Advances in Neural Information Processing Systems 17, 1409-1416 (2005)

Spike-Timing Dependent Plasticity and Mutual
Information Maximization for a Spiking Neuron
Model
Taro Toyoizumi†‡ ,Jean-Pascal Pfister‡Kazuyuki Aihara§ ∗,Wulfram Gerstner‡† Department of Complexity Science and Engineering,
The University of Tokyo, 153-8505 Tokyo, Japan‡ Ecole Polytechnique Fédérale de Lausanne (EPFL),
School of Computer and Communication Sciences andBrain-Mind Institute, 1015 Lausanne, Switzerland§ Graduate School of Information Science and Technology,
The University of Tokyo, 153-8505 Tokyo, Japantaro@sat.t.u-tokyo.ac.jp,
aihara@sat.t.u-tokyo.ac.jp,

jean-pascal.pfister@epfl.chwulfram.gerstner@epfl.ch

Abstract
We derive an optimal learning rule in the sense of mutual information
maximization for a spiking neuron model. Under the assumption ofsmall fluctuations of the input, we find a spike-timing dependent plasticity (STDP) function which depends on the time course of excitatory
postsynaptic potentials (EPSPs) and the autocorrelation function of thepostsynaptic neuron. We show that the STDP function has both positive
and negative phases. The positive phase is related to the shape of the
EPSP while the negative phase is controlled by neuronal refractoriness.

1

Introduction

Spike-timing dependent plasticity (STDP) has been intensively studied during the last
decade both experimentally and theoretically (for reviews see [1, 2]). STDP is a variantof Hebbian learning that is sensitive not only to the spatial but also to the temporal correlations between pre- and postsynaptic neurons. While the exact time course of the STDP
function varies between different types of neurons, the functional consequences of thesedifferences are largely unknown. One line of modeling research takes a given STDP ruleand analyzes the evolution of synaptic efficacies [3–5]. In this article, we take a different∗
Alternative address: ERATO Aihara Complexity Modeling Project, JST, 45-18 Oyama, Shibuyaku, 151-0065 Tokyo , Japanapproach and start from first principles. More precisely, we ask what is the optimal synaptic update rule so as to maximize the mutual information between pre- and postsynaptic
neurons.
Previously information theoretical approaches to neural coding have been used to quantify
the amount of information that a neuron or a neural network is able to encode or transmit [6–8]. In particular, algorithms based on the maximization of the mutual information
between the output and the input of a network, also called infomax principle [9], have beenused to detect the principal (or independent) components of the input signal, or to reduce
the redundancy [10–12]. Although it is a matter of discussion whether neurons simply’transmit’ information as opposed to classification or task-specific processing [13], strategies based on information maximization provide a reasonable starting point to constructneuronal networks in an unsupervised, but principled manner.
Recently, using a rate neuron, Chechik applied information maximization to detect static
input patterns from the output signal, and derived the optimal temporal learning window;
the learning window has a positive part due to the effect of the postsynaptic potential andhas flat negative parts with a length determined by the memory span [14].
In this paper, however, we employ a stochastic spiking neuron model to study not only
the effect of postsynaptic potentials generated by synaptic input but also the effect of therefractory period of the postsynaptic neuron on the shape of the optimal learning window.
We discuss the relation of mutual information and Fisher information for small input variance in Sec. 2. Optimization of the Fisher information by gradient ascent yields an optimallearning rule as shown in Sec. 32Model assumptions

