How adaptation currents change threshold, gain, and variability of
neuronal spiking
Josef Ladenbauer,
1,2
Moritz Augustin,
1,2
and Klaus Obermayer
1,2
1
Neural Information Processing Group, Technische Universität Berlin, Berlin, Germany; and
2
Bernstein Center
for Computational Neuroscience Berlin, Berlin, Germany
Submitted 19 August 2013; accepted in final form 25 October 2013
Ladenbauer J, Augustin M, Obermayer K. How adaptation
currents change threshold, gain, and variability of neuronal spiking. J
Neurophysiol 111: 939–953, 2014. First published October 30, 2013;
doi:10.1152/jn.00586.2013.—Many types of neurons exhibit spike
rate adaptation, mediated by intrinsic slow K
⫹
currents, which effec-
tively inhibit neuronal responses. How these adaptation currents
change the relationship between in vivo like fluctuating synaptic
input, spike rate output, and the spike train statistics, however, is not
well understood. In this computational study we show that an adap-
tation current that primarily depends on the subthreshold membrane
voltage changes the neuronal input-output relationship (I-O curve)
subtractively, thereby increasing the response threshold, and de-
creases its slope (response gain) for low spike rates. A spike-depen-
dent adaptation current alters the I-O curve divisively, thus reducing
the response gain. Both types of an adaptation current naturally
increase the mean interspike interval (ISI), but they can affect ISI
variability in opposite ways. A subthreshold current always causes an
increase of variability while a spike-triggered current decreases high
variability caused by fluctuation-dominated inputs and increases low
variability when the average input is large. The effects on I-O curves
match those caused by synaptic inhibition in networks with asynchro-
nous irregular activity, for which we find subtractive and divisive
changes caused by external and recurrent inhibition, respectively.
Synaptic inhibition, however, always increases the ISI variability. We
analytically derive expressions for the I-O curve and ISI variability,
which demonstrate the robustness of our results. Furthermore, we
show how the biophysical parameters of slow K
⫹
conductances
contribute to the two different types of an adaptation current and find
that Ca
2⫹
-activated K
⫹
currents are effectively captured by a simple
spike-dependent description, while muscarine-sensitive or Na
⫹
-acti-
vated K
⫹
currents show a dominant subthreshold component.
adaptation; gain modulation; Hodgkin-Huxley-like model; integrate-
and-fire model; spike train
ADAPTATION IS A WIDESPREAD phenomenon in nervous systems,
providing flexibility to function under varying external condi-
tions. At the single neuron level, this can be observed as spike
rate adaptation, a gradual decrease in spiking activity following
a sudden increase in stimulus intensity. This type of intrinsic
inhibition, in contrast to the one caused by synaptic interaction,
is typically mediated by slowly decaying somatic K
⫹
currents,
which accumulate when the membrane voltage increases. A
number of slow K
⫹
currents with different activation charac-
teristics have been identified. Muscarine-sensitive (Brown and
Adams 1980; Adams et al. 1982) or Na
⫹
-dependent K
⫹
chan-
nels activate at subthreshold voltage values (Schwindt et al.
1989; Kim and McCormick 1998), whereas Ca
2⫹
-dependent
K
⫹
channels activate at higher, suprathreshold values (Brown
and Griffith 1983; Madison and Nicoll 1984; Schwindt et al.
1992). Such adaptation currents, for example, mediate fre-
quency selectivity of neurons (Fuhrmann et al. 2002; Benda et
al. 2005; Ellis et al. 2007), where the preferred frequency
depends on the current activation type (Deemyad et al. 2012).
They promote network synchronization (Sanchez-Vives and
McCormick 2000; Augustin et al. 2013; Ladenbauer et al.
2013) and are likely involved in the attentional modulation of
neuronal response properties by acetylcholine (Herrero et al.
2008; Soma et al. 2012; McCormick 1992). It has been hy-
pothesized that these complex effects are produced by chang-
ing the relationship between synaptic input and spike rate
output (I-O curve) (Deemyad et al. 2012; Benda and Herz
2003; Soma et al. 2012; Reynolds and Heeger 2009). For
example, changing the I-O curve of a neuron subtractively
sharpens stimulus selectivity, whereas a divisive change down-
scales the neuronal response but preserves selectivity (see
Wilson et al. 2012 in the context of synaptic inhibition). It was
also suggested that adaptation currents affect the neural code
via their effect on the interspike interval (ISI) statistics
(Prescott and Sejnowski 2008). So far, the effects of adaptation
currents on I-O curves have been studied considering constant
current inputs disregarding input fluctuations (Prescott and
Sejnowski 2008; Deemyad et al. 2012) and it has remained
unclear how different types of an adaptation current affect ISI
variability. Therefore, in this contribution we systematically
examine how voltage-dependent subthreshold and spike-de-
pendent adaptation currents change neuronal I-O curves as well
as the ISI distribution for typical in vivo like input statistics and
how the biophysical parameters of slow K
⫹
conductances
contribute to the two types of adaptation current.
We address these questions by studying spike rates and ISI
distributions of model neurons with subthreshold and spike-
triggered adaptation currents, subject to fluctuating in vivo like
inputs, and we compare the results to those induced by synaptic
inhibition. Specifically, we use the adaptive exponential inte-
grate-and-fire (aEIF) neuron model (Brette and Gerstner 2005),
which has been shown to perform well in predicting the
subthreshold properties (Badel et al. 2008) and spiking activity
(Jolivet et al. 2008; Pospischil et al. 2011) of cortical neurons.
To analytically demonstrate the changes of I-O curves and ISI
variability we derive explicit expressions for these properties
based on the simpler perfect integrate-and-fire neuron model
(see, e.g., Gerstein and Mandelbrot 1964) with adaptation
(aPIF). Finally, using a detailed conductance-based neuron
model we quantify the subthreshold and spike-triggered com-
Address for reprint requests and other correspondence: J. Ladenbauer,
Technische Universität Berlin, Neural Information Processing Group, March-
J Neurophysiol 111: 939–953, 2014.
First published October 30, 2013; doi:10.1152/jn.00586.2013.
939Licensed under Creative Commons Attribution CC-BY 3.0: ©the American Physiological Society. ISSN 0022-3077.www.jn.org
by 10.220.33.2 on October 25, 2017http://jn.physiology.org/Downloaded from
ponents of various slow K
⫹
currents and compare the effects of
specific K
⫹
channels on the I-O curve and ISI variability.
MATERIALS AND METHODS
aEIF neuron with noisy input current. We consider an aEIF model
neuron receiving synaptic input currents. The subthreshold dynamics
of the membrane voltage Vis given by
CdV
dt ⫽Iion(V)⫹Isyn(t), (1)
where the capacitive current through the membrane with capacitance
Cequals the sum of ionic currents I
ion
and the synaptic current I
syn
.
Three ionic currents are taken into account,
Iion(V):⫽⫺gL(V⫺EL)⫹gL⌬Texp
冉
V⫺VT
⌬T
冊
⫺w.(2)
The first term on the right-hand side describes the leak current with
conductance g
L
and reversal potential E
L
. The exponential term with
threshold slope factor ⌬
T
and effective threshold voltage V
T
approx-
imates the fast Na
⫹
current at spike initiation, assuming instantaneous
activation of Na
⫹
channels (Fourcaud-Trocmé et al. 2003). wIs the
adaptation current that reflects a slow K
⫹
current. It evolves according
to
w
dw
dt ⫽a(V⫺Ew)⫺w,(3)
with adaptation time constant
w
. Its strength depends on the sub-
threshold membrane voltage via conductance a.E
w
denotes its rever-
sal potential. When Vincreases beyond V
T
, a spike is generated due
to the exponential term in Eq. 2. The downswing of the spike is not
explicitly modeled, instead, when Vreaches a value V
s
ⱖV
T
, the
membrane voltage is reset to a lower value V
r
. At the same time, the
adaptation current wis incremented by a value of b, implementing
the mechanism of spike-triggered adaptation. Immediately after the
reset, Vand ware clamped for a refractory period T
ref
, and subse-
quently governed again by Eqs. 1–3.
The aEIF model can reproduce a wide range of neuronal subthresh-
old dynamics (Touboul and Brette 2008) and spike patterns (Naud et
al. 2008). We selected the following parameter values to model
cortical neurons: C⫽1
F/cm
2
,g
L
⫽0.05 mS/cm
2
,E
L
⫽⫺65 mV,
⌬
T
⫽1.5 mV, V
T
⫽⫺50 mV,
w
⫽200 ms, E
w
⫽⫺80 mV, V
s
⫽
⫺40 mV, V
r
⫽⫺70 mV, and T
ref
⫽1.5 ms (Badel et al. 2008;
Destexhe 2009; Wang et al. 2003). The adaptation parameters aand
bwere varied within reasonable ranges, a僆[0, 0.06] mS/cm
2
,b僆[0,
0.3]
A/cm
2
.
The synaptic input consists of a mean
(t) and a fluctuating part
given by a Gaussian white noise process
(t) with
␦
-autocorrelation
and standard deviation
(t),
Isyn(t)⫽C[
(t)⫹
(t)
(t)]. (4)
Equation 4 describes the total synaptic current received by K
E
excit-
atory and K
I
inhibitory neurons, which produce instantaneous post-
synaptic potentials (PSPs) J
E
⬎0 and J
I
⬍0, respectively. For
synaptic events (i.e., presynaptic spike times) generated by indepen-
dent Poisson processes with rates r
E
(t) and r
I
(t), the infinitesimal
moments
(t) and
(t) are expressed as
(t)⫽JEKErE(t)⫹JIKIrI(t), (5)
(t)2⫽JE
2KErE(t)⫹JI
2KIrI(t), (6)
assuming large numbers K
E
,K
I
, and small magnitudes of J
E
,J
I
(Tuckwell 1988; Renart et al. 2004; Destexhe and Rudolph-Lilith
2012). This diffusion approximation well describes the activity in
many cortical areas (Shadlen and Newsome 1998; Destexhe et al.
