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A Brief History of Time (Constants)
That the cerebral cortex processes information at prodigious speeds
cannot be doubted. Yet the passive time constant •*„ of neurons, often
thought of as a measure of the neuron's "response time" to synaptic
input is relatively long. In the 1950s, T O was estimated to be only a
few milliseconds for mammalian central neurons; with improvement
in recording techniques, its estimated value grew over the years and
it now stands near 20-100 msec. However, as we will argue here, the
functional meaning of tm is ambiguous. On the basis of a newly introduced definition of local delay, we show that the time window for
synaptic integration in passive dendritic trees can be much smaller
than the time constant We argue that the voltage response to very
brief synaptic inputs is essentially independent of •:„ We discuss how
ta can change dynamically with the global activity of the network, as
well as the difficulties of defining a time constant in structures with
voltage-dependent elements. We conclude that the classically defined
Ta only provides a very rough estimate, typically an overestimate, of
the response time of neurons and that alternative measures are required to capture the dependency of the time course of the membrane
potential on ligand-gated and/or voltage-dependent membrane conductances.
In the following pages, we will review what we know about
time constants in neurons. Why, it can be asked, should this
topic be of any great relevance to readers of Cerebral Cortex?
First, and most important, time constants help us to understand the dynamics of the membrane potential Vm(t) in nerve
cells. Time constants should help us to answer such questions
as how fast VJf) can rise and fall in response to synaptic
input or current injections, what is the time window for integration of two or more synaptic inputs, or what influence
the dendritic geometry has on the dynamics of the membrane
potential.
To give a concrete example (from T. Poggio, personal communication): in the dark, few cells are active in visual cortex
and the membrane time constant im is long, enabling the cell
to integrate the input over a long time. Conversely, for bright,
high-contrast visual stimuli, the overall network activity may
be much higher, causing many more synaptic inputs to impinge spontaneously onto the cell, transiently increasing the
local membrane conductance. As shown by Bernander et al.
(1991) and Rapp et al. (1992), this additional conductance can
lower the membrane time constant by an order of magnitude,
reducing the integration times for these bright stimuli. This
sort of adaptive gain-control strategy is somewhat akin to that
used by the network of coupled rods in the retina (Detwiler
et al., 1978).
We will discuss the theoretical status of various definitions
of time constants and their experimentally measured values.
Second, time constants should tell us something about the
computational abilities of the nervous system. It is clear that
how fast neurons can respond to synaptic input will significantly influence how fast the entire organism can respond to
some stimulus in the world. And how do such time constants
affect the ability of the organism— and of individual nerve
cells— to discriminate very fine time differences? Although
Christof Koch,' Moshe Rapp,2 and Idan Segev2
'Computation and Neural Systems Program, California
Institute of Technology, Pasadena, California 91125 and
department of Neurobiology, Institute of Life Sciences and
Center for Neural Computation, Hebrew University,
Jerusalem 91904, Israel
we will introduce the reader to these last two issues, we do
not have space in these pages to fully develop the link between the dynamics of the membrane potential in neurons
and overall system performance.
Classical Definition of the Passive Time Constant
How does the membrane potential across a small and isopotential patch of passive neuronal membrane evolve over time?
If the membrane is characterized by a single membrane capacitance Cm (in units of jjiF/cm2) in series with a single voltage-independent membrane "leak" resistance Rm (in units of
Clan2}, Ohm's law tells us that the dynamics of the potential
Vm across this circuit in response to a current injection I(f)
changes as
with the time constant Tm = RmCm and the input resistance
Rm given by Rm times the surface area of the patch of membrane being considered. If a current step of amplitude /„ is
injected at time t = 0, the charge flows onto the capacitance
and the potential rises as
VJf) = V.d - <?-«*-).
(2)
This time course is governed by exponential decay toward
the steady-state Vm = R,J0, as illustrated in the lower trace of
Figure 1.
What happens in the simplest spatially extended structure,
an infinite cable of constant diameter d and constant membrane properties? Here, the dynamics of Vm is governed by
the well-known linear cable equation. We refer the reader to
the standard monograph by Jack et al. (1975) for the relevant
equation (3-24; first solved by Hodgkin and Rushton, 1946).
For a current step, the voltage response at the site of current
injection (upper trace in Fig. 1) is governed by
Vm(O = V. X erf(V«/O,
(3)
V» = RJ0; the input resistance in the infinite cable R^ =
(#mff()1/7(7rc*3/2); Rlt the specific intracellular resistivity in units
of Ocm; and erfOc), the error function. The dynamics of the
membrane potential in the cable is also shown in Figure 1.
To understand these curves it should be pointed out that
Vm(f) is plotted in normalized units (relative to its steady-state
value VL, = RJ^> in both cases. Equivalently, it can be assumed
that the input resistance for the membrane patch is identical
to the input resistance in the infinite cable. Under these conditions, the potential in the infinite cable will reach its steadystate value faster than the potential in the patch of membrane.
