PHYSICAL REVIEW E 81, 011916 共2010兲
Time scales of spike-train correlation for neural oscillators with common drive
Andrea K. Barreiro,* Eric Shea-Brown, and Evan L. Thilo
Department of Applied Mathematics, University of Washington, P.O. Box 352420, Seattle, Washington 98195, USA
共Received 22 July 2009; published 27 January 2010兲
We examine the effect of the phase-resetting curve on the transfer of correlated input signals into correlated
output spikes in a class of neural models receiving noisy superthreshold stimulation. We use linear-response
theory to approximate the spike correlation coefficient in terms of moments of the associated exit time problem
and contrast the results for type I vs type II models and across the different time scales over which spike
correlations can be assessed. We find that, on long time scales, type I oscillators transfer correlations much
more efficiently than type II oscillators. On short time scales this trend reverses, with the relative efficiency
switching at a time scale that depends on the mean and standard deviation of input currents. This switch occurs
over time scales that could be exploited by downstream circuits.
DOI: 10.1103/PhysRevE.81.011916
PACS number共s兲: 87.18.Sn, 05.10.Gg, 84.35.⫹i
I. INTRODUCTION
Throughout the nervous system, neurons produce spike
trains that are correlated from cell-to-cell. This correlation or
synchrony has received major interest because of its impact
on how neural populations encode information. For example,
correlations can strongly limit the fidelity of a neural code as
measured by the signal-to-noise ratio of homogeneous populations 关1–4兴. However, the presence or stimulus dependence
of correlations can also enhance coding strategies that rely
on discriminating among competing populations 关5–7兴; in
general, the effects of correlation on coding are complex and
can be surprisingly strong 关5–16兴. In addition, stimulusdependent correlations can modulate or carry information directly 关17–24兴. Correlations also play a major role in how
signals are transmitted from layer-to-layer in the brain
关25–27兴.
What is the origin of correlated spiking? One natural
mechanism is the overlap in the inputs to different neurons—
these common inputs can drive common output spikes. This
poses the question: how does the process of transferring of
input correlations to spike train correlations depend on the
nonlinear dynamics of individual neurons? Such correlation
transfer has been modeled in integrate-and-fire-type neurons
关20,27–31兴 and, very recently, in phase reductions in neural
oscillators 关32,33兴.
In particular, 关32,33兴 contrast the correlated activity
evoked in neural oscillators with type I 共i.e., always positive兲
vs type II 共i.e., positive and negative兲 phase-resetting curves
共PRCs兲. When correlations are measured via equilibrium
probability distributions of pairs of neuron phases or via
cross-correlation functions of these phases over time, type II
oscillators display relatively higher levels of correlation
关32,33兴. These results for phase correlation imply a similar
finding for spike train correlation in the limit of very short
time scales 共the connection arises because the spike train
cross-correlation function at ␶ = 0 can be related to the probability that phases will be nearly coincident for the two cells
关34兴.兲
*[email protected]
1539-3755/2010/81共1兲/011916共13兲
In this study, we also contrast correlation transfer in type
I and type II oscillators 共as well as in a continuum of intermediate models兲. The primary extension that we make is to
study spike train correlation over a range of different time
scales. Specifically, we measure the correlation coefficient ␳T
between the number of spikes produced by a pair of neurons
in a time window of length T:
␳T =
cov共n1,n2兲
冑var共n1兲冑var.共n2兲 .
共1兲
Here, n1 and n2 are the numbers of spikes output by neurons
1 and 2, respectively, in the time window; see Fig. 1 for an
illustration.
We first derive a tractable expression for ␳T in the long
time scale limit T → ⬁ 共cf. 关20,30兴兲. This can be given in
terms of moments of an associated exit time problem. This
reveals a dramatically higher level of long-time scale correlation transfer in type I vs type II neural oscillator models,
the opposite of what was found in the earlier studies over
short time scales 共see Fig. 1兲. Next, we study ␳T for successively shorter T, recovering the earlier findings of 关32,33兴,
and noting the critical time scale below which type II neurons become more efficient at transferring correlations. Additional results on how correlation transfer depends on neurons’ operating range—that is, their spike rate and coefficient
of variation 共CV兲 of spiking—are developed as we go along.
II. MODELS OF NEURAL OSCILLATORS
AND CORRELATION TRANSFER
A. Phase reductions
Neural oscillators can be classified into two types based
on their intrinsic dynamics. Both types have the feature that
as applied inputs 共or “injected current”兲 increases, the system
transitions from a stable rest state to periodic firing through a
bifurcation, the nature of which defines the type 关35兴. Here,
as is often taken to be the case, type I neurons undergo a
saddle node on invariant circle bifurcation, in which two
fixed points collide and disappear, producing a periodic orbit
that can have arbitrarily low frequency. Type II neurons undergo either a subcritical or supercritical Hopf bifurcation, in
011916-1
©2010 The American Physical Society
PHYSICAL REVIEW E 81, 011916 共2010兲
BARREIRO, SHEA-BROWN, AND THILO
FIG. 1. 共Color online兲 共a兲 A schematic of the setup and correlation metric used in this study. Two cells receive both common and
independent input, here modeled as white noise. Correlation of output spike trains is measured as the normalized covariance of spike counts
over a finite time window T 关see Eq. 共5兲兴. 共b兲 The principal result of our study; that correlation transfer efficiency depends on both internal
dynamics and time scale of readout, with relative efficiency between type I 共red or light gray兲 and type II 共blue or dark gray兲 switching as
readout time scale changes. The graph shows ␳T for a particular set of model parameters 共␻ , ␴兲 vs the measurement time window T 共T is
shown on a logarithmic scale兲. Insets show the phase-resetting curves used for the type I and type II models.
which a periodic orbit emerges at a nonzero minimum frequency.
Once the oscillator has passed through this bifurcation, it
can be described by a single equation for its phase, in which
inputs are mediated through a PRC which indicates the degree to which an input advances or delays the next spike. A
PRC is derived for a mathematical model by phase reduction
methods and determined experimentally by repeated perturbations of a system by input “kicks” 关36兴. Several investigators have demonstrated a connection between the type of
bifurcation and the shape of the PRC: a type I oscillator PRC
is everywhere positive so that positive injected current advances the time of the next spike 关37兴, whereas a type II PRC
has both positive and negative regions, so positive inputs
advance or delay the next spike depending on their timing
共关38兴; see also 关39兴兲. Moreover, the form of type I and type
II phase-resetting curves near the bifurcation are given by
共1 − cos ␪兲 and −sin共␪兲 respectively. We investigate these two
PRCs, together with a family of PRCs given by a linear
combination of these prototypical examples:
Z共␪兲 = − ␣ sin共␪兲 + 共1 − ␣兲关1 − cos共␪兲兴,
0 ⱕ ␣ ⱕ 1.
共2兲
Here ␣ homotopes the PRC from “purely” type I to type II;
along the way, intermediate PRCs more representative of
phase reductions of biophysical models are encountered
关37,39,40兴. For ␣ = 0, the phase model is exactly equivalent
to the theta model and to the superthreshold quadratic integrate and fire model 关38兴.
The question of how oscillators with different PRCs synchronize when they are coupled has been the subject of extensive study. Here, we ask about a different mechanism by
which such oscillators can become correlated. Specifically,
we consider an uncoupled pair of neurons receiving partially
correlated noise. The dynamics have been reduced to a phase
oscillator; each neuron is represented by a phase only. Each
phase ␪i is governed by the stochastic differential equation
d␪i = ␻dt + ␴Z共␪i兲 ⴰ 共冑1 − cdWit + 冑cdWct 兲,
␪i 苸 共0,2␲兲,
共3兲
where ␻ , ␴ ⬎ 0, ⴰ denotes the Stratonovich integral, and
Z共␪兲 = − ␣ sin共␪兲 + 共1 − ␣兲关1 − cos共␪兲兴.