2003; Compte et al. 2003; Maimon and Assad 2009). The parameter
values were J
E
⫽0.15 mV, J
I
⫽⫺0.45 mV, K
E
⫽2000, K
I
⫽500,
and r
E
,r
I
were varied in [0, 50] Hz. In addition, we directly varied
and
over a wide range of biologically plausible values.
Membrane voltage distribution and spike rate. In the following we
describe how we obtain the distribution of the membrane voltage p(V,
t) and the instantaneous spike rate r(t) of a single neuron at time tfor
a large number Nof independent trials. Note that by trial we refer to
a solution trajectory of the system of stochastic differential equations
(Eqs. 1–4) for a realization of
(t).
First, to reduce computational demands and enable further analysis,
we replace the adaptation current win Eqs. 2 and 3by its average over
trials, w(t):⫽1/N 冱
i⫽1
N
w
i
(t), where iis the trial index (Gigante et al.
2007a). Neglecting the variance of wacross trials is valid under the
assumption that the dynamics of the adaptation current is substantially
slower than that of the membrane voltage, which is supported by
empirical observations (Brown and Adams 1980; Sanchez-Vives and
McCormick 2000; Sanchez-Vives et al. 2000; Stocker 2004). The
instantaneous spike rate at time tcan be estimated by the average
number of spikes in a small interval [t,t⫹⌬t],
r⌬t(t):⫽1
N⌬t兺
i⫽1
N
兰
t
t⫹⌬t兺
k
␦
(s⫺ti
k)ds,(7)
where
␦
is the delta function and t
i
k
denotes the k-th spike time in trial
i. In the limit N¡⬁,⌬t¡0, the probability density p(V,t) obeys the
Fokker-Planck equation (Risken 1996; Tuckwell 1988; Renart et al.
2004),
⭸
⭸tp(V,t)⫹⭸
⭸Vq(V,t)⫽0, (8)
with probability flux q(V,t) given by
q(V,t):⫽
冉
Iion(V;w)
C⫹
(t)
冊
p(V,t)⫺
(t)2
2
⭸
⭸Vp(V,t). (9)
I
ion
(V;w) denotes the sum of ionic currents (cf. Eq. 2) where wis
replaced by the average adaptation current w, which evolves accord-
ing to
w
dw
dt ⫽a
共
具V典p(V,t)⫺Ew
兲
⫺w⫹
wbr(t). (10)
具·典pindicates the average with respect to the probability density p
(Brunel et al. 2003; Gigante et al. 2007b). To account for the reset of
the membrane voltage, the probability flux at V
s
is reinjected at V
r
after the refractory period has passed, i.e.,
lim
VnVr
q(V,t)⫺lim
VmVr
q(V,t)⫽q(Vs,t⫺Tref). (11)
The boundary conditions for this system are reflecting for V¡⫺⬁
and absorbing for V⫽V
s
,
lim
V→⫺⬁
q(V,t)⫽0, p(Vs,t)⫽0, (12)
and the (instantaneous) spike rate is obtained by the probability flux
at V
s
,
r(t)⫽q(Vs,t). (13)
Note that p(V,t) only reflects the proportion of trials where the neuron
is not refractory at time t, given by P(t)⫽兰
⫺⬁
V
sp(v,t)dv [⬍1
for T
ref
⬎0 and r(t)⬎0]. The total probability density that the membrane
voltage is Vat time tis given by p(V,t)⫹p
ref
(V,t), with refractory
density p
ref
(V,t)⫽[1 ⫺P(t)]
␦
(V⫺V
r
). Since p(V,t) does not
integrate to unity in general, the average in Eq. 10 is calculated as
具V典p共V,t兲⫽兰
⫺⬁
V
svp(v,t)dv/P(t). The dynamics of the average adap-
tation current w(t) reflecting the nonrefractory proportion of trials
is well captured by Eq. 10 as long as T
ref
is small compared with
w
. In this (physiologically plausible) case w(t) can be considered
940 EFFECTS OF ADAPTATION CURRENTS ON SPIKE TRAINS
J Neurophysiol •doi:10.1152/jn.00586.2013 •www.jn.org
by 10.220.33.2 on October 25, 2017http://jn.physiology.org/Downloaded from
equal to the average adaptation current over the refractory propor-
tion of trials.
Steady-state spike rate. We consider the membrane voltage distri-
bution of an aEIF neuron with noisy synaptic input, described by the
Eqs. 8–13, has reached its steady-state p
⬁
.p
⬁
obeys ⭸p
⬁
(V)/⭸t⫽0or
equivalently,
⭸
⭸Vq⬁(V)⫽0, (14)
with steady-state probability flux q
⬁
given by
q⬁(V)⫽
冉
Iion(V; w)
C⫹
冊
p⬁(V)⫺
2
2
⭸
⭸Vp⬁(V), (15)
subject to the reset condition,
lim
VnVr
q⬁(V)⫺lim
VmVr
q⬁(V)⫽q⬁(Vs), (16)
and the boundary conditions,
lim
V→⫺⬁
q⬁(V)⫽0, p⬁(Vs)⫽0. (17)
The steady-state spike rate is given by r
⬁
⫽q
⬁
(V
s
) and the steady-
state mean adaptation current reads w
⬁
⫽a共具V典⬁⫺Ew兲⫹
w
br
⬁
.We
multiply both sides of Eq. 14 by Vand integrate over the interval (⫺⬁,
V
s
], assuming that p
⬁
(V) tends sufficiently quickly toward zero for
V¡⫺⬁ (Brunel 2000; Brunel et al. 2003), to obtain an equation that
relates the steady-state spike rate and mean membrane voltage,
r⬁⫽
a⫺gL
冋
具V典⬁⫺EL⫹⌬
T冓exp
冉
V⫺VT
⌬T
冊
冔⬁
册
⁄C
⌬V⫹
wb⁄C,(18)
where
a
:⫽
⫺a共具V典⬁⫺Ew兲/C,⌬V:⫽V
s
⫺V
r
(here and in the
following) and 具·典⬁denotes the average with respect to the density
p
⬁
(V). The spike rate r
⬁
is given by Eq. 18 only for nonnegative
values of the numerator (i.e.,
a
⫺g
L
[...]/Cⱖ0); otherwise, r
⬁
is
defined to be zero. For simplicity, the refractory period T
ref
is omitted
here. Note that the steady-state spike rate for T
ref
⫽0 can be
calculated as r
⬁
/(1 ⫹r
⬁
T
ref
). We cannot express p
⬁
(V) explicitly and
thus the expressions for the averages with respect to p
⬁
(V)inEq. 18
are not known. However, in the case g
L
⫽0, which simplifies the
aEIF model to the aPIF model, an explicit expression for 具V典⬁can be
derived. We multiply Eq. 14 by V
2
and integrate over (⫺⬁,V
s
]on
both sides [assuming again that p
⬁
(V) quickly tends to zero for V¡
⫺⬁] to obtain
具V典⬁⫽1
2a
冋
A⫹aVs⫹Vr
2⫺冑
冉
A⫺aVs⫹Vr
2
冊
2
⫹B
册
,(19)
where A⫽
C⫹aE
w
and B⫽2a
2
C[1 ⫹
w
b/(C⌬V)].
I-O curve. The I-O curve is specified by the spike rate as a function
of input strength. Here we consider two types of I-O curves: a
time-varying (adapting) I-O curve and the steady-state I-O curve. In
particular, we obtain the adapting I-O curve as the instantaneous
spike rate response to a sustained input step (with a small baseline
input) as a function of step size. This curve changes (adapts) over
time, and it eventually converges to the steady-state I-O curve. As
arguments of these (adapting and steady-state) I-O functions we
consider presynaptic spike rates (see Figs. 2Cand 4Band Eq. 38),
input mean and standard deviation
1
(see Figs. 2Dand 4Band Eq.
36), and input mean for fixed values of input standard deviation
(see Fig. 8A).
ISI distribution. We calculate the ISI distribution for an aEIF
neuron that has reached a steady-state spike rate r
⬁
:⫽lim
t¡⬁
r(t)by
solving the so-called first passage time problem (Risken 1996; Tuck-
well 1988). Consider an initial condition where the neuron has just
emitted a spike and the refractory period has passed. That is, the
membrane voltage is at the reset value V
r
and the adaptation current,
which we have replaced by its trial average (see above), takes the
value w
0
, where w
0
will be determined self-consistently (see
below). In each of N(simultaneous) trials, we follow the dynamics
of the neuron given by dV
i
/dt ⫽[I
ion
(V
i
;w)⫹I
syn
(t)]/C,dw/dt ⫽
[a(1/N冱
i⫽1
N
V
i
⫺E
w
)⫺w]/
w
, until its membrane voltage crosses
the value V
s
and record that spike time T
i
. The set of times T
i
⫹T
ref
then gives the ISI distribution. Finally, we determine w
0
by imposing
that the mean ISI matches with the known steady-state spike rate, i.e.,
1/N冱
i⫽1
N
T
i
⫹T
ref
⫽r
⬁
⫺1
. According to this calculation scheme, the
ISI distribution can be obtained in the limit N¡⬁by solving the
Fokker-Planck system Eqs. 8 and 9with mean adaptation current
governed by
w
dw
dt ⫽a
共
具V典p(V,t)⫺Ew
兲
⫺w,(20)
subject to the boundary conditions (12) and initial conditions p(V,0)⫽
␦
(V⫺V
r
), w(0) ⫽w
0
. Note that the reinjection condition Eq. 11 is
omitted (see also the difference between Eqs. 10 and 20) because here
each trial iends once V
i
(t) crosses the value V
s
. The ISI distribution is
given by the probability flux at V
s
(Tuckwell 1988; Ostojic 2011), taking
into account the refractory period
pISI(T)⫽再q(Vs,T⫺Tref) for
0 for
TⱖTref
T⬍Tref
.(21)
Finally, w
0
is determined self-consistently by requiring 具T典pISI ⫽r
⬁
⫺1
.
