Correlations and synchrony in threshold neuron models.

Abstract

We study how threshold models and neocortical neurons transfer temporal and interneuronal input correlations to correlations of spikes. In both, we find that the low common input regime is governed by firing rate dependent spike correlations which are sensitive to the detailed structure of input correlation functions. In the high common input regime, the spike correlations are largely insensitive to the firing rate and exhibit a universal peak shape. We further show that pairs with different firing rates driven by common inputs in general exhibit asymmetric spike correlations.

Full text

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arXiv:0810.2901v3 [q-bio.NC] 17 Jul 2009
Correlations and Synchrony in Threshold Neuron Models
Tatjana Tchumatchenko,1,2Aleksey Malyshev,3,4Theo Geisel,1Maxim Volgushev,3,5,4and Fred Wolf1
1Max Planck Institute for Dynamics and Self-Organization and
Bernstein Center for Computational Neuroscience G¨ ottingen, Germany
2G¨ ottingen Graduate School for Neurosciences and Molecular Biosciences, Germany
3Inst. of Higher Nervous Activity and Neurophysiology, RAS, Moscow, Russia
4Dep. of Psychology, University of Connecticut, Storrs, USA
5Dep. of Neurophysiology, Ruhr-University Bochum, Germany
(Dated: July 17, 2009)
We study how threshold model neurons transfer temporal and interneuronal input correlations to
correlations of spikes. We find that the low common input regime is governed by firing rate dependent
spike correlations which are sensitive to the detailed structure of input correlation functions. In
the high common input regime the spike correlations are insensitive to the firing rate and exhibit a
universal peak shape independent of input correlations. Rate heterogeneous pairs driven by common
inputs in general exhibit asymmetric spike correlations.
PACS numbers: 87.19.lm, 87.19.ll, 05.40.-a, 87.19.lt, 87.85.dm
Neurons in the CNS exhibit temporally correlated ac-
tivity that can reflect specific features of sensory stim-
uli or behavioral tasks [1, 3]. Recently, the origin, sta-
tistical structure and coding properties of spike corre-
lations in neuronal systems have attracted substantial
attention [2, 4, 5]. How do neurons transfer correlated
inputs into correlated output? This fundamental ques-
tion is vital to understand the structure of network cor-
relations, yet it is unanswered. In the past, most theo-
retical analyses addressing this question utilized coupled
stochastic differential equations and the Fokker Planck
formalism [5, 7]. These approaches are technically very
demanding and are therefore practically restricted to sim-
ple stochastic processes, see e.g. [7]. As a result, explicit
expressions for quantities of interest are often lacking or
obtainable only in special limiting cases.
Here we show that an alternative modeling framework,
based on the theory of smooth random functions [8, 9],
can provide a mathematically transparent and highly
tractable description of spike correlations driven by in-
puts of arbitrary temporal structure and correlation
strength. Our theoretical findings may find applications
beyond neuroscience, e.g.
studies, as the statistics of (upward) level crossings is
a general, long standing problem in physics and engi-
neering [9]. We calculate quantities which have so far
escaped a theoretical description by competing Fokker-
Planck based formalism: 1) peak spike correlation for
arbitrary input correlation strength, 2) rate independent
peak shape in the high correlation regime, 3) complete
spike correlation function for weak correlations, 4) asym-
metric spike correlation function in rate inhomogeneous
pairs. A priori, the simple threshold model we use cannot
be expected to completely capture the complex behavior
of cortical neurons. Remarkably, our results reproduce
and extend previous reports on the firing rate depen-
dence of cortical spike correlations [4, 5]. Furthermore,
in spin ordering, reliability
−60
−20
−40
−20
V in mV


01234
−60
−40
t[s]


r≈1
r≈0.3
B
FIG. 1:
which share a component nC and have a correlation strength
r(r > 0). The correlated currents are injected successively
into neurons. (B) MP traces of two neurons (red and blue)
recorded in response to correlated fluctuating current with
r ≈ 1 and r ≈ 0.3. In all four recordings ν ≈ 5Hz, τs=20ms
and CI(τ) as in Eqs. 1,4; spikes are truncated at −10mV
(A) Paradigm of two fluctuating current traces
we were able to qualitatively confirm all new predictions
in our in vitro experiments.
Framework– We model the membrane potential (MP) of
a neuron by a temporally continuous, stationary Gaus-
sian random function V (t) with temporal correlation
C(τ)=?V (0)V (τ)? and zero mean, which we term Gaus-
sian Pseudo Potentials (GPPs); ?·? denotes the ensemble
average. Assuming a simple leaky integrator model with
a membrane time constant τM, τM˙V (t)=−V (t) + ξ(t).
The voltage and current statistics are linked by:
?CI(ω) =?C(ω)(1 + τ2
Mω2). (1)
where CI(τ) denotes the current correlation function
and ˜CI(ω) its Fourier transform.
tions we use common input.
ξj(t), j = 1,2 (1 and 2 denote noise input to neurons 1
and 2) are derived from three statistically independent
temporally correlated Gaussian processes n1,n2,ncwith
current correlation function CI(τ):
ξj(t) =√1 − rnj(t) +√rnc(t).
njare the individual noise components (as in Fig. 1 (A))
and ncthe shared component. Vjand njcan be related
To induce correla-
Two correlated currents
(2)

