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Volume 17, Number 20,
Issue of October 15, 1997
pp. 7606-7625
Copyright ©1997 Society for Neuroscience
Estimating the Time Course of the Excitatory Synaptic Conductance
in Neocortical Pyramidal Cells Using a Novel Voltage Jump
Method
Michael Häusser1 and
Arnd Roth2
1 Laboratoire de Neurobiologie, Ecole Normale
Supérieure, 75005 Paris, France, and 2 Abteilung
Zellphysiologie, Max-Planck-Institut für Medizinische
Forschung, 69120 Heidelberg, Germany
 ABSTRACT
- INTRODUCTION
- MATERIALS AND METHODS
- RESULTS
- DISCUSSION
- FOOTNOTES
- REFERENCES
ABSTRACT 
We introduce a method that permits faithful extraction of the decay
time course of the synaptic conductance independent of dendritic
geometry and the electrotonic location of the synapse. The method is
based on the experimental procedure of Pearce (1993) , consisting of a
series of identical somatic voltage jumps repeated at various times
relative to the onset of the synaptic conductance. The progression of
synaptic charge recovered by successive jumps has a characteristic
shape, which can be described by an analytical function consisting of
sums of exponentials. The voltage jump method was tested with
simulations using simple equivalent cylinder cable models as well as
detailed compartmental models of pyramidal cells. The decay time course
of the synaptic conductance could be estimated with high accuracy, even
with high series resistances, low membrane resistances, and
electrotonically remote, distributed synapses. The method also provides
the time course of the voltage change at the synapse in response to a
somatic voltage-clamp step and thus may be useful for constraining
compartmental models and estimating the relative electrotonic distance
of synapses. In conjunction with an estimate of the attenuation of
synaptic charge, the method also permits recovery of the amplitude of
the synaptic conductance. We use the method experimentally to determine
the decay time course of excitatory synaptic conductances in
neocortical pyramidal cells. The relatively rapid decay time constant
we have estimated ( ~1.7 msec at 35°C) has important
consequences for dendritic integration of synaptic input by these
neurons.
Key words:
neocortex;
pyramidal cell;
space clamp;
voltage clamp;
cable modeling;
synaptic current;
EPSC
INTRODUCTION
Knowledge of the time course of the
synaptic conductance is of fundamental importance to our understanding
of synaptic transmission. The kinetics of the synaptic conductance
influences neuronal function in many ways, from shaping the resulting
synaptic potential and setting the time window for synaptic integration
to determining the synaptic charge (particularly relevant when a
significant fraction of the current is carried by ions such as
Ca2+). Furthermore, comparing synaptic conductance
time course with receptor channel kinetics provides valuable
information about the processes underlying synaptic transmission.
Synaptic conductance is conventionally measured by recording the
synaptic current with somatic voltage clamp. In cells where all
synapses are electrotonically close to or at the soma, such as
cerebellar granule cells (Silver et al., 1992, 1995), neuroendocrine cells (Schneggenburger and Konnerth, 1992; Borst et al., 1994), unipolar brush cells (Rossi et al., 1995) and neurons in the auditory pathway (Forsythe and Barnes-Davies, 1993; Zhang and Trussell, 1994;
Isaacson and Walmsley, 1995), this method can reliably measure the
conductance time course. Alternatively, one can select for somatic
synapses using cable model predictions (Finkel and Redman, 1983; Nelson
et al., 1986). However, in most neurons, the majority of synapses are
located at a considerable electrotonic distance from the soma, and
therefore somatic voltage clamp of these synapses is associated with
substantial attenuation and distortion of the synaptic current
(Johnston and Brown, 1983; Rall and Segev, 1985; Major, 1993; Spruston
et al., 1993; Mainen et al., 1996). This problem has proved to be
rather intractable, and although several solutions have been proposed
to date (see Discussion), none are completely satisfactory.
Recently an experimental technique was introduced by Pearce (1993),
which uses somatic voltage jumps at various times during the synaptic
conductance to determine how long after the onset of the synaptic
current the synaptic conductance remains active. The principle of the
technique is that a voltage jump that increases the synaptic driving
force will only recover additional synaptic charge if the jump occurs
while the conductance is still active. The technique was used to show
that the GABAergic synaptic conductance generated by activation of
distal synapses in hippocampal CA1 pyramidal neurons has a prominent
slow component; however, a quantitative determination of the
conductance time course was not made. This technique was subsequently
applied to excitatory synapses in various neuronal types, also to
demonstrate that the synaptic conductance at these synapses has a
prolonged component (Barbour et al., 1994; Mennerick and Zorumski,
1995; Rossi et al., 1995; Kirson and Yaari, 1996).
Here we show using simulations in a variety of neuronal models that by
measuring the time course of recovered charge this experimental technique can be used to determine the decay time course
of the synaptic conductance with a high degree of accuracy. A simple
analytical function providing a quantitative description of the results
is presented, and limitations and potential applications of the method
are explored. We use the method to estimate the time course of the
excitatory synaptic conductance in neocortical pyramidal cells.
MATERIALS AND METHODS
Simulations
All simulations were performed using NEURON (Hines, 1993)
running on Sun Sparcstations (Sun Microsystems, Mountain View, CA). The
integration time step was 10 µsec. The synaptic conductance consisted
of a sum of two or three exponentials, one for the rise (always 0.2 msec, unless otherwise indicated) and one or two for the decay. A
"delta pulse" synaptic conductance was simulated using a 1 nS
conductance with duration of 0.1 msec. Except for the equivalent
cylinder simulations and the simulations shown in Figure 8, synaptic
contacts were placed at the head of explicitly modeled spines. The
series resistance of the recording pipette was always 0.5 M , except
where otherwise indicated, which is achievable in experiments using the
neuronal types shown here (5 M compensated by 90%). Unless
otherwise indicated, the decay time constant of synaptic currents
recorded at the soma was fit using a single exponential function,
starting at the time point when the current had decayed to ~90% of
the peak amplitude.