Each ␪i receives independent white noise, dWit, and common
white noise dWct is received by both. The noises are weighted
so that the total variance of the noise terms in Eq. 共3兲 is ␴2.
For the remainder of the paper, we treat the equivalent Itô
integral
冉
d␪i = ␻ +
冊
␴2
Z共␪i兲Z⬘共␪i兲 dt + ␴Z共␪i兲dWt,
2
␪i 苸 共0,2␲兲.
共4兲
B. Measuring spike train correlation
We record spike times as those times tki at which ␪i crosses
2␲ 关37,38兴. Because Z共2␲兲 = 0 and ␻ ⬎ 0 for all the models
we consider, ␪ always continues through 2␲ and begins the
next period of interspike dynamics. We consider the output
spike trains y i共t兲 = 兺i␦共t − tki 兲, where tki is the time of the kth
spike of the ith neuron. The firing rate of the ith cell, 具y i共t兲典,
is denoted ␯i. As a quantitative measure of correlation over a
given time scale T, we compute the following statistic:
␳T =
cov共n1,n2兲
冑var共n1兲冑var共n2兲 ,
where n1 and n2 are the numbers of spikes output by neurons
1 and 2, respectively, in a time window of length T; i.e.,
ni共t兲 = 兰t+T
t y i共s兲ds.
By writing ni in terms of a discretized spike train 关i.e.,
␦t
y i共t1兲兴 we can write ␳T in terms of the autocorrelani = 兺tt+T/
1=t
tion and cross-correlation functions of y i; specifically
␳T =
冑冕
T
C12共t兲
兰−T
共T − 兩t兩兲
dt
T
共T − 兩t兩兲
C11共t兲
dt
T
−T
T
冕
共T − 兩t兩兲
C22共t兲
dt
T
−T
T
,
共5兲
where Cij共␶兲 = 具y i共t兲y j共t + ␶兲典 − ␯i␯ j 共see 关41兴; also Appendix A
in 关4兴兲. It will be convenient for us to analyze the system in
the Fourier domain. By the Wiener-Khinchin theorem, we
can write Eq. 共5兲 in terms of the power spectra Pij ⬅ 具ŷ ⴱi ŷ j典 as
011916-2
PHYSICAL REVIEW E 81, 011916 共2010兲
TIME SCALES OF SPIKE-TRAIN CORRELATION FOR…
␳T =
⬁
P12共f兲KT共f兲df
兰−⬁
⬁
⬁
冑兰−⬁
P11共f兲KT共f兲df兰−⬁
P22共f兲KT共f兲df
where the kernel KT is
KT共f兲 =
d␯
⬅ A␻,␴共0兲.
d␮
共6兲
,
The denominator converges to P11共0兲 共assuming the unperturbed oscillators to be statistically identical兲 which for renewal processes is simply CV2␯.
Putting these results together, as T → ⬁, the finite-time
correlations satisfy
冉 冊
4
Tf
sin2
.
Tf 2
2
A. Linear-response theory for ␳T
In this section we recall a derivation for a linear-response
approximation to the spike train correlation; fuller discussion
can be found in 关20,30,42兴. Assume that the fraction of noise
variance c of the correlated input noise is small; then we treat
the system with common noise as a perturbation to the system without common noise. Because the common noise is
small, we will assume that the response of the system can be
treated by linear-response theory; that is in the Fourier domain it can be characterized by a susceptibility function
A␻,␴共f兲 which gives the scaling factor between input and
response at frequency f.
Specifically, we make the ansatz 关42兴 that the Fourier
transform can be written to lowest order in c
as the base spike trains are independent of each other and the
correlated noise Q, and taking the variance of Q to be ␴2.
Then at any finite T, ␳T is linear in c, and 共cf. 关4,30兴兲:
␳T共␻, ␴兲 = cST共␻, ␴兲
冑
共8兲
␯=
CV =
1
,
T1共0兲
冑兵T2共0兲 − 关T1共0兲兴2其
T1共0兲
共11兲
,
1
dT1共0兲
d␯
=−
,
d␮
共T1共0兲兲2 d␮
共12兲
共13兲
where T1共0兲 is the average time to cross ␪ = 2␲, given a start
at ␪0 = 0 共in other words, given a start at the last spike兲, and
T2共0兲 is the second moment of this same quantity.
T1共x兲 and T2共x兲 are given by solutions of the adjoint equation 关43兴;
1
f i共x兲 = A共x兲⳵xTi + B共x兲⳵2x Ti ,
2
共14兲
f 1共x兲 = − 1,
共15兲
f 2共x兲 = − 2T1共x兲,
共16兲
where
Note that ST共␻ , ␴兲 multiplies the input correlation c to yield
the spike train correlation; for this reason we refer to
ST共␻ , ␴兲 as the correlation gain. We recover a simple expression for Eq. 共6兲 in the limit T → ⬁. In the numerator, we
converge to the value of the integrand at 0 as f → ⬁:
A共x兲 = ␻ +
c␴2兩A␻,␴共0兲兩2 .
As A␻,␴共0兲 is the limit of the susceptibility function as the
frequency becomes arbitrarily small, it must be equivalent to
the ratio of the dc response of the system 共that is, the firing
rate ␯兲 to the strength of a constant dc input. Later we will
use the symbol ␮ to include a dc input explicitly for the
purposes of this computation. Therefore we define
共10兲
The quantities ␯, CV, and dd␮␯ are related by moments of
the ISIs and as such can be computed using the associated
exit time problem. Specifically,
共7兲
.
⬅ cS共␻, ␴兲
B. Computing moments of the exit time problem
where y 0,i is the spike output of the neuron without correlated noise, Q̂共f兲 is the Fourier transform of the correlated
noise, and A is the susceptibility function. Then the crossspectrum of the two spike trains, P12共f兲, satisfies
P12共f兲 = c兩A␻,␴共f兲兩2具Q̂ⴱQ̂典 = c␴2兩A␻,␴共f兲兩2
2
where ␯ is the mean output firing rate, CV is the coefficient
of variation in the interspike intervals 共ISIs兲, ␴ is the input
noise amplitude. Each quantity can be computed from statistics of a single oscillator and combined to yield the long-time
correlation gain S共␻ , ␴兲.
ŷ i共f兲 = ŷ 0,i共f兲 + 冑cA␻,␴共f兲Q̂共f兲,
=c
冉 冊
d␯
d␮
lim ␳T = c
CV2␯
T→⬁
␴2
III. CORRELATION IN THE LONG
TIME SCALE LIMIT
⬁
␴2兰−⬁
兩A␻,␴共f兲兩2KT共f兲df
⬁
⬁
兰−⬁
P11共f兲KT共f兲df兰−⬁
P22共f兲KT共f兲df
共9兲
␴2
Z共x兲Z⬘共x兲,
2
B共x兲 = ␴2Z共x兲2 ,
共17兲
共18兲
and boundary conditions are given by Ti共2␲兲 = 0, ⳵Ti / ⳵x
bounded at x = 0. The solution can be obtained by integrating
Eq. 共14兲 twice over the range 关0 , 2␲兴, using the integrating
factor ⌿:
011916-3
PHYSICAL REVIEW E 81, 011916 共2010兲
BARREIRO, SHEA-BROWN, AND THILO
冋冕
x
dx⬘
⌿共x兲 = exp
册
2A共x⬘兲
,
B共x⬘兲
⌿共0兲 = 0.
If ␣ = 0, then B共x兲 has no interior zero and we proceed as
follows:
冋 册
⌿
⳵ Ti ⬘ 2f i共x兲
⌿共x兲,
=
⳵x
B共x兲
1
⳵ Ti
=
⳵ x ⌿共x兲
冕
x
0
1
⳵ Ti
=
⳵ x ⌿共x兲
共19兲
⳵ Ti
共x兲 =
⳵x
冦
冕
冕
1
⌿共x兲
x
共20兲
0
x
␹共␣兲
冧
x ⬍ ␹共␣兲
共22兲
Finally Ti共0兲 is given by a second integration
冕
⳵ Ti
共x兲dx.
