27O 
Correlational Strength and Computational Algebra 
of Synaptic Connections Between Neurons 
Eberhard E. Fetz 
Department of Physiology & Biophysics, 
University of Washington, Seattle, WA 98195 
ABSTRACT 
Intracellular recordings in spinal cord motoneurons and cerebral 
cortex neurons have provided new evidence on the correlational strength of 
monosynaptic connections, and the relation between the shapes of 
postsynaptic potentials and the associated increased firing probability. In 
these cells, excitatory postsynaptic potentials (EPSPs) produce cross- 
correlogram peaks which resemble in large part the derivative of the EPSP. 
Additional synaptic noise broadens the peak, but the peak area -- i.e., the 
number of above-chance firings triggered per EPSP -- remains proportional to 
the EPSP amplitude. A typical EPSP of 100 gv triggers about .01 firings per 
EPSP. The consequences of these data for information processing by 
polysynaptic connections is discussed. The effects of sequential polysynaptic 
links can be calculated by convolving the effects of the underlying 
monosynaptic connections. The net effect of parallel pathways is the sum of 
the individual contributions. 
INTRODUCTION 
Interactions between neurons are determined by the strength and 
distribution of their synaptic connections. The strength of synaptic 
interactions has been measured directly in the central nervous system by two 
techniques. Intracellular recording reveals the magnitude and time course of 
postsynaptic potentials (PSPs) produced by synaptic connections, and cross- 
correlation of extracellular spike trains measures the effect of the PSP's on the 
firing probability of the connected cells. The relation between the shape of 
excitatory postsynaptic potentials (EPSPs) and the shape of the cross- 
correlogram peak they produce has been empirically investigated in cat 
motoneurons 2,4,5 and in neocortical cells 10. 
RELATION BETWEEN EPSP'S AND CORRELOGRAM PEAKS 
Synaptic interactions have been studied most thoroughly in spinal 
cord motoneurons. Figure I illustrates the membrane potential of a 
rhythmically firing motoneuron, and the effect of EPSPs on its firing. An 
EPSP occurring sufficiently close to threshold () will cause the motoneuron 
to fire and will advance an action potential to its rising edge (top). 
Mathematical analysis of this threshold-crossing process predicts that an 
EPSP with shape e(t) will produce a firing probability f(t), which resembles 
 American Institute of Phys. ics 1988 
271 
EPSP 
CROSS- 
CORRELOGRAM 
f(t) 
TIME '1- 
Fig. 1. The relation between EPSP's and motoneuron firing. Top: membrane trajectory of 
rhythmically firing motoneuron, showing EPSP crossing threshold (O) and shortening the 
normal interspike interval by advancing a spike. V(t) is difference between membrane 
potential and threshold. Middle: same threshold-crossing process aligned with EPSP, with 
v(t) plotted as falling trajectory. Intercept (at upward arrow) indicates time of the advanced 
action potential. Bottom: Cross-correlation histogram predicted by threshold crossings. The 
peak in the firing rate fit) above baseline (fo) is produced by spikes advanced from baseline, 
as indicated by the changed counts for the illustrated trajectory. Consequently, the area in 
the peak equals the area of the subsequent trough. 
272 
the derivative of the EPSP 4,8. Specifically, for smooth membrane potential 
trajectories approaching threshold (the case of no additional synaptic noise): 
f(t) = fo + fro/:g) de/dt 
(1) 
where fo is the baseline firing rate of the motoneuron and ' is the rate of 
closure between motoneuron membrane potential and threshold. This 
relation can be derived analytically by tranforming the process to a 
coordinate system aligned with the EPSP (Fig. 1, middle) and calculating the 
relative timing of spikes advanced by intercepts of the threshold trajectories 
with the EPSP 4. The above relation (1) is also valid for the correlogram 
trough during the falling ph. ase of the EPSP, as long as de/dt > -{'; if the EPSP 
falls more rapidly than -v, the trough is limited at zero firing rate (as 
illustrated for the correlogram at bottom). The fact that the shape of the 
correlogram peak above baseline matches the EPSP derivative has been 
empirically confirmed for large E?S?s in cat motoneurons 4. This relation 
implies that the height of the correlogram peak above baseline is proportional 
to the EPS? rate of rise. The integral of this relationship predicts that the area 
between the correlogram peak and baseline is proportional to the EPSP 
amplitude. This linear relation further implies that the effects of 
simultaneously arriving EPSPs will add linearly. 