Fig. 8.
Simulation of a distributed inhibitory
conductance in a CA3 pyramidal cell. A unitary connection made by a
presynaptic "bitufted" inhibitory neuron is modeled, based on the
work of Miles et al. (1996, their Fig. 2). The locations of the 8 individual contacts on apical and basal dendritic shafts are shown
using dots in A. Each synaptic contact
had an identical synaptic conductance, with a peak conductance of 1 nS
and a reversal potential of 0 mV. The rising time constant was 0.2 msec
in all cases, and the decay time constant was either a single
exponential of 5 msec (B-D) or a double
exponential of 5 msec (80%) and 30 msec (20%). B and E compare the somatic clamp current with the perfectly
clamped EPSC. The 20-80% rise times of the currents were 1.66 msec in B and 1.89 msec in E. The decay of the
somatic clamp current could be fit by a single exponential with = 9.5 and 22.2 msec, respectively. C, F,
Recovered currents from successive voltage jumps from  65 mV.
D, G, Charge recovery curves, which have been fit with
the analytical function. For the monoexponentially decaying
conductance, the best fit of the analytical function was with the
following parameters: v1 = 3.24 msec (57%);
v2 = 10.93 msec (43%); rise = 0.66 msec; and dec = 5.22 msec; fitting the decay of the
charge recovery curve with a single exponential gave dec = 5.16 msec. For the conductance with a biexponential decay the best
fit was with the following parameters: v1 = 3.26 msec (59%); v2 = 11.11 msec (41%);
rise = 0.48 msec; dec1 = 5.17 msec
(77%); and dec2 = 30.54 msec (23%). A
double-exponential fit to the decay of the charge recovery gave
dec1 = 5.02 msec (66%); and
dec2 = 30.48 msec (34%).
[View Larger Version of this Image (21K GIF file)]
Equivalent cylinder model. The geometry used in the
equivalent cylinder simulation was as follows (see Fig.
1A): soma, 10 µm long, 10 µm diameter, 10 segments; and dendrite, 500 µm long, 1.2 µm diameter, 100 segments. Electrical parameters were:
Ri = 150 cm; Rm = 50,000 cm2; and Cm = 1.0 µF cm 2, giving an electrotonic length of the
dendrite of L = 0.5. The passive reversal potential was
65 mV.
Fig. 1.
Space-clamp errors affecting the measurement of
dendritic synaptic conductances. All traces in B are
from the same equivalent cylinder shown schematically in
A (soma not to scale), with L = 0.5 and with synapses at three different electrotonic locations on the
cable (X = 0, 0.15, and 0.5). The peak synaptic
conductance was 1 nS in each case, consisting of the sum of a rising
( = 0.2 msec) and a decaying ( = 1, 3, or 10 msec) exponential.
B, In each panel the voltage at the synaptic location
(Vsyn) is shown as the top trace.
The bottom traces show the current recorded at the soma
(thick line), the current actually flowing at the synapse (thin line), and the synaptic current expected
under perfect voltage-clamp conditions (dashed line).
The numbers at the right of each panel
show the relative magnitude of the peak (pk), the decay time constant (), and the charge (Q) of
the somatic current versus the perfectly clamped synaptic current. The
scale bar at the bottom right applies to all
panels.
[View Larger Version of this Image (19K GIF file)]
CA3 pyramidal cell model. The CA3 pyramidal cell model was
based on cell CA3_15 in the article by Major et al. (1994), which is
from a 19-d-old rat. The morphology was converted from the native
format to that of NEURON using a program written in Mathematica (Wolfram Research, Champaign, IL). The electrotonic length of each
segment was <0.01. The electrical parameters were
Ri = 250 cm; Rm = 180,000 cm2; and Cm = 0.66 µF cm2; with a passive reversal potential of
65 mV. Spine corrections were performed as described by Major et al.
(1994), and the axon was not included in the simulations. The spine at
the excitatory synaptic contact had a neck length of 0.66 µm, a neck
diameter of 0.2 µm, a head length of 0.5 µm, and a head diameter of
0.45 µm.
Neocortical pyramidal cell model. The morphology of the
layer 5 pyramidal cell was taken from the work of Markram et al. (1997) and comes from a postnatal day 14 rat (same neuron as shown in red in Markram et al., their Fig. 13). The electrotonic
length of each segment was <0.02. The values for passive cable
properties were Ri = 150 cm;
Rm = 30 000 cm2; and
Cm = 0.75 µF cm2, and the
passive reversal potential was set to 70 mV (Mainen and Sejnowski,
1996). The measured dendritic membrane area was multiplied by a factor
of 2 to account for spines. The axon was included, but axon collaterals
were omitted. The neck length of the explicitly modeled spines was 1.0 µm, neck diameter was 0.35 µm, and head length and diameter were
both 0.7 µm (Peters and Kaiserman-Abramof, 1970).
Active conductances were added to the model as described in Mainen and
Sejnowski (1996), based on the parameters in their original
NEURON files (available via World Wide Web at
http://www.cnl.salk.edu/CNL/simulations.html). Two changes were
made with respect to the original files of Mainen and Sejnowski (1996):
(1) the reversal potential for Ca2+ was not constant
at +140 mV but updated according to the Nernst equation assuming
[Ca2+]o = 2 mM; and 2) the
time step was 10 µsec instead of 25 µsec.