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using the membrane filter in Eq. 1. The corresponding
correlated voltages V1,V2 mimic the neuronal traces of
two neurons subject to common input. r modulates be-
tween full synchrony and asynchrony, 0≤r<1.
Model spike statistics– The simplest conceivable model
of spike generation from a fluctuating MP identifies the
spike times with upward threshold crossings of V (t). The
spike times tjare then given by the spike measure
?
s(t) =δ(t − tj) = δ(V (t) − ψ0)|
˙
V (t)|θ(˙V (t)), (3)
where ψ0is the distance to threshold relative to ?V (0)?,
δ(·) and θ(·) are the Dirac delta and Heaviside theta func-
tions, respectively. In contrast to the classical integrate-
and-fire (IF) model this model has no reset [7] but ex-
hibits an intrinsic silence period after a spike. We as-
sume a smooth C(τ) such that the variance ?˙V (t)˙V (t)? =
−C′′(0) exist and V (t) has a finite rate of thresh-
old crossings.The differential correlation time τs =
?C(0)/|C′′(0)| describes the decay of the correlation
function near τ=0. For numerical simulations and exper-
imental testing we choose the following selfcharacteristic,
exponentially decaying correlation function:
c(τ) = 1/cosh(τ/τs),?V (t)V (t + τ)? = σ2
Vc(τ). (4)
However, most results are independent of the particular
shape of C(τ). All spike correlations between two neu-
rons can be obtained via the covariance matrix C and the
joint probability density of?V =V1(0),˙V1(0),V2(τ),˙V2(τ)
p(?V )=exp(−?VTC−1?V /2)/(4π2√DetC) where



C′
C =

σ2
0
V1
0C12(τ)
−C′
σ2
0
C′
12(τ)
12(τ)
0
σ2
˙V2
σ2
˙V1
12(τ)
12(τ)
12(τ) −C′′
V2
C12(τ) −C′
12(τ) −C′′




(5)
Here C′
poral derivative of C1,2(τ), respectively. Note that Eq. 5
implies that all pairwise correlations can be expressed
as a functional of C12(τ),C′
correlation function C12(τ) is:
12(τ) and C′′
12(τ) denote the first and second tem-
12(τ),C′′
12(τ). Voltage cross
C12(τ) = ?V1(0)V2(τ)? = rσV1σV2c(τ).
The firing rate ?s(t)? of one neuron is then:
ν = ?s(t)? = exp(−ψ2
The firing rate in Eq. 7 depends only on two parame-
ters: the correlation time and the threshold-to-variance
ratio ψ2
V, but not on the specific choice of the cor-
relation function [9]. Hence, processes with the same
τs but different form of C(τ) will have the same ν, de-
spite different temporal spike statistics. The threshold-
to-variance ratio basically determines the probability of
threshold crossings. A decrease in ψ0/σV leads to an in-
crease in ν. Injections of constant currents shift the mean
(6)
0/(2σ2
V))/(2πτs) (7)
0/σ2
potential and thus decrease the distance to threshold ψ0
resulting in a higher ν. ν is also increasing with decreas-
ing τs, because faster fluctuations lead to higher rate of
threshold crossings. This model has a maximal firing
rate ˜ ν = 1/(2πτs), which corresponds to the upward zero
crossings of the random process. It should therefore be
used in the fluctuation driven, low firing rate ν<˜ ν regime
which is prevalent in cortical neurons [6].
Experimental test– To access spike correlations in real
neurons in vitro, we made whole-cell recordings from
layer 2/3 pyramidal neurons (N=19) in neocortical slices
from rats (PND 22-27). Correlated inputs to neurons
were mimicked by injection of digitally synthesized sets
of fluctuating currents with CI(τ) (Eqs. 1,4), varying the
correlation parameter (r ∈ {0.3,1}), and time constants
τM=20ms, τs={20,100}ms.
generated by specifying the Fourier spectrum and trans-
ferring to the time domain as described in [9], such that
the voltage correlation function C(τ) of the neurons was
similar to the model correlation function (Eq. 4). During
the recording (10s or 5s episodes), we targeted two dif-
ferent firing rates ν1=5Hz (T1=10s), ν2=10Hz (T2=5s),
by injection of an additional constant current. The aver-
age firing rate obtained is denoted by νm. We obtained
a total of N=281 recordings for ν1, N=299 recordings
for ν2. For identical noise injection (r=1) we recorded
N=80 at the target rate ν1, N=81 at ν2, and N=21 at
a target rate of 3Hz (T3=T1). Fig. 1 shows examples of
recorded voltage traces for large and small r. We calcu-
lated νcond(τ) using a Gaussian filter kernel with σ=5ms
and 95% Jackknife confidence intervalls for N random
subsamples each containing N/2 recordings. Experimen-
tal results are compared with model predictions in high
and low correlation regimes.
Correlated currents were
0 0.510
200
0 0.5
r
1
0
50
100
νcond (0) [Hz]
A
ν
τs
0
(4πτs)-1
ν
(2πτs)-1
0
(8τs)-1
(4τs)-1
g(0)
A
B
νmax
gmax
01020
0
0.5
1
1.5
τ[ms]
rg(τ) [Hz]
ν=15 Hz
ν=3 Hz
ν=1Hz
C
0 1020
0
0.5
1
1.5
τ[ms]