Indeed, in an infinite cable, Vm rises to 84% of its steady-state
value in one time constant compared to 63% in the isopotential case. The reason for this nonintuitive behavior is that, in
a cable, some of the injected current charges up neighboring
regions in addition to charging the local membrane (as in the
uniform case).
A different way to understand this is by considering the
Cerebral Cortex Mar/Apr 1996;6:95-101; 1047-32ll/96/$4.00
in a tree with uniform membrane properties and sealed ends
(and without a shunt or a voltage clamp), the slowest time
constant T0 is always equal to the membrane time constant
T0 = Tm = RmCm. Second, all time constants scale with the
specific membrane capacitance Cm. Third, introducing either
a shunt conductance at the soma—mimicking the effect of
an intracellular electrode— or a voltage clamp changes (typically reduces) all the time constants, including T0. In other
words, under realistic experimental conditions, the slowest
measured time constant represents a lower bound on the actual value of Tra. Fourth, the T,S have a particularly -pleasing
and simple interpretation for a single finite passive cable with
sealed-end boundary conditions that can be used to estimate
the electrotonic length of the cable (dendritic tree) being
recorded from (Rail, 1969).
The Experimental Determination of T .
Figure 1. The voltage dynamics in two simple model cases: the evolution of the membrane potential Wl in an infinite, passive cable of constant diameter at the site of
current injection (Eq. 3) and in a patch of passive membrane (Eq. 2) in response to a
current step at t = 0. Time is expressed in units of im and the membrane potential in
terms of the steady-state potential, K, We here assume that the input resistance /?„
of the membrane patch is the same as /?„ for the infinite cable. In the cable, the voltage
rises faster to its final value than in a patch of membrane. Indeed, in the infinite cable
it reaches 84% of the maximal value at f = Tm and only 63% of the maximum in the
case of an isopotential patch of membrane {dashed line).
input resistance. While we here assumed that the steady-state
input resistance Rta is the same in both model systems, the
effective input resistance at small times will be higher in the
cable (since the current only "sees" a small part of the cable
early on) than the resistance of the membrane patch, explaining why the same current causes a larger initial voltage rise
in the distributed system than in the isopotential one.
The same argument holds for an electrode injecting current into the soma of a neuron with an extended dendritic
tree. A significant portion of the injected current will flow
onto the extensive dendritic membrane surface, and as a result the buildup (and decay) of the somatic voltage is faster
compared to the isopotential patch of membrane. This effect
was first recognized by Rail (1957). Coombs et al. (1955) had
fitted the experimentally observed membrane transients at
the motoneuron soma with a single exponential with rm = 2
msec. Rail argued that the cable properties of the dendrites
needed to be incorporated and—on the basis of Equation 3—
estimated Tm = 4 msec (for a detailed exposition of this, see
. Segev et al., 1994).
What is the voltage response to current input in spatially
complex structures? Rail (1969) showed that the voltage in
an arbitrary passive cable structure (whether a single finite
cylinder or a large dendritic tree) •with uniform electrical parameters throughout the tree can always be expressed as an
infinite sum of exponential terms:
. . . , (4)
VJjx, f) = VJx)
where the Bfi depend both on the initial conditions and on
x, and Vw is the steady-state component of the voltage. The
T,S can be thought of as equalizing time constants that are
associated with the rapid flow of current (and reduction of
voltage differences) between different regions of the cable.
They are independent of the site of the current input, the site
of recording, or the initial voltage distribution in the tree. In
general, solving for these T,S involves extracting the root of a
recursively denned transcendental equation (for more recent
work on this, see Holmes et al., 1992; Major et al., 1993)Several important points need to be mentioned here. First,
94 A Brief History of Time (Constants) • Koch et al.
The appeal of determining more than just the first equalizing
time constant T0 = -rm lies in the fact that the T,S provide us
insight into certain aspects of the electroanatomy of the neuron, hi other words, a local measurement reveals something
about the global geometry. Direct measurements of im in neurons depended on the development of the intracellular electrode, introduced by Graham and Gerard (1946) for muscles
and first applied to central, mammalian neurons (spinal motoneurons) by Woodbury and Patton (1952) and Brock et al.
(1952). As discussed above, these early studies neglected the
cable properties of the dendrites and estimated Tm = 2 msec
(a further source of error was very noisy film records that
contaminated the "tail" where the time constant is "buried").
A common numerical technique for extracting time constants
involves the peeling of low-order exponents from a semilogarithmic plot of the voltage decay (Rail, 1969; see also Fig. 2;
for a history of all of this, see the commentaries in Segev et
al., 1994).
Peeling was first applied to the problem of extracting the
V i n a motoneurons by Burke and Ten Bruggengate (1971).
They estimated that Tm ranges from 5 to 7 msec. To remove
noise, they averaged 50 voltage transients, thereby obtaining
a relatively smooth tail. Indeed, the smoothed transients allowed them to "peel" another time constant, the first equalizing time constant, T, of about 1 msec. The ratio between
Tra and T, was then used in a formula derived by Rail (1969)
to estimate the electrotonic length (L, in units of \, the space
constant) of the motoneuron dendrites. After this work, peeling was applied to many vertebrate and invertebrate neurons.