Experiments
Whole-cell patch-clamp recordings were made from the soma of
visually identified thick tufted layer 5 pyramidal cells in slices of
rat neocortex as described previously (Stuart et al., 1993; Markram et
al., 1997). Wistar rats (14-18 d) were killed by decapitation, and
sagittal neocortical slices (250-300 µm) were cut on a Vibratome (Dosaka) in ice-cold extracellular solution containing (in
mM): 125 NaCl, 2.5 KCl, 25 glucose, 25 NaHCO3, 1.25 NaH2PO4,
2 CaCl2, and 1 MgCl2. The slices were
incubated at 34°C for 45 min and then kept at room temperature before
transfer to the recording chamber. With the use of an upright
microscope (Axioskop, 40×-W/0.75 numerical aperture water-immersion
objective; Zeiss, Oberkochen, Germany) and infrared differential
interference contrast videomicroscopy (Stuart et al., 1993), layer 5 pyramidal neurons were easily identified by their large somata,
prominent axon initial segment, and thick apical dendrites projecting
to higher layers.
Recordings were made using an Axopatch 200A amplifier (Axon
Instruments, Foster City, CA). The internal patch pipette solution contained (in mM): 100 potassium gluconate, 20 KCl, 10 HEPES, 10 EGTA, 4 Na2-ATP, and 4 MgCl2 (295 mOsm, pH adjusted to 7.3 with KOH); in most experiments internal
solutions also included 1 mM QX-314 (Alomone Laboratories)
to block voltage-gated channels (particularly sodium channels)
(Strichartz, 1973) and 0.5 mM ZD 7288 (Tocris) to block the
hyperpolarization-activated cation current (Harris and Constanti,
1995). NMDA and GABAA receptors were blocked using 30 µM D-APV, 50 µM picrotoxin, and
50 µM bicuculline methiodide, and CaCl2 and
MgCl2 were increased to 3 mM to reduce polysynaptic activity. Membrane potentials were not corrected for the
liquid junction potential. Currents were filtered at a bandwidth of 2 kHz (3 dB) using an eight-pole low-pass Bessel filter and sampled at
20 kHz using pCLAMP software (Axon Instruments). Series resistance
(3-20 M; overall mean, 9.8 ± 1.2 M) was monitored continuously and compensated by 85-90%. All experiments were
performed at 35 ± 1°C.
Excitatory synaptic currents were evoked by a stimulation pipette
filled with extracellular solution located 100-300 µm from the soma
of the neuron being recorded from, usually near its primary apical
dendrite. Care was taken to select inputs without detectable polysynaptic contributions and with minimal "jitter" in the timing of individual currents. The peak amplitude of the EPSCs was typically 10-15 times that of spontaneously occurring EPSCs. Voltage jumps from
70 to 90 mV were alternated with voltage jumps combined with
synaptic stimulation. Jumps at different times relative to the onset of
the conductance were randomized and interleaved to mitigate the effects
of systematic changes in the experimental conditions over time (e.g.,
synaptic "rundown" or increases in series resistance). The
stimulation rate was 0.25-0.33 Hz.
Residual synaptic currents were obtained by subtracting the response to
voltage jumps applied without synaptic stimulation from the response to
jumps with stimulation. From 10 to 42 individual subtracted currents
were averaged for each time point on the charge recovery curve (see
Results). Synaptic charge was measured over an interval of 20-50 msec
after the onset of the synaptic current. Sweeps that contained large
spontaneous events were excluded from analysis. Charge recovery curves
with the lowest noise levels were selected for analysis. Noise levels
were quantified by dividing the SD of the fit residuals of the charge
recovery curve by the difference between the maximum and minimum values
of the fit curve; only complete charge recovery curves for which the
value of this "noise index" was  0.11 were accepted
(n = 8 of 18 experiments). Statistical errors
attributable to synaptic and instrumental noise were estimated by Monte
Carlo simulation of synthetic charge recovery curves (Press et al.,
1992). Gaussian noise (same noise index as the experimental charge
recovery curves) was added to the charge recovery with mean
experimental parameters. The resulting simulated charge recovery curves
were fit by the same procedure as the experimental charge recovery
curves. All values are given as mean ± SEM.
RESULTS
Attenuation and filtering of synaptic currents under poor
space-clamp conditions
The nature of the problem faced when attempting to voltage
clamp dendritic synaptic currents via a somatic electrode is
illustrated in Figure 1B using the simple equivalent
cylinder model shown in Figure 1A. There are two
closely related components of inadequate space clamp that must be
considered: attenuation of the signal along the cable, and the
reduction in driving force at the synapse caused by local
depolarization or hyperpolarization (also known as "voltage
escape"). The outcome of these two effects is that the current
recorded at the soma from synapses located on the dendrites is a
substantially filtered version of the synaptic current expected under
perfect clamp conditions, with the rise time, peak, and decay being
subject to considerable distortion, dependent on the electrotonic
distance of the synapse from the soma and the kinetics of the
conductance. These features have been described in detail previously
(Johnston and Brown, 1983; Rall and Segev, 1985;
Major, 1993; Major et al., 1993; Spruston et al., 1993), but there are
several aspects of particular relevance to the method that deserve
special emphasis. First, the current flowing at the synapse during
somatic voltage clamp is not identical to the current that would be
flowing during perfect clamp of the synapse. This difference is
attributable to the voltage escape at the synapse, which reduces the
driving force of the synaptic current and distorts its shape. Second,
for a given location and peak conductance the voltage escape, and thus
the distortion of the synaptic current, is greatest for the synaptic
conductances with the slowest kinetics, because they continue to charge
the membrane capacitance for a longer period. The magnitude of this effect on the current recorded at the soma will be mitigated by the
fact that slow conductances suffer less attenuation by the cable,
because attenuation is frequency-dependent in a passive system (Rall,
1967; Jack et al., 1983; Spruston et al., 1994). Third, while the
kinetics and the peak of the synaptic current suffer the most
distortion, the attenuation of synaptic charge is much less severe.
Furthermore, the attenuation of charge at a given location is
relatively independent of the kinetics of the current; in these
simulations, there was <10% difference in the recovered charge for
conductances with different kinetics even for the most distal synapses.
This residual difference is attributable to the greater voltage escape
caused by slower conductances: when the voltage escape converges toward
zero, the attenuation of synaptic charge becomes independent of the
kinetics of the synaptic conductance (Rall and Segev, 1985; Major et
al., 1993).
