Temporally Asymmetric Hebbian Learning, 
Spike Timing and Neuronal Response Variability 
L.F. Abbott and Sen Song 
Volen Center and Department of Biology 
Brandeis University 
Waltham MA 02454 
Abstract 
Recent experimental data indicate that the strengthening or weakening of 
synaptic connections between neurons depends on the relative timing of 
pre- and postsynaptic action potentials. A Hebbian synaptic modification 
rule based on these data leads to a stable state in which the excitatory and 
inhibitory inputs to a neuron are balanced, producing an irregular pattern 
of firing. It has been proposed that neurons in vivo operate in such a 
mode. 
1 Introduction 
Hebbian modification of network interconnections plays a central role in the study of learn- 
ing in neural networks (Rumelhart and McClelland, 1986; Hertz et al., 1991). Most work 
on Hebbian learning involves network models in which the activities of the individual units 
are represented by continuous variables. A Hebbian learning rule, in this context, is spec- 
ified by describing how network weights change as a function of the activities of the units 
that transmit and receive signals across a given network connection. While analyses of 
Hebbian learning along these lines have provided important results, direct application of 
these ideas to neuroscience is hindered by the fact that real neurons cannot be adequately 
described by continuous activity variables such as firing rates. Instead, the inputs and out- 
puts of neurons are sequences of action potentials or spikes. All the information conveyed 
by one neuron to another over any appreciable distance is carried by the temporal patterns 
of action potential sequences. Rules by which synaptic connections between real neurons 
are modified in a Hebbian manner should properly be expressed as functions of the relative 
timing of the action potentials fired by the input (presynaptic) and output (postsynaptic) 
neurons. Until recently, little information has been available about the exact dependence of 
synaptic modification on pre- and postsynaptic spike timing (see however, Levy and Stew- 
ard, 1983; Gustafsson et al., 1987). New experimental results (Markram et al., 1997; Bell 
et al., 1997; Debanne et al., 1998; Zhang et al., 1998; Bi and Poo, 1999) have changed 
70 L. F. Abbott and S. Song 
this situation dramatically, and these allow us to study Hebbian learning in a manner that 
is much more realistic and relevant to biological neural networks. The results may find 
application in artificial neural networks as well. 
2 Temporally Asymmetric LTP and LTD 
The biological substrate for Hebbian learning in neuroscience is provided by long-term 
potentiation (LTP) and long-term depression (LTD) of the synaptic connections between 
neurons (see for example, Malenka and Nicoll, 1993). LTP is a long-lasting strength- 
ening of synaptic efficacy associated with paired pre- and postsynaptic activity. LTD is 
a long-lasting weakening of synaptic strength. In recent experiments on neocortical slices 
(Markram et al., 1997), hippocampal cells in culture (Bi and Poo, 1999), and in vivo studies 
of tadpole tectum (Zhang et al., 1998), induction of LTP required that presynaptic action 
potentials preceded postsynaptic firing by no more than about 20 ms. Maximal LTP oc- 
curred when presynaptic spikes preceded postsynaptic action potentials by less than a few 
milliseconds. If presynaptic spikes followed postsynaptic action potentials, long-term de- 
pression rather than potentiation resulted. These results are summarized schematically in 
Figure 1. 
tpre- tpost (ms) 
Figure 1: A model of the change in synaptic strength A7 produced by paired pre- and postsynaptic 
spikes occurring at times tpre and tpost respectively. Positive changes correspond to LTP and negative 
to LTD. There is an abrupt transition at tpre - tpost = 0. The units for A7 are arbitrary in this figure, 
but data indicate a maximum change of approximately 0.5 % per spike pain 
The curve in Figure 1 is a caricature used to model the weight changes arising from pairings 
of pre- and postsynaptic action potentials separated by various intervals of time. This curve 
resembles the data from all three preparations discussed above, but a couple of assump- 
tions have been made in its construction. The data indicate that there is a rapid transition 
from LTP to LTD depending on whether the time difference between pre- and postsynaptic 
spiking is positive or negative, but the existing data cannot resolve exactly what happens 
at the transition point. We have assumed that there is a discontinuous jump from LTP to 
LTD at this point. In addition, we assume that the area under the LTP side of the curve is 
slightly less than the area under the LTD side. In Figure 1, this difference is imposed by 
making the magnitude of LTD slightly greater than the magnitude of LTP, while both sides 
of the curve have equal exponential fall-offs away from zero time difference. Alternately, 
we could have given the LTD side a slower exponential fall-off and equal amplitude. The 
data do not support either assumption unambiguously, nor do they indicate which area is 
larger. The assumption that the area under the LTD side of the curve is larger than that un- 
der the LTP side is critical if the resulting synaptic modification rule is to be stable against 
uncontrolled growth of synaptic strengths. 
