Channel Noise in Excitable Neuronal 
Membranes 
Amit Manwani;  Peter N. Steinmetz and Christof Koch 
Computation and Neural Systems Program, M-S 139-74 
California Institute of Technology Pasadena, CA 91125 
{quixote, peter, koch} @ klab. caltech. edu 
Abstract 
Stochastic fluctuations of voltage-gated ion channels generate current 
and voltage noise in neuronal membranes. This noise may be a criti- 
cal determinant of the efficacy of information processing within neural 
systems. Using Monte-Carlo simulations, we carry out a systematic in- 
vestigation of the relationship between channel kinetics and the result- 
ing membrane voltage noise using a stochastic Markov version of the 
Mainen-Sejnowski model of dendritic excitability in cortical neurons. 
Our simulations show that kinetic parameters which lead to an increase 
in membrane excitability (increasing channel densities, decreasing tem- 
perature) also lead to an increase in the magnitude of the sub-threshold 
voltage noise. Noise also increases as the membrane is depolarized from 
rest towards threshold. This suggests that channel fluctuations may in- 
terfere with a neuron's ability to function as an integrator of its synaptic 
inputs and may limit the reliability and precision of neural information 
processing. 
1 Introduction 
Voltage-gated ion channels undergo random transitions between different conformational 
states due to thermal agitation. Generally, these states differ in their ionic permeabilities 
and the stochastic transitions between them give rise to conductance fluctuations which 
are a source of membrane noise [1]. In excitable cells, voltage-gated channel noise can 
contribute to the generation of spontaneous action potentials [2, 3], and the variability of 
spike timing [4]. Channel fluctuations can also give rise to bursting and chaotic spiking 
dynamics in neurons [5, 6]. 
Our interest in studying membrane noise is based on the thesis that noise ultimately limits 
the ability of neurons to transmit and process information. To study this problem, we com- 
bine methods from information theory, membrane biophysics and compartmental neuronal 
modeling to evaluate ability of different biophysical components of a neuron, such as the 
synapse, the dendritic tree, the soma and so on, to transmit information [7, 8, 9]. These 
neuronal components differ in the type, density, and kinetic properties of their constituent 
ion channels. Thus, measuring the impact of these differences on membrane noise rep- 
* http://www.klab.caltech.eduFquixote 
144 A. Manwani, P. N. Steinmetz and C. Koch 
resents a fundamental step in our overall program of evaluating information transmission 
within and between neurons. 
Although information in the nervous system is mostly communicated in the form of action 
potentials, we first direct our attention to the study of sub-threshold voltage fluctuations for 
three reasons. Firstly, voltage fluctuations near threshold can cause variability in spike tim- 
ing and thus directly influence the reliability and precision of neuronal activity. Secondly, 
many computations putatively performed in the dendritic tree (coincidence detection, mul- 
tiplication, synaptic integration and so on) occur in the sub-threshold regime and thus are 
likely to be influenced by sub-threshold voltage noise. Lastly, several sensory neurons in 
vertebrates and invertebrates are non-spiking and an analysis of voltage fluctuations can be 
used to study information processing in these systems as well. 
Extensive investigations of channel noise were carried out prior to the advent of the patch- 
clamp technique in order to provide indirect evidence for the existence of single ion chan- 
nels (see [ 1 ] for an excellent review). More recently, theoretical studies have focused on the 
effect of random channel fluctuations on spike timing and reliability of individual neurons 
[4], as well as their effect on the dynamics of interconnected networks of neurons [5, 6]. 
In this paper, we determine the effect of varying the kinetic parameters, such as channel 
density and the rate of channel transitions, on the magnitude of sub-threshold voltage noise 
in an iso-potential membrane patches containing stochastic voltage-gated ion channels us- 
ing Monte-Carlo simulations. The simulations are based on the Mainen-Sejnowski (MS) 
kinetic model of active channels in the dendrites of cortical pyramidal neurons [10]. By 
varying two model parameters (channel densities and temperature), we investigate the rela- 
tionship between excitability and noise in neuronal membranes. By linearizing the channel 
kinetics, we derive analytical expressions which provide closed-form estimates of noise 
magnitudes; we contrast the results of the simulations with the linearized expressions to 
determine the parameter range over which they can be used. 