The presence of additional background synaptic "noise", which is 
normally produced by randomly occurring synaptic inputs, tends to make the 
correlogram peak broader than the duration of the EPSP risetime. This 
broadening is produced by membrane potential fluctuations which cause 
additional threshold crossings during the decay of the EPSP by trajectories 
that would have missed the EPSP (e.g., the dashed trajectory in Fig. 1, 
middle). On the basis of indirect empirical comparisons it has been proposed 
6,7 that the broader correlogram peaks can be described by the sum of two 
linear functions of e(t): 
f(t) = fo + a e(t) + b de/dt 
(2) 
This relation provides a reasonable match when the coefficients (a and b) can 
be optimized for each case 5,7, but direct empirical comparisons 2,4 indicate 
that the difference between the correlogram peak and the derivative is 
typically briefer than the EPSP. 
The effect of synaptic noise on the transform 'between EPSP and 
correlogram peak has not yet been analytically derived (except for the case of 
Gaussian noisel). However the threshold-crossing process has been 
simulated by a computer model which adds synaptic noise to the trajectories 
intercepting the EPSP 1. The correlograms generated by the simulation match 
the correlograms measured empirically for small EPS?'s in motoneurons 2, 
confirming the validity of the model. 
Although synaptic noise distributes the triggered firings over a wider 
peak, the area of the correlogram peak, i.e., the number of motoneuron firings 
produced by an EPSP, is essentially preserved and remains proportional to 
EPSP amplitude for moderate noise levels. For unitary EPSP's (produced by 
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a single afferent fiber) in cat motoneurons, the number of firings triggered per 
EPSP (Np) was linearly related to the amplitude (h) of the EPSP 2: 
Np = (0.1/mv)o h (mv) + .003 
(3) 
The fact that the number of triggered spikes increases in proportion to E?S? 
amplitude has also been confirmed for neocortical neurons 10; for cells 
recorded in sensorimotor cortex slices (probably pyramidal cells) the 
coefficient of h was very similar: 0.07/rev. This means that a typical unitary 
EPSP with amplitude of 100 gv, raises the probability that the postsynaptic 
cell fires by less than .01. Moreover, this increase occurs during a specific 
time interval corresponding to the rise time of the EPS? - on the order of 1 - 2 
msec. The net increase in firing rate of the postsynaptic cell is calculated by 
the proportional decrease in interspike intervals produced by the triggered 
spikes 4. (While the above values are typical, unitary EPSP's range in size 
from several hundred gv down to undetectable levels of several gv., and 
have risetimes of .2 - 4 msec.) 
Inhibitory connections between cells, mediated by inhibitory 
postsynaptic potentials (IPSPs), produce a trough in the cross-correlogram. 
This reduction of firing probability below baseline is followed by a 
subsequent broad, shallow peak, representing the spikes that have been 
delayed during the IPSP. Although the effects of inhibitory connections 
remain to be analyzed more quantitatively, preliminary results indicate that 
small IPSP's in synaptic noise produce decreases in firing probability that are 
similar to the increases produced by EPS?'s 4,5. 
DISYNAPTIC LINKS 
The effects of polysynaptic links between neurons can be understood 
as combinations of the underlying monosynaptic connections. A 
monosynaptic connection from cell A to cell B would produce a first-order 
cross-correlation peak Pi(B[A,t), representing the conditional probability that 
neuron B fires above chance at time t, given a spike in cell A at time t = 0. As 
noted above, the shape of this first-order correlogram peak is largely 
proportional to the E?S? derivative (for cells whose interspike interval 
exceeds the duration of the EPSP). The latency of the peak is the conduction 
time from A to B (Fig. 2 top left). 