2␲ ⳵ x
0
共23兲
Here, the argument of the exponential function in ⌿共x兲 can
A共x兲
. ⌿共0兲 关and ⌿共␹共␣兲兲, if ␣ ⬎ 0兴
be any antiderivative of 2B共x兲
is zero because the adjoint Eq. 共14兲 has an irregular singuA共x兲
larity at x = 0 关and at x = ␹共␣兲兴; consequently 2B共x兲
→ ⬁ and
2A共x兲
+
+
x
兰 dx B共x兲 → −⬁ as x → 0 关and as x → ␹共␣兲 兴. This accounts
for the difference between Eq. 共23兲 and, for example, Eq.
共5.2.157兲 in 关43兴 关44兴. Note that, also because Z共0兲 = 0 and
␻ ⬎ 0, x = 0 is an entrance boundary so any exit must take
place at 2␲ 共i.e., an exit from the interval 共0 , 2␲兲 is equivalent to a spike兲.
For the class of PRCs that we consider, the antiderivative
in Eq. 共19兲 can be evaluated symbolically, therefore ⌿共x兲 can
be evaluated analytically. The integrals in Eqs. 共21兲 and 共23兲
must be evaluated by numerical quadrature.
We next discuss the integrability of Eqs. 共21兲 and 共22兲.
First, we consider the case of the theta neuron 共␣ = 0兲, for
which B共x兲 has only two zeros; at 0 and 2␲. We can confirm
the integrability by checking the following conditions:
lim
冕
x
x→0+ 0
2f共x⬘兲
⌿共x⬘兲dx⬘ = 0,
B共x⬘兲
lim
冕
x
x→2␲− 0
共25兲
2f共x⬘兲
⌿共x⬘兲dx⬘ = ⫾ ⬁,
B共x⬘兲
共26兲
lim ⌿共x兲 = ⬁.
x→2␲−
If these are satisfied then by l’Hôpital’s rule,
⳵ Ti
f i共x兲
= lim
,
x→0 ⳵ x
x→0 A共x兲
共29兲
⳵ Ti
f i共x兲
= lim
,
x→2␲ ⳵ x
x→2␲ A共x兲
共30兲
lim
and we can use a quadrature method that can handle integrable singularities. For ␣ ⬎ 0, B共x兲 has one interior zero,
dividing the domain into two intervals on which Eq. 共14兲 has
irregular singularities at each end. ⌿ must be computed
⳵T
separately on each interval. Again we find that ⳵xi is finite on
each interval, permitting computation of Eq. 共22兲 with a standard quadrature routine.
Finally, we compute the derivative of the firing rate with
respect to a dc input; i.e., ⳵⳵␮␯ for the system
d␪ = ␻dt + Z共␪兲共␮dt + ␴ ⴰ dWt兲,
␪ 苸 关0,2␲兲,
共31兲
which is equivalent to the Itô SDE
冉
d␪ = ␻ +
冊
␴2
Z共␪兲Z⬘共␪兲 + ␮Z共␪兲 dt + ␴Z共␪兲dWt,
2
␪ 苸 关0,2␲兲.
共32兲
We wish to differentiate ␯ with respect to ␮ and evaluate at
␮ = 0. According to Eq. 共13兲 we must evaluate dT1 / d␮共0 , ␮兲
where the drift term A共x , ␮兲 = ␻ + ␴2 / 2Z共x兲Z⬘共x兲 + ␮Z共x兲 is a
function of both x and ␮. In general, T1, T2, and ⌿ are also
functions of two arguments 关e.g., T1共x , ␮兲兴, and we have indicated the arguments where needed for clarity. The notation
⳵
⳵x will refer to differentiation with respect to the first argument.
We first consider the case ␣ = 0. Rewriting Eq. 共23兲, we
have
共24兲
lim ⌿共x兲 = 0,
x→0+
共28兲
共21兲
2f i共x⬘兲
⌿共x⬘兲dx⬘ , x ⬎ ␹共␣兲.
B共x⬘兲
Ti共0兲 =
0
2f i共x⬘兲
⌿共x⬘兲dx⬘
B共x⬘兲
lim
2f i共x⬘兲
⌿共x⬘兲dx⬘ .
B共x⬘兲
2f i共x⬘兲
⌿共x⬘兲dx⬘ ,
B共x⬘兲
x
is finite at the end points; in fact
If ␣ ⬎ 0 then B共x兲 has an interior zero at ␹共␣兲, and we integrate separately on 共0 , ␹共␣兲兲 and 共␹共␣兲 , 2␲兲:
1
⌿共x兲
冕
T1共0, ␮兲 =
冕
0
2␲
1
⌿共x, ␮兲
冕
x
0
−2
⌿共x⬘, ␮兲dx⬘dx,
B共x⬘兲
共33兲
where
冋冕
⌿共x, ␮兲 = exp
共27兲
Therefore,
011916-4
x
册
2A共x⬘, ␮兲
dx⬘ .
B共x⬘兲
共34兲
PHYSICAL REVIEW E 81, 011916 共2010兲
TIME SCALES OF SPIKE-TRAIN CORRELATION FOR…
⳵ T1
d
共0, ␮兲 =
⳵␮
d␮
冉冕 冕
0
x
2␲
0
冊冕冕
− 2 ⌿共x⬘, ␮兲
dx⬘dx =
B共x⬘兲 ⌿共x, ␮兲
0
x
2␲
0
− 2 ⌿␮共x⬘, ␮兲⌿共x, ␮兲 − ⌿␮共x, ␮兲⌿共x⬘, ␮兲
dx⬘dx.
B共x⬘兲
⌿共x, ␮兲2
⳵ T1
共0, ␮兲 =
⳵␮
We use the relationship
⌿␮共x, ␮兲 = ⌿共x, ␮兲
冕
x
0
2 ⳵ A共y, ␮兲
dy
B共y兲 ⳵ ␮
共36兲
=
冕冕
冕 冕冕
0
x
2 ⌿共x⬘, ␮兲
B共x⬘兲 ⌿共x, ␮兲
2␲
0
0
x
y
2␲
0
0
冕
2 ⳵ A共y, ␮兲
dydx⬘dx
B共y兲
⳵␮
x⬘
x
2 ⌿共x⬘, ␮兲 2 ⳵ A共y, ␮兲
dx⬘dydx
B共x⬘兲 ⌿共x, ␮兲 B共y兲 ⳵ ␮
共37兲
assuming that the order of integration over x⬘ and y can be
switched. By Tonelli’s theorem, this is valid if the integrand
is single signed. For ␣ = 0, the integrand is always nonnegative 关note that ⳵⳵␮A 共x兲 = Z共x兲 and that B共x兲 is non-negative兴.
Next, we establish the integrability of expression 共37兲.
Rewriting, we have
⳵ ⳵ T1
共x, ␮兲 =
⳵␮ ⳵x
冦
冕冕
冕 冕
x
y
0
0
x
␹共␣兲
=
0
2␲
⫻
to find that
⳵ T1
共0, ␮兲 =
⳵␮
冕
冕
1
⌿共y, ␮兲
0
2␲
␹共␣兲
x
0
y
0
1
⌿共x, ␮兲
2 ⳵A
⌿共y, ␮兲
B共y兲 ⳵ ␮
2
⌿共x⬘, ␮兲dx⬘dydx
B共x⬘兲
x
0
2 ⳵A
⳵ T1
共y兲dydx.
⌿共y, ␮兲
B共y兲 ⳵ ␮
⳵x
As ⳵⳵␮A 共x兲 = Z共x兲 and ⳵T1 / ⳵x are bounded, we can use the
same conditions for integrability unchanged from Eqs.
