A Neural Model of Descending Gain 
Control in the Electrosensory System 
Mark E. Nelson 
Beckman Institute 
University of Illinois 
405 N. Mathews 
Urbana, IL 61801 
Abstract 
In the electrosensory system of weakly electric fish, descending 
pathways to a first-order sensory nucleus have been shown to influ- 
ence the gain of its output neurons. The underlying neural mecha- 
nisms that subserve this descending gain control capability are not 
yet fully understood. We suggest that one possible gain control 
mechanism could involve the regulation of total membrane conduc- 
tance of the output neurons. In this paper, a neural model based 
on this idea is used to demonstrate how activity levels on descend- 
ing pathways could control both the gain and baseline excitation 
of a target neuron. 
I INTRODUCTION 
Certain species of freshwater tropical fish, known as weakly electric fish, possess an 
active electric sense that allows them to detect and discriminate objects in their 
environment using a self-generated electric field (Bullock and Heiligenberg, 1986). 
They detect objects by sensing small perturbations in this electric field using an 
array of specialized receptors, known as electroreceptors, that cover their body sur- 
face. Weakly electric fish often live in turbid water and tend to be nocturnal. These 
conditions, which hinder visual perception, do not adversely affect the electric sense. 
Hence the electrosensory system allows these fish to navigate and capture prey in 
total darkness in much the same way as the sonar system of echolocating bats allows 
them to do the same. A fundamental difference between bat echolocation and fish 
921 
922 Nelson 
"electrolocation" is that the propagation of the electric field emitted by the fish is 
essentially instantaneous when considered on the time scales that characterize ner- 
vous system function. Thus rather than processing echo delays as bats do, electric 
fish extract information from instantaneous amplitude and phase modulations of 
their emitted signals. 
The electric sense must cope with a wide range of stimulus intensities because the 
magnitude of electric field perturbations varies considerably depending on the size, 
distance and impedance of the object that gives rise to them (Bastian, 1981a). 
The range of intensities that the system experiences is also affected by the con- 
ductivity of the surrounding water, which undergoes significant seasonal variation. 
In the electrosensory system, there are no peripheral mechanisms to compensate 
for variations in stimulus intensity. Unlike the visual system, which can regulate 
the intensity of light arriving at photoreceptors by adjusting pupil diameter, the 
electrosensory system has no equivalent means for directly regulating the overall 
stimulus strength experienced by the electroreceptors, x and unlike the auditory 
system, there are no efferent projections to the sensory periphery to control the 
gain of the receptors themselves. The first opportunity for the electrosensory sys- 
tem to make gain adjustments occurs in a first-order sensory nucleus known as the 
electrosensory lateral line lobe (ELL). 
In the ELL, primary afferent axons from peripheral electroreceptors terminate on 
the basal dendrites of a class of pyramidal cells referred to as E-cells (Maler et 
al., 1981; Bastian, 1981b), which represent a subset of the output neurons for the 
nucleus. These pyramidal cells also receive descending inputs from higher brain 
centers on their apical dendrites (Maler et al., 1981). One noteworthy feature is 
that the descending inputs are massive, far outnumbering the afferent inputs in total 
number of synapses. Experiments have shown that the E-cells, unlike peripheral 
electroreceptors, maintain a relatively constant response amplitude to electrosensory 
stimuli when the overall electric field normalization is experimentally altered. This 
automatic gain control capability is lost, however, when descending input to the ELL 
is blocked (Bastian, 1986ab). The underlying neural mechanisms that subserve this 
descending gain control capability are not yet fully understood, although it is known 
that GABAergic inhibition plays a role (Shumway & Maler, 1989). We suggest that 
one possible gain control mechanism could involve the regulation of total membrane 
conductance of the pyramidal cells. In the next section we present a model based 
on this idea and show how activity levels on descending pathways could regulate 
both the gain and baseline excitation of a target neuron. 