The coefficient of variation (CV) of ISIs is then calculated as
CV:⫽兹具T2典pISI ⫺具T典pISI
2
具T典pISI
.(22)
An ISI CV value of 0 indicates regular, clock-like spiking, whereas
for spike times generated by a Poisson process the ISI CV assumes a
value of 1. For a demonstration of the ISI calculation scheme de-
scribed above, see Fig. 1. The results based on the Fokker-Planck
equation and numerical simulations of the aEIF model with fluctuat-
ing input are presented for an increased subthreshold and spike-
triggered adaptation current in separation.
ISI CV for the aPIF model. To calculate the ISI CV we need the
first two ISI moments, cf. Eq. 22. The mean ISI for the aPIF neuron
model is simply calculated by the inverse of the steady-state spike
rate, cf. Eq. 18, derived in the previous section,
具T典pISI ⫽r⬁
⫺1⫽⌬V⫹
wb⁄C
a
,(23)
where we consider
a
⬎0 (here and in the following). We approxi-
mate the second ISI moment by solving the first passage time problem
for the Langevin equation
dV
dt ⫽
a⫺w
0
Cexp(⫺t⁄
w)⫹
(t), (24)
with initial membrane voltage V
r
and boundary voltage V
s
. That is, we
replace 具V典p共V,t兲by its steady-state value 具V典⬁in Eq. 20, which is
justified by large
w
(as already assumed). The first passage time
density (which is equivalent to p
ISI
) and the associated first two
moments for this type of Langevin equation can be calculated as
power series in the limit of small w
0
(Urdapilleta 2011). w
0
is then
determined self-consistently by imposing Eq. 23. Here we approxi-
mate the second ISI moment by using only the most dominant term of
1
Note that because of two arguments we obtain a surface instead of a curve
in this case.
941EFFECTS OF ADAPTATION CURRENTS ON SPIKE TRAINS
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the power series, which yields (the zeroth order approximation)
(Urdapilleta 2011),
具T2典pISI ⫽
2⌬V⫹
a⌬V2
a
3.(25)
Including terms of higher order leads to a complicated expression for
具T2典pISI, which has to be evaluated numerically. We additionally
considered the first order term (not shown) and compared the results
of both approximations (see RESULTS). Effectively, the approximation
above, Eq. 25, is valid for small levels of spike-triggered adaptation
current and mean input, since w
0
increases with band
. Combining
Eqs. 22,23, and 25 the ISI CV reads
CV ⫽兹
2⌬V⁄
a⫺
w
2b2⁄C2⫺2
wb⌬V⁄C
⌬V⫹
wb⁄C.(26)
Neuronal network. To investigate the effects of recurrent (inhibi-
tory) synaptic inputs on the neuronal response properties (spike rates
and ISIs), we consider a network instead of a single neuron, consisting
of N
E
excitatory and N
I
inhibitory aEIF neurons (with separate
parameter sets). The two populations are recurrently coupled in the
following way (see Fig. 4A). Each excitatory neuron receives inputs
from KEE
ext
external excitatory neurons which produce instantaneous
PSPs of magnitude JEE
ext
with Poisson rate rEE
ext
(t). Analogously, each
inhibitory neuron receives inputs from KIE
ext
external excitatory neu-
rons producing instantaneous PSPs of magnitude JIE
ext
with Poisson
rate rIE
ext
(t). In addition, each excitatory neuron receives inputs from
KEI
rec
randomly selected inhibitory neurons of the network with syn-
aptic strength (i.e., instantaneous PSP magnitude) JEI
rec
and each
inhibitory neuron receives inputs from KIE
rec
randomly selected excit-
atory neurons of the network with synaptic strength JIE
rec
. This network
setup was chosen to examine the effects caused by recurrent inhibition
and compare them to the effects produced by external inhibition for
single neurons described above. To reduce the parameter space,
recurrent connections within the two populations in the network were
therefore omitted. The total synaptic current for each neuron of the
network can be described using Eq. 4, where the parameters
(t) and
(t) for excitatory neurons are given by
(t)⫽JEE
extKEE
extrEE
ext(t)⫹JEI
recKEI
recrI
pop(t), (27)
(t)2⫽(JEE
ext)2KEE
extrEE
ext(t)⫹(JEI
rec)2KEI
recrI
pop(t)(28)
and for inhibitory neurons,
(t)⫽JIE
extKIE
extrIE
ext(t)⫹JIE
recKIE
recrE
pop(t), (29)
(t)2⫽(JIE
ext)2KIE
extrIE
ext(t)⫹(JIE
rec)2KIE
recrE
pop(t)(30)
(Brunel 2000; Augustin et al. 2013). rE
pop
(t) and rI
pop
(t) are the spike
rates of the excitatory and inhibitory neurons of the network, respec-
tively. Here we consider large populations of neurons instead of a
large number of trials. In fact, averaging over a large number of trials
in this setting is equivalent to averaging over large populations due to
the random and sparse connectivity. In the limit N
E
,N
I
¡⬁we obtain
a system two coupled Fokker-Planck equations, one for the excitatory
population, described by Eqs. 8–13,27, and 28, and one for the
Fig. 1. Steady-state spike rates and interspike interval (ISI) distributions of single neurons. A,top to bottom: spike times, instantaneous spike rate (r
⌬t
) histogram,
membrane voltage (V
i
), membrane voltage histogram, and adaptation current (w
i
) of an (adapted) adaptive exponential integrate-and-fire (aEIF) neuron with
a⫽0.06 mS/cm
2
,b⫽0(left), and a⫽0, b⫽0.18
A/cm
2
(right) driven by a fluctuating input current with
⫽2.5 mV/ms,
⫽2 mV/兹ms for N⫽5,000
trials. Spike times and adaptation current are shown for a subset of 10 trials, the membrane voltage is shown for one trial. Results from numerical simulations
are shown in grey. Results obtained using the Fokker-Planck equation are indicated by orange lines and include the instantaneous spike rate (r), the membrane
potential distribution (p), and the mean adaptation current ( w). r,p, And wwere calculated from the Eqs. 13,8, and 10, respectively. These quantities have reached
their steady state here. The time bin for r
⌬t
was ⌬t⫽2 ms; for the other parameter values see MATERIALS AND METHODS.B,top: ISI histogram corresponding
to the Ntrials in Aand ISI distribution (p
ISI
, orange line) calculated via the first passage time problem (Eq. 21). B,middle and bottom: membrane voltage and
adaptation current trajectories from 1 trial in Abut rearranged such that just after each spike the time is set to zero. Histograms for the adaptation current just
after the spike times are included. The time-varying mean adaptation current from the first passage time problem (Eq. 20) and the steady-state mean adaptation
current from A(Eq. 10) are indicated by solid and dashed orange lines, respectively. All histograms (in Aand B) represent the data from all Ntrials.
942 EFFECTS OF ADAPTATION CURRENTS ON SPIKE TRAINS
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inhibitory population, given by Eqs. 8–13,29, and 30. Note that r(t)
in Eqs. 10 and 13 is replaced by the spike rates of the excitatory and
inhibitory populations, rE
pop
(t) and rI
pop
(t), respectively. We solve this
system to obtain the steady-state spike rate for each population, rE,⬁
pop
and rI,⬁
pop
. Once these quantities are known, we calculate the ISI
distribution, cf. Eq. 21, for the excitatory population (i.e., for any
neuron of that population) as described above, using Eqs. 27 and 28
for the (steady-state) moments of the synaptic current. The neuron
model parameter values were as above for the single neuron, with a⫽
0.015 mS/cm
2
,b⫽0.1
A/cm
2
for excitatory neurons and a⫽b⫽
0 for inhibitory neurons, since adaptation was found to be weak in
fast-spiking interneurons compared with pyramidal neurons (La Cam-
era et al. 2006). The network parameter values were JEE
ext
⫽JIE
ext
⫽0.15
mV, KEE
ext
⫽KIE
ext
⫽800, constant rEE
ext
僆[0, 80] Hz, JEI
rec
僆[⫺0.75,
⫺0.45] mV, KEI
rec
⫽100, constant rIE
ext
僆[6, 14] Hz, JIE
rec
僆[0.05, 0.2]
mV, and KIE
rec
⫽400.
Numerical solution. We treated the Fokker-Planck equations for the
aPIF model analytically. In case of the aEIF model, we solved these
equations forward in time using a first-order finite volume method on
a nonuniform grid with 512 grid points in the interval [⫺200 mV, V
s
]
and the implicit Euler integration method with a time step of 0.1 ms
for the temporal domain. For more details on the numerical solution,
we refer to Augustin et al. (2013).