共24兲–共27兲.
A parallel derivation to Eqs. 共33兲–共37兲 can be made when
␣ ⬎ 0. In this case we have
⳵ T1
共0, ␮兲 =
⳵␮
冕 冉
冊
⳵ ⳵ T1
共x, ␮兲 dx,
2␲ ⳵ ␮ ⳵ x
0
x ⬍ ␹共␣兲
冧
2 ⌿共x⬘, ␮兲 2 ⳵ A共y, ␮兲
dx⬘dy, x ⬎ ␹共␣兲.
B共x⬘兲 ⌿共x, ␮兲 B共y兲 ⳵ ␮
Here, the switch in the order of integration is justified by the
fact that the integrand ⳵⳵␮A is positive on 共0 , ␹共␣兲兲 and negative on 共␹共␣兲 , 2␲兲, while the range of integration in Eq. 共39兲
is always restricted to lie in one region or the other.
C. Patterns of correlation transfer over long time scales
Having derived and shown how to evaluate formula 共10兲,
we next use it to evaluate spike count correlations ␳. The
results are formally valid in the limits of time scale T → ⬁
and input correlation c → 0 so we also conduct Monte Carlo
simulations to reveal behavior of ␳T for large but finite T and
intermediate input correlation up to c = 0.3 and to test the
applicability of our formula in these regimes.
We compute these spike correlations over a range of ␻
and ␴ values that explores a full dynamical regime of the
model. By this we mean that the values we use span from
dominantly mean-driven firing 共e.g., ␻ = 2.5, ␴ = 0.4 at lowerright white square in Fig. 2兲, to dominantly fluctuation-
共38兲
where we have used the boundary condition T1共2␲ , ␮兲 = 0.
The integrand is given by
2 ⌿共x⬘, ␮兲 2 ⳵ A共y, ␮兲
dx⬘dy,
B共x⬘兲 ⌿共x, ␮兲 B共y兲 ⳵ ␮
y
冕
冕
冕
1
⌿共x, ␮兲
共35兲
共39兲
driven firing 共e.g., ␻ = 0.4, ␴ = 2.4 at upper-left white square兲
and all intermediate possibilities.
The resulting output firing rate ␯ for ␣ = 0, computed via
Eq. 共11兲, ranges from 0 to 0.9 共measured in spikes per time
unit兲; see Fig. 2 共left panel兲. Note that ␯ increases strongly
with ␻ and only weakly with ␴ 共see Sec. III D兲. The CV 关via
Eq. 共12兲兴 ranges from 0 to 0.55 共Fig. 2, right panel兲. As noted
by 关45,46兴, the superthreshold quadratic integrate and fire
model—equivalent to our ␣ = 0 phase model—has an upper
bound on CV of 1 / 冑3 ⬇ .577. By using a time change in the
equations, CV can be seen to depend on the input parameters
␻ and ␴ only through the relationship ˜␴ = ␴ / 冑␻ 共see later in
this section兲 and is therefore invariant on level curves of this
ratio. As Fig. 2 shows, CV increases with this ratio.
We first fix a moderately long time window T = 32, corresponding to 1.6–16 interspike intervals for the parameter
range at hand, and compute ␳T from Monte Carlo simulations
for a range of correlation strengths c 苸 关0 , 0.3兴; see Fig. 3
共right column兲. We see that, for type I models, ␳T is close to
linear in this range of c, as for the linear-response theory. For
011916-5
PHYSICAL REVIEW E 81, 011916 共2010兲
BARREIRO, SHEA-BROWN, AND THILO
FIG. 2. Firing rate 共left兲 and CV 共right兲 of the theta model neuron 共␣ = 0兲 over a range of intrinsic phase speed ␻ and noise variance ␴2.
Sample spike trains are illustrated for four sets of 共␻ , ␴兲 values 共see white squares兲, notably mean-driven 共high ␻, low ␴-bottom-most spike
␴
train兲 and fluctuation-driven 共low ␻, high ␴-top-most spike train兲. Level sets of ˜␴ ⬅ 冑␻ are plotted 共see text in Sec. III C兲.
FIG. 3. 共Left兲 Susceptibility S共␻ , ␴兲 for global model parameter ␣ = 0 共top兲, ␣ = 0.5 共middle兲, and ␣ = 1 共bottom兲 over a range of intrinsic
phase speed ␻ and noise variance ␴2; lines are level sets of ˜␴ ⬅ ␴ / 冑␻. 共Right兲 Spike count correlation ␳T vs fraction of correlated noise c
from Monte Carlo simulations for ␣ = 0 共top兲, ␣ = 0.5 共middle兲, and ␣ = 1 共bottom兲. Error bars indicate the standard deviation over eight trials.
The specific 共␻ , ␴兲 values plotted are those shown in Fig. 2. ␳T is measured over a time window of T = 32 ms.
011916-6
PHYSICAL REVIEW E 81, 011916 共2010兲
TIME SCALES OF SPIKE-TRAIN CORRELATION FOR…
冉
d␪␶ = 1 +
冊
␴2
␮
␴
Z共␪兲Z⬘共␪兲 + Z共␪兲 d␶ +
冑␻ Z共␪兲dW␶
2␻
␻
共40兲
冉
= 1+
冊
˜␴2
␮
Z共␪兲Z⬘共␪兲 + Z共␪兲 d␶ + ˜␴Z共␪兲dW␶ .
2
␻
共41兲
Each exit time must scale identically under this transformation so that the exit time moments scale as
FIG. 4. Scatter plots of susceptibility S共␻ , ␴兲 vs firing rate ␯ 共a兲
and susceptibility vs coefficient of variation 共CV兲 共b兲 for global
model parameter ␣ = 0 共light gray兲, ␣ = 0.5 共medium gray兲, and ␣
= 1 共black兲. S共␻ , ␴兲 was computed for the ranges of intrinsic phase
speed ␻ and noise variance ␴2 shown in Fig. 3.
type II models, linearity holds over a decreased range of c.
Additionally, note that the limiting formula Eq. 共10兲 for ␳
gives a close approximation to ␳T=32 for type I models in
more fluctuation-driven regimes. The approximation is worse
for dominantly mean-driven firing, and for type II models,
but the trend that correlations are lower for type II models is
correctly predicted by Eq. 共10兲.
Next, we discuss the trends in ␳ predicted by the linearresponse theory. Two findings stand out in the left hand panels of Fig. 3. First, values of S共␻ , ␴兲 共and hence ␳ ⬇ Sc兲 are
much larger for type I 共␣ = 0兲 than for type II 共␣ = 1兲 models.
Second, S共␻ , ␴兲 is nearly constant as the input mean and
standard deviation ␻ and ␴ vary over a wide range, for both
the type I and type II models. In sum, type I models transfer
⬇66% of their input correlations into spike correlations over
long time scales; type II models transfer none of these input
correlations, producing long-time scale spike counts that are
uncorrelated. Additionally, an intermediate model 共␣ = 1 / 2兲
transfers an intermediate level of correlations, and these levels do depend on ␻ and ␴. We will provide a partial explanation for overall trends in correlation transfer with ␣ in Sec.
III D.
Figure 4 provides an alternative view of these results by
plotting the correlation gain S vs the firing rate and CV that
are evoked by input parameters drawn from the whole range
of ␻ , ␴. First, note that S does not vary with firing rate for
the type I or type II models, as expected from the previous
plots. For the intermediate 共␣ = 1 / 2兲 model, S does not display a clear functional relationship with firing rate, but there
is such a relationship with CV 关Fig. 4共b兲兴. We note that all of
these findings for the phase oscillators under study are in
contrast to the behavior of linear integrate and fire neurons,
which produce a strongly increasing, nearly functional relationship with firing rate 关20,30兴; we revisit this point in the
discussion.
