Expanded Methods for Nature Website (to accompany manuscript S05587A)
NL cell biophysical model:
No voltage clamp data are available upon which to base a quantitative model of NL
neuron voltage gated membrane conductances. Thus, we kept the number of hypothetical
membrane conductances in our model to a minimum, namely, a voltage gated sodium
conductance and a voltage gated potassium conductance. Because the outward
rectification and DC filtering properties of NL neurons studied by current clamp
stimulation were very similar to those recorded in Nucleus Magnocellularis (NM)
neurons, we based the K conductance of our model NL neuron soma on the K current
measured in voltage clamp in NM cells by Reyes et al. (1994) 1. This current consists
mainly of a “low threshold” K current but may consist partly of the separate “high
threshold” K current later described in NM neurons by Rathouz and Trussell (1998) 2.
Near 37 C NL action potentials are too small to significantly activate the high threshold
K current (Reyes et al. 1996). Although the Rathouz and Trussell (1998) paper provided a
more detailed description of the NM cell K currents than Reyes et al. (1994), the data
from the latter were used for the K current model because those experiments were done
near 37 C, whereas Rathouz and Trussell (1998) 2 worked at room temperature and no
Q10 value was provided to enable the rate constants to be scaled to 37 C. Temperature
was important because the hypothetical conductances were adjusted to reproduce the
variety of NL cell behaviors observed during current clamp stimulations by Reyes et al.
(1996) 3 working near 37 C. The ability of the model to reproduce responses to current
clamp stimulation was the criterion used to decide if the hypothetical model conductances
were acceptable.
The voltage dependence of both steady state activation and time constants of the K
conductance were derived from Figs. 5 C,D of Reyes et al. (1994) 1 assuming 4th order
kinetics, as is consistent with the observed delayed onset of the current (Fig.5 A, inset in
Reyes et al. 1994 1). To reproduce the outward rectification observed in NL neurons
(Reyes et al. 1996 3) it was necessary to make minor modifications in the NM neuron K
conductance, namely, a slightly less steep activation characteristic and an increased K
channel density.
The model’s hypothetical somatic sodium conductance was chosen to reproduce the
height, width, voltage threshold and rheobase current needed to evoke an action potential
at various stimulation frequencies, the inability to fire during sustained (DC)
depolarization, and the difference in spike failure during suprathreshold low frequency
versus high frequency stimuli (Fig. 3, middle panels). Fast sodium conductance time
constants were needed to model this spiking behavior. To keep these time constants
within the realm of biological possibility we scaled the activation time constants
measured in rat peripheral nerve node of Ranvier (Schwarz and Eikhof, 1987 4) and the
inactivation time constants measured in rat hippocampal inhibitory neurons (Martina and
Jonas, 1997 5) to 37 C using a Q10 of 2.2 for activation rate constants and 2.9 for
inactivation rate constants (Schwatrz and Eikhof, 1987 4). The time constants used in the
model were kept longer than the minimal values obtained from this temperature scaling.
Because the peak of the somatic NL cell action potential only reached about –15 mV (at
37 C) and action potential amplitude becomes even smaller during high frequency
stimulation (Reyes et al. 1996 3), it is difficult to count spikes (thus, spike frequency) in
an automated way during long computer runs simulating high frequency stimulation. The
axon described below was attached to the model soma simply to conduct a full size action
potential that could be recognized unambiguously. Merely for the sake of convenience
Hodgkin-Huxley-like formulations of a voltage gated sodium current and a voltage-gated
potassium current were incorporated into the nodes. The criteria for these nodal action
potentials were that they be large and overshooting (i.e., easy to count), of fast duration
and able to follow the same high frequencies as the somatic action potentials, and that
they occur if and only if a somatic action potential was triggered by the same stimulation.
The latter requirement was tested by using a variety of somatic stimuli and adjusting the
steady state activation and inactivation characteristics as necessary to meet the
requirement.