With the development of improved intracellular electrodes,
Tm could be better estimated and larger values were obtained
(see below).
Experimentally, im is usually determined by injecting a
brief current pulse into the soma and recording the voltage
response at the same point. The slope of the tail of the decaying phase of Vm, when plotted on a semilogarithmic scale,
is — lA m . Typically, a short hyperpolarizing current pulse is
used, since long current steps tend to activate voltage-dependent components (e.g., the over- and undershoot investigated
in detail by Ito and Oshima, 1965) that interfere with the
measurement of the passive time constant. In a linear system,
the response to a current pulse is given by the temporal derivative of the response to a step. The derivative of Equation
4 leaves an infinite number of exponential terms with the
slowest term decaying as e~lfm. Thus, the "peeling" method is
also directly applicable with the advantage that, now, the
steady-state component of the voltage disappears. Figure 2
shows an example of the voltage decay following a brief hyperpolarizing current transient plotted on a semilogarithmic
scale for a cortical pyramidal cell. One can clearly observe
that the voltage differences over the cell surface equalizes
10
Time (msec)
20
Figure 2. Peeling the voltage transient in a neocortical pyramidal cell. The morphology of this cell, stained with Lucifer yellow, is shown in the inset The cell is from a slice
preparation of the frontal cortex of the guinea pig. A brief (0.6 msec, 2.7 nA) hyperpolarizing current pulse was injected via intracellular electrode at the soma. The decay of the
voltage response, plotted on a logarithmic scale, is shown as a function of time. Due to the electrically distributed (non-isopotential) nature of the dendrites, the voltage initially
decays faster than T , = tm and only later (i.e., toward the tail) does 144 reach the final (slowest) exponential decay, with T 0 = 16 msec. In this cell, it was also possible to "peel"
the next equalizing time constant T, = 1.7 msec. Experiments were conducted in normal, physiological Ringer solution. The resting potential was - 6 5 mV and the temperature was
32°C. The intracellular electrode was filled with 3 M K-acetate (courtesy of Y. Gutfreund; see details in Gutfreund et al., 1995).
Cerebral Cortex Mar/Apr 1996,
V 6 N 2 9 5
after about 5 msec; from this time on the voltage decays exponentially, as in the isopotential case. The slowest time constant T0 (—16 msec) corresponds to the inverse of the slope
at the tail of the voltage decay. Subtracting this component
from the original voltage curve gives rise to a new transient
Cnoisy curve at bottom) whose tail can now again be fitted
with a straight line whose slope is — 1/T,. In this case T, =
1.7 msec. If we assume that the membrane of this cell is
uniform and that the intracellular electrode did not cause a
significant shunt (injury), then T0 = Tm. Our estimate for Tm is
in agreement with Kim and Connors (1993), who report for
the larger, layer 5 pyramidal cells in slice from somatosensory
cortex tm = 16 ± 5.3 msec in the soma.
It is important to emphasize that the quality of the electrical recording depends to a large extent on the tightness of
the resistive seal around the electrode. During the impalement of the neuronal membrane, the intracellular electrode
can rip a large hole in the membrane, allowing ions to flow
through, thereby seriously compromising the quality of the
recording. The introduction of whole-cell recordings of in
vitro cells and the perforated patch clamp technique (Edwards et al., 1989; Spruston and Johnston, 1992) has dramatically improved the situation, with an effective electrode
shunt on the order of 0.1 nS. Indeed, over the last four decades the estimate for Tm in central neurons has grown significantly. As discussed above, in the 1950s it was assumed to
be only a few milliseconds. With improved averaging and recording techniques, the recent estimates of tm from intracellular sharp-electrode recordings range from 20 to 50 msec for
the major types of central neurons (see, e.g., for a motoneurons: Fleshman et al. 1988; Clements and Redman, 1989; for
three types of hippocampal neurons: Brown et al., 1981; for
vagal motoneurons: Nitzan et al., 1990; for cerebellar Purkinje
cells: Rapp et al., 1994). With tight-seal whole-cell recordings,
the estimates have grown even further and are approaching
100 msec in slice preparations (e.g., Andersen et al., 1990,
found time constants ranging from 50 to 140 msec in the
hippocampus; Major et al., 1994, report an average value of
93 msec in CA3 pyramidal cells; see also Spruston et al., 1994).
A notable exception to such high values is the 2 msec time
constant in slices in the avian cochlear nucleus using wholecell recording with tight seal (Reyes et al., 1994). We conclude
this section by noting that Tra estimates depend heavily on
the composition of the physiological solution during the experiment. Adding specific blockers to the solution as is frequently done, such as cesium ions to block K-dependent rectification or aminophosphonovalerate (APV) to block NMDAdependent channels, directly affects estimates of Tm (usually
increasing the estimates).
