Proc Natl Acad Sci USA 102(14):5239-44 (2005).
Generalized Bienenstock-Cooper-Munro rule for spiking neurons that maximizes information transmission.
Taro Toyoizumi, Jean-Pascal Pfister, Kazuyuki Aihara, and Wulfram GerstnerSupporting Text
Evolution of the Average Synaptic Update RuleIn this appendix we evaluate the derivative of Eq. 9 in the main text,i.e., we need to calculateÀ¿∂P (y k |Y k−1 )P (y k |Y k−1 , X k ).
(1)− γ loglog∂wj
P (y k |Y k−1 )P̃ (y k |Y k−1 ) Y k ,X kBefore we start let us recall some notation. The average of an arbitrary
function fw with arguments x and y is by definition
XXhfw (x, y)ix,y =pw (x, y) fw (x, y)(2)
xywhere pw (x, y) denotes the joint probability of the pair (x, y) to occur andthe sum runs over all configurations of x and y. The subscript w indicatesthat both the probability distribution pw and the function fw may depend
on a parameter w.
By definition, we have pw (x, y) = pw (y|x)p(x) where p(x) is a given input
distribution and pw (y|x) the (parameter-dependent) conditional probabilityof generating an output y given x. Hence Eq. 2 can be transformed into*+XXX
hfw (x, y)ix,y =p(x)pw (y|x) fw (x, y) =pw (y|x) fw (x, y)
(3)
xyyx

If we now take the derivative with respect to the parameter w, the productrule yields two terms*+X∂∂
hfw (x, y)ix,y =fw (x, y)(4)
pw (y|x)∂w∂w
yx*+·¸X∂
pw (y|x)+log pw (y|x) fw (x, y)∂w
yx

The first term contains the derivative of the function fw whereas the second
term contains the derivative of the conditional probability pw . We note thatEq. 4 can also be written in the form
¿
À¿·
¸À
∂
∂
∂
hfw (x, y)ix,y =fw (x, y)+log pw (y|x) fw (x, y), (5)∂w∂w∂w
x,yx,y1

i.e., as an average over the joint distribution of x and y. This formulation
will be useful for the problem at hand.
The gradient in Eq. 1 contains several terms and for the moment wepick only one of these.
­ The others will ®then be treated analogously. Letus focus on the term log P (y k |Y k−1 , X k ) Y k ,X k and apply steps completely
analogous to those leading from Eqs. 2-5.
®
∂ ­log P (y k |Y k−1 , X k ) Y k ,X k∂wj
¿À
∂
kk−1
k=log P (y |Y,X )∂wj
Y k ,X k¿·¸À∂kkkk−1k+log P (Y |X ) log P (y |Y,X )∂wj
Y k ,X k(6)We now evaluate the averages using the identity
h·iY k ,X k = hh·iyk |Y k−1 ,X k iY k−1 ,X k . We find that the first term on the righthand side of Eq. 6 vanishes, sinceÀ
¿∂kk−1
klog P (y |Y,X )∂wj
yk |Y k−1 ,X kX ∂ £¤=log P (y k |Y k−1 , X k ) P (y k |Y k−1 , X k )∂wj
y k ∈{0,1}X∂ P (y k |Y k−1 , X k ) = 0(7)=∂wj ky ∈{0,1}because of the normalizationDof probabilities. TheE same argument can be∂
kk−1
repeated to show that 0 = ∂wj log P (y |Y). The reference
yk |Y k−1 ,X kkk−1

distribution P̃ (y |Y) is by definition independent of wj .
Hence the only term that gives a non-trivial contribution on the righthand side of Eq. 6 is the second term. With an analogous argument for theother factors in Eq. 1 we have¿ÀP (y k |Y k−1 , X k )P (y k |Y k−1 )∂log− γ log∂wjP (y k |Y k−1 )P̃ (y k |Y k−1 ) Y k ,X k¶À¿·¸µ∂ log P (Y k |X k )P (y k |Y k−1 )P (y k |Y k−1 , X k )− γ log=(8)log∂wj
P (y k |Y k−1 )P̃ (y k |Y k−1 )
kk
Y ,X2

An identification of the factors C, F , and G in the main text is straightforward. From Eq. 4 in the main text we havelog P (y k |Y k−1 , X k ) = y k log(ρk ) + (1 − y k ) log(1 − ρk )(9)Hence we can evaluate the factorsFkGkρk1 − ρkP (y k |Y k−1 , X k )kk= y log k +(1−y ) log= log
P (y k |Y k−1 )ρ̄1 − ρ̄kP (y k |Y k−1 )1 − ρ̄kρ̄k= log= y k log +(1−y k ) logρ̃1 − ρ̃P̃ (y k |Y k−1 )Furthermore we can calculate the derivative needed in Eq. 8 using the chain
rule from Eq. 6 of the main text, i.e.,kk