Detailed conductance-based neuron model. For validation pur-
poses we used a biophysical Hodgkin-Huxley-type neuron model with
different types of slow K
⫹
current. The membrane voltage Vof this
neuron model obeys the current balance equation
CdV
dt ⫽I⫺IL⫺INa ⫺IK⫺ICa ⫺IKs,(31)
where C⫽1
F/cm
2
is the membrane capacitance and Idenotes the
injected current. The ionic currents consist of a leak current, I
L
⫽
g
L
(V⫺E
L
), a spike-generating Na
⫹
current, I
Na
⫽g
Na
(V)(V⫺E
Na
),
a delayed rectifier K
⫹
current, I
K
⫽g
K
(V)(V⫺E
K
), a high-threshold
Ca
2⫹
current, I
Ca
⫽g
Ca
(V)(V⫺E
Ca
), and a slow K
⫹
current I
Ks
.g
x
Denote the conductances of the respective ion channels and E
x
are the
reversal potentials. We separately considered three types of slow K
⫹
current: a Ca
2⫹
-activated current (I
Ks
⬅I
KCa
) which is associated
with the slow after-hyperpolarization following a burst of spikes
(Brown and Griffith 1983), a Na
⫹
-activated current (I
Ks
⬅I
KNa
)
(Schwindt et al. 1989), and the voltage-dependent muscarine-sensitive
(M type) current (I
Ks
⬅I
M
) (Brown and Adams 1980). The leak
current depends linearly on the membrane potential. All other ionic
currents depend on Vin a nonlinear way as described by the Hodgkin-
Huxley formalism. We adopted the somatic model from (Wang et al.
2003) and included the M current with dynamics described (for the
soma) by (Mainen and Sejnowski 1996). The conductances underly-
ing the currents I
Na
,I
K
,I
Ca
, and I
M
are given by g
Na
⫽g
Na
m
⬁
3
h,
g
K
⫽g
K
n
4
,g
Ca
⫽g
Ca
s
⬁
2
and g
M
⫽g
M
u, respectively, with steady-state
gating variables m
⬁
⫽
␣
m
/(
␣
m
⫹

m
),
␣
m
⫽⫺0.4(V⫹33)/{exp[⫺(V⫹
33)/10] ⫺1},

m
⫽16 exp[⫺(V⫹58)/12], and s
⬁
⫽1/{1 ⫹exp[⫺(V⫹
20)/9]}. The dynamic gating variables x僆{h,n,u} are governed by
dx
dt ⫽
␣
x(1 ⫺x)⫺

xx,(32)
where
␣
h
⫽0.28 exp [⫺(V⫹50)/10],

h
⫽4/{1 ⫹exp[⫺(V⫹
20)/10]},
␣
n
⫽⫺0.04(V⫹34)/{exp[⫺(V⫹34)/10] ⫺1},

n
⫽0.5
exp[⫺(V⫹44)/25],
␣
u
⫽3.209·10
⫺4
(V⫹30)/{1 ⫺exp[⫺(V⫹
30)/9]} and

u
⫽⫺3.209 ·10
⫺4
(V⫹30)/{1 ⫺exp[(V⫹30)/9]}. The
channel opening and closing rates
␣
x
and

x
are specified in ms
⫺1
and
the membrane voltage Vin the equations above is replaced by its
value in mV. The conductance for the Ca
2⫹
-activated slow K
⫹
current I
KCa
is given by g
KCa
⫽g
KCa
[Ca]/([Ca] ⫹
), where the
intracellular Ca
2⫹
concentration [Ca] satisfies
d[Ca]
dt ⫽⫺
␣
CaICa ⫺[Ca]
Ca
(33)
with
␣
Ca
⫽6.67·10
⫺4
M·cm
2
/(
A·ms),
Ca
⫽240 ms, and
⫽0.03
mM. The conductance for the Na
⫹
-activated slow K
⫹
current I
KNa
is
described by g
KNa
⫽g
KNa
0.37/{1 ⫹(|/[Na])
3.5
}, where |⫽38.7
mM and the intracellular Na
⫹
concentration [Na] is governed by
d[Na]
dt ⫽⫺
␣
Na ⫺3
冉
[Na]3
关Na兴3⫹
3⫺
␥
冊
(34)
with
␣
Na
⫽0.3
M·cm
2
/(
A·ms),
⫽0.6
M/ms,
⫽15 mM, and
␥
⫽0.132. We varied the peak conductances of the three slow K
⫹
currents I
KCa
,I
KNa
,I
M
in the ranges g
KCa
僆[2, 8] mS/cm
2
,g
KNa
僆
[2, 8] mS/cm
2
(Wang et al. 2003), and g
M
僆[0.1, 0.4] mS/cm
2
(Mainen and Sejnowski 1996). The remaining parameter values were
C⫽1
F/cm
2
,g
L
⫽0.1 mS/cm
2
,E
L
⫽⫺65 mV, E
Na
⫽55 mV,
E
K
⫽⫺80 mV, and E
Ca
⫽120 mV (Wang et al. 2003).
The differences of the slow K
⫹
currents (I
KCa
,I
KNa
, and I
M
)is
effectively expressed by their steady-state voltage dependence and
time constants. Therefore, we further considered a range of biologi-
cally plausible steady-state conductance-voltage relationships and
timescales using the generic description of a slow K
⫹
current, I
Ks
⫽
g
Ks
(V)(V⫺E
K
), with peak conductance g
Ks
and gating variable
(V) given by
d
dt ⫽
⬁(V)⫺
,(35)
where
⬁
(V)⫽1/{1 ⫹exp[⫺(V⫺
␣
)/

]}. The shape of the
steady-state curve
⬁
(V) was changed by the parameters
␣
僆[⫺40,
⫺10] mV (half-activation voltage),

僆[6, 12] mV (inverse steep-
ness), and the time constant
was varied in [100, 300] ms. The
model equations were solved using a second order Runge-Kutta
integration method with a time step of 10
s.
To examine the effects of slow K
⫹
currents on the I-O curve and
ISI variability for noisy input, we additionally considered the synaptic
current described by Eq. 4 for the detailed neuron model, i.e., we used
I⬅I
syn
in Eq. 31.
Subthreshold and spike-triggered components of biophysical slow
K
⫹
currents. To assess how the relative levels of subthreshold
adaptation conductance (parameter a) and spike-triggered adaptation
current increments (parameter b) in the aEIF model reflect different
types of slow K
⫹
current, we quantified their subthreshold and
spike-triggered components using the detailed conductance-based
neuron model. First, we fit the steady-state adaptation current w
⬁
⫽
a(V⫺E
w
) from the aEIF model to the respective K
⫹
current I
Ks
of the
Hodgkin-Huxley-type model in steady-state over a range of sub-
threshold values for the membrane voltage, V僆[⫺70, ⫺60] mV.
Thereby we obtained an estimate for a. In the second step, we
measured the absolute and relative change of I
Ks
elicited by one spike.
This was done by injecting a slowly increasing current ramp into the
detailed model neuron and measuring I
Ks
just before and after the first
spike that occurred. Specifically, the absolute change of current
caused by a spike was given by ⌬I
Ks
:⫽I
Ks
(t
s
post
)⫺I
Ks
(t
s
pre
), where the
time points t
s
pre
and t
s
post
were defined by the times at which the
membrane potential crosses a value close to threshold (we chose ⫺50
mV) during the upswing and downswing of the spike, respectively.
⌬I
Ks
provides an estimate for b. The relative change of K
⫹
current
was ⌬I
Ks
rel
:⫽⌬I
Ks
/I
Ks
(t
s
pre
). Here we only fitted the parameters aand b
of the aEIF model. For an alternative fitting procedure which com-
prises all model parameters, we refer to (Brette and Gerstner 2005).
RESULTS
Spike rate adaptation, gain, and threshold modulation in
single neurons. We first examine the responses of single aEIF
neurons with and without an adaptation current, receiving
943EFFECTS OF ADAPTATION CURRENTS ON SPIKE TRAINS
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inputs from stochastically spiking presynaptic excitatory and
inhibitory neurons. The compound effect of the individual
synaptic inputs is represented by an ongoing fluctuating input
current whose mean and standard deviation depend on the
synaptic strengths and spike rates of the presynaptic cells (cf.
Eqs. 4–6 in MATERIALS AND METHODS and Fig. 2A). The neurons
naturally respond to a sudden increase in spike rate of the
presynaptic neurons (an input step) with an abrupt increase in
spike rate and mean membrane voltage (see Fig. 2B). Without
an adaptation current, both quantities remain unchanged after
that increase. In case of a purely subthreshold adaptation
current (a⬎0, b⫽0 in the aEIF model), which is present
already in absence of spiking, the rapid increase of mean
membrane voltage causes the mean adaptation current to build
up slowly, which in turn leads to a gradual decrease in spike
rate and mean membrane voltage. Note that the mean mem-
brane voltage is decreased in the absence of spiking (before the
increase of input) compared with the neuron without adapta-
tion. In case of a purely spike-triggered adaptation current (a⫽
0, b⬎0 in the aEIF model), the sudden increase in spike rate
leads to an increase of mean adaptation current, which again
causes the spike rate and mean membrane voltage to decrease
gradually.