2 
NEURAL CIRCUITRY FOR DESCENDING GAIN 
CONTROL 
Figure i shows a schematic diagram of neural circuitry that could provide the basis 
for a descending gain control mechanism. This circuitry is inspired by the circuitry 
found in the ELL, but has been greatly simplified to retain only the aspects that 
In principle, this could be achieved by regulating the strength of the fish's own electric 
discharge. However, these fish maintain a remarkably stable discharge amplitude and such 
a mechanism has never been observed. 
A Neural Model of Descending Gain Control in the Electrosensory System 923 
(CONTROL) 
descending 
excitatory 
descending 
inhibitory 
excitatory synapse 
inhibitory synapse 
inhibitory 
interneuron 
pyramidal cell (OUTPUT) 
primary 
(INPUT) afferent ) 0 
Figure 1: Neural circuitry for descending gain control. The gain of the pyramidal 
cell response to an input signal arriving on its basilar dendrite can be controlled 
by adjusting the tonic levels of activity on two descending pathways. A descending 
excitatory pathway makes excitatory synapses (open circles) directly on the pyra- 
midal cell. A descending inhibitory pathway acts through an inhibitory interneuron 
(shown in gray) to activate inhibitory synapses (filled circles) on the pyramidal cell. 
are essential for the proposed gain control mechanism. The pyramidal cell receives 
afferent input on a basal dendrite and control inputs from two descending pathways. 
One descending pathway makes excitatory synaptic connections directly on the 
apical dendrite of the pyramidal cell, while a second pathway exerts a net inhibitory 
effect on the pyramidal cell by acting through an inhibitory interneuron. We will 
show that under appropriate conditions, the gain of the pyramidal cell's response 
to an input signal arriving on its basal dendrite can be controlled by adjusting 
the tonic levels of activity on the two descending pathways. At this point it is 
worth pointing out that the spatial segregation of input and control pathways onto 
different parts of the dendritic tree is not an essential feature of the proposed gain 
control mechanism. However, by allowing independent experimental manipulation 
of these two pathways, this segregation has played a key role in the discovery and 
subsequent characterization of the gain control function in this system (Bastian, 
1986ab). 
The gain control function of the neural circuitry show in Figure 1 can best be un- 
derstood by considering an electrical equivalent circuit for the pyramidal cell. The 
equivalent circuit shown in Figure 2 includes only the features that are necessary 
to understand the gain control function and does not reflect the true complexity of 
ELL pyramidal cells, which are known to contain many different types of voltage- 
dependent channels (Mathieson & Maler, 1988). The passive electrical properties 
of the circuit in Figure 2 are described by a membrane capacitance C,,, a leak- 
age conductance geak, and an associated reversal potential E,ak. The excitatory 
descending pathway directly activates excitatory synapses on the pyramidal cell, 
giving rise to an excitatory synaptic conductance gex with a reversal potential 
924 Nelson 
am 
g leak 
1 
ex 
Figure 2: Electrical equivalent circuit for the pyramidal cell in the gain control 
circuit. The excitatory and inhibitory conductances, g,x and gi,h, are shown are 
variable resistances to indicate that their steady-state values are dependent on the 
activity levels of the descending pathways. 
The inhibitory descending pathway acts by exciting a class of inhibitory interneu- 
rons which in turn activate inhibitory synapses on the pyramidal cell with inhibitory 
conductance gin and reversal potential Elna. In this model, the excitatory and in- 
hibitory conductances g,x and girth represent the population conductances of all 
the individual excitatory and inhibitory synapses associated with the descending 
pathways. Although individual synaptic events give rise to a time-dependent con- 
ductance change (which is often modeled by an ( function), we consider the regime 
in which the activity levels on the descending pathways, the number of synapses 
involved, and the synaptic time constants are such that the summed effect can be 
well described by a single time-invariant conductance value for each pathway. The 
input signal (the one under the influence of the gain control mechanism) is modeled 
in a general form as a time-dependent current I(t). This current can represent ei- 
ther the synaptic current arising from activation of synapses in the primary afferent 
pathway, or it can represent direct current injection into the cell, such as might 
occur in an intracellular recording experiment. 