P (Y |X ) =kY

P (y l |Y l−1 , X l )(10)

l=1which yieldsk∂ X∂ log P (Y k |X k )log P (y l |Y l−1 , X l )=∂wj∂wj l=1¸k · l
XX1 − yly0lρ²(tl − tn ) xnj−=llρ
1−ρnl=1(11)(12)∂We note that in Eq. 8 the factor ∂wlog P (Y k |X k ) has to be multiplied withjF k or with Gk before taking the average. Multiplication generates termsof the form hy l y k iY k ,X k = hhy l y k iY k |X k iX k For any given input X k , theautocorrelation hy l y k iY k |X k with l < k of the postsynaptic neuron will havea trivial value

hy l y k iY k |X k = hy l iY k |X k hy k iY k |X kfor k − l > ka

where ka ∆t is the width of the autocorrelation. As a consequence
¸À
¿· l
¢
1 − yl ¡ kyk−F − γG= 0 for k − l > ka
ρl 1 − ρ l
Y k ,X k
3(13)

(14)

P
Hence, for k > ka , we can truncate the sum over l in Eq. 8, i.e., kl=1 →Pkl=k−ka which yields exactly the coincidence measure Cj introduced in themain text; cf. Eq. 11 in the main text, and which we repeat here forconvenience· l¸kXX
y1 − ylk0l²(tl − tn ) xnj(15)Cj =ρ−llρ
1−ρnl=k−ka

From Averages to an Online Rule
The coincidence measure Cjk counts coincidences in a rectangular timewindow. If we replace the rectangular time window by an exponential
one
Pkwith time constant τC and go to continuous time, the summation l=k−ka . . .
Rtin Eq. 15 turns into an integral −∞ dt0 exp[−(t − t0 )/τC ] . . . which can betransformed into a differential equation£¤dCj (t)Cj (t−δ) X(f )=−+²(t−tj )S(t) δ(t− t̂−δ)−g(u(t))R(t) ;dt
τCf(16)

cf. Eq. 15 in the main text. Based on the considerations in the previousparagraph, the time constant τC should best be chosen in the range ka ∆t ≤τC ≤ 10 ka ∆t.
Similarly, the average firing rate ρ̄(t) = ḡ(t) R(t) can be estimated usinga running averageτḡdḡ(t)= −ḡ(t) + g(u(t))
dt(17)

with time constant τḡ .
In Fig. 6, we compare the performance of three different update schemesin numerical simulations. In particular, we show that (i ) the exact value ofthe truncation of the sum in Eq. 15 is not relevant, as long as ka ∆t is larger
than the width of the autocorrelation; and (ii ) that the online rule is a good
approximation to the exact solution.
To do so we take the scenario from Fig. 3 of the main text. For eachsegment of 1 s, we simulate one hundred pairs of input and output spiketrains. We evaluate numerically Eq. 8 by averaging over the 100 samples.
After each segment of 1 second (=1,000 time steps) we update the weights
4using a rule without truncation in the sum of Eq. 15. We call this the fullbatch update; compare. Fig. 6 (Top).
Second, we use the definition of Cjk with the truncated sum and repeat
the above steps; Fig. 6 (Middle). The truncation is set to ka ∆t = 200ms
which is well above the expected width of the autocorrelation function of thepostsynaptic neuron. We call this the truncated batch rule.
Third, we use the online rule discussed in the main body of the paperwith τC = 1s; Fig. 6 (Bottom).
Comparison of top and center graphs of Fig. 6 shows that there is nodifference in the evolution of mean synaptic efficacies, i.e., the truncation of
the sum is allowed, as expected from the theoretical arguments. A further
comparison with Fig. 6 Bottom shows that updates based on the online rule
add some fluctuations to the results, but its trend captures nicely the evolution of the batch rules.
Supplement to the Pattern Detection ParadigmIn Fig. 3 we presented a pattern detection paradigm where patterns
defined by input rates were chosen randomly and applied for one second.
After learning, the spike count over one second is sensitive to the index of
the pattern. Fig. 7A shows the histogram of spike counts for each pattern.
Optimal classification is achieved by choosing for each spike count the patternwhich is most likely. With this criterion 81 percent of the patterns will beclassified correctly.
The update of synaptic efficacies depends on the choice of the parameterγ in the learning rule. According the the optimality criterion in Eq. 8 of
the main text, a high level of γ implies a strong homeostatic control of thefiring rate of the postsynaptic neuron whereas a low level of γ induces only
a weak homeostatic control. In order to study the role of γ, we repeatedthe numerical experiments for the above pattern detection paradigm with avalue of γ = 100 instead of our standard value of γ = 1. Fig. 7B shows thatthe output firing rate is still modulated by the pattern index, the modulation
at γ = 100 is, however, weaker than that at γ = 1. As a results, patterndetection is less reliably with 45 percent correct classification only. We notethat this is still significantly higher than the chance level of 25 percent.

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10time [min]Figure 6: Evolution of the synaptic efficacies for the pattern detection paradigm ofFig. 3 during the first 10 minutes of simulated time. Red: mean synaptic efficacy of the 25synapses that received pattern-dependent input rates. Blue: mean synaptic efficacy of theremaining 75 synapses. The batch update rule (top), the truncated batch rule (middle)
and the online rule (bottom) yield comparable results.

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Figure 7: Pattern detection. A Histograms of spike counts nsp over one second (horizontal axis, bin size 2) during presentation of pattern 1 (dark blue), pattern 2 (light blue),pattern 3 (yellow), and pattern 4 (red). Vertical scale: number of trials n with a given
spike count divided by total number Np of trials for that pattern. B Spike count during
one second (mean and variance) for each of the four patterns with a parameter value γ = 1(light blue) and γ = 100 (dark blue). The values for γ = 1 are redrawn from Fig. 3.

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