The adapting I-O curve of neurons with and without an
adaptation current, that is, the time-varying spike rate response
to a step in presynaptic spike rates as a function of the step size,
is shown in Fig. 2C. Interestingly, the two types of adaptation
current affect the spike rate response in different ways. A
subthreshold adaptation current shifts the I-O curve subtrac-
tively and thus increases the threshold for spiking. In addition,
it decreases the response gain for low (output) spike rates. If
the adaptation current is driven by spikes on the other hand, the
I-O curve changes divisively, that is, the response gain is
reduced over the whole range of spike rate values but the
response threshold remains unchanged. It can be recognized
that for a given type of adaptation current the adapting I-O
Fig. 2. Spike rate adaptation, gain, and threshold modulation in single neurons. A: cartoon of a single neuron visualizing the input parameters and output
quantities. B: instantaneous spike rate r(top), mean membrane voltage 具V典p(middle, squares), and mean adaptation current w(middle, solid lines) of an aEIF
neuron without adaptation, a⫽b⫽0(left), and with either a purely subthreshold adaptation current, a⫽0.06 mS/cm
2
,b⫽0(middle), or a spike-triggered
adaptation current, a⫽0, b⫽0.3
A/cm
2
(right), in response to a sudden increase in synaptic drive (bottom). C: input-output relationship (I-O curve) of the
neurons in B, i.e., spike rate ras a function of presynaptic spike rates r
E
,r
I
. Here, r
E
⫽r
I
, but excitation is stronger than inhibition, due to the coupling parameter
values (see MATERIALS AND METHODS). The I-O curves represent the spike rate response of the neurons to a sudden increase of r
E
and r
I
, measured in steps of
50 ms after that increase (light to dark colors). Dots indicate the evolution of the spike rate corresponding to the input in B.D: steady-state spike rate r
⬁
as a
function of the mean
and standard deviation
of the fluctuating input. Note that
and
are determined by the number of presynaptic neurons, their (Poisson)
spike rates and synaptic strengths, cf. Eqs. 5 and 6. Dashed lines in Dindicate the values of
and
that correspond to the presynaptic spike rates in C, and
circles mark the values of the moments corresponding to the increased input in B.
944 EFFECTS OF ADAPTATION CURRENTS ON SPIKE TRAINS
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curve evaluated shortly after the input steps and the steady-
state I-O curve are changed qualitatively in the same way.
Thus, for the following parameter exploration and analytical
derivation, we focus on (changes of) the steady-state I-O
relationship.
We next explore the effects of an adaptation current on the
steady-state spike rate for a wide range of input statistics, that
is, different values of the mean
and the standard deviation
of the fluctuating total synaptic input (see Fig. 2D). If excit-
atory and inhibitory inputs are approximately balanced, the
standard deviation
of the compound input is large compared
with its mean
. The spike rate increases with an increase of
either
or
or both. A subthreshold adaptation current
increases the threshold for spiking in terms of
as well as
.
A spike-triggered adaptation current, however, does not change
the threshold for spiking but reduces the gain of the spike rate
as a function of
or
. Thus the differential effects of both
types of adaptation current are robust across different input
configurations. Note that the I-O curve as a function of mean
input
changes additively for increased levels of standard
deviation
while its slope (i.e., gain) decreases, particularly
for small values of
. This can be recognized by the contour
lines in Fig. 2Dand is most prominent for increased subthresh-
old adaptation. Consequently, this type of adaptation current
increases the sensitivity of the steady-state spike rate to noise
intensity for low spike rates.
To analytically demonstrate the differential effects of sub-
threshold and spike-triggered adaptation currents on the
(steady-state) I-O curve, we consider the aPIF neuron model,
which is obtained by neglecting the leak conductance (g
L
⫽0)
in the aEIF model. This allows to derive an explicit expression
for the steady-state spike rate,
r⬁⫽
⫺a
共
具V典⬁⫺Ew
兲
⁄C
⌬V⫹
wb⁄C,(36)
where the mean membrane voltage 具V典⬁with respect to the
steady-state distribution p
⬁
(V) is given by Eq. 19 and ⌬V:⫽
V
s
⫺V
r
is the difference between spike and reset voltage; r
⬁
⫽
0 for
⬍a(具V典⬁⫺Ew)/C(see MATERIALS AND METHODS).
Equation 36 mathematically demonstrates the subtractive com-
ponent of the effect a subthreshold adaptation current (a⬎0)
produces when the mean membrane voltage is larger than the
reversal potential E
w
of the (K
⫹
) adaptation current. Taking the
derivative of Eq. 36 with respect to
further reveals that an
increase of areduces the gain when the input fluctuations (
)
are large compared with the mean (
). A spike-triggered
adaptation current (b⬎0), however, produces a purely divisive
effect that can be pronounced even for small current incre-
ments bif the adaptation timescale
w
is large.
Differential effects of adaptation currents on spiking
variability. We next investigate how adaptation currents affect
ISIs for different input statistics. For that reason we calculate
the distribution of times at which the membrane voltage of an
aEIF neuron crosses the threshold V
s
for the first time, which is
equivalent to the distribution of ISIs (see MATERIALS AND METH-
ODS). These ISI distributions are shown in Fig. 3Afor neurons
with different levels of subthreshold or spike-triggered adap-
tation and a given input. An increase of either type of adapta-
tion current (via parameters aand b) naturally increases the
mean ISI. Interestingly, while subthreshold adaptation leads to
Fig. 3. Changes of spiking variability in single neurons. A: ISI distribution (p
ISI
) of a single aEIF neuron in response to a fluctuating input with mean
⫽0.75
mV/ms and standard deviation
⫽3.25 mV/兹ms, for a⫽0, 0.03, 0.06 mS/cm
2
,b⫽0(top) and a⫽0, b⫽0, 0.15, and 0.3
A/cm
2
(bottom). B: ISI coefficient
of variation (CV) as a function of
and
, for a neuron without adaptation, a⫽b⫽0(left), and with either a subthreshold adaptation current, a⫽0.06 mS/cm
2
,
b⫽0(middle), or a spike-triggered adaptation current, a⫽0, b⫽0.3
A/cm
2
(right). Circles indicate the values of
and
used in A.C: change of ISI CV
caused by a subthreshold (left) or spike-triggered (right) adaptation current as a function of
and
. White regions in Band Cindicate the parameter values
for which the ISI CV was not computed, because r
⬁
⬍1 Hz.
945EFFECTS OF ADAPTATION CURRENTS ON SPIKE TRAINS
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ISI distributions with long tails, spike-triggered adaptation
causes ISI distributions with bulky shapes. These differential
effects on the shape of the ISI distribution lead to opposite
changes of the CV (cf. Eq. 22), which quantifies the variability
of ISIs. An increase of subthreshold adaptation current pro-
duces an increase of CV, whereas an increase of spike-trig-
gered adaptation current leads to a decreased ISI variability.
How these effects on the CV of ISIs depend on the statistics (
and
) of the fluctuating input is shown in Fig. 3, Band C.
With or without an adaptation current, if the mean
is large,
that is, far above threshold, and the standard deviation
is
comparatively small, the neuronal dynamics is close to deter-
ministic and the firing is almost periodic; hence, the CV is
small. In contrast, if
is close to the threshold and
is large
(enough), the ISI distribution will be broad as indicated by the
large CV. A subthreshold adaptation current either leads to an
increased CV or leaves the ISI variability unchanged. In case
of a spike-triggered adaptation current the effect on the CV
depends on the input statistics. This type of adaptation current
causes a decrease of the high ISI variability in the region (of
the
,
-plane) where the mean input
is small and an increase
of the low ISI variability for larger values of
.
We analytically derived an approximation of the ISI CV for
the aPIF model, which emphasizes the opposite effects of the
two types of adaptation current. It is obtained as
CV ⫽兹
2⌬V⁄
a⫺
w
2b2⁄C2⫺2
wb⌬V⁄C
⌬V⫹
wb⁄C(37)
(same as Eq. 26), where
a
:⫽
⫺a[具V典⬁⫺E
w
]/Cis the
effective mean input which is again assumed to be positive and
takes into account the counteracting subthreshold adaptation
current. The steady-state mean membrane voltage 具V典⬁is given
by Eq. 19 (see MATERIALS AND METHODS). Equation 37 mathe-
matically demonstrates that an increase of subthreshold adap-
tation curent (a⬎0) causes an increase of CV as long as 具V典⬁
is larger than E
w
, that is, the mean membrane voltage is not too
hyperpolarized. An increase of spike-triggered adaptation cur-
rent (b⬎0) on the other hand leads to a reduction of ISI
variability. Note that this approximation is only valid for small
values of mean input (
) and adaptation current increment (b).
It does not account for the increase of CV caused by spike-
triggered adaptation for large levels of
(cf. Fig. 3C). Both
(input dependent) effects of spike-triggered adaptation on the
ISI variability can be captured by a refined approximation of
the CV compared with Eq. 37 (not shown, see MATERIALS AND
METHODS for an outline), which requires numerical evaluation.
Differential effects of synaptic inhibition on I-O curves. Here
we examine how synaptic input received from a population of
inhibitory neurons affect gain and threshold of spiking. We
consider that the neuron we monitor belongs to a population of
excitatory neurons which are recurrently coupled to neurons
from an inhibitory population, as depicted in Fig. 4A: Each
neuron of the network receives excitatory synaptic input from
external neurons and additional synaptic input from a number
of neurons of the other population. The specific choice of the
monitored excitatory neuron does not matter because of iden-
tical model parameters within each population and sparse
random connectivity (see MATERIALS AND METHODS). Figure 4B
shows how the steady-state I-O curve of excitatory neurons,
i.e., the spike rate r
E,⬁
pop
as a function of the external (input) spike
rate r
EE
ext
, is changed by external excitation to the inhibitory
neurons (via r
IE
ext
) and by the strengths of the recurrent excit-
atory and inhibitory synapses (J
IE
rec
and J
EI
rec
), respectively. An
increase of external excitation to the inhibitory population (via
J
IE
ext
) changes the I-O curve subtractively, thus increasing the
response threshold, while an increase of recurrent excitation to
Fig. 4. Gain and threshold modulation
caused by network interaction. A: cartoon of
the network visualizing the coupling param-
eters. B,top: steady-state spike rate of excit-
atory aEIF neurons, rE,⬁
pop
(solid lines) and
inhibitory aEIF neurons, rI,⬁
pop
(dashed
lines), as a function of rEE
ext
, for rIE
ext
⫽6, 10,
and 14 Hz (left); JIE
rec
⫽0.05, 0.1, and 0.2 mV
(middle); and JEI
rec
⫽⫺0.45, ⫺0.6, and ⫺0.75
mV (right). Insets: cartoons visualizing the
varied parameters as specified on the top left.
