Frequency-Dependent Synaptic Potentiation, Depression,
and Spike Timing Induced by Hebbian Pairing
in Cortical Pyramidal Neurons
Murat
Okatan and Stephen Grossberg
[1]
Department
of Cognitive and Neural Systems and Center for Adaptive Systems
Boston University, 677
Beacon St, Boston, MA 02215
E-mail: okatan@cns.bu.edu, steve@bu.edu
Neural Networks, in
press
Technical Report
CAS/CNS-2000-003
Boston, MA: Boston
University
[1] Acknowledgements: MO and SG were supported in part by the Defense Advanced Research Projects Agency and the Office of Naval Research (ONR N00014-95-1-0409) and the National Science Foundation (NSF IRI-97-20333). The authors would like to thank Diana Meyers for her assistance in arranging the article for the web pages.
Abstract¾Experiments
by Markram and Tsodyks (1996) have suggested that Hebbian pairing in cortical
pyramidal neurons potentiates or depresses the transmission of a subsequent
presynaptic spike train at steady-state depending on whether the spike train is
of low frequency or high frequency, respectively. The frequency above which
pairing induced a significant decrease in steady-state synaptic efficacy was as
low as about 20 Hz and this value depends on such synaptic properties as
probability of release and time constant of recovery from short-term synaptic
depression. These characteristics of cortical synapses have not yet been fully
explained by neural models, notably the decreased steady-state synaptic
efficacy at high presynaptic firing rates. This article suggests that this
decrease in synaptic efficacy in cortical synapses was not observed at
steady-state, but rather during a transition period preceding it whose duration
is frequency-dependent. It is shown that the time taken to reach steady-state
may be frequency-dependent, and may take considerably longer to occur at high
than low frequencies. As a result, the pairing-induced decrease in synaptic
efficacy at high presynaptic firing rates helps to localize the firing of the
postsynaptic neuron to a short time interval following the onset of high
frequency presynaptic spike trains. This effect may "speed up the time
scale" in response to high frequency bursts of spikes, and may contribute
to rapid synchronization of spike firing across cortical cells that are bound
together by associatively learned connections.
Key Words: synaptic potentiation, synaptic
depression, frequency-dependent synaptic plasticity, cortical pyramidal cells,
Hebbian pairing, cortical synchronization
1. INTRODUCTION
The simple model of a synapse as a
frequency-independent gain element has received wide acceptance and provided
the basis for several correlation-based theories and neural models of learning
and memory. Experimental evidence has accumulated, however, suggesting that
synaptic function requires a more complicated model to account for changes in
synaptic efficacy such as short-term synaptic depression (STD) and
frequency-dependent synaptic potentiation (FDP). STD refers to the
use-dependent short-term decrease in synaptic efficacy that results from axonal
transmission and synaptic release events. This decrease in efficacy may be due
to factors such as decrease in presynaptic action potential (AP) amplitude and
transmission failure of APs at axonal branch points during repetitive stimulation (Brody and Yue, 2000),
vesicle depletion (Stevens and Tsujimoto, 1995), inactivation of release machinery
(Matveev and Wang, 2000) and postsynaptic receptor desensitization (Markram,
1997; Trussell et al., 1993). FDP is a special form of synaptic potentiation
that is induced by Hebbian pairing. It has long been assumed that synaptic
potentiation increased the amplitude of the postsynaptic signal regardless of
the frequency of the inducing presynaptic spike train. However, recent data
have reported that Hebbian pairing increased or decreased the amplitude of the
postsynaptic signal depending on the frequency of the inducing presynaptic
spike train (Markram and Tsodyks, 1996). Specifically,
the data suggested that Hebbian pairing potentiated low frequency stimuli and
depressed high frequency stimuli. This article offers a possible explanation
and quantitative simulations of these surprising data. Henceforth the term FDP
will be restricted to this phenomenon.
~
STD has been studied and
characterized for over five decades in several different neuron types and
across species (Abbott et al., 1997; Feng, 1941; Galarreta and Hestrin, 1998;
Liley and North, 1953; Markram and Tsodyks, 1996; Parnas and Atwood, 1966;
Pinsker et al., 1970; Thomson and Deuchars, 1994; Varela et al., 1997), and has
been an integral part of some neural models for over four decades (Abbott et
al., 1997; Carpenter and Grossberg, 1981; Chance et al., 1998; Francis et al.,
1994; Francis and Grossberg, 1996a, 1996b; Grossberg, 1968, 1969, 1972, 1975,
1984; Liley and North, 1953; Markram et al., 1998a, 1998b, 1998c; Ögmen, 1993;
Tsodyks and Markram, 1997; Varela et al., 1997; Wang, 1999). Recently, neural
models featuring synapses that exhibit FDP have been proposed (Carpenter, 1994,
1996, 1997). These models preceded the experimental report of FDP by Markram
and Tsodyks (1996) and its modeling by Tsodyks and Markram (1997) and Markram
et al. (1998a, 1998b, 1998c), and predicted that FDP should be useful for
stable learning of distributed codes.