A New Functional Definition for T—The Local Delay
One problem with the standard definition of im (and the associated equalizing time constants T,) is that they assume the
convenient fiction of a steady-state input. Thus, they only tell
us how slowly the membrane can respond. However, synaptic
input can be much faster than rm, and we need a measure
that tells us how rapidly the membrane can respond to such
physiological input. Furthermore, it is unclear how the local
geometry, that is, whether the current injection occurs at a
soma or in a distal dendrite or spine, affects the dynamic of
voltage. These problems were addressed and solved (for passive membranes) by Agmon-Snir and Segev (1993; see also
Zador et al., 1995).
Agmon-Snir and Segev (1993) introduced a simple yet
powerful method to calculate delays associated with arbitrary
current and voltage signals in passive cable structures. They
base their method on the use of moments to characterize the
properties of the signal. In particular, the centroid, or "center
96 A Brief History of Time CConstants) • Koch et al.
of mass," of the signaiy(x,0 at location x is defined as the ratio
of the first to the zero-th moment of a transient signal,fix,ty.
L
f.
tf(x, f) dt
(5)
f(x, 0 dt
Here/can be either a current or a voltage of (practically) any
shape. The input or local delay, D^ is defined as the difference between the centroid of the input current and the centroid of the voltage response at the same location:
A* = K - t'x
(6)
The same approach can be used to define the transfer delay
as the difference between the centroid of the induced voltage
change at location y and the centroid of the current injected
at location x. However, we will not pursue this topic here.
Agmon-Snir and Segev (1993) prove that the local delay is
always positive and independent of the shape of the transient
input current; in other words, D^ is a property of the passive
structure and not of the input and provides a measure for the
time window for synaptic integration. It is straightforward to
show that for an isopotential neuron, D^ = im; the centroid
of an EPSP (or an IPSP) occurs exactly one time constant later
then the centroid for the current underlying this potential. In
this case, synaptic input arriving with temporal dispersion
smaller than Tm will affect each other (integrate); for larger
temporal dispersion the inputs do not interact. In an infinite
(or semi-infinite) cylinder, a portion of the injected charge
flows along the cable, and therefore the potential at the input
cite rises and decays faster, as we saw already in Figure 1.
Indeed, for an infinite cylinder, the input delay between any
current input and the corresponding local potential is only
half a time constant, namely, £)„ = T ^ 2 . In this case, the effective time window for synaptic interaction is smaller than
The method of moments can be applied to calculate analytically the local delay at any point in an arbitrary dendritic
tree. Figure 3 shows the local delays computed for a detailed
compartmental model of a layer 5 pyramidal cell reconstructed during in vivo experiments in the primary visual cortex
of an adult cat (Douglas et al., 1991). Each of the four panels
shows the depolarizing current input waveform (lower trace)
and the resultant local EPSP (upper trace). Since the dendritic
tree, and in particular, the basal dendrites making up a very
large part of the cell, is very compact from the point of view
of the soma, £>„ = 17.7 msec at the soma (lower left panel
in Fig. 3), not very far from the £>„ = tm relationship expected
of an isopotential membrane patch. For distal, dendritic locations, additional current sinks provided by the rest of the
dendritic tree reduce £>„. The local delay is very small for
distal basal sites in layer 5 (lower right panel in Fig. 3), since
the current can "escape" very quickly because of the proximity to the large current sink at the soma. This effect is less
pronounced in the distal apical tree (upper right panel in Fig.
3), since the conductance load imposed by the apical tree
and the distant soma is relatively less significant. Indeed, D^
(here 5.5 msec) is closer to what is expected at the sealed
end of an semi-infinite cable (j^2; Agmon-Snir and Segev,
1993). As is visualized in this figure, small values of D^ imply
that the associated EPSPs decay very rapidly (see also Rinzel
and Rail, 1974). Thus, local synaptic inputs have to be very
well synchronized to interact. Moving synaptic input from a
dendrite onto a spine reduces £>„ by an amount depending
on the exact electroanatomy (H. Agmon-Snir, personal communication).
Figure 3. Local delay Da in a passive
model of a pyramidal cell. A brief depolarizing "alpha"-shaped current input (1
nS peak conductance change at 0.5
msec after onset) was applied at different sites (lower traced, and the voltage
response at these sites is depicted by
the corresponding upper traces. Arrows
denote the centroids of the corresponding transients; the value of the centroid
relative to the onset of synaptic input at
t = 1 msec is shown below each transient At the soma [lower left traces) the
value of the centroid of the voltage response is close to Tn = 20 msec. Note
the decrease in the time course of the
voltage response (i.e., the smaller value
of the centroid) as the input is moved
away from the soma. This layer 5 pyramidal cell from the visual cortex of the
cat was reconstructed by J. Anderson
and R. Douglas. The numerical results
were obtained using NEURON, the compartmental model simulator developed
by M. Hines (Hines, 1989). Model parameters: flm = 20,000 ncm2, fl; = 200 ncm,
and Cm = 1
14.6 msec
18.7 msec
I
I
J
I 1 msec
From a functional view point, the local delay £>„ can be
used for defining what w e mean by simultaneously
arriving
inputs. At the soma, the time window for input integration is
on the order of the membrane time constant, while the time
window for synaptic inputs to arrive simultaneously in distal
dendritic sites is much narrower. In other words, the soma of
a cortical neuron behaves more as an integrator whereas distal dendritic arbors behave more like coincidence detectors
(Koch et al., 1983; Segev and Parnas, 1983; Softky and Koch,
1993; see, in particular, Softky, 1994).