Hebb (1949) postulated that a synapse should be strengthened when the presynaptic neuron 
is frequently involved in making the postsynaptic neuron fire an action potential. Causality 
is an important element in Hebb's statement; synaptic potentiation should occur only if 
there is a causal relationship between the pre- and postsynaptic spiking. The LTP/LTD rule 
summarized in Figure 1 imposes causality through a tight timing requirement. The narrow 
Hebbian Learning and Response Variability 71 
windows for LTP and LTD seen in the data, and the abrupt transition from potentiation to 
depression near zero separation between pre- and postsynaptic spike times impose a strict 
causality condition for LTP induction. 
3 Response Variability 
What are the implications of the synaptic modification rule summarized in Figure 1 ? To ad- 
dress this question, we introduce another topic that has been discussed extensively within 
the computational neuroscience community in recent years, the origin of response vari- 
ability (Softky and Koch, 1992 & 1994; Shadlen and Newsome, 1994 & 1998; Tsodyks 
and Sejnowski, 1995; Amit and Brunel, 1997; Troyer and Miller, 1997a & b; Bugmann 
et al., 1997; van Vreeswijk and Sompolinsky, 1996 & 1998). Neurons can respond to 
multiple synaptic inputs in two different modes of operation. Figure 2 shows membrane 
potentials of a model neuron receiving 1000 excitatory and 200 inhibitory synaptic inputs. 
Each input consists of an independent Poisson spike train driving a synaptic conductance. 
The integrate-and-fire model neuron used in this example integrates these synaptic conduc- 
tances as a simple capacitor-resistor circuit. To generate action potentials in this model, 
we monitor the membrane potential and compare it to a threshold voltage. Whenever the 
membrane potential reaches the threshold an action potential is "pasted" onto the mem- 
brane potential trace and the membrane potential is reset to a prescribed value. 
A B 
-54 ................... g -.?,4 ......... 
> -s6 
-s8 > -se 
250 500 750 1000 250 500 750 tOO0 
50 100 150 200 250 500 750 
t (ms) t (ms) 
lOOO 
Figure 2: Regular and irregular firing modes of a model integrate-and-fire neuron. Upper panels 
show the model with action potentials deactivated, and the dashed lines show the action potential 
threshold. The lower figures show the model with action potentials activated. A) In the regular firing 
mode, the average membrane potential without spikes is above threshold and the firing pattern is fast 
and regular (note the different time scale in the lower panel). B) In the irregular firing mode, the 
average membrane potential without spikes is below threshold and the firing pattern is slower and 
irregular. 
Figures 2A and 2B illustrate the two modes of operation. The upper panels of Figure 2 show 
the membrane potential with the action potential generation mechanism of the model turned 
off, and the lower panels show the membrane potential and spike sequences that result when 
the action potential generation is turned on. In Figure 2A, the effect of the excitatory inputs 
is strong enough relative to that of the inhibitory inputs so that the average membrane 
potential, when action potential generation is blocked, is above the spike threshold of the 
model. When the action potential mechanism is turned back on (lower panel of Figure 
2A), this produces a fairly regular pattern of action potentials at a relatively high rate. 
The total synaptic input attempts to charge the neuron above the threshold, but every time 
the potential reaches the threshold it gets reset and starts charging again. In this regular 
72 L. F. Abbott and S. Song 
firing mode of operation, the timing of the action potentials is determined primarily by the 
charging rate of the cell, which is controlled by its membrane time constant. Since this does 
not vary as a function of time, the firing pattern is regular despite the fact that the synaptic 
input is varying. 
Figure 2B shows the other mode of operation that produces an irregular firing pattern. In 
the irregular firing mode, the average membrane is more hyperpolarized than the threshold 
for action potential generation (upper panel of Figure 2B). In this mode, action potentials 
are only generated when there is a fluctuation in the total synaptic current strong enough to 
make the membrane potential cross the threshold. This results in slower and more irregular 
firing (lower panel of Figure 2B). The irregular firing mode has a number of interesting 
features (Shadlen and Newsome, 1994 & 1998; Tsodyks and Sejnowski, 1995; Amit and 
Brunel, 1997; Troyer and Miller, 1997a & b; Bugmann et al., 1997; van Vreeswijk and 
Sompolinsky, 1996 & 1998). First, it generates irregular firing patterns that are far closer 
to the firing patterns seen in vivo than the patterns produced in the regular firing mode. 