2 Monte-Carlo Simulations 
Consider an iso-potential membrane patch containing voltage-gated K + and Na+channels 
and leak channels, 
--7 dVm 
dt = gE (V - EE) + gN (V - ENd) + g (V - E) + hn (1) 
where C is the membrane capacitance and #K (gNa, gL) and EK (JNa, EL) denote the 
K + (Na + , leak) conductance and the K + (Na +, leak) reversal potential respectively. Current 
injected into the patch is denoted by Ira j, with the convention that inward current is nega- 
tive. The channels which give rise to potassium and sodium conductances switch randomly 
between different conformational states with voltage-dependent transition rates. Thus, #K 
and #va are voltage-dependent random processes and eq. 1 is a non-linear stochastic differ- 
ential equation. Generally, ion channel transitions are assumed to be Markovian [11] and 
the stochastic dynamics of eq. 1 can be studied using Monte-Carlo simulations of finite- 
state Markov models of channel kinetics. 
Earlier studies have carried out simulations of stochastic versions of the classical Hodgkin- 
Huxley kinetic model [12] to study the effect of conductance fluctuations on neuronal 
spiking [13, 2, 4]. Since we are interested in sub-threshold voltage noise, we consider 
a stochastic Markov version of a less excitable kinetic model used to describe dendrites of 
cortical neurons [10]. We shall refer to it as the Mainen-Sejnowski (MS) kinetic scheme. 
The K+ conductance is modeled by a single activation sub-unit (denoted by n) whereas 
the Na+conductance is comprised of three identical activation sub-units (denoted by m) 
and one inactivation sub-unit (denoted by/). Thus, the stochastic discrete-state Markov 
models of the K+and Na+channel have 2 and 8 states respectively (shown in Fig. 1). The 
Channel Noise in Excitable Neural Membranes 145 
A 
single channel conductances and the densities of the ion channels (K +,Na +) are denoted 
by (7K,7Na) and (r/K,r/va) respectively. Thus, gx and gv) are given by the products of 
the respective single channel conductances and the corresponding numbers of channels in 
the conducting states. 
gNa(v't): Na [m3h] 
gK(v,t) =K [nl] 
Figure 1: Kinetic scheme for the voltage-gated 
Mainen-Sejnowski K+(A) and Na+(B) channels. 
no and n represent the closed and open states of 
K+channel. rn0-2h represent the 3 closed states, 
rn0-ah0 the four inactivated states and rnah the 
open state of the Na+channel. 
were held at the fixed value corresponding to the 
for details of this procedure). 
-70 -60 V (m,50 -40 
m 
2 
-z 
Figure 2: Steady-state I-V curves for different 
multiples (2v,) of the nominal MS Na+channel 
density. Circles indicate locations of fixed-points 
in the absence of current injection. 
We performed Monte-Carlo simulations 
of the MS kinetic scheme using a fixed 
time step of At = 10 ttsec. During each 
step, the number of sub-units undergo- 
ing transitions between states i and j 
was determined by drawing a pseudo- 
random binomial deviate (bnldov sub- 
routine [14] driven by the ran2 subrou- 
tine of the 2 nd edition) with N equal 
to the number of sub-units in state i 
and p given by the conditional proba- 
bility of the transition between i and j. 
After updating the number of channels 
in each state, eq. 1 was integrated us- 
ing fourth order Runge-Kutta integration 
with adaptive step size control [ 14]. Dur- 
ing each step, the channel conductances 
new numbers of open channels. (See [4] 
Due to random channel transitions, the 
membrane voltage fluctuates around the 
steady-state resting membrane voltage 
Vrest. By varying the magnitude of 
the constant injected current Iinj, the 
steady-state voltage can be varied over a 
broad range, which depends on the chan- 
nel densities. The current required to 
maintain the membrane at a holding volt- 
age Vhota can be determined from the 
steady-state I-V curve of the system, as 
shown in Fig. 2. Voltages for which 
the slope of the I-V curve is negative 
cannot be maintained as steady-states. 
By injecting an external current to offset 
the total membrane current, a fixed point 
in the negative slope region can be ob- 
tained but since the fixed point is unsta- 
ble, any perturbation, such as a stochastic 
ion channel opening or closing, causes 
the system to be driven to the closest sta- 
ble fixed point. We measured sub-threshold voltage noise only for stable steady-state hold- 
ing voltages. A typical voltage trace from our simulations is shown in Fig. 3. To estimate 
the standard deviation of the voltage noise accurately, simulations were performed for a 
total of 492 seconds, divided into 60 blocks of 8.2 seconds each, for each steady-state 
value. 
