Signal Detection in Noisy Weakly-Active 
Dendrites 
Amit Manwani and Christof Koch 
{quixote, koch}@klab. cal tech. edu 
Computation and Neural Systems Program 
California Institute of Technology 
Pasadena, CA 91125 
Abstract 
Here we derive measures quantifying the information loss of a synaptic 
signal due to the presence ofneuronal noise sources, as it electrotonically 
propagates along a weakly-active dendrite. We model the dendrite as an 
infinite linear cable, with noise sources distributed along its length. The 
noise sources we consider are thermal noise, channel noise arising from 
the stochastic nature of voltage-dependent ionic channels (K + and Na +) 
and synaptic noise due to spontaneous background activity. We assess the 
efficacy of information transfer using a signal detection paradigm where 
the objective is to detect the presence/absence of a presynaptic spike from 
the post-synaptic membrane voltage. This allows us to analytically assess 
the role of each of these noise sources in information transfer. For our 
choice of parameters, we find that the synaptic noise is the dominant 
noise source which limits the maximum length over which information 
be reliably transmitted. 
1 Introduction 
This is a continuation of our efforts (Manwani and Koch, 1998) to understand the informa- 
tion capacity of a neuronal link (in terms of the specific nature of neural "hardware") by a 
systematic study of information processing at different biophysical stages in a model of a 
single neuron. Here we investigate how the presence of neuronal noise sources influences 
the information transmission capabilities of a simplified model of a weakly-active dendrite. 
The noise sources we include are, thermal noise, channel noise arising from the stochastic 
nature of voltage-dependent channels (K + and Na +) and synaptic noise due to spontaneous 
background activity. We characterize the noise sources using analytical expressions of their 
current power spectral densities and compare their magnitudes for dendritic parameters re- 
ported in literature (Mainen and Sejnowski, 1998). To assess the role of these noise sources 
on dendritic integration, we consider a simplified scenario and model the dendrite as a lin- 
Signal Detection in Noisy Weakly-Active Dendrites 133 
LSynapse 
V,(x,t) 
Optimal 1 Spike 
Detector I No spie P 
Cable /Measurement 
y Noise Sources x 
Figure 1: Schematic diagram of a simplified dendritic channel. The dendrite is modeled a weakly- 
active 1-D cable with noise sources distributed along its length. Loss of signal fidelity as it propagates 
from a synaptic location (input) !/to a measurement (output) location z is studied using a signal 
detection task. The objective is to optimally detect the presence of the synaptic input I(!/, t) (in the 
form of a unitary synaptic event) on the basis of the noisy voltage waveform Vm (z, t), filtered by the 
cable's Green's function and corrupted by the noise sources along the cable. The probability of error, 
Pe is used to quantify task performance. 
ear, infinite, one-dimensional cable with distributed current noises. When the noise sources 
are weak so that the corresponding voltage fluctuations are small, the membrane voltage 
satisfies a linear stochastic differential equation satisfied. Using linear cable theory, we ex- 
press the power spectral density of the voltage noise in terms of the Green's function of an 
infinite cable and the current noise spectra. We use these results to quantify the efficacy of 
information transfer under a "signal detection" paradigm  where the objective is to detect 
the presence/absence of a presynaptic spike (in the form of an epsc) from the post-synaptic 
membrane voltage along the dendrite. The formalism used in this paper is summarized in 
Figure 1. 
2 Neuronal Noise Sources 
In this section we consider some current noise sources present in nerve membranes which 
distort a synaptic signal as it propagates along a dendrite. An excellent treatment of mem- 
brane noise is given in DeFelice (1981) and we refer the reader to it for details. For a linear 
one-dimensional cable, it is convenient to express quantities in specific length units. Thus, 
we express all conductances in units of S/ttm and current power spectra in units of A2/Hz 
/zm. 
A. Thermal Noise 
Thermal noise arises due to the random thermal agitation of electrical charges in a con- 
ductor and represents a fundamental lower limit of noise in a system. A conductor of 
resistance R is equivalent to a noiseless resistor R in series with a voltage noise source 
Vtt (t) of spectral density Svtt (f) = 2kTR (V2/Hz), or a noiseless resistor R in parallel 
with a current noise source,/ta (t) of spectral density S,rth (f) = 2kT/R (A2/Hz), where k 
is the Boltzmann constant and T is the absolute temperature of the conductor 2. The trans- 
verse resistance r, (units off ttm) of a nerve membrane is due to the combined resistance 
of the lipid bilayer and the resting conductances of various voltage-gated, ligand-gated and 
leak channels embedded in the lipid matrix. Thus, the current noise due to rra, has power 
For sake of brevity, we do not discuss the corresponding signal estimation paradigm as in Man- 
wani and Koch (1998). 