If not indicated otherwise, JEI
rec
⫽⫺0.6 mV,
rIE
ext
⫽10 Hz, and JIE
rec
⫽0.1 mV. For the
other parameter values see MATERIALS AND
METHODS.B,bottom: steady-state spike rate
rE,⬁
pop
as a function of the input parameters
and
for the excitatory neurons. Solid lines
and dots at top correspond to those of equal
color at bottom.
946 EFFECTS OF ADAPTATION CURRENTS ON SPIKE TRAINS
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the inhibitory neurons (via J
IE
rec
) has a purely divisive effect,
that is, the gain is reduced. On the other hand, an increase of
recurrent inhibition to the excitatory neurons (via J
EI
rec
) affects
the I-O curve in both ways.
We demonstrate these effects analytically for a network of
perfect integrate-and-fire (PIF) model neurons (instead of aEIF
neurons). That is, we disregard the adaptation current here for
simplicity (a⫽b⫽0), since it does not change the results
qualitatively. An explicit expression for the steady-state spike
rate of the excitatory neurons, r
E,⬁
pop
, can be derived using Eq. 36
for all the neurons in the network with mean input
given by
Eq. 27 for excitatory neurons and by Eq. 29 for inhibitory
neurons. We solve for r
E,⬁
pop
self-consistently to obtain,
rE,⬁
pop ⫽JEE
extKEE
extrEE
ext⌬V⫹JEI
recKEI
recJIE
extKIE
extrIE
ext
⌬V2⫺JIE
recKIE
recJEI
recKEI
rec .(38)
The equation above states that r
E,⬁
pop
is directly proportional to the
strength of external excitation to the excitatory population, nega-
tively proportional to the strength of external excitation to the
inhibitory population (since J
EI
rec
⬍0) and inversely proportional
to the strength of recurrent excitation, where all proportionalities
include an offset. Equation 38 clearly shows that the effect of
external excitation to the inhibitory population is purely subtrac-
tive (since J
EI
rec
⬍0), the effect of recurrent excitation (to the
inhibitory population) is purely divisive, and the effect of recur-
rent inhibition (to the excitatory population) includes both com-
ponents. For comparison, consider a single (nonadapting) PIF
neuron receiving (external) excitatory and inhibitory input. With
the use of Eq. 36 with mean input
given by Eq. 5, the
steady-state spike rate of this neuron reads r
⬁
⫽(J
E
K
E
r
E
⫹
J
I
K
I
r
I
)/⌬V. Thus, an increase of external inhibition affects the I-O
curve of an excitatory neuron in the same way (subtractively) as
an increase of external excitation to the inhibitory population
within a recurrent network as described above.
Effects of synaptic inhibition on spiking variability. We next
investigate how inhibitory synaptic input changes the ISI variabil-
ity of the neurons (from the excitatory population) in the network
described above. An increase of external excitation to the inhib-
itory neurons (via r
IE
ext
), and the strengths of the recurrent synapses
(J
IE
rec
and J
EI
rec
) individually, leads to an increase of the mean ISI
and an increased tail of the ISI distribution, as shown in Fig. 5A.
Furthermore, an increase of r
IE
ext
or the magnitude of J
IE
rec
or J
EI
rec
,
each causes the coefficient of variation of ISIs (CV
E
pop
) to increase
(see Fig. 5B). Thus an increase of inhibition always leads to an
increase of spiking variability. An increase of external excitation
to the excitatory neurons (via r
EE
ext
), on the other hand, leads to a
decrease of CV
E
pop
.
To demonstrate these effects analytically we derived CV
E
pop
for a network of PIF model neurons using Eqs. 26–28, where
we obtained the steady-state spike rate of the inhibitory neu-
rons, r
I,⬁
pop
, analogously to r
E,⬁
pop
(as described above). Below, we
express CV
E
pop
as a function of either r
IE
ext
,J
IE
rec
,orJ
EI
rec
, and lump
together all other fixed parameters in a number of constants,
CVE
pop ⫽
冦
(c1rIE
ext ⫹c2)⁄(c3⫺c4rIE
ext)
c5JIE
rec ⫹c6
(c7(JEI
rec)2⫺c8JEI
rec)⁄(c9⫹c10JEI
rec).
(39)
The constants c
1
,...,c
10
in Eq. 39 are nonnegative functions
of the fixed parameters. Clearly, an increase of r
IE
ext
or the
magnitudes of J
IE
rec
and J
EI
rec
each produce an increase of CV
E
pop
(since J
EI
rec
⬍0). Considering a single PIF neuron receiving
(external) excitatory and inhibitory input for comparison, we
use Eq. 37 with mean
and standard deviation
of the input
given by Eqs. 5 and 6, respectively, to express the CV as
CV ⫽冑JE
2KErE⫹JI
2KIrI
⌬V(JEKErE⫹JIKIrI).(40)
Fig. 5. Changes of spiking variability caused
by network interaction. A: ISI distributions
(p
ISI
) of excitatory aEIF neurons for rEE
ext
⫽
50 Hz. JEI
rec
⫽⫺0.6 mV, rIE
ext
⫽10 Hz, and
JIE
rec
⫽0.1 mV if not indicated otherwise.
B: ISI CV for excitatory neurons (CVE
pop
)as
a function of rEE
ext
. Color code as in A. Dots
indicate the input and ISI CV values for the
ISI distributions in A.Insets: ISI CV as a
function of the input parameters
and
for
the excitatory neurons. Lines and dots (in-
sets) correspond to those of equal color in B.
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Note that Eq. 40 is only valid for positive mean input (J
E
K
E
r
E
⫹
J
I
K
I
r
I
⬎0). Again, ISI variability increases with inhibition. The
effect of inhibition on spiking variability can be understood
intuitively as follows. Inhibitory synaptic input reduces the mean
total synaptic input
and increases its standard deviation
for the
target neuron (population), which in turn causes an increase of ISI
variability.
Subthreshold and spike-triggered components of slow K
⫹
currents. Here we examine how the two types of an adaptation
current in the aEIF model reflect different slow K
⫹
currents in
a detailed conductance-based neuron model. First, we consider
three prominent slow K
⫹
currents: a Ca
2⫹
-activated after-
hyperpolarization current (I
KCa
),aNa
⫹
-activated current
(I
KNa
) and the voltage-dependent M current (I
M
). Figure 6A
shows how the conductances associated with these K
⫹
currents
depend on the membrane voltage in the steady state, compared
with the steady-state spike-generating Na
⫹
conductance. The
threshold membrane voltage at which a spike is elicited in
response to a slowly increasing input current is primarily
determined by the conductance-voltage relationship for Na
⫹
.
The threshold value lies in the interval where this curve has a
positive slope (the precise value depends on the peak conduc-
tances of all currents and on the input). The curve g
Na
,
⬁
(V) thus
indicates the subthreshold and suprathreshold membrane volt-
age ranges. In the subthreshold voltage range the conductance
g
KCa,⬁
is almost zero, while the conductances g
KNa,⬁
and g
M,⬁
reach significant values close to the voltage threshold. Thus the
curves in Fig. 6Aindicate that I
KCa
is activated by spikes, while
I
M
and particularly I
KNa
can be increased in the absence of
spiking.
The results of the fitting procedure in Fig. 6, Band C, show
the absolute and relative amounts of current triggered by a
spike vs. its subthreshold level quantified by the voltage
independent conductance a.I
KCa
has a dominant spike-trig-
gered component as expected, while I
KNa
shows a very small
increment caused by a spike compared with the subthreshold
component. I
M
, on the other hand, shows significant levels of
both components. Note, however, that the amount of I
M
elic-
ited by a spike is smaller compared with the level of I
M
that can
be caused by subthreshold membrane depolarization without
spiking (since ⌬I
Ks
rel
⬍1 for I
Ks
⬅I
M
, see Fig. 6C).
We further considered a range of biologically plausible slow
K
⫹
currents. That is, we varied the steady-state conductance-
voltage relationship for K
⫹
,g
Ks,⬁
(V), within a realistic range,
as shown in Fig. 7A, and quantified the subthreshold and
spike-triggered components for each of these K
⫹
currents (see
Fig. 7, Band C). The value of subthreshold conductance a
naturally increases with the fraction of K
⫹
conductance present
Fig. 6. Subthreshold and spike-triggered components of I
KCa
,I
KNa
, and I
M
.A: conductances for the slow K
⫹
currents I
Na
,I
KCa
,I
KNa
, and I
M
in steady state as
a function of the membrane voltage, normalized to a peak value of 1 mS/cm
2
.Band C: Subthreshold conductance aand spike-triggered absolute increment ⌬I
Ks
(B) and relative increment ⌬I
Ks
rel
(C) obtained from the fitting procedure (see MATERIALS AND METHODS) for the conductance-based model neurons with g
KCa
僆
[2, 8] mS/cm
2
and g
KNa
⫽g
M
⫽0 (dots), g
KNa
僆[2, 8] mS/cm
2
and g
KCa
⫽g
M
⫽0 (squares), and g
M
僆[0.1, 0.4] mS/cm
2
and g
KCa
⫽g
KNa
⫽0 (diamonds).
Darker symbols indicate larger conductance values.
Fig. 7. Subthreshold and spike-triggered
components of a range of slow K
⫹
currents.