τs=5 ms
τs=10 ms
τs=20 ms
FIG. 2: Simulated and analytical spike cross correlations of
two correlated neurons (ν1=ν2). (A) νcond(0) vs. r as in Eq. 9;
νcond(0) for a fixed τs=10ms and ν=0.1Hz(red), ν=1Hz(blue),
ν=10Hz(black). (inset) νcond(0) for a fixed ν=5Hz and
τs=10ms(red), τs=5ms(blue), τs=1ms(black). Circles denote
simulated results. (B) g(0) vs. ν for fixed τs and small r. In
red νmax and gmax, simulation results from (C) are shown as
corresponding symbols. (C) rg(τ) vs. τ as in Eq. 12 and sim-
ulated results for r=0.05 (left) ν=15,3,1Hz, τs=10ms. (right)
τs=5,10,20ms, ν=5Hz.

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Spike correlations– To quantify the temporal spike cross
correlations between neuron 1 and 2 we used the con-
ditional firing rate νcond(τ), which is the firing rate of
neuron 2 triggered on the spikes of neuron 1:
νcond(τ) = (ν1ν2)−1/2?s1(t)s2(t + τ)?
This quantity measures correlations on all time scales
and is equivalent to the spike count correlation coeffi-
cient ρ (ρ≈T(νcond(0)−ν)) for small time bin T. For a
pair of identical neurons νcond(τ) is a symmetrical func-
tion which approaches ν as τ increases and maximally
deviates from ν at τ=0. We obtain νcond(0) by solving
the Gaussian integrals in Eq. 3,5 for any r:
?ν
˜ ν
(8)
νcond(0) = ˜ ν
?R
[1 + 2rarctan(
√
R−1)/
?
1 − r2], (9)
where R = (1 − r)/(1 + r). Eq. 9 predicts a superlinear
increase of νcond(0) with r, see Fig. 2(A).
Strong cross correlations r≈1– In this limit we find that
the peak spike correlations are independent of ν and of
the particular shape of C12(τ).
νcond(0) ≈ 1/(2√2√1 − rτs)
Do these predictions hold for neuronal spike correlations?
Fig. 3 (left) depicts recorded νcond(τ) in the high r regime
for different firing rates. The correlation peaks for 3,5.3
and 8.5Hz are essentially identical (Fig. 3 (left)). These
recordings confirm the prediction that the amplitude is
insensitive to the firing rate. Additionally, the peak form
suggests that there is a rate independent universal corre-
lation peak shape. To assess this possibility theoretically,
we calculate νcond(τ) by solving the Gaussian integrals in
Eq. 3,5 for r ≈ 1 and τ ≪ τs. We obtain:
νcond(τ) = 1/(2τ∗
(r ≈ 1) (10)
s)(1 − 3/2ˆ τ2+ 30/16ˆ τ4) + O(τ6) (11)
where ˆ τ=τ/τ∗with the time constant τ∗
For r = 1 the δ-peak present at the origin of the auto
conditional firing rate is recovered. Eq. 11 indeed demon-
states the existence of a rate independent universal peak
s=√2√1 − rτs.
−2000200
0
5
10
15
20
25
τ [ms]
νcond(τ) [Hz]
νm =8.5±2 Hz
νm =5.3±1.5 Hz
νm =3±0.7 Hz
−2000 200 100−100
5
10
15
20
25
τ [ms]
νcond(τ) [Hz]
ν=6 Hz
ν=3Hz
FIG. 3: Firing rate independence of spike correlations for
strong input correlations, theory and experiment.
νcond(τ) vs.τ calculated for identical current injections