2Since the power spectra of real signals are even functions of frequency, we choose the double- 
sided convention for all power spectral densities. 
134 A. Manwani and C. Koch 
spectral density, 
2kT 
$,rtn(f) = (1) 
B. Channel Noise 
Neuronal membranes contain microscopic voltage-gated and ligand-gated channels which 
open and close randomly. These random fluctuations in the number of channels is another 
source of membrane noise. We restrict ourselves to voltage-gated K + and Na + channels, 
although the following can be used to characterize noise due to other types of ionic channels 
as well. In the classical Hodgkin-Huxley formalism (Koch, 1998), a K + channel consists 
of four identical two-state sub-units (denoted by n) which can either be open or closed. 
The K + channel conducts only when all the sub-units are in their open states. Since the 
sub-units are identical, the channel can be in one of five states; from the state in which 
all the sub-units are closed to the open state in which all sub-units are open. Fluctuations 
in the number of open channels cause a random K + current I: of power spectral density 
(DeFelice, 1981) 
4 
= - EK) no.  (1 -- ',,oo) ',% . 
i= 1 + 4r2 f2(O,/i)2 (2) 
where r/:, '),: and E: denote the K + channel density (per unit length), the K + single 
channel conductance and the K + reversal potential respectively. Here we assume that the 
membrane voltage has been clamped to a value Vm. noo and 0, are the steady-state open 
probability and relaxation time constant of a single K + sub-unit respectively and are in 
general non-linear functions of V, (Koch, 1998). When V, is close to the resting potential 
Vrest (usually between -70 to -65 mV), noo << 1 and one can simplify SK (f) as 
2 V - 2 4 
SK(f)  IK'YK( rest -- (3) 
E:) noo(1 noo) 4 2 0,/4 
1 q- 4rr2f2(O./4)2 
Similarly, the Hodgkin-Huxley Na + channel is characterized by three identical activation 
sub-units (denoted by m) and an inactivation sub-unit (denoted by h). The Na + channel 
conducts only when all the m sub-units are open and the h sub-unit is not inactivated. Thus, 
the Na + channel can be in one of eight states from the state corresponding to all m sub- 
units closed and the h sub-unit inactivated to the open state with all m sub-units open and 
the h sub-unit not inactivated. moo (resp. boo) and 0, (resp. 0h) are the corresponding 
steady-state open probability and relaxation time constant of a single Na + m (resp. h) 
sub-unit respectively. For Vm -, Vrest, moo << 1, boo  1 and 
2 20m/3 
Sva(f) m Wvaq'v(Vrest Ev):m3(1 a : 
- -moo) hoo 1 +4rr2f2(Om/3)2 (4) 
where r/v, ')'v and Eva denote the Na + channel density, the Na + single channel con- 
ductance and the sodium reversal potential respectively. 
C. $ynaptic Noise 
In addition to voltage-gated ionic channels, dendrites are also awash in ligand-gated synap- 
tic receptors. We restrict our attention to fast voltage-independent (AMPA-like) synapses. 
A commonly used function to represent the postsynaptic conductance change in response 
to a presynaptic spike is the alpha function (Koch, 1998) 
ga(t) -- gpeak e t e -t/tvea, 0 < t < oo (5) 
tpea k -- 
where gpeak denotes the peak conductance change and tpeak the time-to-peak of the con- 
ductance change. We shall assume that for a spike train s(t) = -.j 6(t - tj), the postsy- 
naptic conductance is given gSy, (t) -- y'j ga (t -- t j). This ignores inter-spike interaction 
Signal Detection in Noisy Weakly-Active Dendrites 135 
r i 
Cm 
EK-- I-- ENd-- Esyn T EL T 
Figure 2: Schematic diagram of the equivalent electrical circuit of a linear dendritic cable. The 
dendrite is modeled as an infinite ladder network. ri (units of f/tt m) denotes the longitudinal cyto- 
plasmic resistance; Cm (units of F/it m) and g, (units of S/it m) denote the transverse membrane ca- 
pacitance and conductance (due to leak channels with reversal potential E,) respectively. The mem- 
brane also contains active channels (K +, Na +) with conductances and reversal potentials denoted by 
(gc, gva) and (Ec, Eva) respectively, and fast voltage-independent (AMPA-like) synapses with 
conductance gs and reversal potential Es. 