146 A. Manwani, P N. Steinmetz and C. Koch 
-64 
-65 - 
-66  
-67  
1 O0 200 300 400 500 
Time (msec) 
Figure 3: Monte-Carlo simulations 
of a 1000 /zm 2 membrane patch with 
stochastic Na + and deterministic K + 
channels with MS kinetics. Bottom 
record shows the number of open Na + 
channels as a function of time. Top 
trace shows the corresponding fluctua- 
tions of the membrane voltage. Sum- 
mary of nominal MS parameters: G, = 
0.75 F/cm 2 r/: = 1.5 channels/m 2 
r/N, = 2 channels//.m 2, E: = -90 mV, 
EN. = 60 mV, Es = -70 mV, g5 = 0.25 
PS//zm 2 , 'YK = 3'Na = 20 pS. 
3 Linearized Analysis 
The non-linear stochastic differential equation (eq. 1) cannot be solved analytically. How- 
ever, one can linearize it by expressing the ionic conductances and the membrane voltage 
as small perturbations (6) around their steady-state values: 
-C d6V, 
dt - (g: + gva + g)6V, + (V  - EK)6gK + (V  - ENa)6gNa (2) 
where g: and 9Va denote the values of the ionic conductances at the steady-state voltage 
V . G = g + gva + g is the total steady-state patch conductance. Since the leak channel 
conductance is constant, 6g = 0. On the other hand, 6gK and 6gNa depend on 6V and t. 
It is known that, to first order, the voltage- and time-dependence of active ion channels can 
be modeled as phenomenological impedances [15, 16]. Fig. 4 shows the linearized equiv- 
alent circuit of a membrane patch, given by the parallel combination of the capacitance C, 
the conductance G and three series RL branches representing phenomenological models of 
K + activation, Na + activation and Na + inactivation. 
= K(EK -- ) + -- ) (3) 
represents the current noise due to fluctuations in the channel conductances (denoted by 
0K and ONe) at the membrane voltage V (also referred to as holding voltage Vhota). 
The details of the linearization are provided [16]. The complex admittance (inverse of the 
impedance) of Fig. 4 is given by, 
1 1 
(f) = G + j2rfC + + 
rn + j2xfln rm + j2xflm 
The variance of the voltage fluctuations or} can be computed as, 
+ (4) 
rh + j2rfl 
(5) 
where the power spectral density of In is given by the sum of the individual channel current 
noise spectra, Sn(f) = $K(f) + SNa(f). 
For the MS scheme, the autocovariance of the K + current noise for patch of area A, clamped 
at a voltage V, can be derived using [1, 11], 
CK(t) = A rlK ,y, (V  - Ex )2 nm (1 -nm) e -ltl/ (6) 
where n and r respectively denote the steady-state probability and time constant of the 
K+ activation sub-unit at V. The power special density of the K+cuent noise SzK(f) 
can be obtained from the Fourier transform of Cx (t), 
SiK(f) : 2 A V  (V - EK)2n r (7) 
1 + (2Wfrn) 2 
Channel Noise in Excitable Neural Membranes 147 
I rn rh rm 
In Inn 
Figure 4: Linearized circuit of the 
membrane patch containing stochastic 
voltage-gated ion channels. G denotes 
the membrane capacitance, G is the sum 
of the steady-state conductances of the 
channels and the leak. ri's and/i's de- 
note the phenomenological resistances 
and inductances due to the voltage- and 
time-dependent ionic conductances. 
Thus, $zK(f) is a single Lorentzian spectrum with cut-off frequency determined by 
Similarly, the auto-covariance of the MS Na+current noise can be written as [1], 
where 
c, Na(t) = AnNa 
m(t) : mm + (1- mm)e -t/Tin, h(t) -- ha + (1- h)e -t/Th (9) 
As before, mo (hm) and r, (rh) are the open probability and the time constant of 
Na+activation (inactivation) sub-unit. The Na+current noise spectrum SN(f) can be 
expressed as a sum of Lorentzian spectra with cut-off frequencies corresponding to the 
seven time constants rm, rh, 2 rm, 3 rra, rra + rI, 2 rra + rh and 3 rm + rh. The details of 
the derivations can be found in [8]. 