A: steady-state K
⫹
conductance g
Ks,⬁
(V)⫽
g
Ks
⬁
(V) as a function of the membrane
voltage, for the generic Hodgkin-Huxley-
type description of a slow K
⫹
current (see
MATERIALS AND METHODS), with half-activa-
tion voltage
␣
⫽⫺40 mV (left curves);
␣
⫽
⫺10 mV (right curves); inverse steepness

⫽6, 9, and 12 mV; and peak conductance
g
Ks
⫽1 mS/cm
2
. The dashed curve indicates
the Na
⫹
conductance g
Na,⬁
(V) of the con-
ductance-based model, normalized to a max-
imum value of 1 mS/cm
2
.B: subthreshold
conductance aobtained from the fitting pro-
cedure for different values of the parameters
␣
and

.C: absolute and relative spike-
triggered increments ⌬I
Ks
(top) and ⌬I
Ks
rel
(bottom), respectively, as a function of
␣
, for
⫽100 ms (left) and
⫽300 ms (right).
948 EFFECTS OF ADAPTATION CURRENTS ON SPIKE TRAINS
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at subthreshold voltage values. For the quantification of spike-
triggered current increments we also considered different K
⫹
time constants
. The absolute value of current increment
⌬I
Ks
decreases with increasing
and changes only slightly
with changes of the shape of the conductance-voltage curve
g
Ks,⬁
(V) (via the parameters
␣
,

). However, the current
increment caused by a spike relative to the amount of current
already present in the absence of spiking (⌬I
Ks
rel
) is strongly
determined by g
Ks
(V). ⌬I
Ks
rel
increases with an increase of
half-activation voltage (parameter
␣
), steepness (via parameter

) and with decreasing time constant (
).
Effects of slow K
⫹
currents on I-O curve and ISI variability.
Here we examine how the different types of slow K
⫹
current
affect the I-O curve and spiking variability of uncoupled
conductance-based model neurons subject to noisy inputs and
compare the effects to those caused by subthreshold and
spike-triggered adaptation in aEIF neurons. Without a slow K
⫹
current, the spike rate I-O curve does not change over time (see
Fig. 8A). An increase of I
KCa
has a purely divisive effect on the
I-O curve while an increase of I
M
changes this curve in a
mostly subtractive and slightly divisive way. For both types of
slow K
⫹
current the adapting spike rates reach their steady-
state values in ⬍500 ms. These effects are consistent with our
results based on the aEIF model, given that I
KCa
predominantly
depends on spikes and I
M
includes both, subthreshold as well
as spike-triggered, components (Fig. 6B). In case of increased
I
KNa
, on the other hand, the steady-state I-O curve is signifi-
cantly altered in both ways (subtractively and divisively), and
the spike rate adapts very slowly, that is, steady-state rates are
reached after several seconds. At first sight, this seems contra-
dictory to the effect predicted above for subthreshold adapta-
tion, considering that the amount of I
KNa
triggered by a spike
is small compared with its subthreshold level. Since the time-
scale of I
KNa
is very large (Fig. 8Aand Wang et al. 2003) even
a small spike-triggered component leads to a significant divi-
sive change of the steady-state I-O curve, cf. Eq. 36. This
divisive effect is caused by K
⫹
current building up slowly
because of small current increments triggered repeatedly by
repetitive spiking and very slow decay between spikes due to
the large timescale of the current.
Considering ISI variability, an increase of I
KCa
reduces the
CV for small values of mean input
and increases the CV for
larger values of
(see Fig. 8B). An increase of each of the
other slow K
⫹
currents, I
KNa
, and I
M
, leads to an increase of ISI
CV in general. These effects are consistent with those caused
by subthreshold and spike-triggered adaptation currents in the
aEIF model, considering the subthreshold and spike-triggered
components of I
KCa
,I
KNa
, and I
M
, respectively (Fig. 6). Thus,
the results from the detailed conductance-based neuron model
are in agreement with the results based on the adaptive IF
models presented above.
DISCUSSION
In this study, we have systematically examined how adap-
tation currents and synaptic inhibition modulate the threshold
and gain of spiking as well as ISI variability in response to
fluctuating inputs resulting from stochastic synaptic events.
Based on a simple neuron model with subthreshold and spike-
triggered adaptation components, we used analytical and nu-
merical tools to describe spike rates and ISIs for a wide range
of input statistics. We then measured subthreshold and spike-
triggered components of different types of a slow K
⫹
current
using detailed conductance-based model neurons, and we val-
Fig. 8. Effects of I
KCa
,I
KNa
, and I
M
on I-O
curve and ISI variability. A: spike rate of a
conductance-based model neuron without
slow K
⫹
currents, g
KCa
⫽g
KNa
⫽g
M
⫽0
(black), and with either type of slow K
⫹
current included, g
KCa
⫽8 mS/cm
2
(red),
g
KNa
⫽8 mS/cm
2
(blue), g
M
⫽0.4 mS/cm
2
(green), in response to a sudden increase of
mean input
, measured in four subsequent
time intervals of 250 ms after that increase
(light to dark colors). The baseline mean
input was
⫽0.05 mV/ms and the input
standard deviation was
⫽0.5 mV/兹ms.
Average values over 50 independent trials
are shown. The adapting I-O curve of the
neuron with increased I
KNa
(g
KNa
⫽8 mS/
cm
2
) converges very slowly to the steady-
state curve (dashed blue) measured 20 s after
the increase in
.B: ISI CV of the neurons
in A as a function of mean input
for low
(left), medium (middle), and high (right)
noise intensity (
⫽1, 1.5, and 2 mV/
兹ms), respectively. The ISIs were col-
lected over an interval of 10 s after the
steady-state spike rates were reached, in 50
independent trials.
949EFFECTS OF ADAPTATION CURRENTS ON SPIKE TRAINS
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idated our (analytical) results from the simple neuron model by
numerical simulations of the detailed model.
We have shown that a purely subthreshold voltage-depen-
dent adaptation current increases the threshold for spiking and
reduces the gain at low spike rates in the presence of input
fluctuations. This type of current produces a long-tailed ISI
distribution and thus leads to an increase of variability for a
broad range of input statistics. A spike-triggered adaptation
current, on the other hand, causes a divisive change of the I-O
curve, thereby reducing the response gain but leaving the
response threshold unaffected, irrespective of the input noise
intensity. This type of current decreases the ISI CV for fluc-
tuation-dominated inputs but increases the CV when the mean
input is strong, i.e., it reduces the sensitivity of spiking vari-
ability to the mean input. For comparison, an increase of
external inhibition leads to a subtractive shift of the I-O curve
while an increase of recurrent inhibition changes it divisively.
The ISI variability, however, is increased by both types of
synaptic inhibition.
We have further demonstrated that the Ca
2⫹
-activated after-
hyperpolarization K
⫹
current is effectively captured by a
simple description based on spike-triggered increments, while
the muscarine-sensitive and Na
⫹
-activated K
⫹
currents, re-
spectively, have dominant subthreshold components. Despite
its small spike-triggered component, the Na
⫹
-dependent K
⫹
current also substantially affects the neuronal gain, due to its
large timescale.
Methodological aspects. Our approach involves the diffu-
sion approximation and Fokker-Planck equation, both of which
have been widely applied to analyze the spike rates of scalar IF
type neurons in a noisy setting (see, e.g., Amit and Brunel
1997; Brunel 2000; Fourcaud-Trocmé et al. 2003; Burkitt
2006; Roxin et al. 2011). Our assumption of separated time-
scales between slow adaptation and fast membrane voltage
dynamics has also been frequently used in such a setting
(Brunel et al. 2003; La Camera et al. 2004; Gigante et al.
2007b; Richardson 2009; Augustin et al. 2013). While most of
these previous studies concentrated on spike rate dynamics,
here we focused on asynchronous (nonoscillatory) activity. To
examine ISI distributions we extended the method described
previously for scalar IF models, which is based on the first
passage time problem (Tuckwell 1988; Ostojic 2011), to the
aEIF model, accounting for the dynamics of the adaptation
current between spikes. Furthermore, we analytically derived
an expression for the steady-state spike rate (i.e., steady-state
I-O relationship) based on Brunel et al. (2003) and an approx-
imation of the ISI CV using recent results from Urdapilleta
(2011) for the perfect IF model with two types of an adaptation
current (aPIF model). The I-O functions we calculated can be
used to relate (adaptive) spiking neuron models to linear-
nonlinear cascade models, which describe the instantaneous
spike rate of a neuron by applying to the stimulus signal
successively a linear temporal filter and a static nonlinear
function (Ostojic and Brunel 2011). Such cascade models have
proven valuable for studying how sensory inputs are mapped to
neuronal activity (see, e.g., Schwartz et al. 2006; Pillow et al.
2008).
It is worth noting that our approach further allows to easily
calculate the power spectrum Pand (normalized) autocorrela-
tion function Aof the neuronal spike train once the ISI
distribution has been obtained, via the relation
P(
)⫽A
^(
)⫽r⬁Re
冉
1⫹p
^ISI(
)
1⫺p
^ISI(
)
冊
,(41)
where A
^and p
^ISI ISI denote the Fourier transforms of the
autocorrelation function and ISI distribution, respectively (see
Gerstner and Kistler 2002). Equation 41 strictly applies to
memoryless (so-called renewal) stochastic processes and an
adaptation mechanism usually leads to a violation of this
requirement for a model neuron subject to fluctuating input.
Here we have derived a renewal process (V
i
(t), w(t)) from the
original nonrenewal process (V
i
(t), w(t)) by averaging the
adaptation current and self-consistently determining its reset
value (see ISI distribution). An alternative approach that allows
for the application of the above relationship Eq. 41 to adapting
model neurons has recently been described in Naud and Ger-
stner (2012).
Modulation of spike rate threshold and gain. Purely subtrac-
tive and divisive changes of the I-O curve by subthreshold and
spike-triggered adaptation, respectively, have previously been
shown for model neurons considering constant current inputs
but neglecting input fluctuations (Prescott and Sejnowski 2008;
Ladenbauer et al. 2012). These theoretical results describe the
effects shown in recent in vitro experiments that involved
blocking the low-threshold M current and a Ca
2⫹
-activated K
⫹
current separately [Deemyad et al. 2012 (Fig. 3); see also
Alaburda et al. 2002 (Fig. 3), Smith et al. 2002, and Miles et al.