2.1

Neuron model

The model we are considering is a stochastic neuron with refractoriness. The instantaneous
firing rate ρ at time t depends on the membrane potential u(t) and refractoriness R(t):(1)ρ(t) = g(βu(t))R(t),where g(βu) = g0 log2 [1+eβu ] is a smoothed piecewise linear function with a scaling varit̂−τabs )2
Θ(t −able β and a constant g0 = 85Hz. The refractory variable is R(t) = τ 2 (t−+(t−t̂−τ )2
refr

abs

t̂ − τabs ) and depends on the time elapsed since the last firing time t̂, the absolute refractory period τabs = 3 ms, and the time constant of relative refractoriness τrefr = 10 ms. The
Heaviside step function Θ takes a value of 1 for positive arguments and zero otherwise.
The postsynaptic potential depends on the input spike trains of N presynaptic neurons. A
presynaptic spike of neuron i ∈ {1, 2, . . . , N } emitted at time tfi evokes a postsynaptic
potential with time course ²(t − tfi ). The total membrane potential is
u(t)

=

N
X
i=1

wi

X
f

²(t − tfi ) =

N
X
i=1

wi

Z

²(s)xi (t − s)ds(2)

P
where xi (t) = f δ(t − tfi ) denotes the spike train of the presynaptic neuron i. The above
model is a special case of the spike response model with escape noise [2]. For vanishingrefractoriness τrefr → 0 and τabs → 0, the above model reduces to an inhomogeneous
Poisson process.
For a given set of presynaptic spikes in an interval [0, T ], hence for a given time course ofmembrane potential {u(t)|t ∈ [0, T ]}, the model generates an output spike trainXy(t) =
δ(t − tf )(3)fwith firing times {tf |f = 1, . . . , n} with a probability density"Z#
T
P (y|u) = exp(y(t) log ρ(t) − ρ(t)) dt .

(4)0where ρ(t) is given by Eq. (1), i.e., ρ(t) = g(βu(t)) R(t). Since the refractory variable R
depends on the firing time t̂ of the previous output spike, we sometimes write ρ(t|t̂) insteadof ρ(t) in order to make this dependence explicit.
Equation (4) can then be re-expressed inR− t̂t ρ(s|t̂)ds
and the interval distribution Q(t|t̂) =terms of the survivor function S(t|t̂) = eρ(t|t̂)S(t|t̂) in a more transparent form:nYQ(tf |tf −1 ) S(T |tn ),P (y|u) = 
(5)
f =1

where t0 = 0 and n is the number of postsynaptic spikes in [0, T ]. In words, the probability
that a specific output spike train y occurs can be calculated from the interspike intervals
Q(tf |tf −1 ) and the probability that the neuron ‘survives’ from the last spike at time tn totime T without further firing.

2.2

Fisher information and mutual information

Let us consider input spike trains with stationary statistics. These input spike trains generate
an input potential u(t) with an average value u0 and standard deviation σ. Assuming aweak dependence of g on the membrane potential u, i.e., for small β, we expand g aroundg0 = g(0) to obtain g(βu(t)) = g0 + g00 βu(t) + g000 [βu(t)]2 /2 + O(β 3 ) where g0 is thevalue of g in the absence of input and the next terms describe the influence of the input.
Here and in the following, all calculations will be done to order β 2 .
In the limit of small β, the mutual information is given by [15]
Z T
Z
β2 T
I(Y ; X) =dt0 Σ(t − t0 )J0 (t − t0 ) + O(β 3 ),
dt
2 0
0

(6)

with the autocovariance function of the membrane potential
Σ(t − t0 )=h∆u(t)∆u(t0 )iX ,(7)with ∆u(t) = u(t) − u0 and Fisher information*+¯∂ 2 log P (y|u) ¯¯0J0 (t − t ) = −∂βu(t)∂βu(t0 ) ¯β=0,(8)