Second, responses to changes in the synaptic input are much more rapid in this mode, being 
limited only by the synaptic rise time rather than the membrane time constant. Finally, the 
timing of action potentials in the irregular firing mode is related to the timing of fluctuations 
in the synaptic input rather than being determined primarily by the membrane time constant 
of the cell. 
^ B 
' 1.2 ' 1.2 
{3' {3' 
.>_ 
o. o 
-20 -10 0 10 20 -20 -10 0 10 20 
tpre - tpost (ms) tpre - tpost (ms) 
Figure 3: Histograms indicating the relative probability of finding pre- and postsynaptic spikes 
separated by the indicated time interval. A) Regular firing mode. The probability is essentially flat 
and at the chance level of one. B) Irregular firing mode. There is an excess of presynaptic spike 
shortly before a postsynaptic spike. 
An important difference between the regular and irregular firing modes is illustrated in the 
cross-correlograms shown in Figure 3 (Troyer and Miller, 1997b; Bugmann et al. 1997). 
These indicate the probability that an action potential fired by the postsynaptic neuron is 
preceded or followed by an presynaptic spike separated by various intervals. The histogram 
has been normalized so its value for pairings that are due solely to chance is one. The 
histogram when the model is in the regular firing mode (Figure 3A) takes a value close to 
one for almost all input-output spike time differences. This is a reflection of the fact that the 
timing of individual action potentials in the regular firing mode is relatively independent 
of the timing of the presynaptic inputs. In contrast, the histogram for a model neuron in 
the irregular firing mode (Figure 3B) shows a much larger excess of presynaptic spikes 
occurring shortly before the postsynaptic neuron fires. This excess reflects the fluctuations 
in the total synaptic input that push the membrane potential up to the threshold and produce 
a spike in the irregular firing mode. It indicates that, in this mode, there is a tight temporal 
correlation between the timing of such fluctuations and output spikes. 
For a neuron to operate in the irregular firing mode, there must be an appropriate balance 
between the strength of its excitatory and inhibitory inputs. The excitatory input must be 
weak enough, relative to the inhibitory input, so that the average membrane potential in the 
absence of spikes is below the action potential threshold to avoid regular firing. However, 
excitatory input must be sufficiently strong to keep the average potential close enough to 
Hebbian Learning and Response Variability 73 
the threshold so that fluctuations can reach it and cause the cell to fire. How is this balance 
achieved? 
4 Asymmetric LTP/LTD Leads to an Irregular Firing State 
A comparison of the LTP/LTD synaptic modification rule illustrated in Figure 1, and the 
presynaptic/postsynaptic timing histogram shown in Figure 3, reveals that a temporally 
asymmetric synaptic modification rule based on the curve in Figure 1 can automatically 
generate the balance of excitation and inhibition needed to produce an irregular firing state. 
Suppose that we start a neuron model in a regular firing mode by giving it relatively strong 
excitatory synaptic strengths. We then apply the LTP/LTD rule of Figure 1 to the excitatory 
synapse while holding the inhibitory synapse at constant values. Recall that Figure 1 has 
been adjusted so that the area under the LTD part of the curve is greater than that under the 
LTP part. This means that if there is an equal probability of a presynaptic spike to either 
precede or follow a postsynaptic spike the net effect will be a weakening of the excitatory 
synapses. This is exactly what happens in the regular firing mode, where the relationship 
between the timing of pre- and postsynaptic spikes is approximately random (Figure 3A). 
As the LTP/LTD rule weakens the excitatory synapses, the average membrane potential 
drops and the neuron enters the irregular firing mode. In the irregular firing mode, there is 
a higher probability for a presynaptic spike to precede than to follow a postsynaptic spike 
(Figure 3B). This compensates for the fact that the rule we use produces more LTD than 
LTP. Equilibrium will be reached when the asymmetry of the LTP/LTD modification curve 
of Figure 1 is matched by the asymmetry of the presynaptic/postsynaptic timing histogram 
of Figure 3B. The equilibrium state corresponds to a balanced, irregular firing mode of 
operation, and it is automatically produced by the temporally asymmetric learning rule. 