in recorded neurons. (right) νcond(τ) for simulated pair of
neurons for ν1 = 3Hz (black) and ν2=6Hz (blue), 12.5 ≤
√1 − rτs≤12.8 with 20 ≤τs≤ 25ms and 0.59≤r≤0.75. The
overlap of both families of curves is plotted in red.
(left)
shape and height, determined by τs. Note, that Eq. 11
predicts that correlation peak shape is insensitive to the
functional form of C12(τ).
agreement between theory and experiment is, we com-
puted νcond(τ) from simulated pairs and found a good
qualitative agreement with experimental findings for a
broad range of parameters. The salient correlation peak
structure can be faithfully reproduced by our framework
and the additional weak periodic modulation of the ex-
perimental stationary rates might be due to additional
ion conductances of real neurons. The width and hight
of the common peak can be qualitatively described by the
theoretical curves in Fig. 3 (right). We note, that r≈0.7
in the simulations and in the experiments (0.6<r< 0.9)
were similar and both were close to the typical corre-
lation coefficient (ρ ≈ 0.6 for time bin 40ms) reported
recently for cell pairs subject to identical white noise cur-
rents [Fig.1d in [5]].
Low correlation strength– In this limit, νcond(τ) recov-
ers the rate dependence and shows sensitivity to c(τ).
This is valid for rc(τ) ≪ 1.
ν + rg(τ) + O(r2) by solving the Gaussian integrals in
Eq. 3, 5. Here, g(τ):
To explore how close the
We obtain νcond(τ) =
g(τ) = ν(c(τ)|2log(ν/˜ ν)| − π/2τ2
g(0) = ν(|2log(ν/˜ ν)| + π/2)
Fig. 2(C) illustrates the dependence of νcond(τ) on τsand
ν. Eq. 12 implies that the spikes are typically correlated
on a shorter time scale than the underlying MPs, due
to the admixture of c′′(τ) which has a shorter time scale
than c(τ). For low rates, the contribution of c′′(τ) is
negligible, but already at 20% of ˜ ν the influence of c′′(τ)
cannot be neglected leading to a sharpening of spike cor-
relations. Notably, the spike correlations with temporal
widths much smaller than the underlying voltage cor-
relations have been previously observed in vivo[p. 367
sc′′(τ))(12)
(13)
−200 −1000 100200
0
2
4
τ [ms]
νcond(τ) −ν [Hz]
τs=20 ms
r=0.2
ν=3 Hz
A
−200 −1000100200
0
0.5
1
τ [ms]
( νcond(τ) −ν)/ν
τs=20 ms
r=0.2
ν=3 Hz
−200
νm=7.56 Hz±1.3 Hz
−1000100
τs=100 ms
200
0
2
4
νcond(τ) −ν [Hz]
−400−2000200 400
−1
0
1
2
τ [ms]
νm=2.6 Hz± 0.7 Hz
νm=9.4 Hz±1.5 Hz
νm=2.96 Hz± 0.8 Hz
B
τs=20 ms
−200 −1000100
τs=100 ms
200
0
0.4
(νcond(τ)−ν)/ν
0 200−200 −400400
0
0.4
τ [ms]
C
τs=20 ms
FIG. 4:
small r, theory and experiment. (A) νcond(τ) − ν vs τ and
(νcond(τ) − ν)/ν vs τ for ν = 3Hz,τs = 20ms; simulation re-
sults are superimposed. (B,C) measured νcond(τ) − ν vs. τ
and (νcond(τ) − ν)/ν vs. τ for τs ∈ {20,100}ms; computed
from pairs of patch clamp recordings.
Firing rate dependence of spike correlations for