and synaptic saturation. The synaptic current is given by isu (t) = gsu (t)(Vm - ESu) 
where Esu is the synaptic reversal potential. If the spike train can be modeled as a homo- 
geneous Poisson process with mean firing rate ,, the power spectrum of isu (t) can be 
computed using Campbell's theorem (Papoulis, 1991) 
$sun(f) - rIsun/kn(Vm - Esun) 2 [ G(f) I 2 , (6) 
where r/sun denotes the synaptic density and G (f) = f ga (t) ezp(-j2rrft) dt is the 
Fourier transform ofg (t). Substituting for g (t) gives 
SISyn(f) -- rISyn /n (e gpeaktpeak(Vrn -- ESyn)) 2 
(1 + 4 '2 f2ts2)  (7) 
3 Noise in Linear Cables 
The linear infinite cable corresponding to a dendrite is modeled by the ladder network 
shown in Figure 2. The membrane voltage Vm(z,t) satisfies the differential equation 
(Tuckwell, 1988), 
02 Vm [ O Vm 
02 x -- r i Cm Ot 
-- + ga:(v,,, - Ea:) + g:v.(V,,, - 
+ gSyn(Vrn - Esyn) + gL(Vm -- EL) ] 
(8) 
Since the ionic conductances are random and nonlinearly related to Vm, eq. 8 is a non- 
linear stochastic differential equation. If the voltage fluctuations (denoted by V) around 
the resting potential Vrest are small, one can express the conductances as small deviations 
(denoted by ) from their corresponding resting values and transform eq. 8 to 
_ 02V(x,t) OV(x,t) In 
Ox 2 + r Ot + (1 + (5)V(x,t) = - (9) 
where 2 = 1/(riG) and r = cm/G denote the length and time constant of the mem- 
brane respectively. G is the passive membrane conductance and is given by the sum of 
the resting values of all the conductances. ( - : + O2va + OSyn/G represents the ran- 
dom changes in the membrane conductance due to synaptic and channel stochasticity; ( 
136 A. Manwani and C. Koch 
-36 -16 
0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 
 ) (L s) Fy (m) ( ) 
Figure 3: (a) Comparison of current spectra St(f) of the four noise sources we consider. Synaptic 
noise is the most dominant source source of noise and thermal noise, the smallest. (b) Voltage 
noise spectrum of a 1-D infinite cable due to the current noise sources. $vth (f) is also shown for 
comparison. Summary of the parameters used (adopted from Mainen and Sejnowski, 1998): / - 
40 kf/cm 2, Cm = 0.75/F/cm , ri = 200 f/cm, d (dend. dia.) = 0.75/m, r/: = 2.3/m -x, r/va =  
/m -x, r/s, = 0.1/m -, E: = -95 mV, Ev,, = 50 mV, Es, = 0 mV, EL = Vrest = 70 mV, 7: 
7v = 20 pS. 
Vrest) + Ith denotes the total effective current noise due to the different noise sources. In 
order to derive analytical closed-form solutions to eq. 9, we frether assume that 6 < < 13, 
which reduces it to the familiar one-dimensional cable equation with noisy current input 
(Tuckwell, 1988). For resting initial conditions (no charge stored on the membrane at 
t = 0), V is linearly related to I, and can be obtained by convolving In with the Green's 
function g(z, y, t) of the cable for the appropriate boundary conditions. It has been shown 
that V (z, t) is an asymptotically wide-sense stationary process (Tuckwell and Walsh, 1991) 
and its power spectrum Sv (z, f) can be expressed in terms of the power spectrum of 
Sn(f) as 
$v(x, f) -- $n(f) /_ ' ' 
cs 16(x,x, f)l 2 dx 
oo 
(10) 
where 6(x, x', f) is the Fourier transform ofg(x, x', t). For an infinite cable 
, e -T -(x-x'? X' 
g(X,X ,T) -  e 4- , -oe < X, < oe, 0 <_ T < oe (11) 
where X = x/A, X' = x' /  and T = t / r are the corresponding dimensionless variables. 
Substituting for g(x, x', t) we obtain 
n(f) sin(tan-?rfr) ) 
Sv(f) - 2  G 2 2rfr (1 + (2rfr)2) / (12) 
Since the noise sources are independent, $n(f) = $ltl(f) + $lK(f) + $lNa(f) + 
Ssyn (f). Thus, eq. 12 allows us to compute the relative contribution of each of the noise 
sources to the voltage noise. The current and voltage noise spectra for biophysically rele- 
vant parameter values (Mainen and Sejnowski, 1998) are shown in Figure 3. 
3Using self-consistency, we find the assumption to be satisfied in our case. In general, it needs 
verified on a case-by-case basis. 