Now, we discuss a scaling relationship for the underlying
equations that simplifies the parameter dependence and helps
to explain the plots of S vs ␯ and S vs CV. This symmetry
共which was noted in 关45兴 for the quadratic integrate and fire
model兲, allows us to reduce the free parameters ␻ , ␴ to one
parameter ˜␴ ⬅ ␴ / 冑␻. The stochastic differential equation
冉
d␪t = ␻ +
冊
␴2
Z共␪兲Z⬘共␪兲 + ␮Z共␪兲 dt + ␴Z共␪兲dWt
2
becomes, under a time change ␶ = ␻t,
T̃1 = ␻T1 → ˜␯ =
␯
,
␻
共42兲
T̃2 = ␻2T2 ,
共43兲
d˜␯ 1 d␯
=
.
d␮ ␻ d␮
共44兲
and therefore the CV= 冑T̃2 − T̃21 / T̃1 is invariant under the
time change. Thus, the CV, which is computed with ␮ = 0,
depends only on ˜␴, not on ␻ and ␴ separately. Now consider
the two sets of parameters 共␻ , ␴兲 and 共1 , ˜␴兲, such that ˜␴
= ␴ / 冑␻. According to Eq. 共41兲, the effect of ␮ is scaled by
1 / ␻ so that
Therefore, the susceptibility
S共␻, ␴兲 =
␴2
=
冉 冊
d␯
d␮
CV2␯
␴2
2
冉 冊
1 d˜␯
␻2 d␮
2 ˜␯
CV
␻
冉 冊
d˜␯ 2
␴2 d␮
=
2˜␯
␻ CV
=S共1, ˜␴兲
共45兲
2
共46兲
共47兲
共48兲
is invariant under the change in parameters 共␻ , ␴兲
→ 共1 , ␴ / 冑␻兲; exactly the same change in parameters under
which the CV is conserved. Therefore, if there is a single
contour for each value of CV 共i.e., there are no disconnected
contours兲, we expect that S will be a function of CV, as in
Fig. 4共b兲. Conversely, note that the firing rate will vary
widely over any particular ˜␴ contour so that many different
firing rates can be expected to yield the same S; unless S is
constant, we expect a given firing rate to be associated with
a range of S values, as also seen in Fig. 4共a兲. The uniqueness
of CV level sets for ␣ = 0 is previously known 关46兴.
Finally, in Fig. 5, we demonstrate that the behavior of S
seems to roughly “interpolate” between the ␣ = 0 , 1 / 2 , 1
cases considered here as it varies over the whole range of ␣.
S typically decreases as the total noise variance ␴ increases.
011916-7
PHYSICAL REVIEW E 81, 011916 共2010兲
BARREIRO, SHEA-BROWN, AND THILO
D. Analytical arguments for type I vs type II
difference in correlation gain S
nally, we give an intuitive argument to buttress these calculations.
We now seek to explain the drop in S共␻ , ␴兲 关Eq. 共10兲兴 as
␣ increases, ranging from type I 共␣ = 0兲 to type II 共␣ = 1兲
PRCs. There are two tractable limits in which explicit calculations can be performed. First, we show that S共␻ , ␴兲 = 0 for
“purely” type II models, for any values of ␻ and ␴. Next, for
arbitrary ␣, we derive an S共␻ , ␴兲 valid to first order in
␴2—this reveals a monotonic decrease in S with ␣ and gives
a good description of correlation transfer even for ␴ ⬇ 1. Fi-
⳵ ⳵ T1
=
⳵␮ ⳵x
冦
1
⌿共x, ␮兲
1
⌿共x, ␮兲
冕
冕
x
0
x
␲
We start with ␣ = 1. It is straightforward to see that
d␯ / d␮ = 0 for any ␻ , ␴. Recall that d␯ / d␮ is given by the
integral of a function over the interval 共0 , 2␲兲, in Eq. 共38兲;
we will show this function integrates to zero. Rewriting Eq.
共39兲, using ␹共1兲 = ␲,
2 ⳵A
⳵ T1
共y, ␮兲dy, x ⬍ ␲
共y, ␮兲⌿共y, ␮兲 ⫻
B共y兲 ⳵ ␮
⳵x
冧
共49兲
2 ⳵A
⳵ T1
共y, ␮兲dy, x ⬎ ␲ .
共y, ␮兲⌿共y, ␮兲 ⫻
B共y兲 ⳵ ␮
⳵x
⳵T
⌿共x , ␮兲, B共x兲, and ⳵x1 共x , ␮兲 are each ␲ periodic 关i.e., f共x兲
= f共x + ␲兲兴. ⳵⳵␮A ⬅ Z共x兲 = −sin共x兲, however, is antiperiodic
关f共x兲 = −f共x + ␲兲兴. Then
⳵ ⳵ T1
⳵ ⳵ T1
共x, ␮兲 = −
共x + ␲, ␮兲
⳵␮ ⳵x
⳵␮ ⳵x
1. S = 0 for purely type II models
If we examine the integral in Eq. 共21兲 and the inner integral of Eq. 共35兲 for ␣ = 0 or the integrals in Eq. 共22兲 and Eq.
共39兲 for ␣ ⬎ 0, we see that we can write each of them in the
form
共50兲
⳵T
and integrating ⳵⳵␮ 关 ⳵x1 共x , ␮兲兴 over a full period x 苸 共0 , 2␲兲
yields zero. Plugging this into Eq. 共10兲, we see that
S共␻ , ␴兲 ⬅ 0 for models with type II PRCs Z共␪兲 = −sin共␪兲 共␯
and CV are always nonzero in this paper兲.
2. Evaluation of S(␻ , ␴) in the limit ␴ \ 0
We next derive an analytical expression for S共␻ , ␴兲 in the
limit ␴ → 0. We will show that each relevant term T1共0兲,
⳵T
T2共0兲, and ⳵␮1 共0兲 admits an asymptotic expansion in the
small parameter ␴2. We compute these terms explicitly and
combine them to get the first term of the associated expansion for S共␻ , ␴兲.
1
Z共y兲
冕
再
y
f共x⬘兲exp
a
冎
1
关F共x⬘兲 − F共y兲兴 dx⬘ ,
␴2
共51兲
where F共x⬘兲 is strictly increasing on 共a , y兲. a may be either 0
or ␹␣, depending on the circumstance. This integral admits
an asymptotic expansion in ␴2, with successive terms essentially given by integrating by parts and retaining only the
contribution from the end of the interval where F共x⬘兲 − F共y兲
= 0. In the case of T1共x兲 we have
f共y兲 =
1
.
Z共y兲
In the case of T2共x兲 we have
f共y兲 =
T1共y兲
Z共y兲
and for d␯ / d␮
f共y兲 =
⳵ T1
共y兲.
⳵x
In each case F共x⬘兲 is given by the antiderivative of
2␻ / Z共x⬘兲2; because this is a positive function on 共0 , ␹共␣兲兲
and 共␹共␣兲 , 2␲兲, F共x⬘兲 must clearly be increasing on these
intervals. The integral
冕
b
f共x⬘兲exp共␭␾共x⬘兲兲dx⬘
共52兲
a
FIG. 5. Susceptibility S共1 , ␴兲 for a continuum of models specified by ␣ 苸 关0 , 1兴.
admits the following expansion for ␭ → ⬁, if ␾⬘ ⬎ 0 on 关a , b兴
and f meets certain conditions 共see, for example, 关47兴, 兲:
011916-8
PHYSICAL REVIEW E 81, 011916 共2010兲
TIME SCALES OF SPIKE-TRAIN CORRELATION FOR…
3. Argument for general PRCs
We close this by section by noting that, for arbitrary PRCs
Z共␪兲 and small ␴,
d␯/d␮ ⬀
冕
2␲
Z共␪兲d␪ .