In contrast, several types of disynaptic linkages between A and B, 
mediated by a third neuron C, will produce a second-order correlation peak 
between A and B. A disynaptic link may be produced by two serial 
monosynaptic connections, from A to C and from C to B (Fig. 2, bottom left), 
or by a common synaptic input from C ending on both A and B (Fig. 2, 
bottom right). In both cases, the second-order correlation between A and B 
produced by the disynaptic link would be the convolution of the two first- 
order correlations between the monosynaptically connected cells: 
P2(BIA) = P(BIC) OP(CIA) (4) 
274 
As indicated by the diagram, the cross-correlogram peak P2(BIA,t) would be 
smaller and more dispersed than the peaks of the underlying first-order 
correlation peaks. For serial connections the peak would appear to the right 
of the origin, at a latency that is the sum of the two monosynaptic latencies. 
The peak produced by a common input typically straddles the origin, since its 
timing reflects the difference between the underlying latencies. 
Monosynaptic connection --> First-order correlation 
. : (AIB,t) = P(BIA,-t) 
: (B ,A, t) 
 , 
Disynaptic connection 
Serial connection 
Second-order correlation 
Common input 
. (C IA) 
: 
% 
(AIC) 
I% 
I \ 
' \./1 c) 
:  
I 
_/N. P2 (BIA) 
Fig. 2. Correlational effects of monosynaptic and disynaptic links between two neurons. 
Top: monosynaptic excitatory link from A to B produces an increase in firing probability of B 
after A (left). As with all correlograms this is the time-inverted probability of increased firing 
in A relative to B (right). Bottom: Two common disynaptic links between A and B are a 
serial connection via C (left) and a common input from C. In both cases the effect of the 
disynaptic link is the convolution of the underlying monosynaptic links. 
275 
This relation means that the probability that a spike in cell A will 
produce a correlated spike in cell B would be the product of the two 
probabilities for the intervening monosynaptic connections. Given a typical 
N of 01/EPSP, this would reduce the effectiveness of a given disynaptic 
p  
linkage by two orders of magnitude relative to a monosynaptic connection. 
However, the net strength of all the disynaptic linkages between two given 
cells is proportional to the number of mediating interneurons {C}, since the 
effects of parallel pathways add. Thus, the net potency of all the disynaptic 
linkages between two cells could approach that of a monosynaptic linkage if 
the number of mediating interneurons were sufficiently large. It should also 
be noted that some interneurons may fire more than once per E?S? and have 
a higher probability of being triggered to fire than motoneurons 11. 
For completeness, two other possible disynaptic links between A and B 
involving a third cell C may be considered. One is a serial connection from B 
to C to A, which is the reverse of the serial connection from A to B. This 
would produce a P2(B[A) with peak to the left of the origin. The fourth 
circuit involves convergent connections from both A and B to C; this is the 
only combination that would not produce any causal link between A and B. 
The effects of still higher-order polysynaptic linkages can be computed 
similarly, by convolving the effects produced by the sequential connections. 
For example, trisynaptic linkages between four neurons are equivalent to 
combinations of disynaptic and monosynaptic connections. 
The cross-correlograms between two cells have a certain symmetry, 
depending on which is the reference cell. The cross-correlation histogram of 
cell B referenced to A is identical to the time-inverted correlogram of A 
referenced to B. This is illustrated for the monosynaptic connection in Fig.2, 
top right, but is true for all correlograms. This symmetry represents the fact 
that the above-chance probability of B firing after A is the same as the 
probability of A firing before B: 
P(BIA, t) = P(AIB,-t) 
(5) 
As a consequence, polysynaptic correlational links can be computed as the 
same convolution integral (Eq. 4), independent of the direction of impulse 
propagation. 