The voltage jump method described in this paper circumvents the
filtering of the synaptic current by the cable and provides a reliable
estimate of the synaptic conductance time course for even the most
electrotonically distal synapses. The method is particularly concerned
with (and is most effective for) fast synaptic conductances, which
suffer the most severe distortions under conditions of inadequate space
clamp.
Measuring charge recovery
The experimental procedure for recovering synaptic charge,
following the method introduced by Pearce (1993), is demonstrated using
a simple equivalent cylinder simulation in Figure
2. According to this procedure the
somatic voltage is held at the apparent synaptic reversal potential,
and a hyperpolarizing voltage jump is made, providing a driving force
to generate synaptic current. The voltage jump is repeated in the
presence and absence of synaptic activation, and the resulting somatic
currents are subtracted, thus eliminating the capacitive transient that
accompanies the voltage jump. This procedure gives a residual synaptic
current with a time course and amplitude that depend on the relative
time of the jump and the onset of the synaptic conductance (see Fig. 3A). If the jump occurs
sufficiently long before the onset of the conductance, then the
residual current will approach identity with the synaptic current
recorded at that potential under steady-state conditions. On the other
hand, if the jump occurs a sufficiently long time after the onset of
the synaptic conductance, then it will eventually recover no current at
all, because the synaptic conductance will have terminated. The current
resulting from each jump therefore results from an interaction between
the time course of the increase in driving force at the synapse and the
kinetics of the conductance itself.
Fig. 2.
Experimental protocol for measuring charge
recovery. Same equivalent cylinder as in Figure 1; synapse at
X = 0.15; peak conductance, 1 nS; rise and decay
time constants, 0.2 and 3.0 msec, respectively. Top,
20 mV voltage jump applied at the soma via the somatic electrode. The
somatic holding potential is set to 4.10 mV, making the voltage at the
synapse equal to the reversal potential (0 mV). The somatic voltage-clamp command is shown in the top trace; the
voltage at the synapse is shown in the middle trace; and
the (truncated) somatic clamp current is shown in the bottom
trace. Middle, The synaptic conductance is
activated 1 msec before the same voltage jump. The time course of the
synaptic conductance is shown by the dashed line, with
the amplitude equal to that of the perfectly clamped synaptic current.
The somatic clamp current in the presence (solid line)
and absence (dotted line) of the synaptic conductance is
shown. Bottom, Residual synaptic current (thick
trace) after subtraction of somatic clamp current under the two
conditions. The synaptic current expected under perfect voltage clamp
at a constant holding potential of 20 mV is superimposed as a
dashed line.
[View Larger Version of this Image (11K GIF file)]
Fig. 3.
Charge recovery depends on the time of the voltage
jump. Same conditions as in Figure 2. A, 20 superimposed
sweeps of somatic voltage jumps (Vcom,
top traces) at different times relative to the onset of
the synaptic conductance. The interval between jump traces is 1 msec;
the earliest jump is 7 msec before the onset of the synaptic
conductance, and the latest is 12 msec after onset of the conductance.
Also shown are the voltage at the synapse (Vsyn), the time course of the synaptic
conductance (gsyn), and the
"recovered" somatic currents (Isoma)
obtained by subtracting the somatic clamp current in the presence and
absence of the synaptic current for each jump. B, Plot
of the charge associated with the recovered somatic synaptic currents
(Isoma) versus time of the somatic voltage jump;
0.5 msec jump intervals.
[View Larger Version of this Image (16K GIF file)]
The synaptic charge associated with each residual current is plotted
against the time of the respective jump in Figure 3B. The
resulting "charge recovery curve" has a sigmoidal shape consisting of an exponential "onset" and "offset" with a transition at
~0 msec, i.e., at the beginning of the synaptic conductance. The determinants of the two components of the curve will be examined in the
following section.
Charge recovery after the onset of the synaptic conductance is
determined by the conductance time course
Figure 4 shows several charge
recovery curves from a synapse at the same location as in Figures 2 and
3 with a range of kinetics for the synaptic conductance. It is clear
from Figure 4 that the portion of the charge recovery curve that
follows the onset of the synaptic conductance is determined by the
decay time constant of the synaptic conductance; when the decay of the
conductance is effectively instantaneous, as with the delta pulse, then
no charge is recovered after t = 0 msec. For the more
realistic synaptic conductances in Figure 4B-D, the
decay of the charge recovery closely matches the actual decay time
course of the synaptic conductance. This finding holds for the
condition rise  decay of the
conductance, as is true for most synaptic conductances found to date.
Generally, it was found that for a monoexponentially decaying synaptic
conductance, the later the start time of the fit, the better the
correspondence between the fit decay and actual decay, because starting
the fit at later times helps avoid potential distortions attributable to voltage escape (see below). Of course, when the synaptic conductance time course is unknown it may be an oversimplification to assume that
it has a single exponential decay (e.g., see Pearce, 1993).
Fig. 4.
Charge recovery after the onset of the synaptic
conductance is determined by the synaptic decay. A-D,
Charge recovery plots for synaptic conductances with different
kinetics: a delta pulse (A) or a
double-exponential function with the same rising exponential (0.2 msec)
and different decay time constants (1, 3, and 10 msec in
B-D, respectively). Peak conductance 1 nS in each case;
all synapses were located at X = 0.15 using the
same equivalent cylinder as in Figure 1. A single-exponential decay has
been fit to the decay of the charge recovery in B-D;
note the close correspondence with the decay time constant
(dec) of the original synaptic conductance in each case.
E, Each charge recovery curve has been normalized by its
value at the onset of the synaptic conductance and superimposed. The
individual points of each curve have been
joined by a line for clarity.