Markram
and Tsodyks (1996) suggested that the effects of Hebbian pairing on the
amplitude of the excitatory postsynaptic potentials could be characterized at
steady-state (e.p.s.p_{stat}). Using their experimental results (Figure
1) they noted that "the increase in the amplitude of... e.p.s.p_{stat}
for very low-frequency stimulation..., and the lack of an effect on the
amplitude of e.p.s.p_{stat} for a high frequency train, indicates that
the potentiation is conditional on the presynaptic spike frequency. The effect
of pairing on e.p.s.p_{stat} at several different frequencies was
therefore examined." They found that "potentiation of synaptic
responses... only occurred when the presynaptic frequency was below 20 Hz"
(Figure 2c). In support of this finding they also reported that "e.p.s.p_{stat}
is not changed in the 40-Hz trace but is increased in the 5-Hz trace" due
to pairing (Figure 2a, b).
Tsodyks and Markram (1997) proposed
a phenomenological model of STD and FDP, named the TM model by Markram (1997).
Using the TM model, Markram et al. (1998b) concluded that the effect of pairing
is “to selectively regulate low frequency synaptic transmission” (Figure 3).
Similarly, Markram et al. (1998a) reported that pairing “results in a selective
change in low-frequency synaptic transmission, leaving high-frequency
transmission unaffected.”
Contrary to the above
interpretations of the Markram and Tsodyks (1996) data and the predictions of
the TM model, the decrease below the 100% level apparent in Figure 2c suggests
that paired-activity does have an effect on steady-state synaptic efficacy at
high frequencies, and this effect is to further decrease it below the 100%
baseline level. We suggest that this
decrease in synaptic efficacy was observed not at steady-state but during a
transition period preceding it. In addition, our analysis proposes that the
frequency-dependent decrease in synaptic efficacy induced by Hebbian pairing
may help to localize the firing of the postsynaptic neuron to a short time
interval following the onset of high frequency presynaptic spike trains. This
localization of firing may help the system to speed up its processing rate
under high-frequency conditions. In some neural systems, such as the
transduction of light by turtle cones (Baylor, Hodgkin, and Lamb, 1974a, 1974b;
Carpenter and Grossberg, 1981), and the rate of key pecking in pigeons (Grossberg and Schmajuk, 1989; Wilkie,
1987), increasing stimulus intensity is observed to speed up the time scale of
neural activity and also to time-localize it. Our results suggest that an
increase in the level of learning has similar temporal effects in the activity
of cortical neurons. We also explain why the TM model prediction in Figure 3,
which shows no frequency-dependent decrease below the 100% baseline, differs
from the data in Figure 2c. In particular, we show that the time taken to reach
steady-state in these systems may also be frequency-dependent. Unless one
compensates for this frequency-dependent settling time, important cellular
properties, such as frequency-dependent depression, may not be correctly
understood.
2. MATERIALS AND METHODS
2.1 STD Experiment
Markram
and Tsodyks (1996) characterized STD in fast-depressing excitatory synapses
between tufted pyramidal neurons in somatosensory cortical layer 5 of the rat
(named TL5 neurons in Markram, 1997). Their results are illustrated in Figure
1. In this experiment they induced a presynaptic TL5 neuron to fire an AP by
injecting a 2 nA, 5 ms current pulse into its soma. They administered seven
such injections at 23 Hz in each stimulus sweep. A new sweep was started every
5 s. The postsynaptic membrane potential trace (Post V_{m}) before
pairing reflects the average of 58 sweeps and the one after pairing reflects
the average of 59 sweeps in the same synaptic connection (Figure 1a).
The pairing method consisted of
injecting sustained current pulses of 200 ms duration into visually identified
individual presynaptic and postsynaptic TL5 neurons. The current intensity was
adjusted to evoke 4–8 spikes and the current pulse in the postsynaptic neuron
was delayed (1–5 ms) to ensure that the postsynaptic neuron discharged after
onset of synaptic input. No attempt was made to control subsequent spikes. The
procedure was repeated 30 times every 20 s.
The effect of paired-activity on
e.p.s.p amplitudes is illustrated in Figure 1b, where the e.p.s.p amplitudes
measured from Figure 1a are plotted. e.p.s.p amplitudes were measured from the
voltage immediately before the onset of the e.p.s.p to the peak of the e.p.s.p.
Markram and Tsodyks (1996) noted that “the presynaptic train of action
potentials results in depression of the synaptic response, until a stationary
level of e.p.s.p amplitude is reached (defined here as e.p.s.p_{stat}).”
This definition of e.p.s.p_{stat} implies that it is used to denote the
steady-state e.p.s.p amplitude. The e.p.s.p_{stat} was computed as the
average of the last 20% of the single exponential fit to the e.p.s.p amplitudes
in Figure 1b, and was “roughly equivalent to the average of the last 2
e.p.s.ps.” (Markram and Tsodyks, 1996). In these responses, Markram and Tsodyks
(1996) also distinguished the initial and the transition e.p.s.ps. They noted
that “...whereas the amplitude of e.p.s.p_{init} was increased... the
average amplitude of e.p.s.p_{stat} was unaffected.” In Figure 1b, the
amplitude of the transition e.p.s.ps decreases due to pairing. Based on these
results they concluded that “the effect reported here is not an unconditional
potentiation of the efficacy of the synaptic connection; instead, it is a
redistribution of the existing efficacy between spikes in a train.” Figure 1c
illustrates the effect of Ca^{2+} concentration on the differential
effect of pairing on the initial and the stationary e.p.s.ps.