A major drawback of using the local delay is that in the
presence of voltage-dependent components, £>„ can rapidly
become zero or even negative (even for very small inputs).
This results from the activation of potassium-dependent conductances that cause the EPSP to undershoot. Although this
hyperpolarization may never be very deep, it can be quite
prolonged, rendering the definition of the voltage centroid
less useful. In this case, one should only consider the positive
phase of the EPSP for computing £>„.
Fast Local EPSPs Are Independent of T S
It is important to understand what aspects of the voltage dynamics are, and what aspects are not, constrained by the
membrane time constant.
Given the very fast rise times of synaptic currents at voltage-independent AMPA receptors (Hestrin et al., 1990; Hestrin,
1992; Jonas et al., 1993; Trussell et al., 1993), synchronized
synaptic input can easily deliver a large and powerful current
within a narrow time •window to the postsynaptic membrane
(Softky, 1994). Depending on the location of the synapse, this
inward current can rapidly depolarize the local membrane. It
can be shown that for fast inputs the rise time of the EPSP is
independent of Rm but does depend on Cm and R,. At first
glance this appears counterintuitive, since tm = RmCm should
govern the dynamics of the voltage. However, early on, the
injected current primarily charges up the membrane capacitance, both at the site of current injection and at neighboring
locations (via the axial resistance R^>. The larger R, or the
smaller Cm, the faster the voltage will change. This is illustrated in a dramatic fashion in Figure 4, where the EPSP due to
1 msec
a fast, excitatory synaptic input in the dendritic tree is plotted
for Rm = 100,000 flcm2 and for an Rm value 10 times lower.
Although the time constant changes also by a factor of 10,
the rise time and peak amplitude of the local EPSP are barely
affected. Because of the distributed capacitance between the
site of the input in the distal dendritic tree and the soma, the
signal becomes slower while the EPSP propagates toward the
soma. For EPSPs with slow time course, more current flows
across the membrane, and Rm becomes more relevant. For a
larger Rm value, less current leaks out through the membrane
and the somatic EPSP is larger and slower.
It should be noted here that over the last several years,
experimental estimates of the intracellular resistivity have increased several-fold over the widely adopted value of Rt = 70
flcm and now stand around 150-300 flcm (Stratford et al.,
1989; Major et al., 1994; Rapp et al., 1994). A major reason for
this trend may lie in the observation that a significant fraction
of the cross-sectional area of dendrites is occluded by cytoskeletal and cellular organelles, increasing the value of R( compared to that obtained in aqueous solution. In contrast, estimates of the membrane capacitance Cm have slowly crept
downward from the early estimate of 1 (iF/cm2 to 0.7-0.8
(i-F/cm2 (most likely caused by an underestimate of the membrane area; Benz et al., 1975; Major et al., 1994).
We conclude that at distal dendritic sites and for brief synaptic inputs, the time course of locally evoked synaptic potentials, as well as the temporal window within which they
can interact, is not constrained by Rm but rather by the geometry of the tree and by Cm and Rf. On the other hand, the
dynamics of synaptic evoked potentials that propagate from
one site to another, such as toward the soma, are inexorably
shaped by im (Fig. 4).
The Effect of the Synaptic Background Activity on T O
Neurons do not exist in isolation but are embedded within a
tightly interwoven network of nerve cells. A typical neocortical pyramidal cell is the recipient of between 5,000 and
20,000 synapses (Larkman, 1991; Anderson et al., 1994). In
lightly anesthetized or in behaving animals, these cells are
spontaneously active, with spike frequencies ranging from
Cerebral Cortex Mar/Apr 1996, V 6 N 2 97
A)
Time Constant
Local potential
Rm-100.000Rm = 10,000 -
100
V3
(mV)
10
L
Passive neuron
^Single cable
Active neuron
t (msec)
B)
Somatic potential
2
Vm
(mV)
40
60
80
4
6
10
Background Activity (Hz)
R m - 100,000 R m - 10.000 -
100
Figure 5. The membrane time constant varies with synaptic background activity: illustration of the effect of varying the spontaneous synaptic background activity fb on tm
as estimated in the soma of the layer 5 pyramidal cell shown in Figure 3 in the presence
of seven voltage-dependent somatic membrane conductances {Active neuron). Notice
the logarithmic scale for T . In the passive neuron curve, all active conductances in
the soma were removed. These numerical results were compared against a highly
simplified model of the cell, consisting of a single finite cable {Single cable; with sealedend boundary conditions). All three models show the same effect as fb increases, the
membrane becomes more and more "leaky" and Tm (as well as the input resistance,
not shown) decreases. In a typical neocortical slice preparation, little spontaneous
activity exists, corresponding to ft ~ 0, while in behaving animals, fb can be anywhere
from 2 to 20 Hz. From Bemander (1933).
