Neurocomputing 5254 (2003) 949 954

www.elsevier.com/locate/neucom

The spread of rate and correlation in stationary

cortical networks
Tom Tetzla$ , Michael Buscherm(ohle, Theo Geisel,
Markus Diesmann
Department of Nonlinear Dynamics, Max-Planck-Institut fur Stromungsforschung, Gottingen, Germany

Abstract
The analysis of the spatial and temporal structure of spike cross-correlation in experimental
data is an important tool in the exploration of cortical processing. Recent theoretical studies
investigated the impact of correlation between a$erents on the spike rate of single neurons
and the e$ect of input correlation on the output correlation of pairs of neurons. Here, this
knowledge is combined to a model simultaneously describing the spatial propagation of rate
and correlation, allowing for an interpretation of its constituents in terms of network activity.
The application to an embedded feed-forward network provides insight into the mechanisms
stabilizing its asynchronous mode.
c 2003 Elsevier Science B.V. All rights reserved.

Keywords: Rate model; Cross-correlation; Integrate-and-7re; Syn7re chain

1. Introduction
Equilibrated activity states in models of cortical networks are often described in
terms of 7ring rate or related measures. It has been shown (e.g. Refs. [5,7]) that
the transmission of 7ring rates by a single neuron is modulated by temporal relationships between its a$erents. Correlations between converging inputs generally
a$ect the magnitude of post-synaptic current <uctuations. Since spiking is driven by
both mean and variance of the membrane potential the output rate is not only determined by the 7ring rate of the inputs but also by the degree of correlation. Depending
on the network architecture the sets of a$erents of two di$erent neurons exhibit a

Corresponding author.
E-mail address: tom@chaos.gwdg.de (Tom Tetzla$).

c 2003 Elsevier Science B.V. All rights reserved.

0925-2312/03/$ - see front matter

doi:10.1016/S0925-2312(02)00854-8

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Tom Tetzla% et al. / Neurocomputing 5254 (2003) 949 954

certain overlap. These common input channels induce correlations between the total
inputs of the two neurons depending on the size of the overlap, the a$erent 7ring
rates, and on correlations inside each set of a$erents. As shown e.g. in Ref. [7]
such common input correlations are to some degree transmitted by the neuron pair.
In summary, there is a complex interplay between 7ring rates and correlations. Therefore, it appears questionable whether pure rate models can reveal a realistic picture
of network dynamics. In the present study we attempt to construct a model describing both variables simultaneously. While rate transmission is treated analytically, the
dependence of output correlation on input rate and input correlation is obtained by
two-neuron simulations. As an example application we investigate to what extent the
rate propagation in a feed-forward network is distorted if correlations are taken into
account.

2. Rate transmission
Both analytics and numerics are based on a leaky-integrate-and-7re neuron model
with
-function shaped synaptic currents [3]. Since amplitudes of post-synaptic potentials h(t) (PSPs) are chosen to be small compared to the distance # between reset
potential and spike threshold, membrane potential dynamics can be considered as a
di$usion process. In this framework the output rate out is approximately determined
by the mean  and variance 2 of the stationary membrane potential distribution (e.g.
Ref. [4]). Because of the linearity of the subthreshold dynamics,  and  can be calculated by linear 7lter theory, using h(t) as the

systems impulse response. This way the
statistical properties of the superposition X = iA xi of all processes xi arriving from
the set A of a$erents can be easily mapped onto those of the membrane potential.
Here, we assume each process xi to be stationary Poisson with rate i

. Hence, the mean
weighted number of spikes
X  in a time interval Ft is given by iA gi i Ft. For
simplicity, the synaptic weights gi are considered to be completely determined by the
pre-synaptic neuron. Denoting the coeHcient of correlation between processes xi and
xj by rij , the variance of X is given by



gi2 i Ft +
rij gi gj i j Ft:
(1)
Var[X ] =
i

i=j

Thus, the output rate out of a neuron depends on both the rates i of all its a$erents and the correlations rij between all a$erent pairs. The dependence of the output
rate out on the rate E of the total excitatory input is shown in Fig. 1A for a 7xed
Poisson inhibitory input. The curves refer to di$erent variance-to-mean ratios F (Fano
factors) of the excitatory input process. The solid line represents the Poisson case
(F = 1). An increase in variance, e.g. due to correlations between the input channels,
raises the probability of reaching spike threshold at lower mean membrane potentials.
Hence, the output rate starts rising at lower E (dashed curve: F = 2, dotted curve:
F = 3).