Figure 1. The effect of pairing
on postsynaptic responses elicited by 23 Hz test stimuli consisting of seven
APs. (a) The average postsynaptic
membrane potential (Post V_{m}) is obtained as the mean of 58 sweeps
before pairing (left) and 59 sweeps 20 min after pairing. (b) The amplitudes of the e.p.s.ps in (a) are plotted. The amplitude of an e.p.s.p was computed as the
difference between the membrane potential at the peak of an e.p.s.p and the
membrane potential at the onset of that e.p.s.p. Fitted curves are single
exponentials. The average value of the last 20% of the fitted curves, which was
roughly equal to the average of the sixth and seventh e.p.s.p amplitudes, was
used as the stationary e.p.s.p amplitude (e.p.s.p_{stat}). Pairing
increased the amplitude of the initial e.p.s.p (e.p.s.p_{init}),
decreased those of the transition e.p.s.ps (e.p.s.p_{transition}) and
left e.p.s.p_{stat} unaffected. (c)
Pairing increases the amplitude of the initial e.p.s.p while leaving e.p.s.p_{stat}
unaffected, both at experimental level of extracellular Ca^{2+}
concentration (2 mM) and at the physiological level for a rat of this age (1.5
mM). (Reprinted with permission from Markram and Tsodyks, 1996).
Based on these results, Markram and
Tsodyks (1996) stated that “the increase in the amplitude of e.p.s.p_{init},
which actually represents e.p.s.p_{stat} for very low-frequency
stimulation (<0.25 Hz), and the lack of an effect on the amplitude of
e.p.s.p_{stat} for a high-frequency train, indicates that the
potentiation is conditional on the presynaptic spike frequency. The effect of
pairing on e.p.s.p_{stat} at several different frequencies was
therefore examined.” The next section describes this experiment.
2.2 FDP Experiment
The results of
the Markram and Tsodyks (1996) experiment that characterized FDP in TL5 neurons
are illustrated in Figure 2.
In
this experiment, they elicited e.p.s.ps using the same procedure as in the STD
experiment, but at several different firing rates ranging from 0.067 Hz to 40
Hz. At all frequencies, they used trains of six APs as test stimuli. Figure 2a
shows the postsynaptic potential traces elicited by stimuli of 40 Hz and 5 Hz.
Figure 2b shows these traces after pairing. They noted that “e.p.s.p_{init}
is increased to the same extent for both frequencies, e.p.s.p_{stat} is
not changed in the 40-Hz trace but is increased in the 5-Hz trace (compare for
example the amplitude of the last 2 e.p.s.ps in the 40-Hz train before and after
pairing).” Markram and Tsodyks (1996) summarized the change in e.p.s.p_{stat}
by computing the ratio of the post-pairing e.p.s.p_{stat} to the
pre-pairing e.p.s.p_{stat} at different test frequencies. This ratio is
plotted as a function of test frequency in Figure 2c, where the fitted curve is
a single exponential (Henry Markram, personal communication). Based on the
results shown in Figure 2, they concluded that “in the same neuron, e.p.s.p_{init}
was found to increase equally for low- and high-frequency trains, whereas
e.p.s.p_{stat} was increased only when the frequency was low (Fig. 3a,
b; here Figure 2a, b). Potentiation
of synaptic responses therefore only occurred when the presynaptic frequency
was below 20 Hz (Fig. 3c; here Figure 2c).”
They also noted that “the physiological implications of redistribution of
synaptic efficacy are also entirely different from unconditional potentiation
or depression and are partly predicted by the frequency-dependent potentiation
seen in Fig.3 (here Figure 2).”
Unlike Figure 1b which shows the data collected from a single synapse, Figure
2c represents the average data obtained from 33 synaptic connections (1–4
frequencies tested per synaptic connection). Note that the e.p.s.p amplitudes
corresponding to the 7^{th} response number in Figure 1b would yield an
amplitude ratio comparable to the data point shown at 23 Hz in Figure 2c, where
pairing-induced decrease in synaptic efficacy below the 100% baseline at
high-frequencies is not yet noticeable.
Figure 2. Frequency-dependence
of synaptic potentiation in Markram and Tsodyks (1996). The average
postsynaptic membrane potential (Post V_{m}) traces were elicited by
using presynaptic spike trains consisting of six APs at 5 Hz and 40 Hz, before (a) and after (b) paired-activity to assess the induced changes in synaptic
efficacy. (c) After paired-activity,
e.p.s.p_{stat} (See STD
Experiment in Materials and Methods) is plotted as a function of test
frequency. Control refers to the pre-pairing value of e.p.s.p_{stat}.
Superimposed is a single exponential fit (Henry Markram, personal
communication). Data were collected from 33 synaptic connections (1–4
frequencies tested per connection). The leftmost data point represents the
average of measurements obtained at 0.067 Hz and 0.25 Hz. All other data points
were collected at a single test frequency. Note that pairing increased e.p.s.p_{stat}
at low frequencies but decreased it at high frequencies. The article proposes
an explanation for this decrease below. (Reprinted with permission from Markram
and Tsodyks, 1996).