[View Larger Version of this Image (19K GIF file)]
Charge recovery before the onset of the synaptic conductance is
determined by the electrotonic distance of the synapse
Figure 5 demonstrates that the early
component of the charge recovery, before the onset of the synaptic
conductance, reflects the time course of the voltage change at the
synapse produced by the somatic voltage command. This was shown by
placing a delta pulse synaptic conductance at various distances from
the recording site, thereby eliminating the influence of synaptic
kinetics on the charge recovery. Under these conditions, the charge
recovery curve for a synapse located at the soma was essentially a step function, whereas the curve for more distal synapses became
progressively more rounded. The same was true for the voltage response
to a somatic voltage jump at different distances. The symmetry between the time course of the two curves is demonstrated by overlaying the
scaled voltage response on top of the charge recovery, as shown in
Figure 5D.
Fig. 5.
Charge recovery before the onset of the
synaptic conductance reflects the voltage change at the synapse caused
by the somatic voltage command. All simulations are from the same
equivalent cylinder as in Figure 1. A-C, Left panels,
Synaptic voltage (Vsyn) in response to a somatic
voltage-clamp step (of arbitrary amplitude) at three different
locations; right panels, charge recovery curves for a
synaptic delta pulse (1 nS peak conductance) at the same three
locations. D, Superimposition of the synaptic voltage
responses on the respective charge recoveries; both the charge
recoveries and the voltage responses have been normalized by their
respective maxima, and the time axis of the voltage response has been
inverted. Note the exact correspondence of the voltage time course and
the charge recovery in each case.
[View Larger Version of this Image (16K GIF file)]
A simple analytical function describes the charge
recovery curve
In a linear system, the voltage response at the synapse to a
somatic voltage step can always be described by a sum of exponentials (Rall, 1969; Major et al., 1993) [we follow the convention of Major et
al. (1993) in setting resting membrane potential and the reversal
potential of the synaptic conductance to zero]. This sum is often
dominated by a single exponential, with time constant v
(see Fig. 3A):
|
(1)
|
where  is the steady-state attenuation factor of the voltage
command, Vcom, at the soma,  is the
Heaviside step function:
and s is the time of the voltage step with respect to
the onset (t = 0) of the synaptic conductance,
g(t).
For simplicity, we first choose a  function synaptic
conductance:
|
(2)
|
The resulting current flowing at the synapse (neglecting voltage
escape):
|
(3)
|
can be integrated over time to give the synaptic charge:
|
(4)
|
which of course depends on the time of jump, s (see
Fig. 3B). The charge recovered at the somatic voltage clamp
electrode:
|
(5)
|
is a constant fraction of the total synaptic charge (Redman,
1973; Rinzel and Rall, 1974; Carnevale and Johnston, 1982; Jack et al.,
1983; Rall and Segev, 1985; Major et al., 1993).
The assumption of a function synaptic conductance is unrealistic,
and therefore we repeat the calculation in Equation 4 with a synaptic
conductance that rises instantaneously to a peak at t = 0 and then decays exponentially with time constant
dec:
|
(6)
|
which yields a recovered charge:
|
(7)
|
that changes exponentially with a single time constant equal to
v for voltage jumps occurring before the onset of the
synaptic conductance and a single time constant equal to
dec afterward (compare Figs. 5 and 4). The ratio of the
amplitudes of the onset and offset phases of the charge recovery is
equal to v/dec. Because
integration is a linear operation, the integral in Equation 4 can still
be evaluated if both the voltage response at the synapse and the
synaptic conductance are described by sums of exponentials. The time
constants of the charge recovery for s 0 are given by the time constants of the voltage response, and the time constants of the charge recovery for s > 0 are given by the time
constants of the synaptic conductance. We illustrate this for the case
that the voltage response at the synapse is a sum of two
exponentials:
|
(8)
|
and the synaptic conductance is represented by three exponentials
(one for the rise and two for the decay):
|
(9)
|
In this case the recovered charge is:
|
(10)
|
To allow well conditioned fits of charge recovery data, the
amplitudes av1 and
av2 in Equation 10 were normalized
according to av1 + av2 = 1. The factors
2, Vcom,
 1 and 2
were combined in two overall amplitudes of the fit function,
g1* = 2Vcom1
and g2* = 2Vcom2,
which were free parameters of the fit. Constant offsets in s
and Qsoma(s) can also be introduced
to allow latency variations and jumps from other potentials than the
apparent reversal potential of the synaptic conductance.
In practice it may not always be necessary (or possible) to fit the
entire analytical function. As demonstrated above, the charge recovery
can be separated into two components, with the second determined by the
kinetics of the conductance (see Fig. 4 and Eqs. 7 and 10). This can be
exploited experimentally in situations in which the time of recording
is limited or in which only the decay of the synaptic conductance is of
interest. By making a series of jumps at different times after the
onset of the synaptic conductance and then fitting the decay of the
recovered charge with an exponential function, an estimate can be made
of the decay of the conductance (assuming that rise
decay). It is also possible to fit multiple
exponential functions to the decay; in this case, the time constants
will be extracted faithfully, but the relative amplitudes of the faster
components will be underestimated. This "shortcut" could in
principle allow the voltage jump method to be applied to spontaneous
synaptic currents, by triggering voltage jumps (using a software or
hardware trigger) with a variable delay after the synaptic current
crosses a threshold amplitude. Some jitter will be introduced in the
time of the jump if the spontaneous currents have widely different
amplitudes and/or rise times; this can be corrected for by later
normalizing the time of each jump to a reference point on the rise. As
with evoked synaptic conductances, the mean decay time course of the
underlying conductances can then be estimated from the charge recovery
curve.
The voltage jump method also works in current-clamp mode
In principle, a change in driving force at the synapse can be
generated either with a voltage command under voltage clamp or
by injecting a fixed amount of current to generate a
reproducible voltage change in current-clamp mode. Because the
analytical solutions for both the "time integral" of synaptic
potentials and the synaptic charge in voltage clamp depend only on the
charge flowing at the synapse (Major et al., 1993), one can fit the
curve of the time integral of the synaptic potentials obtained after a
series of identical square current pulses with Equation 7 or 10. In
this case, the measured v will be determined by the
membrane time constant m, because
m determines the dendritic voltage response to a square
current step (neglecting the faster equalization time constants, which
generally have much smaller amplitudes for a long current step).