Signal Detection in Noisy Weakly-Active Dendrites 13 7 
4 Signal Detection 
The framework and notation used here are identical to that in Manwani and Koch (1998) 
and so we refer the reader to it for details. The goal in the signal detection task is to 
optimally decide between the two hypotheses 
Ho  y(t) = n(t), 0 < t < T Noise 
H : y(t) = a(t) * s(t) + n(t), 0 < t < T Signal + Noise (13) 
where n(t), g(t) and s(t) denote the dendritic voltage noise, the Green's function of the 
cable (function of the distance between the input and measurement locations) and the epsc 
waveform (due to a presynaptic spike) respectively. The decision strategy which minimizes 
= (1 - Po) are the prior 
the probability of error Pe = PoPf + PiPre, where po and Pl 
probabilities of Ho and H respectively, is 
H 
o 
Ho 
(14) 
where A(y) = P[yIH]/P[yIHo] and /20 = po/(1 - Po). PI and P, denote the 
false alarm and miss probability respectively. Since n(t) arises due to the effect of sev- 
eral independent noise sources, by invoking the Central Limit theorem, we can assume 
H 
that n(t) is Gaussian, for which eq. 14 reduces to r <> fl. r = f y(t) ha(-t) dt 
Ho 
is a correlation between y(t) and the matched filter ha(t), given in the Fourier do- 
main as Ha(f) = e-J2'fr6*(f)S*(f)/S(f). 6(f) and S(f) are Fourier transforms 
of g(t) and s(t) respectively and S,(f) is the noise power spectrum. The conditional 
means and variances of the Gaussian variable r under Ho and H are tz0 = 0, tZl = 
df and ao 2 = tr = a 2 = / respectively. The error prob- 
abilities are given by Pf = f P[rlHo] dr and P, = f_no. P[rIH1 ] dr . The op- 
timal value of the threshold r/ depends on a and the prior probability Po. For equi- 
probable hypotheses (Po = 1 - Po = 0.5), the optimal r/ = (tto + lz)/2 = a2/2 and 
Pe = 0.5 Erfc[a/2V/]. One can also regard the overall decision system as an effective 
binary channel. Let M and D be binary variables which take values in the set {Ho, H } 
and denote the input and output of the dendritic channel respectively. Thus, the system 
performance can equivalently be assessed by computing the mutual information between 
M and D, I(M; D) = (po (1- Pro) + (1-po) Pf)-po7-t(P,)- (1-po)7-t(Pf (Cover 
and Thomas, 1991) where 7(x) is the binary entropy function. For equi-probable hypothe- 
ses, I(M; D) - 1 - 7-t (P,) bits. It is clear from the plots for P, and I(M; D) (Figure 4) 
as a function of the distance between the synaptic (input) and the measurement (output) 
location that an epsc. can be detected with almost certainty at short distances, after which, 
there is a rapid decrease in detectability with distance. Thus, we find that membrane noise 
may limit the maximum length of a dendrite over which information can be transmitted 
reliably. 
5 Conclusions 
In this study we have investigated how neuronal noise sources might influence and limit the 
ability of one-dimensional cable structures to propagate information. When extended to re- 
alistic dendritic geometries, this approach can help address questions as, is the length of the 
apical dendrite in a neocortical pyramidal cell limited by considerations of signal-to-noise, 
which synaptic locations on a dendritic tree (if any) are better at transmitting information, 
what is the functional significance of active dendrites (Yuste and Tank, 1996) and so on. 
Given the recent interest in dendritic properties, it seems timely to apply an information- 
theoretic approach to study dendritic integration. In an attempt to experimentally verify 
138 A. Manwani and C. Koch 
Figure 4: Information loss in signal detection. (a) Probability of Error (Pc) and (b) Mutual informa- 
tion (I(M;D)) for an infinite cable as a function of distance from the synaptic input location. Almost 
perfect detection occurs for small distances but performance degrades steeply over larger distances 
as the signal-to-noise ratio drops below some threshold. This suggests that dendritic lengths may be 
ultimately limited by signal-to-noise considerations. Epsc. parameters: gpe, = 0.1 nS, t,e, = 1.5 
msec and Es, = 0 mV. Ns, is the number of synchronous synapses which activate in response to 
a pre-synaptic action potential. 
the validity of our results, we are currently engaged in a quantitative comparison using 
neocortical pyramidal cells (Manwani et al, 1998). 
Acknowledgements 
This research was supported by NSF, NIMH and the Sloan Center for Theoretical Neuro- 
science. We thank Idan Segev, Elad Schneidman, Mild London, YosefYarom and Fabrizio 
Gabbiani for illuminating discussions. 
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