2.3 The TM Model
The TM model,
which was proposed by Tsodyks and Markram (1997) and further developed in Markram et al. (1998a, 1998b, 1998c) is a
phenomenological model of short-term synaptic plasticity. It applies to both
facilitating excitatory synapses from TL5 neurons to bipolar inhibitory
interneurons, and to the fast-depressing excitatory synapses between TL5
neurons. It can be used to compute the amplitude of the e.p.s.ps elicited by an
arbitrary presynaptic spike train (Tsodyks and Markram, 1997). The equations
describing this model in Markram et al. (1998b) are:
_{}, |
(1) |
_{}, |
(2) |
_{}. |
(3) |
In (1) E_{n}
is the amplitude of the n^{th} e.p.s.p averaged across trials, where
the subscript “n” corresponds to the response number shown in the abscissa of
Figure 1b. Function u_{n} in (1)–(3) denotes the running value of the
utilization of synaptic efficacy. Its minimum value is U, and it is transiently
elevated following each AP. It then decays to U with the facilitation time
constant t_{facil}. Function
R_{n} in (1) and (3) denotes the fraction of available synaptic efficacy immediately before the
n^{th} AP. It is temporarily decreased after each AP and it converges to its maximum value of
1 with recovery time constant t_{rec}. Parameter A is the absolute synaptic efficacy and denotes
the maximal e.p.s.p amplitude that is obtained when R and U are each equal to
1. Dt is the interspike interval and is equal to the inverse of
the firing rate for constant frequency stimuli. Markram et al. (1998b) note
that “synaptic connections displaying depression are characterized by
negligible values of t_{facil} [3 ms in Tsodyks and
Markram (1997)] and
hence u_{n} = U.” Consequently, u_{n} = U in (1)–(3) for
depressing synapses. In short, the equations describing STD in the TM model
are:
_{}, |
(4) |
_{}. |
(5) |
Markram et al. (1998b) proposed that
U represents the probability of transmitter release. In their model, the effect
of paired-activity is simulated by raising the value of U. U is constrained to
lie in [0, 1] and is kept constant during presynaptic activity alone. Markram
et al. (1998b) used this hypothesis to simulate FDP. Their prediction for FDP
at steady-state is illustrated in Figure 3.
Figure 3. Frequency-dependence of
synaptic potentiation predicted by the TM model at theoretical steady-state.
Markram et al. (1998b) used the TM model to simulate frequency-dependence of
synaptic potentiation (Markram and Tsodyks, 1996). In the TM model of STD
(Equations (4)–(5)) the effect of pairing is simulated by increasing the value
of the parameter U, which is proposed to represent the probability of release.
Here, the steady-state e.p.s.p amplitude (EPSP_{st} amp) is computed
using a test stimulus of infinite duration (n ® ¥ in Equation (6)),
before (U = 0.4, control) and after (U = 0.8) pairing. The post-to-pre ratio of
EPSP_{st} amplitude is plotted as a function of the test stimulus
frequency. The frequency-dependence of this ratio is different from the
experimentally determined dependence shown in Figure 2c since it does not exhibit
the decrease below the 100% level at high presynaptic firing rates. A = 1 and t_{rec} = 800 ms in Equations
(4)–(5). (Reprinted with permission from Markram et al., 1998b).
Based
on Figure 3, Markram et al. (1998b) stated that “The phenomenon produced when
Pr changes is readily distinguished from virtually all other types of synaptic changes
since changing Pr is a mechanism to selectively regulate low frequency synaptic
transmission.” Here, “Pr” means probability of release, which is denoted by U
in Equations (4)–(5), and “other types of synaptic changes” include changes in
the absolute synaptic efficacy, A, or the recovery time constant, t_{rec}. Also, Markram et al. (1998a) noted that
“...changes in U... result in changes in the frequency dependence of
transmission, also referred to as ‘redistribution of synaptic efficacy’
(Markram & Tsodyks, 1996). Specifically, changing U results in a selective
change in low-frequency synaptic transmission, leaving high-frequency
transmission unaffected.”
Figure
3
To simplify the computation of the
trial-averaged e.p.s.p amplitude E_{n} at a given response number, a
non-iterative expression for E_{n} was obtained from the iterative
equations in (4) and (5) (see Appendix):
_{}, where |
(6) |
_{}. |
(7) |
We
used Equations (6) and (7) to simulate the STD and FDP data of Markram and
Tsodyks (1996) and to explain the pairing-induced decrease in synaptic efficacy
below the 100% baseline in Figure 2c. Below we describe the computational
procedures employed to fit the data. The implications of the fits are explained
in the Results section.
2.4 Determination of
Parameter Values
STD data We
determined the optimal parameter values in (6)–(7) by minimizing the
root-mean-squared-error (rmse) from the data:
_{}, |
(8) |
where
e_{n}
denotes the difference between E_{n-experiment} and E_{n-predicted}. The parameters that minimize (8) can be
determined by generating the error surface e(U, t_{rec}) and finding its global minimum.