Y |β=0with h·iY |β=0 = · P (y|β = 0)dy and h·iX = · P (x)dx. Note that the Fisher
information (8) is to be evaluated at the constant g0 , i.e., at the value βu = 0, whereas
the autocovariance in Eq. (7) is calculated with respect to the mean membrane potentitalu0 = hu(t)iX which is in general different from zero. The derivation of (6) is basedon the assumption that the variability of the output signal is small and g(βu) does notdeviate much from g0 , i.e., it corresponds to the regime of small signal-to-noise ratio.
It is well known that the information capacity of the Gaussian channel is given by thelog of the signal-to-noise ratio [16], and the mutual information is proportional to theRRsignal-to-noise ratio when it is small. The relation between the Fisher information, themutual information, and optimal tuning curves has previously been established in theregime of large signal-to-noise ratio [17].
We introduce the following notation: Let µ0 = hy(t)iY |β=0 = hρ(t)iY |β=0 be the spon00taneous firing rate in the absence of input and µ−10 hy(t)y(t )iY |β=0 = δ(t − t ) + µ0 [1 +0
φ(t − t )] be the postsynaptic firing probability at time t given a postsynaptic spike at t 0 ,i.e., the autocorrelation function of Y . From the theory of stationary renewal processes [2]
·Z
¸−1
µ0 =
s Q0 (s)ds,Zµ0 [1 + φ(s)] = Q0 (|s|) + Q0 (s0 )µ0 [1 + φ(|s| − s0 )] Θ(|s| − s0 )ds0 ,(9)where Q0 (s) = g0 R(s)e−g0 [(s−τabs )−τrefr arctan(s−τabs )/τrefr ] is the interval distribution for
constant g = g0 . The interval distribution vanishes during the absolute refractory time τabs ;cf. Fig. 1.
(A)

(B)

0.05

0.2

0.04

0

PSfrag replacements

−0.2

φ(s)

Q0 (s)

0.03
0.02

−0.6

0.01

PSfrag replacements −0.8

0

φ(s)

−0.01
0

−0.4

20

40

60

s [ms]80

Q0 (s)
100

−1
0

10

20

30

40

50

s [ms]

Figure 1: Interspike interval distribution Q0 and normalized autocorrelation function φ.
The circles show numerical results, the solid line the theory.
The Fisher information of (8) is calculated from (4) to beµ 0 ¶2g0
J0 (t − t0 ) = δ(t − t0 )hρ0 (t)iY |β=0
g0with the instantaneous firing rate ρ0 (t) = g0 R(t). Hence the mutual information isµ ¶2 Z T
β 2 g00
dt µ0 σ 2I(Y ; X) =
2 g0
0
µ ¶2
β 2 g00
T µ0 σ 2 .
=
2 g0

(10)

(11)
(12)

For an interpretation of Eq. (11) we note that σ 2 = Σ(0) is the variance of the membrane potential and depends on the statistics of the presynaptic input whereas µ 0 is the
spontaneous firing rate which characterizes the output of the postsynaptic neuron. Hence,
Equation (11) contains both pre- and postsynaptic factors.

3

Results: Optimal spike-timing dependent learning rule

In the previous section we have calculated the mutual information between presynaptic
input spike trains and the output of the postsynaptic neuron under the assumption of small

fluctuations of g. The mutual information depends on parameters of the model neuron, inparticular the synaptic weights that characterize the efficacy of the connections between
pre- and postsynaptic neurons. In this section, we will optimize the mutual informationby changing the synaptic weights in an appropriate fashion. To do so we will proceed inseveral steps.
First, based on gradient ascent we derive a batch learning rule of synaptic weights thatmaximizes the mutual information. In a second step, we transform the batch rule into an
online rule that reduces to the batch version when averaged. Finally, in subsection 3.2, wewill see that the online learning rule shares properties with STDP, in particular a biphasic
dependence upon the relative timing of pre- and postsynaptic spikes.
3.1

Learning rule for spiking model neuron

In order to keep the analysis as simple as possible, we suppose that the input spike trains
are independent Poisson trains, i.e., h∆xi (t)∆xj (t0 )iX = νi δ(t − t0 )δij , where ∆xi (t) =xi (t) − νi with rate νi = hxi (t)iX . Then we obtain the variance of the membrane potentialX
wj2 νj(13)
σ 2 = h[∆u(t)]2 iX = ²2jwith ²2 =R

2² (s)ds.