A B 
5 
4 
2 
1 
0 
+ 
J + ++ 
3 
-80 -60 -,0 -20 -80 -60 
Vhoid (mY) Vhoid (mY) 
-20 
Figure 5: Standard deviation of the voltage noise av in a 1000/rn 2patch as a function of the 
holding voltage Vhota. Circles denote results of the Monte-Carlo simulations for the nominal MS 
parameter values (see Fig. 3). The solid curve corresponds to the theoretical expression obtained 
by linearizing the channel kinetics. (A) Effect of increasing the sodium channel density by a factor 
(compared to the nominal value) of 2 (pluses), 3 (asterisks) and 4 (squares) on the magnitude of 
voltage noise. (B) Effect of increasing both the sodium and potassium channel densities by a factor 
of two (pluses). 
4 Effect of Varying Channel Densities 
Fig. 5 shows the voltage noise for a 1000/m 2 patch as a function of the holding voltage 
for different values of the channel densities. Noise increases as the membrane is depolar- 
ized from rest towards -50 mV and the rate of increase is higher for higher Na+densities. 
The range of Vaotd for sub-threshold behavior extends up to -20 mV for nominal densities, 
148 A. Manwani, P N. Steinmetz and C. Koch 
but does not exceed -60 mV for higher Na+densities. For moderate levels of depolariza- 
tion, an increase in the magnitude of the ionic current noise with voltage is the dominant 
factor which leads to an increase in voltage noise; for higher voltages phenomenological 
impedances are large and shunt away the current noise. Increasing Na+density increases 
voltage noise, whereas, increasing K+density causes a decrease in noise magnitude (com- 
pare Fig. 5A and 5B). We linearized closed-form expressions provide accurate estimates 
of the noise magnitudes when the noise is small (of the order 3 mV). 
5 Effect of Varying Temperature 
Fig. 6 shows that voltage noise decreases with 
temperature. To model the effect of temperature, 
transition rates were scaled by a factor Qr/io 
(Q10 = 2.3 for K +, Q10 = 3 for Na+). Tem- 
perature increases the rates of channel transitions 
and thus the bandwidth of the ionic current noise 
fluctuations. The magnitude of the current noise, 
on the other hand, is independent of temperature. 
Since the membrane acts as a low-pass RC fil- 
ter (at moderately depolarized voltages, the phe- 
nomenological inductances are small), increasing 
the bandwidth of the noise results in lower volt- 
age noise as the high frequency components are 
filtered out. 
6 Conclusions 
1.5 
1 
020 25 35 
T (Celsius) 
Figure 6: av as a function of tem- 
perature for a 1000 /m 2 patch with 
MS kinetics (Vhotd = -60 mV). Circles 
denote Monte-Carlo simulations, solid 
curve denotes linearized approximation. 
We studied sub-threshold voltage noise due to stochastic ion channel fluctuations in an iso- 
potential membrane patch with Mainen-Sejnowski kinetics. For the MS kinetic scheme, 
noise increases as the membrane is depolarized from rest, up to the point where the phe- 
nomenological impedances due to the voltage- and time-dependence of the ion channels 
become large and shunt away the noise. Increasing Na+channel density increases both the 
magnitude of the noise and its rate of increase with membrane voltage. On the other hand, 
increasing the rates of channel transitions by increasing temperature, leads to a decrease 
in noise. It has previously been shown that neural excitability increases with Na+channel 
density [17] and decreases with temperature [15]. Thus, our findings suggest that an in- 
crease in membrane excitability is inevitably accompanied by an increase in the magnitude 
of sub-threshold voltage noise fluctuations. The magnitude and the rapid increase of volt- 
age noise with depolarization suggests that channel fluctuations can contribute significantly 
to the variability in spike timing [4] and the stochastic nature of ion channels may have a 
significant impact on information processing within individual neurons. It also potentially 
argues against the conventional role of a neuron as integrator of synaptic inputs [ 18], as 
the the slow depolarization associated with integration of small synaptic inputs would be 
accompanied by noise, making the membrane voltage a very unreliable indicator of the 
integrated inputs. We are actively investigating this issue more carefully. 
When the magnitudes of the noise and the phenomenological impedances are small, the 
non-linear kinetic schemes are well-modeled by their linearized approximations. We have 
found this to be valid for other kinetic schemes as well [19]. These analytical approxi- 
mations can be used to study noise in more sophisticated neuronal models incorporating 
realistic dendritic geometries, where Monte-Carlo simulations may be too computationally 
intensive to use. 
Channel Noise in Excitable Neural Membranes 149 
Acknowledgments 
This work was funded by NSF, NIMH and the Sloan Center for Theoretical Neuroscience. 
We thank our collaborators Michael London, Idan Segev and Yosef Yarom for their invalu- 
able suggestions. 
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