2005 (Fig. 1) for experimental evidence of either effect]. Here
we have shown that a subthreshold adaptation current also
causes a reduction of response gain (in addition to an increase
of response threshold) when the fluctuations of the input are
strong compared with its mean. On the other hand, a spike-
triggered adaptation current decreases the response gain over
the whole input range, irrespective of the level of input fluc-
tuations. These results apply for adapting as well as the adapted
(steady) states.
2
When considering the onset I-O curve, i.e., the
immediate response to a sudden increase of input, an increased
level of spike-triggered adaptation current due to preadaptation
has been shown to produce a rather subtractive change (Benda
et al. 2010). This, however, does not contradict our results. On
the contrary, either type of adaptation current (subthreshold or
spike-triggered) naturally leads to a subtractive change of the
onset I-O curve for neurons which are preadapted to an
increased input (not shown).
Notably, when considering conductance-based noisy synap-
tic inputs, an increase in balanced synaptic background activity
can also reduce the spike rate gain (Chance et al. 2002; Burkitt
et al. 2003) and external inhibition can reduce the gain and
increase the response threshold at the same time (Mitchell and
Silver 2003). This means that the response gain can change due
to external inputs that are independent of the activity of the
target neuron, which can be understood as follows. An increase
of noisy (excitatory or inhibitory) synaptic conductance leads
to an increase of total membrane conductance, which causes a
purely subtractive change of the I-O curve, and an increase in
synaptic current noise, which causes an additive change of the
I-O curve and decreases its slope (particularly for small input
2
Note that in case of a very large adaptation timescale a (small) spike-
triggered adaptation current has a negligible effect on the adapting I-O curve,
evaluated shortly after the input steps, but a significant effect on the steady-
state I-O curve (see Fig. 8A).
950 EFFECTS OF ADAPTATION CURRENTS ON SPIKE TRAINS
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strengths) (Chance et al. 2002) (Fig. 3). Both effects combined
lead to the observed change of response gain. The two separate
components are included in our results. An increase of mem-
brane conductance (represented by g
L
in the aEIF model)
subtracts from the spike rate response, see Eq. 18, and the
abovementioned effects of an increase of noise intensity
have
been described in RESULTS (see Fig. 2D).
Modulation of response gain is an important phenomenon,
particularly in sensory neurons, because neuronal sensitivity to
changes in the input is amplified or downscaled without chang-
ing input selectivity. A spike-dependent adaptation current thus
represents a cellular mechanism by which this is achieved. For
example, neuronal response gain increases during selective
attention (McAdams and Maunsell 1999). It has been shown in
vivo that the neuromodulator acetylcholine (ACh) contributes
substantially to attentional upregulation of spike rates (Herrero
et al. 2008). Cholinergic changes of neuronal excitability and
response gain (Soma et al. 2012) in turn are likely produced via
downregulation of slow K
⫹
currents (Madison et al. 1987;
McCormick 1992; Sripati and Johnson 2006). Together with
our results, these observations suggest that excitability and
response gain of cortical neurons are controlled by neuromodu-
latory substances through (de)activation of subthreshold and
spike-triggered K
⫹
currents, respectively.
We have further shown that external inhibitory synaptic
inputs change the I-O curve subtractively, which is consistent
with the results of a previous numerical study using a conduc-
tance based neuron model without consideration of noise
(Capaday 2002). Recurrent synaptic (feedback) inhibition,
which is a function of the neuronal spike rate, on the other
hand, reduces the response gain. This is in agreement with the
results obtained by (Sutherland et al. 2009) based on IF type
neurons subject to noisy inputs. Recent in vivo recordings from
mouse visual cortex have shown that distinct types of inhibi-
tory neurons produce these differential effects (i.e., subtractive
and divisive changes of I-O curves) at their target neurons
(Wilson et al. 2012). Functional connectivity analysis suggests
that the inhibitory neurons that changed the I-O curve of their
target neurons subtractively were less likely connected recur-
rently to the recorded targets than the inhibitory neurons that
changed the responses of the targets divisively (Wilson et al.
2012). By application of our results based on the simple
network model, the observed differential effects caused by the
two types of inhibitory cells can thus be explained by their
patterns of connectivity with the target cells.
Effects on ISI variability. We have shown that a spike-
triggered adaptation current reduces high ISI variability at low
spike rates (when input fluctuations are strong compared with
the mean) and increases low ISI variability at high spike rates
(caused by a large mean input). This result is in agreement with
a previous numerical simulation study (Liu and Wang 2001)
but seems to disagree with other theoretical work (Wang 1998;
Prescott and Sejnowski 2008; Schwalger et al. 2010) at first
sight. Wang (1998) and later Prescott and Sejnowski (2008)
showed that spike-driven adaptation reduces the ISI CV at
low spike rates but they did not find an increase of ISI CV at
higher spike rates in their simulation studies. The reason for
this is that the ISI CVs of adapting and nonadapting neurons
were compared at equal spike rates (i.e., at equal mean ISIs)
but different input statistics. That is, the input to the adapting
neurons was adjusted to compensate for the change of spike
rate (or mean ISI) caused by the adaptation currents. Increasing
the mean input to the adapting neurons to achieve equal mean
ISIs, however, decreases its ISI CV (cf. Eq. 37). Here we
compare the ISI statistics across different neurons for equal
inputs. On the other hand, Schwalger et al. (2010) analyzed the
ISI statistics of perfect IF model neurons with spike-triggered
adaptation and found that this type of adaptation always leads
to an increase of ISI CV in response to a noisy input current.
Their approach is similar to the one presented here but differs
in that the dynamics of the adaptation current was neglected in
Schwalger et al. (2010); see Fig. 1B,bottom, for a visualization
of that difference. Assuming a stationary adaptation current
leads to a reduced effective mean input to the neuron, leaving
the input variance unchanged, which always causes increased
ISI variability (cf. Eq. 37). Together with theoretical work
showing that a spike-dependent adaptation current causes neg-
ative serial ISI correlations (Prescott and Sejnowski 2008;
Farkhooi et al. 2011), our results suggest that spike rate coding
is improved by such a current for low-frequency inputs
(Prescott and Sejnowski 2008; Farkhooi et al. 2011).
In contrast, an adaptation current that is predominantly
driven by the subthreshold membrane voltage usually leads to
an increase of ISI CV, as we have demonstrated. This seems to
be not consistent with a previous study (Prescott and Sejnowski
2008) where subthreshold adaptation was found to produce a
small decrease of ISI variability. The apparent discrepancy is
caused by differences in the presentation of the data: Prescott
and Sejnowski (2008) compared the ISI CVs for equal spike
rates as explained above. That is, the mean input was adjusted
to obtain equal mean ISIs for adapting and nonadapting neu-
rons but the input variance remained unchanged. However,
increasing the mean input (
in Eq. 37) to the adapting neuron
counteracts the effect of subthreshold adaptation on the effec-
tive mean input (
a
in Eq. 37). Consequently, one cannot
observe an increased ISI CV in neurons with subthreshold
adaptation currents when the mean input to these neurons is
increased. Note that our results do not contradict those in
(Prescott and Sejnowski 2008) but instead reveal that an
increase of a subthreshold adaptation current always causes an
increase of ISI CV for given input statistics and an increase of
a spike-dependent adaptation current leads to an increase of ISI
CV if the mean input is large.
Finally, we have shown that an increase in synaptic inhibi-
tion increases the ISI variability, regardless of whether this
inhibition originates from an external population of neurons or
from recurrently coupled ones. An intuitive explanation for this
effect is that increased inhibitory input reduces the mean input
but increases the input variance (see Eqs. 5 and 6). The reason
why recurrent synaptic inhibition and spike-triggered adapta-
tion change the ISI variability in opposite ways in a fluctua-
tion-dominated input regime could be the different timescales.
Synaptic inhibition usually acts on a much faster timescale than
adaptation currents whose time constants range from about 100
ms to seconds. Thus recurrent synaptic inhibition in contrast to
spike-triggered adaptation cannot provide a memory trace of
past spiking activity (over a duration of several ISIs) that could
shape the ISI distribution. Notably, our results on ISIs in a
network setting strictly apply to networks in asynchronous
states. Recurrent synaptic inhibition, however, can also medi-
ate oscillatory activity (Brunel 2000; Brunel et al. 2003;
951EFFECTS OF ADAPTATION CURRENTS ON SPIKE TRAINS
J Neurophysiol •doi:10.1152/jn.00586.2013 •www.jn.org
by 10.220.33.2 on October 25, 2017http://jn.physiology.org/Downloaded from
Isaacson and Scanziani 2011; Augustin et al. 2013) where the
variability of ISIs might be affected differently.
ACKNOWLEDGMENTS
We thank Maziar Hashemi-Nezhad for helpful comments on the manu-
script.
GRANTS
This work was supported by Deutsche Forschungsgemeinschaft in the
framework of Collaborative Research Center SFB910.
DISCLOSURES
No conflicts of interest, financial or otherwise, are declared by the author(s).
AUTHOR CONTRIBUTIONS
Author contributions: J.L. and K.O. conception and design of research; J.L.
and M.A. performed experiments; J.L. and M.A. analyzed data; J.L., M.A., and
K.O. interpreted results of experiments; J.L. and M.A. prepared figures; J.L.
drafted manuscript; J.L., M.A., and K.O. edited and revised manuscript; J.L.,
M.A., and K.O. approved final version of manuscript.
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