Applying gradient ascent to (11) with an appropriate learning rate α, we obtain the batchlearning rule of synaptic weights asµ ¶2 Z T∂σ 2∂I(Y ; X)β 2 g00
dt µ0∆wi = α
≈α
.
(14)
∂wi
2 g0
∂wi
0
The derivative of µ0 with respect to wi vanishes, since µ0 is the spontaneous firing rate in
the absence of input. We note that both µ0 and σ 2 are defined by an ensemble averages, as
is typical for a ‘batch’ rule.
While there are many candidates of online learning rule that give (14) on average, we
are interested in rules that depend directly on neuronal spikes
P rather than mean rates. To
2
2
proceeditisusefultowrite
σ=h[∆u(t)]
i
with
∆u
=
X
i wi ∆²i (t) where ∆²i (t) =R²(s)∆xi (t − s)ds. In this notation, one simple form of an online learning rule that
depends on both the postsynaptic firing statistics and presynaptic autocorrelation isµ 0 ¶2dwi
g0y(t)∆²i (t)∆u(t),(15)= αβ 2dt
g0Hence weights are updated with each postsynaptic spike with an amplitude proportional
to an online estimate of the membrane potential variance calculated as the product of∆u and ∆²i . Indeed, to order β 0 , the input and the output spikes are independent;hy(t)∆²i (t)∆u(t)iY,X = hy(t)iY |β=0 h∆²i (t)∆u(t)iX and the average of (15) leads backto (14).
3.2

STDP function as a spike-pair effectApplication of the online learning rule (15) during a trial of duration T , yields a totalchange of the synaptic efficacy which depends on all the presynaptic spikes via the factor∆²i ; on the postsynaptic potential via the factor ∆u; and on the postsynaptic spike trainy(t). In order to extract the spike pair effect evoked by a given presynaptic spike at t preiand a postsynaptic spike at tpost , we average over x and y given the pair of spikes. The
spike pair effect up to the second order of β is therefore described asµ 0 ¶2 Z T
g0
pre
post
2
∆wi (t− ti ) = αβ, (16)
dthy(t)iY |tpost ,β=0 h∆²i (t)∆u(t)iX|tpre
i
g0
0

R
R
= dx · P (x|tpre
where h·iY |tpost ,β=0 = dy · P (y|tpost , β = 0) and h·iX|tpre
i ).
iNote that the leading factor of Eq. (16) is already of order β 2 , so that all other factors have to be evaluated to order β 0 . Suppressing all terms containing β, we obtainpost
P (y|tpost , u) ≈ P (y|tpost , β = 0) and from the Bayes formula P (x|tpre) =i ,tP (tpost |x,tpre)
prei
P (x|tprei ) ≈ P (x|ti ).
hP (tpost |x,tpre)iX|tpreii

In order to see the contribution of tpre
and tpost , we think of separating the effects caused byipost
spikes at tpre
,
t
from
the
mean
weightevolution caused by all other spikes. Thereforei=we insert hy(t)iY |tpost ,β=0 = δ(t−tpost )+µ0 [1+φ(t−tpost )] and h∆²i (t)∆u(t)iX|tpreiwi [²2 (t − tpre ) + ²2 νi ] into Eq. (16) and decompose ∆wi (tpost − tpre
)
into
the
followingi³ 0 ´20
2 g0four terms: the drift term ∆wi = αβ g0 T µ0 ²2 wi νi of the batch learning (14) that
³ 0 ´2
pre
post
2 g0
µ0 ²2 widoes not depend on tpre
or
t
;
the
presynaptic
component
∆w
=
αβ
i
i
g0

post
=
that is triggered by the presynaptic spike at tpre
i ; the postsynaptic component ∆wi³ 0 ´2 h
i
R
T
2 g0
post
αβ g0
)dt ²2 wi νi that is triggered by the postsynaptic spike
1 + µ0 0 φ(t − t

at tpost ; and the correlation component#µ 0 ¶2 "
Z T
g0
pre
pre
corr
2
2 postpost 2wi ² (t∆wi= αβ− ti ) + µ 0
φ(t − t)² (t − ti )dt (17)g00that depends on the difference of the pre- and postsynaptic spike timing.
(A)