t (msec)
Figure 4. Independence of fast, local EPSP on -<„ the dendritic {A) and somatic (8)
membrane potential in response to a single, fast excitatory synapse located in the distal,
apical tree of the reconstructed layer 5 pyramidal cell. The upper voltage trace in both
panels corresponds to Rm = 100,000, and the lower voltage trace, to Rm = 10,000 ncm 2 .
The synaptic input current peaks at 0.5 msec and is over within 2 msec. This is so fast
relative to the membrane time constant of either 10 or 100 msec that the rising phase
as well as the peak dendritic response is more-or-less independent of /?„ Due to the
low-pass filtering between the synapse and the soma, the potential at the cell body
does show the effect of decreasing the electrotonic distance and increasing the time
constant via changes in /?„ This shows the fallacy of using Tm as an indicator for how
rapidly the membrane can respond to synaptic input
low values of 0.3-2.5 spikes/sec in the anesthetized cat visual
cortex to 5-10 spikes/sec in various extrastriate cortical areas
(Gilbert, 1977; Leventhal and Hirsch, 1978; Woody et al., 1984;
Abeles, 1991). This "spontaneous" firing activity, combined
with massive convergence of synaptic input onto cortical
cells, implies that neurons are subject to continuous synaptic
bombardment. This is in sharp contrast to the situation prevailing in in vitro slice or culture preparations used extensively for biophysical studies, where very little spike activity
exists in the absence of direct stimulation. What consequences does this have for the measurement of neuronal im
(Abbott, 1991; Bemander et al., 1991; Amit and Tsodyks, 1992;
Rapp et al., 1992)?
Because synaptic input transiently increases the membrane conductance to particular ions, physical intuition tells
us that massive synaptic background activity will decrease the
effective membrane resistance Rm. Note that this is true both
for excitatory and for inhibitory synapses. While a single 0.5
nS peak conductance change for 2-5 msec will not make a
large difference, the joint activity of 10,000 synapses, each
independently firing several times each second, can cause •
large overall conductance changes. Figure 5 shows what happens in a model of a layer 5 pyramidal cell (shown in Fig. 3)
as the "spontaneous" firing frequency fb is varied for 4000
excitatory and 1000 inhibitory synapses distributed through-
98 A Brief History of Time CConstants) • Koch et al.
out the cell (each firing independently). Increasing/,, from 0
to 10 Hz effectively decreases the time constant measured at
the cell body by a factor of 50 (from 100 to 2 msec). As fb
increases, the cell becomes more and more sensitive to differences in the arrival times of synaptic input (i.e., to their
temporal dispersion), with possible important functional consequences for the single-cell performance (Bemander et al.,
1991; Rapp et al., 1992) as well as for the performance of the
network (Abbott, 1991). Because typically/ = 0 in slice or
cultured cells, im may be much higher than in an equivalent
in vivo preparation (Bindman et al., 1988).
These results imply that Tm should not be thought of as a
fixed constant that the cell is stuck with for the rest of its
life, but rather as a dynamic variable that can be rapidly modulated by the overall network activity.
xm in the Presence of Voltage-Dependent Conductances
In all of the above considerations we assumed that at subthreshold voltages, the dendritic membrane is essentially passive. Yet, as already discussed by Ito and Oshima (1965) for
the spinal motoneuron and recently by Markram and Sakmann (1994; see also Stuart and Sakmann, 1994) for cortical
pyramid cells, a perturbation as small as a few millivolts at
the dendrites can lead to a nonlinear response (manifest as
under- and overshoots in the voltage response). This nonlinear behavior is attributed to the activation of low-threshold,
voltage-gated dendritic channels. These effects can be regarded as yet another important source for modulating the effective value of Tm.
As briefly discussed above, the presence of even relatively
small amounts of potassium currents can lead to a dramatic
reduction of D^ even causing it to be negative (due to the
presence of a long-lasting hyperpolarization). Under these
conditions, what operational definition for rra or other measures can we adopt in the presence of voltage-dependent currents? This is a difficult question that has not received suffi-
-10
•30
-50
20
Time (msec)
Figure 6. Powerful synaptic input can cause a spike to occur within a fraction of T ^
the somatic membrane, potential in our compartmental model of the layer 5 pyramidal
cell shown in Figure 3. Here 135 excitatory, voltage-independent AMPA-type synaptic
inputs located on layer 4 (i.e., within the first 450 \m of the apical trunk) were simultaneously activated at t = 1 msec. Each input corresponds to a brief conductance
change (peaking at f = 1.5 msec with a maximal conductance change of 0.5 nS) in
series with a reversal potential of 65 mV relative to the resting potential. The soma and
axon hillock were endowed with Hodgkin and Huxley-like active channels (gN, = 120
mS/cm2; & = 36 mS/cm2; gL = 0.3 mS/cm2). The time constants for Na inactfvation and
K activation were fivefold slower than in the original Hodgkin and Huxley model (for T
= 20°C). Following this powerful input the threshold for spike firing at the soma is
reached after a delay of only 2.75 msec. The "resting" time constant Tm = 20 msec.