0
FIG. 6. lim␴→0+ S共1 , ␴兲 for a continuum of models specified by
␣ 苸 关0 , 1兴 共dashed black line兲. This is a good approximation for
small 共␴ = 0.2; dark gray line兲 and moderate 共␴ = 1; light gray line兲
values of ␴.
⬁
I共␭兲 ⬃ 兺
n=0
再
冋
共− 1兲n
1 d
exp关␭␾共x兲兴
n+1
␭
␾⬘共x兲 dx
册 冎
n
f共x兲
␾⬘共x兲
b
.
a
共53兲
If ␾共x兲 → −⬁ as x → a, as is the case in our use of the formula, only the right-hand end point makes a contribution to
the integral.
Substituting this contribution into the outer integral and
evaluating these quantities to the required order, we find
T1共0兲 =
T2共0兲 =
2␲
+ O共␴4兲,
␻
冉
共54兲
冊
4␲2
2 ␲
共3 − 6␣ + 4␣2兲 + O共␴4兲,
2 +␴
␻
␻3
共55兲
2␲
dT1
共0兲 = −
共1 − ␣兲 + O共␴4兲.
d␮
␻
共56兲
In passing, we note that the firing rate gain, d␯ / d␮, is given
by
1 dT1
d␯
=−
共0兲
d␮
T1共0兲2 d␮
=
1−␣
+ O共␴4兲.
2␲
2共1 − ␣兲2
+ O共␴2兲.
3 − 6␣ + 4␣2
IV. CORRELATION OVER SHORTER
TIME SCALES
Figure 7 shows ␳T computed for a sequence of finite time
windows T. We can characterize the nondimensional T in
terms of its length in terms of a typical interspike interval
共ISI兲 of the oscillator. The time window T = 1 varies from
0.05–0.5 ISI, roughly, from left to right; the time window
T = 32 varies from 1.6–16 ISI.
We see a striking dependence of transferred correlations
on T. ␳32 is larger for type I than for type II oscillators for
most parameter values, consistent with our long time results.
However, ␳1共␻ , ␴兲 is smaller for type I than for type II for
95% of parameter pairs 共␻ , ␴兲. This is consistent with recent
results on the response of phase oscillators to correlated
noise 关32,33兴; in particular 关32兴, study the distribution of
phase difference ⌬␪ ⬅ ␪1 − ␪2 between two oscillators driven
by common noise and find that the probability that ⌬␪ = 0 is
greater for type II than for type I. As noted in the Introduction, this metric can be shown to have a direct relationship
with our ␳T as T → 0. To summarize, the “switch” in ST共␻ , ␴兲
共from higher correlation gain in type II models to higher
correlation gain in type I兲 occurs at time scales T over which
each cell fires several spikes; such time scales are biologically relevant, as we discuss in the next section.
共57兲
共58兲
Putting these results together we see that
S共␻, ␴兲 =
The calculations showing this are in the appendix. While
S共␻ , ␴兲 = ␴2共d␯ / d␮兲2 / CV2␯ has other terms that depend on
the PRC, this calculation does suggest that S is likely to be
smaller for PRCs with lower means in general, and lends
some intuition to what drives the decrease in S with ␣.
共59兲
It can be readily checked that this function decreases monotonically from a value of 2/3 at ␣ = 0, to a value of 0 at ␣
= 1.
While this calculation is in the limit ␴ → 0+, it in fact
remains a good approximation for moderate ␴, in fact, even
for ␴ ⬃ ␻. Figure 6 shows the limiting value as well as computed S values at small 共relative to ␻ = 1; ␴ = 0.2兲 and moderate 共␴ = 1兲 values of ␴. The limiting value remains a good
approximation throughout this range.
V. SUMMARY AND DISCUSSION
We asked how correlated input currents are transferred
into correlated spike trains in a class of nonlinear phase models that are generic reductions in neural oscillators. Linearresponse methods, asymptotics, and Monte Carlo simulations
gave the following answers:
共1兲 Over long time scales, type I oscillators transfer 66%
of incoming current correlations into correlated spike counts,
while type II oscillators transfer almost none of their input
correlations into spike count correlations. Models with intermediate phase response curves transfer intermediate levels of
correlations.
共2兲 Over long time scales, correlation transfer in type I
and type II models is independent of the rate and coefficient
of variation 共CV兲 of spiking. For intermediate models, correlation transfer decreases with CV and shows no clear dependence on rate.
共3兲 That there is a time scale T beneath which these results reverse: type II neurons become more efficient at trans-
011916-9
PHYSICAL REVIEW E 81, 011916 共2010兲
BARREIRO, SHEA-BROWN, AND THILO
FIG. 7. Spike count correlation ␳T共␻ , ␴兲 as measured from Monte Carlo simulations with global model parameter ␣ = 0 共top row兲, ␣
= 0.5 共middle row兲, and ␣ = 1 共bottom row兲. Measurement time intervals are, from the leftmost column, T = 1, T = 2, T = 8, and T = 32 共ms兲.
␳T is computed for a range of intrinsic phase speed ␻ and noise variance ␴2. The fraction of noise variance that is received by both oscillators
is c = 0.1; therefore, to recover the approximate susceptibility ST共␻ , ␴兲 ⬇ ␳T共␻ , ␴兲 / c, multiply by 10. As T increases, ST approaches the T
→ ⬁ limit illustrated in Fig. 3.
ferring correlations than type I, there is an increasing dependence of correlation transfer on spike rate, and the strong
dependence on CV weakens.
We note that results 共1兲 and 共2兲 are highly distinct from
findings for the leaky integrate-and-fire neuron model, for
which up to 90%–100% of correlations are transferred over
long time scales, with this level depending strongly on firing
rate but very weakly on CV 关20,30兴. This demonstrates a
strong role for subthreshold dynamics in determining correlation transfer in the oscillatory regime 共as seen for the quadratic integrate-and-fire model in 关30兴兲.
What time scales of spike count correlation actually matter in a given application? This depends on the circuit that is
“downstream” of the pair 共or, similarly, layer兲 of neural oscillators that we have studied in this paper; in other words,
on what system is reading out the neurons we study here.
Clearly, different neurons and networks are sensitive to input
fluctuations over widely varying time scales. For example,
some single neurons and circuits can respond only to events
in which many of the cells that provide inputs spike nearly
simultaneously. This is the coincidence detector mode of operation 共cf. 关48兴 and references therein兲 and can result from
fast membrane time constants 共as occur in high-conductance
states 关49兴兲; circuit mechanisms, such as feed-forward inhibition 关50兴, can also play a role. For such systems, short-time
scale correlations among upstream cells are relevant—small
window lengths T. On the opposite extreme, networks operating as neural integrators will accumulate inputs over arbitrarily long time scales 共see 关51兴 and references therein兲; in
this case, spike-time correlations over large windows T are
reflected in circuit activity. In general there is a range of
possible behaviors, and the time scales over which inputs are
integrated can differ among various components of a network, among components of an individual cell 关48,50兴 or
among different times in a cell lifetime, depending on background input characteristics 关49兴.
One domain in which different levels of correlation
transfer—and different dependences of this correlation transfer on neurons’ operating ranges—can have a strong effect is
the population coding of sensory stimuli. For example, if
neurons are read out over long time scales, then type I vs
type II populations offer a choice between relatively high
and low levels of correlation across the population. Depending on heterogeneity of the population response to the stimulus at hand, one or the other of these choices can yield dramatically greater 共Fisher兲 information about the encoded
stimulus 关1,5,15兴. The opposite choice of neuron type would
be preferred for readout over short time scales, where trends
in correlation transfer reverse. Beyond averaged levels of
correlation, a separate question that can affect encoding is
whether correlations depend on the stimulus 关24,52兴. A natural way that this can occur is when correlations depend on
the evoked rate or CV of firing. We demonstrate that such
dependencies are present in phase models over short time
scales and, over long time scales, that they are present in
intermediate but not “purely” type-I or type-II models. Once
again, depending on details of stimulus encoding, these dependencies can either enhance or degrade encoding. Overall,
the picture that emerges is that correlation transfer is another
factor to consider in asking which nonlinearities allow neu-
011916-10
PHYSICAL REVIEW E 81, 011916 共2010兲
TIME SCALES OF SPIKE-TRAIN CORRELATION FOR…
ron models to best encode stimuli, and that there will be
different answers for different stimuli.