PARALLEL PATHS AND FEEDBACK LOOPS 
In addition to the simple combinations of pair-wise connections 
between neurons illustrated above, additional connections between the same 
cells may form circuits with various kinds of loops. Recurrent connections 
can produce feedback loops, whose correlational effects are also calculated by 
convolving effects of the underlying synaptic links. Parallel feed-forward 
paths can form multiple pathways between the same cells. These produce 
correlational effects that are the sum of the effects of the individual 
underlying connections. 
The simplest feedback loop is formed by reciprocal connections 
between a pair of cells. The effects of excitatory feedback can be computed by 
276 
successive convolutions of the underlying monosynaptic connections (Fig. 3 
top). Note that such a positive feedback loop would be capable of sustaining 
activity only if the connections were sufficiently potent to ensure 
postsynaptic firing. Since the probabilities of triggered firings at a single 
synapse are considerably less than one, reverberating activity can be 
sustained only if the number of interacting cells is correspondingly increased. 
Thus, if the probability for a single link is on the order of .01, reverberating 
activity can be sustained if A and B are similarly interconnected with at least 
a hundred cells in parallel. 
Connections between three neurons may produce various kinds of 
loops. Feedforward parallel pathways are formed when cell A is 
monosynaptically connected to B and in addition has a serial disynaptic 
connection through C, as illustrated in Fig. 3 (bottom left); the correlational 
effects of the two linkages from A to B would sum linearly, as shown for 
excitatory connections. Again, the effect of a larger set of cells {C} would be 
additive. Feedback loops could be formed with three cells by recurrent 
connections between any pair; the correlational consequences of the loop 
again are the convolution of the underlying links. Three cells can form 
another type loop if both A and B are monosynaptically connected, and 
simultaneously influenced by a common interneuron C (Fig. 3 bottom right). 
In this case the expected correlogram between A and B would be the sum of 
the individual components -- a common input peak around the origin plus a 
delayed peak produced by the serial connection. 
Feedback loop 
] .1 (aim) P2(ata ) 
'..,.i "-.....' 
, ?'., 
I / " ." ' 
Li "'" 
I 
Parallel feedforward path 
P1 (BIA)+P2(BIA) 
Common input loop 
I 
P1 (BIA) +P2 (BIA) 
Fig. 3. Correlational effects of parallel connections between two neurons. Top: feedback 
loop between two neurons A and B produces higher-order effects equivalent to convolution 
of monosynaptic effects. Bottom: Loops formed by parallel feedforward paths (left) and by a 
common input concurrent with a monosynaptic link (right) produce additive effects. 
277 
CONCLUSIONS 
Thus, a simple computational algebra can be used to derive the 
correlational effects of a given network structure. Effects of sequential 
connections can be computed by convolution and effects of parallel paths by 
summation. The inverse problem, of deducing the circuitry from the 
correlational data is more difficult, since similar correlogram features may be 
produced by different circuits 9. 
The fact that monosynaptic links produce small correlational effects on 
the order of .01 represents a significant constraint in the mechanisms of 
information processing in real neural nets. For example, secure propagation 
of activity through serial polysynaptic linkages requires that the small 
probability of triggered firing via a given link is compensated by a 
proportional increase in the number of parallel links. Thus, reliable serial 
conduction would require hundreds of neurons at each level, with 
appropriate divergent and convergent connections. It should also be noted 
that the effect of interneurons can be modulated by changing their activity. 
The intervening cells need to be active to mediate the correlational effects. As 
indicated by eq. 1, the size of the correlogram peak is proportional to the 
firing rate (fo) of the postsynaptic cell. This allows dynamic modulation of 
polysynaptic linkages. The greater the number of links, the more susceptible 
they are to modulation. 
Acknowledgements: The author thanks Mr. Garrett Kenyon for stimulating 
discussions and the cited colleagues for collaborative efforts. This work was 
supported in part by NIH grants NS 12542 and RR00166. 
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