For comparison, the same input was applied to a corresponding passive model where
the active channels were replaced by passive channels, keeping the same time constant
in both cases. The voltage response (the somatic EPSP) in this passive model is indicated by the lower curve. The threshold for spike initiation is reached in a fraction of
TOT much sooner than the centroid of the underlying EPSP in the passive case {Da =
17.7 msec at the soma; see the arrow at 18.7 msec).
cient treatment in the literature. In principle, any increase in
the membrane conductance, whether via a change in the
membrane voltage or induced by a chemically bound ligand,
effectively decreases Tm. However, given the rich repertoire of
voltage-gated channels available to neurons, an activity-dependent decrease in membrane conductance can also be expected under certain conditions, leading to an effective increase
in Tm. Indeed, when voltage-gated channels are considered, the
effective Tm becomes a dynamical parameter" that depends on
both membrane voltage and on time. Another point to emphasize is that several voltage-dependent conductances have
very slow kinetics (e.g., low-threshold Ca2+ current and
NMDA-modulated currents). When these currents are activated, the time course of the voltage response will be dominated
primarily by the slow kinetics of the active current and will
be practically independent of Tm. A perfect example of this is
the role of the transient, inactivating potassium current /A in
delaying the onset of spikes by many hundreds of milliseconds in mollusk neurons (Connor and Stevens, 1971b) as well
as in cortical pyramidal cells.
An important point to consider is the extent to which im
influences the delay between the onset of synaptic input and
the spike generation. Depending on the amount and the degree of synchronization (Bernander et al., 1994), the threshold
for spike generation can be reached in a fraction of -xm. This
case is demonstrated in Figure 6, where 135 fast, synchronized, non-NMDA, excitatory synapses located in the layer 4
portion of the apical tree of the cell shown in Figure 3 cause
the cell to generate a spike within 4 msec of the onset of
synaptic activity (for comparison, the somatic EPSP in the corresponding passive case is also shown). Increasing the num-
ber of afferent synapses or moving them closer to the soma—
a situation occurring for excitatory input onto smooth inhibitory interneurons—can reduce this time even further. If the
input is temporally and/or spatially dispersed (in particular
toward distal sites), the delay between onset of activity and
spike generation will begin to be more and more influenced
by Tm (Agmon-Snir and Segev, 1993; Cauller and Connors,
1994).
It is difficult to make general statements about the influence of voltage-dependent properties on j m . A case in point
that demonstrates that the value of the passive TOT does determine active properties is the conduction velocity of the
action potential in unmyelinated axons. As already discussed
by Rushton (1937) and more recently by Jack et al. (1975,
p. 428), spike propagation velocity is proportional to the reciprocal of Tro (and directly proportional to the space constant, X.). The intuitive explanation for this relationship is that
the flow of current ahead of the traveling action potential,
and therefore its velocity, is partially governed by the passive
properties of the (essentially passive) cable that lies in front
of the traveling spike. The important message is diat, for particular cases, the passive properties of the neuron do play an
important role in determining the behavior of active phenomena.
So, What Does the Passive Neuronal Time Constant Really Mean?
As mentioned above, the problem with im and associated measures is not so much that they differ by a factor of 2 or 3 for
cells with voltage-dependent components (i.e., for almost all
neurons), but that they characterize the behavior of the cell
in an operating regime of little physiological relevance. Computing Tm or £>„ relies on measuring the dynamics of the
membrane potential in response to a subthreshold input, such
as a sub-nanoampere current injection or a single EPSP. However, when a neuron in the cortex of the cat responds to a
visual input, say, a running mouse, it receives input from hundreds or more presynaptic cells and responds with a barrage
of spikes. Such a cell is far removed from the subthreshold
regime used to assess the response time of the membrane.
Could one argue that Tm therefore represents a lower
bound on the neuronal response time? Unfortunately, even
this is not true. It is straightforward to construct systems that
spike with arbitrarily low frequencies. As mentioned already
above, a well-known example of this is the very low frequency behavior (on the order of one spike per second) seen in
mollusk neurons (Connor and Stevens, 1971a,b). Here the
transient potassium current, /A—although endowed with time
constants less or equal to 100 msec—can space successive
action potentials much more •widely than could be achieved
with a combination of fast channels and a passive membrane
(Connor, 1978). The moral here is that Tra does have its uses
in characterizing the behavior of the cell under subthreshold
conditions, yet it does not tell us much about the dynamics
of neurons in their normal operating range.