In closing, we note that we have studied only simple 共but
widely-used兲 one-dimensional neural models here. However,
preliminary simulations suggest that the trends for correlation transfer found here also hold in some standard type-I vs
type-II conductance-based neuron models 关35兴. The situation
is more complex, as global features of the neural dynamics
can be involved, and will be explored in future work.
APPENDIX
再
⌿ = 兩1 − cos共x兲兩exp −
We can verify Eq. 共24兲 as before.
3. Small ␴ argument for general PRC
Here we replicate the small ␴ value of d␯ / d␮, using a
perturbation expansion valid for an arbitrary PRC.
We consider the stationary densities p共␪兲 and p̂共␪兲 of two
different processes,
In this appendix we give some more details about calculation of T1共0兲, T2共0兲, and dT1 / d␮共0兲 for specific values of
␣.
d␪ = a共␪兲dt + b共␪兲dWt ,
共A8兲
d␪ˆ = â共␪兲dt + b共␪兲dWt ,
共A9兲
where
1. ␣ = 1 and numerical details
For Z共x兲 = −sin共x兲 共␣ = 1兲,
␴2
Z共␪兲Z⬘共␪兲,
2
共A10兲
␴2
Z共␪兲Z⬘共␪兲 + ␮Z共␪兲,
2
共A11兲
a共␪兲 = ␻ +
␴
sin共x兲cos共x兲,
2
2
A共x兲 = ␻ +
B共x兲 = ␴ 关sin共x兲兴
2
共A1兲
â共␪兲 = ␻ +
共A2兲
2
are periodic on 关0 , ␲兴 with B共x兲 = 0 at 0共2␲兲 and ␲.
⌿共x兲 is defined as an antiderivative as in Eq. 共19兲; we can
compute this symbolically yielding
冉
⌿共x兲 = sin共x兲exp −
冊
2␻
cot共x兲 .
␴2
共A3兲
We can check that limx→0+ ⌿共x兲 = 0, limx→␲− ⌿共x兲 = ⬁, and
that ⌿ / B is not integrable at either 0 or ␲.
For the interval ␲ to 2␲ we use the fact that A共x兲 and B共x兲
are ␲ periodic; ⌿ repeats on this second interval; that is,
冉
2␻
⌿共x兲 = 兩sin共x兲兩exp − 2 cot共x兲
␴
冎
2␻ 关2 − cos共x兲兴sin共x兲
. 共A7兲
3␴2 关1 − cos共x兲兴2
冊
p共␪兲 satisfies the stationary Fokker-Planck equation
−
2. ␣ = 0
ap −
␯=
␯ˆ =
B共x兲 = ␴2关1 − cos共x兲兴2
共A6兲
are periodic on 关0 , 2␲兴 with B共x兲 = 0 at 0共2␲兲. Again we can
integrate 2A共x兲 / B共x兲 symbolically, finding
共A13兲
1
2␲
冕
1
2␲
冕
2␲
a共␪兲p共␪兲d␪ ,
共A14兲
â共␪兲p̂共␪兲d␪ .
共A15兲
0
2␲
0
The dc input response, d␯ / d␮, can be computed by differentiating ␯ˆ with respect to ␮ and evaluating at ␮ = 0. Therefore
we expand ␯ˆ as a series in ␮ and take the first-order term
For the theta model, Z共x兲 = 1 − cos共x兲 or ␣ = 0,
共A5兲
⳵
共bp兲 = ␯
⳵␪
and therefore
We find that
␴2
A共x兲 = ␻ + 关1 − cos共x兲兴sin共x兲,
2
共A12兲
This can be integrated and the constant of integration is
equal to the firing rate:
共A4兲
is an expression that is valid everywhere Z共␪兲 ⫽ 0.
To compute values of ⳵T1 / ⳵x on a uniform mesh, we use
an adaptive quadrature routine to evaluate Eqs. 共22兲; in particular Simpson’s rule implemented via the MATLAB routine
quad. We have already demonstrated the integrability of Eqs.
共22兲 in Sec. III B. T1共0兲 is then computed using Simpson’s
three-point rule.
To compute values of ⳵T2 / ⳵x, we use adaptive quadrature
as well, with the caveat that ⳵T1 / ⳵x is evaluated in between
mesh points by linear interpolation.
⳵
⳵2
共ap兲 + 2 共bp兲 = 0
⳵␪
⳵␪
1
d␯
= ␯1 =
d␮
2␲
p̂ = p + ␮ p1 + O共␮2兲,
共A16兲
␯ˆ = ␯ + ␮␯1 + O共␮2兲.
共A17兲
冋冕
2␲
0
p共␪兲Z共␪兲d␪ +
冕
2␲
0
册
p1共␪兲a共␪兲d␪ .
共A18兲
We consider Eq. 共A18兲 more carefully. We will see that
all but one term are multiplied by ␴2; for small ␴, the term
that remains significant is the mean of the phase-resetting
curve.
011916-11
PHYSICAL REVIEW E 81, 011916 共2010兲
BARREIRO, SHEA-BROWN, AND THILO
We rewrite Eq. 共A18兲 as follows:
1
␯1 =
2␲
冋冕 冉
2␲
+ ␴2
冕
0
2␲
0
1
=
2␲
冋冕
+ ␴2
冕
2␲
0
2␲
0
冊
p0 =
1
+ p̃共␪兲 Z共␪兲d␪
2␲
1
Z共␪兲Z⬘共␪兲p1共␪兲d␪
2
1
Z共␪兲d␪ +
2␲
冕
2␲
1
Z共␪兲Z⬘共␪兲p1共␪兲d␪
2
册
共A19兲
冉
d␪ = ␻ +
册
共A20兲
冊
␴
Z共␪兲Z⬘共␪兲 dt + ␴Z共␪兲dWt .
2
The stationary density p and firing rate J satisfy
冉
␻+
冊
冉
2
冊
If ␴ = 0 then the stationary density is
expand p and J in powers of ␴2;
and
p共␪兲 = p0 + p̃
J0 = 2␻␲ .
We
冕
共A24兲
J = J0 + ␴2J1 + O共␴4兲.
共A25兲
共A27兲
冊
2␲
J1 = 2␲J1 = −
0
共A28兲
1
4␲
冕
2␲
ZZ⬘d␪ = 0
共A29兲
0
as ZZ⬘ is the perfect derivative of a periodic function. For
general order, we have
p̃n =
冋
冉 冊册
1
1
⳵ Z2
p̃n−1
Jn − ZZ⬘ p̃n−1 +
2
␻
⳵␪ 2
1
4␲
冕
2␲
ZZ⬘ p̃n−1d␪ ,
,
共A30兲
共A31兲
0
where we no longer expect Jn to be zero.
Let’s return to Eq. 共A20兲. The first integral is the mean of
Z共␪兲. The second integral, in our expansion, first appears at
fourth-order in ␴. To see this, we examine
冕
共A23兲
=p0 + ␴2 p̃1 + O共␴4兲,
冊
1
1
J1 +
ZZ⬘d␪ ,
␻
4␲
Jn =
共A22兲
p0 = 21␲
冉
冉
1
Z2 ⳵ p0 1
J1 +
+ ZZ⬘ p0
2 ⳵␪ 2
␻
J1 is determined so that p is a probability density at any
order:
共A21兲
⳵ ␴ 2
␴
Z共␪兲Z⬘共␪兲 p共␪兲 −
Z 共␪兲p共␪兲 = J.