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951

Fig. 1. (A) Response rate out as a function of excitatory input rate E for di$erent Fano factors F = 1
(Poisson, solid curve), F = 2 (dashed) and F = 3 (dotted). Inhibitory input is a Poisson process with a
7xed rate I = 30:24 kHz. (B) Measured output correlation r (10; 000 spikes=neuron) depending on input
correlation R for three di$erent excitatory Poissonian inputs with rates E = 34:4 kHz (), 36 kHz ( )
and 39 kHz (+). The inset summarizes the result of the linear regression r =  R (error bars indicate 95%
con7dence interval). I = 30:24 kHz, # = 15 mV, m = 10 ms, ref = 2 ms, membrane capacity: 250 pF,
PSP amplitude: 0:14 mV, PSP rise time: 1:7 ms, correlation width F = 5:1 ms.

3. Correlation transmission
Given the sets Ak , Al of a$erent channels of two di$erent neurons k and l, we
de7ne the set Ckl of common inputs as the intersection Ak Al . We call the complements Bkl := Ak \ Ckl the sets of background processes. The total input processes
Xk=l can now

be subdivided into a common
and a background fraction: Xk=l =Ckl +Bkl=lk


(with Ckl := iCkl xi and Bkl=lk := iBkl=lk xi ). The coeHcient of correlation Rkl between Xk and Xl is given by
Rkl =

Var[Ckl ]
Cov[Xk ; Xl ]
=
:
Var[Xk ]Var[Xl ] Var[Ckl ] + Var[Bkl ]

(2)

For reasons of clarity, here we assumed that the three processes Ckl , Bkl and Blk
are pairwise independent and that the backgrounds are statistically equivalent, i.e.
Var[Bkl ] = Var[Blk ]. The degree of correlation induced by common input is thus determined by the ratio of the variances of the total common and background process. As
shown in Eq. (1) these variances in turn depend on the rates and correlations inside
the a$erent populations.
Since the dynamics of our model neuron is completely deterministic, the input correlations Rkl = 0 and 1 are mapped 1:1 to the output correlation rkl , independently of the
statistics of the individual inputs. It is tempting to assume that this is true for all values
of Rkl . However, several studies revealed that the gain of correlation is in general rather
low and is modulated by the mean and variance of the input processes [6,7]. With the

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Tom Tetzla% et al. / Neurocomputing 5254 (2003) 949 954

help of two-neuron simulations we measured the correlation transmission Rkl  rkl

and its dependence on a$erent 7ring rates. Here, both neurons k and l receive stationary Poisson inputs from a common excitatory channel C of rate MC and individual
background channels Bk=l . The latter are formed by superpositions of excitatory and inE
I
and Bk=l
. Their spike rates MBE and MBI are chosen to be identical
hibitory processes Bk=l
for both neurons. Excitatory and inhibitory synaptic weights di$er only in sign. Under
these conditions Eq. (2) reads R = MC =(MC + MBE + MBI ). Therefore, the degree of input
correlation can be tuned by adjusting the rates. Focussing on the dependence of the
correlation transmission on the total excitatory input rate ME = MC + MBE the inhibitory
input rate MI = MBI is kept constant. Simulations are performed on a temporal grid of
0:1ms resolution. Fig. 1B shows the measured output correlations r as a function of the
input correlation R. At low input correlations R the output r is well approximated by
the linear relation r =R. The inset in Fig. 1B shows the dependence of the 7tted slope
 on the excitatory input rate E . Results are consistent with those of former studies
[6,7]. It has to be emphasized that the maximum correlation gain max 0:3 is small
in the observed range. The 7ndings in Ref. [7] revealed that correlation transmission
is enhanced by increasing the magnitude of <uctuations of the stimuli. Our analysis is
restricted to Poisson inputs with -type cross-correlations.