NL model cell geometry:
The model NL cell consisted of a soma (a 20 m sphere) with 18 dendrites (each 44.4 m
long, 2 m in diameter) and an axon initial segment (20 m long) connected to a
myelinated axon with 3 internodes (each 200m long) and 3 nodes of Ranvier (each 2
m long, 1.7 m in diameter). Nine dendrites received input from the ipsilateral ear and 9
received input from the contralateral ear. Synapses (up to 8 per dendrite) were distributed
uniformly along the length of each dendrite. In spite of the presence of dendrites this
model was equivalent to a point neuron because negligible attenuation of EPSPs occurred
from dendrite to soma (unpublished observations; dendritic electro tonic length: 0.09).
NL cell model membrane conductances:
The specific resistance (Rm) of all neuronal membrane was 4712 -cm, resulting in a
hyperpolarizing input resistance of 73 M(Reyes et al. 1996 3)The myelinated
internodes had 100 times the Rm and 1/100th the specific capacitance as the value (1
F/cm2) for neuronal membrane.
The same type of voltage gated conductances were placed in the soma, axon initial
segment and dendrites. These differed from the conductances on the nodes of Ranvier as
described below. The density of the dendritic sodium and potassium currents were 1/18th
and 1/10th that of the soma, respectively.
Ionic currents were calculated from Hodgkin-Huxley-like descriptions of the underlying
whole-cell conductances. For sodium currents,
INa = GNa*act3*inact* (V-ENa)
and for potassium currents,
(1)
IK = Gk*act4*(V-Ek)
(2)
where V is membrane potential, GNa and Gk are the maximum whole cell conductance
values, act is the steady state activation characteristic, inact is the steady state
inactivation characteristic and ENa and Ek are the sodium and potassium Nernst
equilibrium potentials, respectively.
For each conductance the steady state activation and inactivation characteristics were
calculated from the Boltzmann relations,
act = 1/(1+exp((V1/2-V)/k))
(3)
inact = 1/(1+ exp((V-V1/2)/k))
(4)
where V1/2 is the potential of half activation or inactivation and k is a slope factor.
Time constants (tau) of activation and inactivation for each conductance were calculated
from the bell-shaped relation,
tau = 2*[B + A/(exp(-(V+Vh1/2)/kh) + exp((V+Vd1/2 )/kd))]
(5)
where A,B Vh1/2,Vd1/2, kh and kd are constants.
The values of the constants used in Eqns 1-5 for each conductance are given below in
Table 1.
1. Reyes, A. D., Rubel, E. W & Spain, W. J. Membrane properties underlying the firing
of neurons in the avian cochlear nucleus. J Neurosci 14, 5352-5364 (1994).
2. Rathouz, R. & Trussell L. Characterization of Outward Currents in Neurons of the
Avian Nucleus Magnocellularis J Neurophysiol 80, 2824–2835 (1998).
3. Reyes, A. D., Rubel, E. W & Spain, W. J. In vitro analysis of optimal stimuli for
phase-locking and time-delayed modulation of firing in avian nucleus laminaris neurons.
J Neurosci 16, 993-1007 (1996).
4. Schwarz, J. R. & Eikhof, G. Na currents and action potentials in rat myelinated nerve
fibres at 20 and 37 degrees C. Pflugers Arch 409, 569-577. (1987).
5. Martina, M. & Jonas, P. Functional differences in Na+ channel gating between fastspiking interneurones and principal neurones of rat hippocampus. J Physiol 505, 593-603.
(1997).
Table 1. Conductance parameters of NL cell membrane model specified by Eqns 1-5
Parameter:
soma Na
soma K
node Na
node K
Max conductance, S/cm2
Equilibrium potential, mV
Steady state activation
V1/2, mV
k, mV
Steady state inactivation:
V1/2, mV
k, mV
Activation tau:
A, ms
B, ms
Vh1/2, mV
kh, mV
Vd1/2, mV
kd, mV
Inactivation tau:
A, ms
B, ms
Vh1/2, mV
kh, mV
Vd1/2, mV
kd, mV
0.04
40
0.006
-100
0.5
40
0.02
-100
-42
4.5
-54
9
-26
9.3
-51.1
5.1
-50.5
4.5
-
-50
7.1
-
0.05
0.0030
40
19
40
19
0.8
0
50
15
50
15
0.0244
0
26
26.3
26
26.3
1.5
0
54
7.5
54
7.5
4
0.01
46
4.5
46
4.5
-
1.145
0
50
13.3
50
13.3
-