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4
−100−50050100
0
2
4
τ [ms]
νcond(τ)−(ν1⋅ν2)1/2
−5.5 ms
τs=20 ms
−200−1000100200
−1
0
1
τ [ms]
−100−50
−44 ms
050 100
0
2
4
νcond(τ)−(ν1ν2)1/2
τs=100 ms
τs=20 ms
−8 ms
ν1=2.96 Hz
ν2=9.4 Hz
ν1=2.6 Hz
ν2=7.56 Hz
FIG. 5: Difference in firing rate leads to asymmetric spike
correlations. (left) νcond(τ)−√ν1ν2 vs. τ, simulation results
and theoretical second order prediction for r=0.2, ν1=2.65Hz,
ν2=3·2.65Hz and τs=20ms. Arrows denote the peak position.
(right) νcond(τ)−√ν1ν2 vs. τ. Experimental results for τs =
20,100ms for different rates.
in [3]]. Eq. 13 also implies an increase of correlation of
spikes with ν (Fig. 2 B). However, the percentage of si-
multaneous spikes ((νcond(τ) − ν)/ν) is higher for lower
rates, because at low firing rates (ψ0≫σV) the additional
common component nc(t) is more critical for reaching
the threshold, than ni(t). g(0) attains a maximal value
gmax=2νmax for the rate νmax=(2πτs)−1exp(π/4 − 1)
(Fig. 2,A). This simple model qualitatively captures the
salient correlation peak for low firing rates recorded in
neurons subject to weak correlated input(Fig. 2(C)), the
presence of weak damped periodic modulation might be
due to additional ion conductances of real neurons. The
ν-dependence predicted by Eq. 13 is also evident in Fig. 4
(B), even though νM = 9.4Hz escapes direct compar-
ison (νm>˜ ν): with increasing ν, νcond(0)−ν increases
and (νcond(0) − ν)/ν is decreasing with ν.
sults are consistent with recent reports (Fig.1c in [5] and
Fig.3(B,C) in [4]).
Rate asymmetry– The correlation matrix in Eq. 5 in-
cludes c′(τ), which so far did not enter νcond(τ).
c′(τ) is an antisymmetric function it is conceivable to
assume that a broken symmetry (ν1 ?= ν2) will lead to
asymmetric νcond(τ).In the low r regime, we obtain
g(τ)=(νcond(τ)−√ν1ν2)/r by solving the Gaussian inte-
grals in Eq. 3,5. g(τ) then is:
g(τ) =√ν1ν2(c(τ)e1e2− π/2τ2
τPeak= τs∆/(e1e2+ πc(4)(0)τ4
where ei=ψ0/σViand ∆=?π/2(e2−e1). The peak posi-
tion is no longer at τ=0 but is shifted to τPeak. c′(τ) leads
to a temporal delay of spikes indicating that the spikes
of the higher rate neuron precede spikes of the lower rate
neuron, despite the perfect synchrony of the common in-
put. The asymmetry increases with increasing difference
of threshold-to-variance ratios and increasing τs. The-
oretically predicted asymmetric νcond(τ) (Fig. 5 (left))
is in good agreement with experimental results (Fig. 5
(top right)). Measured νcond(τ) also shows an increase
of the temporal shift with τs. This τs dependence is in
qualitative agreement with Eq. 15, despite the fact that
experiments with τs=100ms escape direct quantitative
Both re-
As
sc′′(τ) − c′(τ)τs∆) (14)
s/2).(15)
comparison as νm>˜ ν≈1.6Hz. Notably, shifted correla-
tions are well known in the biological literature and are
often interpreted as indications of synaptic connections or
the presence of delayed inputs [10]. However, our frame-
work reveals a potential mechanism for the occurrence
of asymmetric correlations in pairs with synchronous in-
puts [3].
Discussion– We presented a framework for the descrip-
tion of the auto- and cross correlations of upward level
crossings with arbitrary functional form of input corre-
lations. Our results confirm previous reports on the rate
dependence of spike correlations [4, 5]. This behavior,
however, holds for weak correlations only. With strongly
correlated inputs, spike correlations become independent
of the firing rate but depend on the correlation time of
voltage fluctuations. In cell pairs with rate differences
the temporal symmetry of spike correlations is lost. Fi-
nally, let us stress that input correlations modeled in our
framework do not imply a particular connectivity as they
can arise from common input or reciprocal connections.
Identifying self-consistent choices of c(τ), r, ψ0,i, σV,iin
a network of prescribed connectivity will be a fruitful di-
rection of future research.
Acknowledgments We thank E. Nikitin for help conduct-
ing the experiments, I. Fleidervish, M.Gutnick, A. Witt,
M. Huang, W. Wei, B. Kriener, W. Keil for fruitful dis-
cussions and A. Witt for help with statistical analysis.
We thank BMBF, GIF and Max Planck Society for sup-
port.
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