The behavior of the membrane potential V(t) for this nodel system is described by 
av(t) 
q- gteak(V(t) - Eteak) q- gex(V(t) -/ex) q- ginh(V(t) -- /inh) -- I(t) (1) 
In the absence of an input signal (I - 0), the system will reach a steady-state 
(dV/dt -- O) membrane potential Vs given by 
Vss(I = O) = gteakEtek + gexEex + ginhEinh 
gteak q- gex + ginh 
(2) 
A Neural Model of Descending Gain Control in the Electrosensory System 925 
If we consider the input I(t) to give rise to fluctuations in membrane potential U(t) 
about this steady state value 
then (1) can be rewritten as 
v(t) = v(t)- , 
dU(/) 
Cm d---5-- + gtotU(t)= I(t) (4) 
where grot is the total membrane conductance 
grot -- gteak q- gex q- ginh (5) 
Equation (4) describes a first-order low-pass filter with a transfer function G(s), 
from input current to output voltage change, given by 
/tot 
G(s) = (6) 
rs+ 1 
where s is the complex frequency (s = ivy), Rtot is the total membrane resistance 
(ltot -- 1/grot), and r is the RC time constant 
Cm 
7' -- Rtot Cm -- -- (7) 
grot 
The frequency dependence of the response gain IG(iov)l is shown in Figure 3. For low 
frequency components of the input signal (ovr << 1), the gain is inversely propor- 
tional to the total membrane conductance grot, while at high frequencies (ovr >> 1), 
the gain is independent of grot. This is due to the fact that the impedance of the 
RC circuit shown in Figure 2 is dominated by the resistive components at low fre- 
quencies and by the capacitive component at high frequencies. Note that the RC 
time constant r, which characterizes the low-pass filter cutoff frequency, varies in- 
versely with grot. For components of the input signal below the cutoff frequency, 
gain control can be accomplished by regulating the total membrane conductance. 
In electrophysiological terms, this mechanism can be thought of in terms of regu- 
lating the input resistance of the neuron. As the total membrane conductance is 
increased, the input resistance is decreased, meaning that a fixed amount of current 
injection will cause a smaller change in membrane potential. Hence increasing the 
total membrane conductance decreases the response gain. 
In our model, we propose that regulation of total membrane conductance occurs 
via activity on descending pathways that activate excitatory and inhibitory synaptic 
conductances. For this proposed mechanism to be effective, these synaptic conduc- 
tances must make a significant contribution to the total membrane conductance of 
the pyramidal cell. Whether this condition actually holds for ELL pyramidal cells 
has not yet been experimentally tested. However, it is not an unreasonable assump- 
tion to make, considering recent reports that synaptic background activity can have 
926 Nelson 
20 
0 
-20 
-80 
gtot = gleak 
gtot = 10 gDak 
gtot = 100 gleak 
-r = 100 msec 
 = 10 ms 
 = l msec  
(rad/sec) 
Figure 3: Gain as a function of frequency for three different values of total membrane 
conductance grot. At low frequencies, gain is inversely proportional to #tot. Note 
that the time constant r, which characterizes the low-pass cutoff frequency, also 
varies inversely with grot. Gain is normalized to the maximum gain: G,,ax =  
Gain(dB) = 20 logic (6--)' 
a significant influence on the total membrane conductance of cortical pyramidal cells 
(Bernander et al., 1991) and cerebellar Purkinje cells (Rapp et al., 1992). 
3 CONTROL OF BASELINE EXCITATION 
If the only functional goal was to achieve regulation of total membrane conductance, 
then synaptic background activity on a single descending pathway would be suffi- 
cient and there would be no need for the paired excitatory and inhibitory control 
pathways shown in Figure 1. However, the goal of gain control is regulate the total 
membrane conductance while holding the baseline level of excitation constant. In 
other words, we would like to be able to change the sensitivity of a neuron's response 
without changing its spontaneous level of activity (or steady-state resting potential 
if it is below spiking threshold). By having paired excitatory and inhibitory control 
pathways, as shown in Figure 1, we gain the extra degree-of-freedom necessary to 
achieve this goal. 