Figure 4A shows a transition from a regular to an irregular firing state mediated by the tem- 
porally asymmetric LTP/LTD modification rule. The irregularity of the postsynaptic spike 
train has been quantified by plotting the coefficient of variation (CV), the standard devia- 
tion over the mean of the interspike intervals, of the model neuron as a function of time. 
Initially, the neuron was in a regular firing state with a low CV value. After the synaptic 
modification rule reached an equilibrium state, the CV took a value near one indicating that 
the neuron has been transformed into an irregular firing mode. The solid curve in Figure 4B 
shows that temporally asymmetric LTP/LTD can robustly generate irregular output firing 
for a wide range of input firing rates. 
1.2- 
1.0- 
> 0.6- 
o 
' 04- 
._e 
(,-) 
'"-- 0.2- 
o 
0.0 
0.8- 
0,6- 
04- 
0.2- 
0 0 I0 0 0 5 
time steps input rate (Hz) 
Figure 4: Coefficient of variation (CV) of the output spike train of the model neuron. A) Transi- 
tion from a regular to an irregular firing state as temporally asymmetric LTP/LTD modifies synaptic 
strengths. The units of time in this plot'are arbitrary because they depend on the magnitude of LTP 
and LTD used in the model. B) Equilibrium CV values as a function of the firing rates of excitatory 
inputs to the model neuron. The solid curve gives the results when temporally asymmetric LTP/LTD 
is active. The dashed curve shows the results if the synaptic strengths that arose for 5 Hz inputs are 
left unmodified. 
74 L. F. Abbott and S. Song 
5 Discussion 
Temporally asymmetric LTP/LTD provides a Hebbian-type learning rule with interesting 
properties (Kempter et al., 1998). In the past, temporally asymmetric Hebbian learning 
rules have been studied and applied to problems of temporal sequence generation (Manai 
and Levy, 1993), navigation (Blum and Abbott, 1996; Gerstner and Abbott, 1997), motor 
learning (Abbott and Blum, 1996), and detection of spike synchrony (Gerstner et al., 1996). 
In these studies, two different LTP/LTD window sizes were assumed: either of order 100 ms 
(Manai and Levy, 1993; Blum and Abbott, 1996; Gerstner and Abbott, 1997; (Abbott and 
Blum, 1996) or around 1 ms (Gerstner et al., 1996). The new data (Markram et al., 1997; 
Bell et al., 1997; Zhang et al., 1998; Bi and Poo, 1999) give a window size of order 10 
ms. For a 1 ms window size, temporally asymmetric LTP/LTD is sensitive to precise spike 
timing. When the window size is of order 100 ms, changes in stimuli or motor actions on a 
behavioral level become relevant for LTP and LTD. A window size of 10 ms, as supported 
by the recent data, suggests that LTP and LTD are sensitive to firing correlations relevant 
to neuronal circuitry, such as input-output correlations, which vary over this time scale. 
Temporally asymmetric LTP/LTD has some interesting properties that distinguish it from 
Hebbian learning rules based on correlations or covariances in pre- and postsynaptic rates. 
We have found that the rule used here is not sensitive to input firing rates or to variability 
in input rates. If we split the excitatory inputs of the model into two groups and give these 
two input sets different rates, we see no difference in the distribution of synaptic strengths 
arising from the learning rule. Similarly, if one group is given a steady firing rate and 
the other group has firing rates that vary in time, no difference in synaptic strengths is 
apparent. The most effective way to induce LTP in a set of inputs is to synchronize some of 
their spikes. Inputs with synchronized spikes are slightly more effective at firing the neuron 
than unsynchronized spikes. This means that such inputs will preceded postsynaptic spikes 
more frequently and thus will get stronger. This suggests that spike synchrony may be a 
signal that marks a set of inputs for learning. Even when this synchrony has no particular 
functional effect, so that it has little impact on the firing pattern of the postsynaptic neuron, 
it can lead to dramatic shifts in synaptic strength. Thus, spike synchronization may be a 
mechanism for inducing LTP and LTD. 
Acknowledgments 
Research supported by the National Science Foundation (DMS-9503261), the Sloan Cen- 
ter for Theoretical Neurobiology at Brandeis University, a Howard Hughes Predoctoral 
Fellowship, and the W.M. Keck Foundation. 
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