Although the kinetics of the synaptic conductance can be extracted
reliably as described above, because m  decay for most neurons and synaptic conductances, the
amplitude of the time integral curve will be dominated by the component
attributable to m (the onset). Therefore, for
determining the time course of the synaptic conductance it is always
preferable to use voltage clamp rather than current clamp, because
v for voltage clamp will always be smaller than
m [except in the limiting case, in which they are
identical (Major et al., 1993)] and thus will provide better
signal-to-noise ratios for extracting rise and
decay. Voltage clamp will also reduce the voltage
excursion at the synapse (although only slightly for some synapses) and
thus also distortion in the synaptic current. For these reasons all
subsequent simulations as well as the experiments were done in
voltage-clamp mode.
Effect of voltage escape at the synapse
The analytical function derived above assumes that the voltage
escape associated with the synaptic current at the synaptic site is
negligible. Because some voltage escape will inevitably be associated
with somatic voltage clamp of dendritic synapses, it is therefore
necessary to test how voltage escape affects the accuracy of the
method. This was done using the equivalent cylinder model by
progressively increasing the magnitude of the peak synaptic conductance
at a given location. The results of such simulations are shown in
Figure 6. As the synaptic conductance is
increased, the voltage escape at the synapse progressively approaches
the synaptic reversal potential, causing substantial distortions both in the current flowing at the synapse as well as in the current recorded at the soma. The charge recovery curves obtained from the same
synapses show a progressive distortion and slowing after t = 0. When comparing the decay time constant fit to
the charge recovery curve with the actual time course of decay of the
conductance (Fig. 6F), serious errors (>10%) were
found only for the largest conductances ( 20 nS). These errors could
be reduced further by changing the fit range; fits with a later onset
produced greater accuracy (although, as pointed out above, this is not
feasible for conductances that may contain a slow component). By
contrast, the time constants fit to the decay of the current measured
at the soma were seriously in error for all conductance values chosen; delaying the onset of the fit produced little improvement in
accuracy.
Fig. 6.
Effects of local depolarization (voltage escape)
on the reliability of the charge recovery method. A-C,
Simulations from the same equivalent cylinder as in Figure 1, with the
synapse at a constant location (X = 0.15) and with
a range of peak synaptic conductances as indicated (rising and decaying
time constants, 0.2 and 3.0 msec respectively). A,
Voltage at the synapse (Vsyn); B,
current flowing at the synapse (Isyn);
C, Current recorded at the soma
(Isoma). The charge recovery plots from the
various conductances are shown unscaled in D and scaled
by the peak charge in E. The graph in
F compares the decay time constant obtained by fitting
either the somatic current or the charge recovery curve (fit; fit beginning 7 msec after onset of the conductance in each case) with the actual decay time constant of the synaptic conductance (syn). Note that the time constant estimated by
the charge recovery is relatively faithful to the actual synaptic decay
time constant except at very high values of peak conductance.
[View Larger Version of this Image (19K GIF file)]
These findings suggest that the voltage jump method can reliably
extract the decay time course of the synaptic conductance over a wide
range of magnitudes of the conductance, but that the substantial
voltage escape associated with very large, highly localized synaptic
conductances may reduce its accuracy. The amplitude of the voltage
escape will depend not only on the magnitude of the conductance but
also on the geometry of the cell as well as its electrical properties.
To test the method rigorously, it is therefore of great importance to
carry out simulations in compartmental models of real neurons, with
realistic values for the membrane parameters and the synaptic
conductance.
Application to pyramidal cell geometries
CA3 pyramidal cell
Figure 7 shows a test of the voltage
jump method in a detailed compartmental model of a CA3 pyramidal cell
(Major et al., 1994). As shown previously (Major et al., 1994), a
synaptic input placed on the distal apical dendrites is substantially
filtered and attenuated by space-clamp errors (Fig. 7B). The
voltage jump protocol was performed at a holding potential of 65 mV.
Because the analytical function assumes that the system is passive, it should not matter from which holding potential the jumps are made or
which voltage is jumped to, as long as there is a change in synaptic
driving force; the charge recovery curve is simply shifted downward on
the y-axis by the difference in synaptic charge at the two
holding potentials. By fitting the charge recovery with Equation 10, it
was possible to extract the decay of the synaptic conductance with high
accuracy (<5% error; for details, see legend to Fig. 7). To determine
the effect of high membrane conductance on the accuracy of the method,
Rm was decreased from 180,000 to 20,000 cm2 (which reduced the input resistance from 305 to 43.4 M). Under these conditions, as might be expected to occur
in vivo because of tonic synaptic bombardment, the method
extracted the decay time course of the conductance to within 2% error
(data not shown). The method also maintained high accuracy under
conditions of high series resistance (20 M; Fig. 7E-G).
Note that in these simulations, the time course of the initial phase of
the charge recovery (and the v values extracted by
fitting the analytical function) were much slower than with low series
resistance, consistent with the greater effective electrotonic distance
of the synapse in the high series resistance condition.
Fig. 7.
Simulations of the voltage jump method in a CA3
pyramidal cell model. A, Morphology of the CA3 pyramidal
cell with which the simulations were performed showing the location of
the simulated synapse, which was placed on a spine head
(filled circle). B-G, A synaptic
conductance (peak, 0.5 nS) consisting of a double-exponential function
(rise = 0.2 msec; dec = 2.5 msec) was
used; the conditions in B-D and E-G are
identical, except that the series resistance of the somatic pipette was
0.5 M in B-D and 20 M in E-G.
B, E, Somatic clamp current resulting from activation of
the synaptic conductance (thick trace) as well as the
synaptic current expected under conditions of perfect space clamp. The
20-80% rise times of the currents were 1.00 msec in B
and 1.71 msec in E. The decay time course of the somatic
clamp current could be fit with a single exponential function with = 6.44 msec in B and 12.90 msec in E.