Equation (2) provided us with a relationship between the synaptic conductances in 
our model and the steady-state membrane potential. In order to change the gain 
of a neuron, without changing its baseline level of excitation, the excitatory and 
inhibitory conductances must be adjusted in a way that achieves the desired total 
membrane conductance grot and maintains a constant Vs. Solving equations (2) 
and (5) simultaneously for g,x and gi, h, we find 
A Neural Model of Descending Gain Control in the Electrosensory System 927 
gtot(Vss - Jinh) -- g/eak(E/eak -- /inh) 
gez -- (j[7e x _ j[7inh ) 
(8) 
gtot(Vss- j[7ex ) -gteak(J[7teak -- /7ex ) 
girth -- (inh- Eex) 
(9) 
For example, consider a case where the reversal potentials are Et,a = -70 mV, 
Eel, = 0 mV, and Einh -' --90 mY. Assume want to find values of the steady-state 
conductances, g, and git, h, that would result in a total membrane conductance 
that is twice the leakage conductance (i.e. gtot= 2g&a) and would produce a 
steady-state alepolarization of 10 mV (i.e. V - -60 mV). Using (8) and (9) we 
_ 5 
find the required synaptic conductance levels are g, = gteak and girth -- 5gteak. 
4 DISCUSSION 
The ability to regulate a target neuron's gain using descending control signals would 
provide the nervous system with a powerful means for implementing adaptive signal 
processing algorithms in sensory processing pathways as well as other parts of the 
brain. The simple gain control mechanism proposed here, involving the regulation 
of total membrane conductance, may find widespread use in the nervous system. 
Determining whether or not this is the case, of course, requires experimental ver- 
ification. Even in the electrosensory system, which provided the inspiration for 
this model, definitive experimental tests of the proposed mechanism have yet to 
be carried out. Fortunately the model provides a straightforward experimentally 
testable prediction, namely that if activity levels on the presumed control path- 
ways are changed, then the input resistance of the target neuron will reflect those 
changes. In the case of the ELL, the prediction is that if descending pathways 
were silenced while monitoring the input resistance of an E-type pyramidal cell, one 
would observe an increase in input resistance corresponding to the elimination of 
the descending contributions to the total membrane conductance. 
We have mentioned that the gain control circuitry of Figure i was inspired by the 
neural circuitry of the ELL. For those familiar with this circuitry, it is interesting to 
speculate on the identity of the interneuron in the inhibitory control pathway. In the 
gymnotid ELL, there are at least six identified classes of inhibitory interneurons. For 
the proposed gain control mechanism, we are interested in the identifying those that 
receive descending input and which make inhibitory synapses onto pyramidal cells. 
Four of the six classes meet these criteria: granule cell type 2 (GC2), polymorphic, 
stellate, and ventral molecular layer neurons. While all four classes may participate 
to some extent in the gain control mechanism, one would predict, based on cell 
number and synapse location, that GC2 (as suggested by Shumway &: Maler, 1989) 
and polymorphic cells would make the dominant contribution. The morphology of 
GC2 and polymorphic neurons differs somewhat from that shown in Figure 1. In 
addition to the apical dendrite, which is shown in the figure, these neurons also 
have a basal dendrite that receives primary afferent input. GC2 and polymorphic 
neurons are excited by primary afferent input and thus provide additional inhibition 
to pyramidal cells when afferent activity levels increase. This can be viewed as 
providing a feedforward component to the automatic gain control mechanism. 
928 Nelson 
In this paper, we have confined our analysis to the effects of tonic changes in de- 
scending activity. While this may be a reasonable approximantion for certain ex- 
perimental manipulations, it is unlikely to be a good representation of the dynamic 
patterns that occur under natural conditions, particularly since the descending path- 
ways form part of a feedback loop that includes the ELL output neurons. The full 
story in the electrosensory system will undoubtably be much more complex. For 
example, there is already experimental evidence demonstrating that, in addition 
to gain control, descending pathways influence the spatial and temporal filtering 
properties of ELL output neurons (Bastian, 1986ab; Shumway & Maler, 1989). 
Acknowledgements 
This work was supported by NIMH 1-R29-MH49242-01. Thanks to Joe Bastian and 
Lenny Maler for many enlightening discussions on descending control in the ELL. 
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