For the STD experiment, E_{n-experiment
}is measured from Figure 1b for each of the seven e.p.s.ps before and
after pairing. The pre-pairing and
post-pairing data are pooled to compute the rmse in (8). To remove the
dependency on A in Equation (6), all e.p.s.ps are then divided by the amplitude
of the initial e.p.s.p measured after pairing. According to the TM model, E_{1}
= AU, as can be obtained from Equation (6). Thus, the ratio of the initial
e.p.s.p amplitudes after and before pairing yields the ratio of the after
pairing to before pairing values of U. This ratio is computed to be 1.956 in
Figure 1b, and constrains the parameter search.
Markram (1997) reported that the
values of U and t_{rec} were observed to lie in the ranges 0.1–0.95 and 0.5 s–1.5
s, respectively. Equation (6) is used to compute the error surface for values
of t_{rec} ranging
from 0.2 s to 2 s with steps of 10 ms, and with U in the range from 0.1 to 0.95
with steps of 0.01. The global minimum of e(U, t_{rec}) was found in the region of the parameter space defined by
these ranges. The optimal parameters and fits are shown in the Results section.
FDP data The
experimentally determined amplitude ratios that characterize FDP can be
measured from Figure 2c. Since the test stimuli consisted of only six APs at
each test frequency and since e.p.s.p_{stat} was roughly equivalent to
the average of the sixth and seventh e.p.s.ps in Figure 1b, in the current
article the simulated amplitude ratios were computed based on the amplitude of
the sixth e.p.s.p. Thus, Equation (6) was used to compute E_{6}(U_{post},
t_{rec})/E_{6}(U_{pre},
t_{rec})
at each test frequency and the root-mean-squared-error from the experimental
data of Figure 2c was computed. Markram and Tsodyks (1996) reported that the
leftmost data point in Figure 2c represents the average measurement from two
different low-frequency test stimuli at 0.067 Hz and 0.25 Hz. All other data
points were measured at a single test frequency. Accordingly, the simulated
leftmost data point was computed as the average of the e.p.s.p ratios computed
at 0.067 Hz and 0.25 Hz.
As discussed in the previous
section, according to the TM model, the amplitude ratio at the initial e.p.s.p
reflects the ratio of the post-pairing to pre-pairing values of U. At very low
frequencies, the synapse recovers almost completely between consecutive spikes
and thus the amplitude of the sixth e.p.s.p can be considered equal to that of
the initial e.p.s.p. Thus the amplitude ratio at the sixth e.p.s.p also
reflects the ratio of the post-pairing to pre-pairing values of U at very low
firing rates. Markram and Tsodyks (1996) reported that this ratio is found to
be 1.665 in Figure 2c. This ratio constrains the parameter search, as in the
previous section.
3. RESULTS
3.1 Optimal parameters and
fits
STD data We have conducted an exhaustive search, as described in the
Methods section, to find the optimal parameters in the physiologically
plausible intervals for U and t_{rec}. This method differs from the method of Markram et al.
(1998b) who iteratively changed the model parameters A, U, and t_{rec}, to minimize Equation (8). Although
different optimal parameter values may be found by the two methods, this does
not affect any of our conclusions about the impact of Hebbian pairing on
synaptic transmission at high frequencies.
The parameter search revealed that a global minimum exists at the
point (U_{pre}, t_{rec}) = (0.363, 0.65 s). The post-pairing value of U is found to
be 0.71 = 0.363´1.956. These
parameter values lie in the range of experimentally observed values reported by
Markram (1997). The curves in Figure 4 illustrate the prediction of the TM
model using the optimal parameters. The data points represent the
experimentally determined e.p.s.p amplitudes measured from Figure 1b and
normalized to the post-pairing amplitude of the initial e.p.s.p.
Figure 4. Simulation of short-term synaptic depression using the TM model. The mean-squared-error between the TM model’s prediction and the experimental data provided in Figure 1b was at a global minimum for the parameter values U = 0.363 and t_{rec} = 0.65 s. The curves in the top and bottom panels illustrate the simulation results using these parameter values. The data points represent the experimental results shown in Figure 1b after normalization to the post-pairing amplitude of the initial e.p.s.p.
FDP data The
analysis revealed that there is one global minimum and one local minimum in the
domain defined by the intervals [0.1, 0.95] for U and [0.2 s, 2 s] for t_{rec}. The global minimum was found at the
point (U_{pre}, t_{rec}) = (0.18, 0.87 s) and the local minimum was found at (U_{pre},
t_{rec})
= (0.5, 0.44 s). As mentioned before, the data in Figure 2c were collected from
33 synaptic connections (1–4 frequencies tested per connection). Thus the
optimal values of U and t_{rec} do not necessarily belong to any particular synapse. The
rmse at the local minimum was 17% larger than the one at the global minimum. In
either case, the post-pairing value of U is obtained by multiplying the
pre-pairing value by 1.665. The globally optimal parameter values lie in the
range reported by Markram (1997). But the locally optimal value of 0.44 s is
slightly smaller than the lower bound of 0.5 s of the range of observed values
of t_{rec}.
The curve in Figure 5 illustrates
the prediction of the TM model using the globally optimal parameter set. The
data points and error bars represent the experimental results that are measured
from Figure 2c.
Figure
5.