C, F, Currents recovered by a series of 20 mV voltage
jumps from 65 mV (1 msec interval between jumps). D,
G, Charge recovery curves measured from the traces in
C and F together with the best fit of the
analytical function (Eq. 10). Note the different onset of the two
curves. For the low series resistance condition the best fit was with
the following parameters: v1 = 1.58 msec (40%); v2 = 8.53 msec (60%); rise = 0.22 msec; and dec = 2.55 msec (here and wherever
appropriate, Eq. 10 was modified such that dec1 = dec2 = dec). For the high
series resistance condition the best fit was with
v1 = 2.15 msec (5%); v2 = 12.75 msec (95%); rise = 0.19 msec; and
dec = 2.56 msec. A single-exponential fit to the decay
of the charge recovery curve gave dec = 2.54 msec in
both cases. H-J, An NMDA receptor-mediated synaptic
conductance was simulated at the same location (peak, 0.1 nS;
rise = 5.0 msec; dec = 40 msec) with 0.5 M series resistance, assuming zero external Mg2+.
H compares the perfectly clamped synaptic current with
the measured somatic current. The 20-80% rise time of the somatic
current in H was 7.17 msec, and current was fit with a
double-exponential function with rise = 9.6 msec and
dec = 39.8 msec. The currents recovered by voltage jumps
from 65 to 85 mV are shown in I, and the respective
charge recovery curve is shown in J. The values of the
best fit of the analytical function were v1 = 1.45 msec (41%); v2 = 8.58 msec (59%);
rise = 5.15 msec; and dec = 40.4 msec. A
single-exponential fit to the decay of the charge recovery curve gave
dec = 41.2 msec.
[View Larger Version of this Image (21K GIF file)]
To test whether it is also possible to extract accurately the rise time
of a slow synaptic conductance, an NMDA receptor-mediated EPSC (Kirson
and Yaari, 1996) was simulated at the same synaptic location in Figure
7H-J. Although the decay of this synaptic current was not
significantly distorted because of its slow time course, the rise time
was slowed substantially (from rise = 5.0 to 9.6 msec).
The analytical function was able to extract the rise time (as well as
the decay) to within 3% of its original value, indicating that the
method may also be useful for this purpose.
To examine the effectiveness of the method for distributed conductances
in the CA3 pyramidal cell, an inhibitory connection was simulated (Fig.
8), with the location of the contacts
based on a reconstructed connection between an interneuron and a
simultaneously recorded CA3 pyramidal cell (Miles et al., 1996, their
Fig. 2). Either single- or double-exponentially decaying conductances
were simulated at each contact (Pearce, 1993). When the decay of the synaptic conductance was double-exponential, the fast component of the
decay was filtered more heavily than the slow component, such
that the synaptic current measured at the soma could be fit with a
single exponential with a intermediate to the two time constants of
the conductance decay. Because the synapses in this simulation were at
widely distributed electrotonic locations, when applying a somatic
voltage jump each synapse experienced voltage transients with a
different time course. This caused slight distortions of the rise time
extracted with the analytical function. The decay appeared to be
relatively little affected by this nonuniformity, as with both the
single- and double-exponentially decaying conductances, it was possible
to extract the time constants and their relative amplitudes to a high
degree of accuracy (<5% error). To test the effect of the synaptic
conductance kinetics on the accuracy of the method, we also performed
simulations under the same conditions with a conductance decay time
constant of 1 msec. The decay time constant extracted by the method was
1.01 msec (data not shown), confirming that high accuracy could be
maintained even with rapid input kinetics.
Neocortical pyramidal cell
The most stringent tests of the method were performed using a
detailed compartmental model of a layer 5 pyramidal cell (Markram et
al., 1997), a cell type that has one of the most extensive dendritic
trees of any neuron in the brain. A morphologically reconstructed
unitary input made by an adjacent, simultaneously recorded layer 5 pyramidal cell was simulated (Markram et al., 1997), which made eight
contacts at widely dispersed electrotonic locations (mean
X = 0.71; range, 0.063-1.4). When this distributed input was activated, the analytical function extracted the decay time
constant of the synaptic conductance to within 5% error, despite
substantial filtering of the synaptic current waveform (Fig.
9B-D). Errors remained small
(<5%) when the magnitude of the conductance at each contact was
quadrupled to 4 nS, when the decay time constant of the synaptic
conductance was reduced to 1 msec, and when the series resistance was
increased to 5 M (not shown).
Fig. 9.
Simulation of a distributed synaptic connection
in an active layer 5 pyramidal cell model. A reconstructed synaptic
connection made by a single presynaptic layer 5 pyramidal neuron is
simulated, with 8 contacts (marked by dots in
A) distributed on apical and basal dendritic spines
(Markram et al., 1997). All synaptic conductances are identical
(rise = 0.20 msec; dec = 2 msec). The
model either was passive (B-D) or contained
active conductances (E-J), as described in
Materials and Methods. The peak synaptic conductance at each contact
was either 1 or 4 nS; the kinetics of the currents and charge
recoveries obtained from the 1 and 4 nS passive simulations was nearly
identical, and therefore only the results from the 1 nS simulation are
shown. B, E, and H compare the somatic
clamp current at a holding potential of 65 mV with the perfectly
clamped EPSC for the passive and active model. Insets in
E and H compare the clamp current in the
active model with that of the corresponding simulation in the passive
model (same period as in the main panels; scale bars apply to the
larger traces). Note that in the simulations with 1 nS peak
conductance, the active and passive models produce a virtually
identical EPSC, whereas in the 4 nS simulation the EPSC in the active
model clearly shows an additional current component in the tail of the
EPSC. The 20-80% rise times of the somatic EPSCs were 0.36 msec in
each case. The decay of the somatic EPSCs could be fit by a single
exponential with = 3.3 msec in the passive simulations as well as
in the active 1 nS simulation, and with = 3.7 msec in the 4 nS
active simulation. C, F, I, Recovered currents from
successive 20 mV hyperpolarizing voltage jumps from a holding
potential of 65 mV. D, G, J, Respective charge
recovery curves measured from the recovered currents. In D and G, the curves have been fit with
the analytical function. The best fit in the passive model gave
v1 = 0.36 msec (69%); v2 = 11.3 msec (31%); rise = 0.22 msec; and
dec = 2.02 msec; whereas in the active model the values
were v1 = 0.23 msec (69%); v2 = 11.6 msec (31%); rise = 0.34 msec; and
dec = 1.90 msec. A single-exponential fit to the decay
of the charge recovery gave dec = 2.00 msec in both
cases. Because of the distortion of the charge recovery in the 4 nS
active simulation, a fit of the analytical function was not possible.