2
⳵␪ 2
2
p̃1 =
=
using the fact that p1共␪兲 must average to 0 so that p̂ remains
a probability density. We have also written p共␪兲, the stationary density for the process with ␮ = 0, as the sum of a uniform density and a deviation p̃共␪兲.
Consider the process
2
共A26兲
as already stated. At O共␴2兲
p̃共␪兲Z共␪兲d␪
0
J0
␻
2␲
0
p̃1Z共␪兲d␪ =
1
4␲
=0.
冕
2␲
Z2共␪兲Z⬘共␪兲d␪
共A32兲
0
共A33兲
At O共1兲 we have
The third term is second-order in ␴. Thus the dominant term
in Eq. 共A20兲 is the first one, and we have shown the desired
result.
关1兴 E. Zohary, M. Shadlen, and W. Newsome, Nature 共London兲
370, 140 共1994兲.
关2兴 K. O. Johnson, J. Neurophysiol. 43, 1793 共1980兲.
关3兴 K. H. Britten, M. N. Shadlen, W. T. Newsome, and J. A. Movshon, J. Neurosci. 12, 4745 共1992兲.
关4兴 W. Bair, E. Zohary, and W. T. Newsome, J. Neurosci. 21, 1676
共2001兲.
关5兴 L. F. Abbott and P. Dayan, Neural Comput. 11, 91 共1999兲.
关6兴 B. B. Averbeck, P. E. Latham, and A. Pouget, Nat. Rev. Neurosci. 7, 358 共2006兲.
关7兴 R. Romo, A. Hernandez, A. Zainos, and E. Salinas, Neuron
38, 649 共2003兲.
关8兴 Y. Chen, W. S. Geisler, and E. Seidemann, Nat. Neurosci. 9,
1412 共2006兲.
关9兴 J. Poort and P. R. Roelfsema, Cereb. Cortex 共to be published兲.
关10兴 M. W. Oram, P. Földiák, D. I. Perrett, and F. Sengpiel, Trends
Neurosci. 21, 259 共1998兲.
关11兴 P. Seriès, P. E. Latham, and A. Pouget, Nat. Neurosci. 7, 1129
共2004兲.
关12兴 A. Kohn, M. A. Smith, and J. A. Movshon, Computational and
Systems Neuroscience 共Cold Spring Harbor, New York, 2004兲.
关13兴 M. Shamir and H. Sompolinsky, Neural Comput. 16, 1105
共2004兲.
关14兴 M. Shamir and H. Sompolinsky, Neural Comput. 18, 1951
共2006兲.
关15兴 H. Sompolinsky, H. Yoon, K. Kang, and M. Shamir, Phys.
Rev. E 64, 051904 共2001兲.
关16兴 E. Schneidman, M. J. Berry, R. S. II, and W. Bialek, Nature
共London兲 440, 1007 共2006兲.
关17兴 R. C. deCharms and M. M. Merzenich, Nature 共London兲 381,
610 共1996兲.
关18兴 J. M. Samonds, J. D. Allison, H. A. Brown, and A. B. Bonds,
J. Neurosci. 23, 2416 共2003兲.
关19兴 A. Kohn and M. A. Smith, J. Neurosci. 25, 3661 共2005兲.
011916-12
PHYSICAL REVIEW E 81, 011916 共2010兲
TIME SCALES OF SPIKE-TRAIN CORRELATION FOR…
关20兴 J. de la Rocha, B. Doiron, E. Shea-Brown, K. Josic, and A.
Reyes, Nature 共London兲 448, 802 共2007兲.
关21兴 C. M. Gray, P. K. A. K. Engel, and W. Singer, Nature 共London兲 338, 334 共1989兲.
关22兴 J. Biederlack, M. Castelo-Branco, S. Neuenschwander, D. W.
Wheeler, W. Singer, and D. Nikolić, Neuron 52, 1073 共2006兲.
关23兴 M. J. Chacron and J. Bastian, J. Neurophysiol. 99, 1825
共2008兲.
关24兴 K. Josic, E. Shea-Brown, B. Doiron, and J. de la Rocha, Neural Comput. 21, 2774 共2009兲.
关25兴 E. Salinas and T. Sejnowski, J. Neurosci. 20, 6193 共2000兲.
关26兴 A. Kuhn, A. Aertsen, and S. Rotter, Neural Comput. 15, 67
共2003兲.
关27兴 T. Tetzlaff, S. Rotter, E. Stark, M. Abeles, A. Aertsen, and M.
Diesmann, Neural Comput. 20, 2133 共2008兲.
关28兴 M. D. Binder and R. K. Powers, J. Neurophysiol. 86, 2266
共2001兲.
关29兴 R. Moreno-Bote and N. Parga, Phys. Rev. Lett. 96, 028101
共2006兲.
关30兴 E. Shea-Brown, K. Josić, J. de la Rocha, and B. Doiron, Phys.
Rev. Lett. 100, 108102 共2008兲.
关31兴 M. Shadlen and W. Newsome, J. Neurosci. 18, 3870 共1998兲.
关32兴 S. Marella and G. Ermentrout, Phys. Rev. E 77, 041918
共2008兲.
关33兴 R. F. Galan, G. B. Ermentrout, and N. N. Urban, Phys. Rev. E
76, 056110 共2007兲.
关34兴 B. Pfeuty, G. Mato, D. Golomb, and D. Hansel, Neural Comput. 17, 633 共2005兲.
关35兴 J. Rinzel and G. Ermentrout, in Methods in Neuronal Modeling, edited by C. Koch and I. Segev 共MIT Press, Cambridge,
MA, 1998兲, pp. 251–291.
关36兴 A. Winfree, The Geometry of Biological Time 共Springer, New
York, 2001兲.
关37兴 G. Ermentrout, Neural Comput. 8, 979 共1996兲.
关38兴 G. Ermentrout and N. Kopell, SIAM J. Math. Anal. 15, 215
共1984兲.
关39兴 E. Brown, J. Moehlis, and P. Holmes, Neural Comput. 16, 673
共2004兲.
关40兴 D. Hansel, G. Mato, and C. Meunier, Europhys. Lett. 23, 367
共1993兲.
关41兴 D. Cox and P. Lewis, The Statistical Analysis of a Series of
Events 共John Wiley, London, 1966兲.
关42兴 B. Lindner, B. Doiron, and A. Longtin, Phys. Rev. E 72,
061919 共2005兲.
关43兴 C. Gardiner, Handbook of Stochastic Methods, Series in Synergetics, 3rd ed. 共Springer, New York, 2004兲.
关44兴 The expression we obtain is identical to the evaluation of this
expression in the case of a reflecting boundary at 0.
关45兴 B. Lindner, A. Longtin, and A. Bulsara, Neural Comput. 15,
1761 共2003兲.
关46兴 R. Vilela and B. Lindner, J. Theor. Biol. 257, 90 共2009兲.
关47兴 N. Bleistein and R. Handelsman, Asymptotic Expansions of
Integrals 共Dover, New York, 1986兲.
关48兴 M. Rudolph and A. Destexhe, J. Comput. Neurosci. 14, 239
共2003兲.
关49兴 A. Destexhe, M. Rudolph, and D. Paré, Nat. Rev. Neurosci. 4,
739 共2003兲.
关50兴 F. Pouille and M. Scanziani, Science 293, 1159 共2001兲.
关51兴 C. Brody, R. Romo, and A. Kepecs, Curr. Opin. Neurobiol.
13, 204 共2003兲.
关52兴 S. Panzeri, S. R. Schultz, A. Treves, and E. T. Rolls, Proc. R.
Soc. London, Ser. B 266, 1001 共1999兲.
011916-13