Even though a single neuron is unlikely to response much
faster than a few milliseconds (witness Fig. 6), it is known
that animals can discriminate times that are much, much
briefer. Behavioral studies show that the electric fish Eigenmannia is able to effectively discriminate temporal differences as small as 0.4 (jusec (Rose and Heiligenberg, 1985). Another instance is the barn owl, a bird of prey sensitive to
differences in the arrival time of sound at its two ears of 2
jjLsec (corresponding to being able to localize a sound source,
such as a squeaking mouse, with 1.5° precision; Konishi et al.,
1985). Extracellular recordings have shown that individual
cells in the inferior colliculus in the midbrain of the owl,
specialized for processing interaural time differences between
the two ears, have tuning curves with a half-width at half-
Cerebra] Cortex Mar/Apr 1996, V 6 N 2 99
height of about 15 ixsec (Moiseff and Konishi, 1981). The
brain achieves such hyperacuity performances using a combination of neurons whose biophysical properties are optimized to respond very transiently (e.g., neurons in the avian
nucleus magnocellularis have very fast time constants of 2
msec; Reyes et al., 1994), in combination with massive convergence and population coding (Konishi, 1991). Softky
(1994) has shown how a few fast, excitatory synapses located
at the tip of distal basal dendrites in a model pyramidal cell
(recall the low value of D^ « 1 msec in the lower right panel
in Fig. 3) can perform coincidence detection with sub-millisecond resolution (see also Koch and Poggio, 1983; Segev and
Rail, 1988). Thus, although the dynamics of the membrane
potential in neurons might in general be limited to the millisecond range, this does not preclude individual neurons performing temporal resolution on a much finer time scale. Indeed, detecting such temporal differences is primarily limited
by signal-to-noise considerations, rather than by TOT.
When thinking about the membrane time constant as codetermining the "processing speed" of neurons, it is useful to
remember that in such a highly parallel architecture as the
cerebral cortex, very deep (in the sense of requiring many
steps) computations can be executed in a very short time. A
dramatic example of this is evident in the recognition study
by E. E. Cooper and I. Biederman (unpublished observations,
1994). They flash a large number of simple line drawings of
various common objects, such as a truck, a fire hydrant, a cat,
a fruit, and so on, onto a monitor at a rate between 1 and 40
images per second (this paradigm is called rapid serial visual
presentation) and ask subjects to push a button if a particular
image, say a lion, comes up. Most naive and untrained subjects
can perform this tasks effortlessly at 10 Hz, that is, when on
average only 100 msec is available to process each image in
each part of the brain (assuming a pipelined neuronal architecture). Performance at 20 Hz is approximately 90% and does
not become chance (guessing) until 40-50 Hz. Even when
the task is denned by exclusion (e.g., "push the button when
you see the first drawing of an object that can't be used to
as a means of transportation") subjects can perform reasonably well at a 20 Hz presentation rate, a surprising result. A
rate of 20 Hz implies 50 msec, which is only on the order of
2-10 times slower than Tra or D^ or related measures. In other
words, the ratio of switching times of the individual components (i.e., neurons) versus the execution times of the whole
system is on the order of 101, compared to 107-109 for a similar performance on today's so-called "massive parallel" digital
computers (if such performance can be achieved at all). This
is quite a remarkable aspect of nervous systems that never
ceases to amaze.
In conclusion, the rise time of local EPSPs and IPSPs is
limited by the dynamics of the synaptic current (i.e., by how
fast the synaptic current can be pushed through the ionic
channels) and by the membrane capacitance, but not by Tm.
Their decay time is relatively well characterized by the local
delay D^ which can be small in distal structure where the
current can rapidly dissipate but is on the order of Tm at the
cell body. When these postsynaptic potentials propagate to
other locations (such as the cell body), im does play a significant role in shaping their rise times and their long tails. Today,
the passive time constant is estimated to be on the order of
20-100 msec. It has so far not been possible to introduce for
neurons with voltage-dependent membranes any measure
equivalent to D^ Such a measure is sorely needed to more
fully characterize neurons in physiologically relevant situations.
Notes
We thank Ojvind Bernander and Hagai Agmon-Snlr for their detailed
comments and their help in preparing the manuscript. The research
100 A Brief History of Time (Constants) • Koch ct al.
reported here was funded by grants to C.K.fromONR and the NIMH
Center for Neuroscience and by grants to I.S. from ONR and BSE I.S.
acknowledges the hospitality of the Mathematical Research Branch
(NIDDK) at the NTH in Bethesda, and C.K., the hospitality of the Santa
Fe Institute in Santa Fe, New Mexico. Early versions of the manuscript
were written while these two authors spent part of their summer at
these institutions.
Address correspondence to Dr. Christof Koch, Computation and
Neural Systems Program, Caltech, 139-74, Pasadena, CA 91125.
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