Simulation of frequency-dependent synaptic plasticity using the TM model. The
mean-squared-error between the TM model’s prediction and the experimental data
provided in Figure 2c was at a global minimum for the parameter values U = 0.18
and t_{rec} = 0.87 s. The curve
represents the TM model’s prediction using these parameter values and it shows
the post-pairing amplitude of the sixth e.p.s.p (E_{6-post}) in
percents of the pre-pairing amplitude of the sixth e.p.s.p (E_{6-pre}).
The data points and the error bars were measured from Figure 2c. The simulation
fits the experimental results successfully and predicts the pairing-induced
decrease apparent at high-frequencies. E_{6} is computed using Equation
(6).
3.2 Computing e.p.s.p
ratios at the sixth e.p.s.p explains pairing-induced decrease in synaptic
efficacy
How can the
discrepancy be explained between the high-frequency depression that is shown in
the data of Figure 2c (Markram and Tsodyks, 1996) and the theoretical curve of
Markram et al. (1998b) that is shown in Figure 3, which does not include this
depression? In Figure 2c, Markram and Tsodyks (1996) estimated e.p.s.p_{stat}
from the sixth e.p.s.p in the data; see the Methods section for the details.
However, to derive the curve in Figure 3, Markram et al. (1998b) estimated this
value using the theoretical steady-state, which they derived by letting n ® ¥ in Equation (6). By so doing, Markram et al. (1998b)
assumed that the steady-state was essentially reached by the 6^{th}
e.p.s.p in Figure 2. It is, however, shown below that the theoretical steady
state is not achieved at the 6^{th}
spike at high frequencies. In fact,
as frequency increases, it takes more and more spikes for the theoretical
steady-state to be reached. On the other hand, if one computes Figure 5
directly from Equation (6), evaluated at the 6^{th} spike, rather than
as n ® ¥, then the pairing-induced decrease in synaptic efficacy
that is seen in the data above 20 Hz is predicted by the TM model.
The
fact that more spikes are needed to reach the steady-state at high frequencies
can be shown by using Equation (6) to determine the minimum number of APs that
is needed for the e.p.s.p amplitudes to reach a criterion fraction of the
theoretical steady-state level. Since E_{n} converges to the
steady-state level from above as n ® ¥ in Equation
(6), the smallest number n for which E_{n}/E_{¥} is less than or equal to the criterion
fraction must be found. For the synaptic parameters used in Figure 5, this
number is shown as a function of test frequency in Figure 6 for an arbitrarily
chosen criterion fraction of 105%.
Figure 6. Minimum number of
action potentials needed for the e.p.s.p amplitude to reach a criterion
fraction of the theoretical steady-state level. Equation (6) is used to compute
the minimum value of the response number n at which E_{n} enters the
range [1.05E_{¥} E_{¥}]. The criterion
fraction of 105% was chosen arbitrarily. E_{¥} denotes the e.p.s.p
amplitude at theoretical steady-state. E_{n} converges to E_{¥} from above. n is seen
to increase with presynaptic spike frequency. The curve shows that for the
synaptic parameters used in Figure 5, n = 8 at 5 Hz and n = 23 at 40 Hz.
According to
Figure 6, for these values of U and t_{rec}, it takes at least eight APs for the e.p.s.p amplitude to
be less than or equal to 105% of the theoretical-steady state level at 5 Hz,
while this number is 23 at 40 Hz. Thus, if test stimuli consisting of a
constant number of APs are used at different test frequencies, changes in
synaptic efficacy may be compared at different phases of the postsynaptic
response, unless a test stimulus of very long duration is used. Note that in
Figure 2c, stimuli of six APs were used at all firing rates, which apparently
was not enough to reach the steady-state at high frequencies.
3.3 Pairing-induced decrease in synaptic
efficacy is a significant synaptic property
The
previous section suggested that a common ground to compare synaptic responses
of different frequencies may be to compare e.p.s.p amplitudes that are within a
criterion fraction of the theoretical steady-state level. As shown in Figure 6,
however, the test stimuli need to be longer at high frequencies for the e.p.s.p
amplitude to reach a criterion level. The required stimulus duration may even
be longer than the duration of a physiological spike train of a given frequency
at high firing rates. In such a case, the frequency of the presynaptic firing
might change before the postsynaptic response has reached the criterion-level.
In other words steady-state may not be a functionally relevant phase at high
frequencies. This fact was also pointed out by Markram and Tsodyks (1996):
“Under in vivo conditions, neurons
tend to discharge irregularly, which effectively represents a multitude of
spike frequencies persisting for different time periods, indicating that the
effect of pairing on synaptic input generated by an irregular presynaptic spike
train would be complex if redistribution of synaptic efficacy was to occur
(Fig. 4; not shown here). The effect
cannot be predicted as most synaptic responses during such a train have not
reached a stationary level for the given frequency and hence are all transition
e.p.s.ps (see Fig. 2b; here Figure 1b).
As discussed, transition e.p.s.ps could be enhanced, depressed or unchanged
after pairing. Redistribution of synaptic efficacy may therefore serve as a
powerful mechanism to alter the dynamics of synaptic transmission in subtle
ways and hence to alter the content rather than the gain of signals conveyed
between neurons.”
The effect of pairing may be studied
not at a particular criterion level but as a function of stimulus duration.