4. Application to feed-forward networks

In the following example we show how the simultaneous transmission of rate and
correlation yields a model describing the spatial propagation of stationary spiking activity in a syn7re chain [1]. The common input Ci of two neurons a and b in the (i + 1)th
layer of a complete syn7re chain consists of the w channels ci; j (j {1; 2; : : : ; w})
from the preceding group i (see Fig. 2A). In addition, a and b receive independent

Fig. 2. (A) Syn7re link, (B) (; r)-phase-space of a syn7re chain of group size w =200 (log-scaled ordinate).
Gray trajectories indicate the spatial evolution from several initial conditions 0 ; r0 . Black dots mark the
positions of the two attractors. Their basins meet at the separatrix (thick black curve). States below the
dotted curve correspond to excitatory inputs with Fano factors F 2 (parameters as in Fig. 1).

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953

E
I
background inputs Ba=b
and Ba=b
. Keeping the total number of excitatory synapses 7xed
at KE and the number of inhibitory synapses at KI mean and variance of the resulting
input processes Xa=b are given by

Xa=b  = [(KE w)E + wi KI I ]Ft;

Var[Xa=b ] = [(KE w)E + (1 + ri [w 1])wi + KI I ]Ft;

E

(3)

I

Here,
and
denote the 7ring rates of the background neurons and i those of
the neurons in the ith layer. ri is the coeHcient of correlation between each pair
of neurons in that group. Intra-chain connections are purely excitatory. Again, each
individual process is assumed to be Poissonian. The resulting input correlation is


1
(KE w)E + KI I
Ri+1 (i ; ri ) = 1 +
:
(4)
wi (1 + ri [w 1])
Using the approach presented in Section 2 and the simulation results for the correlation
transmission an iterative map (i ; ri )w  (i+1 ; ri+1 )w can be constructed which simultaneously describes the propagation of 7ring rates and correlations along the chain. A
typical result is shown in Fig. 2B. The phase portrait illustrates the spatial evolution
of  and r for several initial conditions 0 ; r0 (gray trajectories). The map reveals two
attractors (black dots): one at a low rate of about 2 Hz and a correlation of almost
zero (we call this the ground state [8]) and one at large rates and higher correlations
(gray area marks its basin of attraction). The thick black line shows the position of the
separatrix. Its near horizontal shape is typical for the investigated system and arises
as a result of the low correlation gain. This 7nding explains the results of a former
study [8] which revealed that the transition from the asynchronous into the synchronous
regime is well determined by a pure rate model (r = 0). In our network model the stability of the ground state is barely a$ected by correlation perturbations and maintained
for a reasonable range of group sizes w (however, cf. [2]). Due to the restrictions on
Poisson processes our model is valid only at low rates and correlations. As a rough
estimate for deviations from the Poisson assumption we considered the Fano factors
F(; r) of the total excitatory input process in the di$erent regions of the (; r)-space
(Fig. 2B).
5. Discussion
In the present study we analyzed the interplay between spike rates and correlations. Transmission characteristics serve as ingredients for a general framework to
explore stationary states in cortical networks. Due to the requirement of Poissonian
spike count statistics the scope of this framework is restricted to low rates and correlations. Since cross-correlations exhibit 7nite temporal widths the reduction to a
single integrated quantitythe coeHcient of correlationconstitutes an important constraint of the present work. In addition, higher-order correlations, e.g. those enforced
by pairwise correlations, may play an important role in network dynamics. The present
study exclusively incorporates pairwise correlations. As suggested by the example, the

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Tom Tetzla% et al. / Neurocomputing 5254 (2003) 949 954

designed theory is nevertheless applicable to certain network structures and produces

reasonable results.
Acknowledgements
We acknowledge stimulating discussions with M. Abeles, Y. Aviel, M. Herrmann,
A. Kuhn, and C. Mehring.
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