(B)

(C)

0.05

15

−5

²2 (s)

tpost − tpre
i [ms]2µ0 (φ ∗ ² )(s)∆wicorr−0.1
tpost − tpre
i [ms]²2 (s) −0.15

0.4
0.2
0
−50

0

PSfrag replacements −0.05−250s [ms]

25∆w50icorr

−0.2
−50

PSfrag replacementss [ms]∆wicorr

0.8

PSfrag replacements 0.6µ0 (φ ∗ ²2 )(s)

1

−25

0

10

5

²2 (s) 0µ0 (φ ∗ ²2 )(s) −5

s [ms]

25

50

x 10

−50

−25

post

0

25

50

− tprei [ms]t
Figure 2: (A) The effect from EPSP: the first term in the square bracket of (17). (B) The
effect from refractoriness: the second term in the square bracket of (17). (C) Temporallearning window ∆wicorr of (17).
In the following, we choose a simple exponential EPSP ²(t) = Θ(s)e−s/τu with a time
constant τu = 10 ms. The parameters are N = 100, νi = 40 Hz for all i, wi = (N τu νi )−1 ,α = 1 and β = 0.1.

Figure 2 shows ∆wicorr of (17). The first term of (17) indicates the contribution of apresynaptic spike at tipre to increase the online estimation of membrane potential varianceat time tpost , whereas the second term represents the effect of the refractory period on
postsynaptic firing intensity, i.e., the normalized autocorrelation function convolved withinthe presynaptic contribution term. Due to the averaging of h·iY |tpost ,β=0 and h·iX|tpre
i(16), this optimal temporal learning window is local in time; we do not need to impose amemory span [14] to restrict the negative part of the learning window.
Figure 3 compares ∆wi of (16) with numerical simulations of (15). We note a good agreement between theory and simulation. We recall, that all calculations, and hence the STDP
function of (17) are valid for small β, i.e., for small fluctuation of g.

−4

3.6

x 10

3.4
3.2

∆wi

3
2.8
2.6
2.4

PSfrag replacements
tpost − tpre
i [ms]

2.2
−50

−25

0

25

50

tpost − tpre
i [ms]
Figure 3: The comparison of the analytical result of (16) ( solid line ) and the numericalsimulation of the online learning rule (15) ( circles ). For the simulation, the conditionaldwi
average h∆wi iX,Y |tpre,tpost is evaluated by integrating dt over 200 ms around spike pairsiprewith the given interval tpost − ti ;4

Conclusion

It is important for neurons especially in primary sensory systems to send information fromprevious processing circuits to neurons in other areas while capturing the essential featuresof its input. Mutual information is a natural quantity to be maximized from this perspective. We introduced an online learning rule for synaptic weights that increases informationtransmission for small input fluctuation. Introduction of the temporal properties of the target neuron enables us to analyze the temporal properties of the learning rule required toincrease the mutual information. Consequently, the temporal learning window is given interms of the time course of EPSPs and the autocorrelation function of the postsynaptic neuron. In particular, neuronal refractoriness plays a major role and yields the negative partof the learning window. Though we restrict our analysis here to excitatory synapses withindependent spike trains, it is straightforward to generalize the approach to a mixture of excitatory and inhibitory neurons with weakly correlated spike trains as long as the synaptic
weights are small enough. The analytically derived temporal learning window is similar to
the experimentally observed bimodal STDP window [1]. Since the effective time courseof EPSPs and the autocorrelation function of output spike trains vary from one part ofthe brain to another, it is important to compare those functions with the temporal learning
window in biological settings.
Acknowledgments
T.T. is supported by the Japan Society for the Promotion of Science and a Grant-in-Aid for
JSPS Fellows; J.-P.P. is supported by the Swiss National Science Foundation. We thank Y.
Aviel for discussions.

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