Figure 7 illustrates FDP as a function of both stimulus duration in number of
APs and stimulus frequency, for the synaptic parameters used in Figure 5.
Figure 7. Dependence of pairing-induced changes in synaptic efficacy on test stimulus frequency and response number. The ratio of post-pairing to pre-pairing values of E_{n} (Equation (6)) is shown for response number n = 1–60 and test frequencies of 0.1 Hz to 100 Hz for the synaptic parameter values used in Figure 5. The black contour lines denote the 60%-160% levels in steps of 20%. The white contour line denotes the 100% level. The dashed line at n = 6 is the TM model’s prediction shown in Figure 5. The solid line illustrates the dependence of the ratio on response number at F = 23 Hz. Around 20 Hz, the ratio passes through the 100% level at the sixth AP. At higher frequencies, this transition occurs at earlier response numbers, thereby causing the ratio to be lower than 100% at the sixth AP.
Figure
7 shows that redistribution of synaptic efficacy (RSE) translates into a
prominent decrease in synaptic efficacy that sets in soon after the onset of
moderate to high frequency spike trains. The white contour line denotes the
100% level on the surface. The decreased efficacy lasts for 9 APs (~390 ms), 17
APs (~420 ms) and 27 APs (~270 ms) at 23 Hz, 40 Hz and 100 Hz, respectively.
Note that at the end of trains of 60 APs, the frequency-dependence of synaptic
potentiation has the same form as in Figure 3. This is because the postsynaptic
response is close to theoretical-steady state level at all frequencies by the
60^{th} AP. The continuous curve at 23 Hz illustrates the e.p.s.p
ratios at that frequency. The dashed curve at response number n = 6 is the
curve shown in Figure 5. Note that it enters the region of decreased efficacy
at about 20 Hz.
In addition to lasting long,
pairing-induced decrease in synaptic efficacy may also reach significant
levels. For instance, pairing decreases the amplitude of the 11^{th}
e.p.s.p to 58% of the pre-pairing level at 100 Hz. The timing and extent of the
decrease in efficacy is seen to depend on the presynaptic firing rate. It also
depends on the parameters U and t_{rec}. Figures 8 and 9 illustrate this dependence at 40 Hz while
the ratio U_{post}/U_{pre} is held constant at 1.665.
Figure 8. Dependence of
pairing-induced changes in synaptic efficacy on the pre-pairing value of the
parameter U. The ratio E_{n-post}/E_{n-pre} (%) was computed
using Equation (6) at 40 Hz for different pre-pairing values of the parameter
U. These values are shown next to the corresponding curves. t_{rec} = 0.87 s and U_{post}/U_{pre}
= 1.665 for each curve.
It
can be seen in Figure 8 that the same proportional increase in U results in a
larger decrease in efficacy that also sets in sooner and lasts longer if the
pre-pairing value of U is high. However, pairing induces a less defined
decrease in efficacy or no decrease at all in synapses with very low release
probability to begin with. It should be noted that the same amount of pairing
may result in different increases in U depending on the pre-pairing value of
the latter. In other words, a given amount of pairing results in an increase in
U that is dependent on U_{pre}. Thus, Figure 8 does not characterize
the effect of a fixed amount of pairing for different pre-pairing values of U.
The dependence of pairing-induced
decrease in synaptic efficacy on the recovery time constant is illustrated in
Figure 9. In synapses that recover slowly, pairing induces a more pronounced
and longer-lasting decrease in synaptic efficacy.
Figure 9. Dependence of
pairing-induced changes in synaptic efficacy on recovery time constant t_{rec}. The ratio E_{n-post}/E_{n-pre}
(%) was computed using Equation (6) at 40 Hz for t_{rec} = 0.5 s and 1.5 s. These values are shown next
to the corresponding curves. U_{pre} = 0.18 and U_{post}/U_{pre}
= 1.665 for each curve.
3.4 Pairing localizes
postsynaptic firing to a short time interval following the onset of moderate to
high frequency presynaptic spike trains
The
dependence of synaptic potentiation on the stimulus duration that is
illustrated in Figure 7 suggests that the overall effect of pairing is to selectively
enhance the transmission of early APs in a train. As a result, presynaptic
activity increases postsynaptic firing probability preferentially near the
onset of stimulation. Pairing hereby decreases the average delay between the
onset of presynaptic spike train and the occurrence of the postsynaptic APs
elicited in response. This trend is observable at all stimulation frequencies
except below about 1 Hz.
At moderate and high frequencies
(above about 20 Hz in Figure 7), the additional feature of pairing-induced
decrease in synaptic efficacy emerges. The exact frequency at which this
feature emerges is dependent on the values of the synaptic parameters U and t_{rec} and also on the extent of
pairing-induced increase in U. Hebbian pairing accentuates the synaptic
depression through RSE, and this results in the exclusive enhancement of the
transmission of the first few APs in a train, while the transmission of subsequent
APs is depressed until steady-state is reached. Thus, pairing sharpens the time
window during which presynaptic stimulation is likely to induce postsynaptic
firing. The end of this window is sharply defined at moderate and high firing
rates by the decrease in synaptic efficacy to below pre-pairing levels, but not
at low firing rates.