However, the decay phase of the charge recovery was fit with a single
exponential of 1.90 msec.
[View Larger Version of this Image (22K GIF file)]
To investigate the influence of active conductances on the method, the
simulations were repeated incorporating an active membrane model of
neocortical layer 5 pyramidal cells containing a variety of
voltage-gated conductances, which reproduces the firing pattern of
these neurons (Mainen and Sejnowski, 1996). Simulations with the active
model at 1 nS peak conductance per contact produced results that were
very similar to those found with the passive model, consistent with the
lack of distortion in the synaptic current (Fig. 9E, inset).
When the peak synaptic conductance was increased to 4 nS/contact,
however, an obvious "boosting" component could be observed in the
decay of the synaptic current (Fig. 9H, inset). The boosting
current arose almost exclusively via activation of sodium and calcium
conductances in the apical tuft branches (not shown); virtually no
boosting was observed at the peak of the synaptic current, primarily
because the measured peak is dominated by current from basal inputs,
which are better clamped.
The extra charge contributed by the active conductances caused clear
distortions in the charge recovery curve, with an extra component
emerging in the onset of the charge recovery, representing jumps made
just before the beginning of the synaptic conductance. The shape of
this extra component results from a highly nonlinear process involving
the increase in the driving force caused by the hyperpolarization,
which is still weak enough at "late" times to permit activation of
voltage-gated channels. Despite this distortion, the charge recovery
after t = 0 msec remained dominated by the decay of the
synaptic conductance; when a single exponential was fit to this
component, the decay was estimated to within 10%. Similar results were
obtained when the decay time constant was reduced to 1 msec (not
shown). In this model, therefore, the errors caused by active
conductances depend on various factors, particularly the size of the
synaptic conductance (Fig. 9E,H) and the holding potential. These findings demonstrate that care must be taken to choose
the appropriate voltage range over which to carry out the voltage
jumps, and that tests must be done to evaluate the possible
contribution of voltage-gated conductances.
Experimental application of the voltage jump method to
neocortical pyramidal cells
The voltage jump method was used to determine
experimentally the time course of excitatory synaptic conductances in
layer 5 neocortical pyramidal cells. We evoked EPSCs resulting from the
activation of only one or a few presynaptic fibers (peak amplitude, 546 ± 50 pA at 70 mV; n = 25). The evoked EPSCs
had an average 20-80% rise time of 0.89 ± 0.03 (range,
0.55-1.20) msec at 70 mV, and their decay could be fit well using a
single exponential function with a time constant of 3.83 ± 0.24 (range, 2.1-6.3) msec. The linearity of the membrane between 70 and
90 mV was examined by recording the membrane currents in response to
a series of depolarizing and hyperpolarizing voltage jumps of different amplitudes starting from a holding potential of 80 mV. When scaled by
the jump amplitude, these currents superimposed well for jumps of
different amplitude (see Fig.
10B). To check for
distortions in the EPSC caused by activation of voltage-gated
conductances, the time course of the EPSC was compared at 70 and 90
mV. At 90 mV, the 20-80% rise time was 0.86 ± 0.04 msec
(p = 0.07, paired t test), and the
decay time constant was 3.74 ± 0.22 msec
(p = 0.06; see Fig. 10C,D). To
confirm that activation of voltage-gated conductances did not affect
the synaptic current and to assess possible distortions caused by
voltage escape, the synaptic conductance was reduced by application of
a submaximal concentration (40 µM) of the noncompetitive
AMPA receptor antagonist GYKI 52466 (Paternain et al., 1995). While the
peak amplitude of the EPSC at 70 mV was reduced to 24 ± 3%
(n = 3) compared with control, the 20-80% rise time
and decay time constant of the EPSC were 106 ± 2%
(p = 0.08) and 108 ± 9%
(p = 0.4) of the control values, respectively. These findings indicate that the EPSCs were not substantially distorted
by voltage escape or by the activation of voltage-gated conductances.
Fig. 10.
Determining the time course of excitatory
synaptic conductances in neocortical pyramidal neurons using the
voltage jump method. All traces taken from a somatic whole-cell
recording of a layer 5 neocortical pyramidal neuron at 35°C; the
internal solution contained 1 mM QX-314 and 0.5 mM ZD 7288. A, The neuron was held at 80
mV, and a series of voltage jumps (from 95 to 65 mV in 5 mV steps)
was given to test for membrane linearity, bracketing the voltage range
used for determining the charge recovery. The resulting currents are
shown below the voltage commands (average of 5 traces
each; the series resistance of 6.0 M was compensated by 90%).
B, The currents were scaled by the command voltage and superimposed to demonstrate linearity. An EPSC was evoked by
stimulation of afferent fibers near the apical dendrite and is shown at
two different holding potentials in C (averages of 25 traces). D, The traces in C have been
scaled by their peak amplitudes and superimposed. The 20-80% rise
times of the currents were 1.15 and 1.13 msec |
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