatively asynchronous signals is still adequately 
described by the analog neuron model. b) Firing synchrony within a layer increases 
with layer depth for initial synchrony levels above  30%. Below this level syn- 
chrony within a layer decreases with layer depth. 
left to right. Figure 3a shows that signals propagate through layers of interneurons 
with little dependence on firing synchrony. The solid line is the prediction from an 
equivalent analog neuron model with a linear input-output relation (eq. 5). At all 
levels of input synchrony, signal propagation is reasonably well approximated by 
the simplified model. 
Firing synchrony between cells in the same layer may be measured using a mass 
correlogram (MCH). The MCH is defined as the auto-correlation of the population 
spike record, which combines the individual spike records of all cells in a given layer. 
Figure 3b shows that for all initial levels of synchrony produced in the input layer, 
the intra-layer firing synchrony increased rapidly with layer depth. 
The simulations were repeated using an identical network, but with the tonic level 
of input reduced sufficiently to fix the cells in the mode II state (fig. 4). In contrast 
with the mode I case, the effect of firing synchrony is substantial. When firing is 
asynchronous only a weak impulse response is present in the third layer (fig. 4a, 
bottom left), as predicted by the analog neuron model (eq. 5). For levels of input 
synchrony above  30%, however, the response in the third layer is substantially 
more prominent. A similar effect occurs for synchrony within a layer. At input 
Effects of Firing Synchrony on Signal Propagation in Layered Networks 147 
8,) 1. ;o 4og b' lg 2o, 4o 
0 4 8o 4 80 4 8 
(msec) 
0 4 8 0 4 8 0 4 8 
(msec) 
Figure 5: Propagation of sinusoidal signals. Similar organization to figs. 3,4. Top 
row shows modulation of input sources. a) Mode I activity. Signal propagation is 
not significantly influenced by the level of firing synchrony. Analog neuron model 
(solid line) gives reasonable prediction of signal tranmission. b) Mode II activity. 
At initial levels of firing synchrony above  30%, signal propagation is amplified. 
The propagation of asynchronous signals is still well described by the analog neuron 
model. Period of applied oscillation - 10 msec. 
synchrony levels below . 30%, firing synchrony between cells in the same layer 
(fig. 4b) falls off in successive layers. Above this level, however, synchrony grows 
rapidly from layer to layer. 
To confirm that our results are not limited to the propagation of signals generated 
by a single impulse, oscillatory signals were produced by sinusoidally modulating 
the firing rates of both the common and independent input sources to the first layer 
(fig. 5). In the mode I state (fig. 5a), we again find that firing synchrony does 
not significantly alter the degree of signal penetration. The solid line shows that 
signal transmission is adequately described by the simplified model (eqs. 4,5). In 
the mode I! case, however, firing synchrony is seen to have an amplifying effect 
on sinusoidal signals as well (fig. 5b). Although the propagation of asynchronous 
signals is well described by the analog neuron model, at higher levels of synchrony 
propagation is enhanced. 
148 Kenyon, Fetz and Puff 
6 Discussion 
It is widely accepted that biological neurons code information in their spike den- 
sity or firing rate. The degree to which the firing correlations between neurons can 
code additional information by modulating the transmission of FM signals, depends 
strongly on dynamical factors. We have shown that for cells whose average mem- 
brane potential is sufficiently below the threshold for firing, spike correlations can 
significantly enhance the transmission of FM signals. We have also shown that the 
propagation of asynchronous signals is well described by analog neuron models with 
linear transforms. These results may be useful for understanding the role played by 
synchronized SEO's in primary visual cortex [1, 2]. Such signals may be propagated 
more effectively to subsequent processing areas as a consequence of their relative 
synchronization. 
These observations may also pertain to the neural mechanisms underlying the in- 
creased levels of synchronous discharge of cerebral cortex cells observed in slow wave 
sleep [7]. Another relevant phenomenon is the spread of synchronous discharge from 
an epileptic focus; the extent to which synchronous activity is propagated through 
surrounding areas may be modulated by changing their level of activation through 
voluntary effort or changing levels of arousal. These physiological phenomena may 
involve mechanisms similar to those exhibited by our network model. 
Acknowledgement s 
This work is supported by an NIH pre-doctoral training grant in molecular bio- 
physics (grant  T32-GM 08268) and by the Office of Naval Research (contract 
 N 00018-89-J-1240). 
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