4. DISCUSSION
The
characterization of FDP by Markram and Tsodyks (1996) has important
implications for neural models. Their results suggested that pairing-induced
synaptic potentiation is not a frequency-independent process in excitatory
synapses between TL5 neurons. The phenomenological model that Tsodyks and
Markram (1997) proposed for depressing synapses is reminiscent of the Liley and
North (1953) model, which also predicts FDP in a similar way, even though this
was not explicitly shown by Liley and North (1953). Grossberg and colleagues
(Fiala, Grossberg, and Bullock, 1996; Grossberg, 1987; Grossberg and Merrill,
1992, 1996; Grossberg and Schmajuk, 1989) have also developed a synaptic model
wherein a depressing variable, like R in (4), multiplies an associative
variable that is influenced by Hebbian pairing, like U in (4). Unlike in Equation (5), in their
model the rate of depression does not depend on the associative variable, but
the associative variable does depend on the rate of depression. This model was
used to explain data about adaptively timed learning processes.
The Markram and Tsodyks (1996) data
about FDP showed that paired-activity induces a decrease in steady-state
synaptic efficacy at high frequencies, as shown in Figure 2c. However, this
decrease was not further analyzed and was treated in Markram and Tsodyks (1996)
and in later studies (Markram et al., 1998a, 1998b, 1998c) as consistent with
no change at all. The current results suggest that the observed decrease was
significant and that it was due to the fact that FDP was not characterized at a
phase close to steady-state at high frequencies. This is because the duration
of the test stimuli, which consisted of six APs in Figure 2, was apparently not
long enough for the e.p.s.p amplitude to reach the same criterion fraction of
the steady-state level that was reached at low frequencies. These results also
explain why the Markram et al. (1998b) simulation of FDP (Figure 3) at
theoretical steady-state using the TM model deviates from the experimental
results of Markram and Tsodyks (1996) at high frequencies (Figure 2c). It is
also shown that steady-state may take a long time to settle at high frequencies
(Figure 6), raising the possibility that the change in steady-state synaptic
efficacy may not be a functionally relevant descriptive feature of FDP at those
frequencies.
An alternative characterization of
FDP is proposed in Figure 7, where the change in synaptic efficacy is
illustrated as a function of stimulus duration in number of APs and stimulus
frequency. Figure 7 illustrates the consequences of RSE as a function of
presynaptic firing rate. Since both potentiation and depression are observable
in Figure 7, it may be more appropriate to use the abbreviation FDP to mean
frequency-dependent synaptic plasticity instead of potentiation. The
characterization of FDP shown in Figure 7 reasserts the existence of the trend
suggested in Figures 1 and 2 that pairing selectively enhances the transmission
of early APs in a train. Such an enhancement may decrease the average delay
between the onset of a presynaptic spike train and the occurrence of the
postsynaptic APs elicited in response, by decreasing the scattering of the
induced postsynaptic spikes in time. This trend is observable at all
stimulation frequencies except below about 1 Hz in Figure 7.
The pairing-induced decrease in
synaptic efficacy that is observable at high frequencies is part of the same
trend and furthermore results in the exclusive enhancement of the transmission
of the first few APs in a train, while depressing the transmission of
subsequent APs during a frequency-dependent time interval. Thus, pairing
sharpens and narrows down the time window during which presynaptic activity is
likely to induce postsynaptic firing. Consequently, further increase in release
probability, which requires paired-activity, is less likely to be triggered
outside a time interval that immediately follows presynaptic activity. The
length of this time interval becomes progressively shorter after each pairing,
as suggested by Figure 8.
The findings of Markram and Tsodyks
(1996) and the current analysis of their results suggest that the excitatory
synapses between TL5 neurons directly participate in temporal signal processing
in cortical networks instead of acting as frequency-independent gain elements.
Current analysis suggest that paired-activity regulates synaptic transmission
not only at low frequencies but also at high frequencies. Pairing-induced
decrease in synaptic efficacy, which is a consequence of RSE, appears to have
an important role in controlling the timing of postsynaptic spiking driven by
presynaptic activity. These results encourage investigating neural models in
which global functional properties are obtained as a result of the
frequency-dependence of pairing-induced changes in synaptic efficacy. In
particular, the present results may clarify how certain cortical circuits can
rapidly synchronize their firing across spatially disjoint cell populations
(Brecht et al., 1998; Eckhorn et al., 1988, Gray and Singer, 1989; Grossberg
and Grunewald, 1997; Grossberg and Somers, 1991), including populations that
may be linked together by associative learning.
Appendix
Derivation
of Equation (6) from Equations (4) and (5):
Equation
(5) is repeated as Equation (A.1):
_{}. |
(A.1) |
This
equation is simplified by using the following notation:
_{} where _{}, _{}. |
(A.2) (A.3) (A.4) |
The
first AP occurs after the synapse has been at rest for a while. Therefore
immediately before the first AP, R = R_{1} = 1. Iterating (A.2) yields:
_{} |
(A.5) |
Thus R_{n} can be written as:
_{}. |
(A.6) |
(A.6)
is then simplified using the expressions for L and G
given in (A.3) and (A.4) to obtain (A.7):
_{}. |
(A.7) |
E_{n} = AUR_{n} is obtained
by multiplying (A.7) by AU.
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