Hebbian Learning and Spike-Timing-Dependent Plasticity - Computational Neuroscience: A Comprehensive Approach - page 318

and

r in A + τ + + τ r in 2 w i / U ( w ) ≈− G ( − ( A − − A + ) +

m

) D

τ + τ

(11.48)

m

r in 2 − 2 ( − G ( A − − A + )

i w j

+ θ r in ( A − − A + )) w i / D .

j =

As we noted in the previous section, U ( 0 ) ≈ U ( g max ) ,so

g max r in A + τ + + τ

m

r in 2

∑ j

m ≈

2 ( A − − A + )( w j − θ r in ) , (11.49)

τ + τ

or

g max GA + τ + + τ

m

r out

m ≈

2 ( A − − A + ) , (11.50)

τ + τ

In the steady state situation, the left-hand term is not affected by the postsynaptic
rate, so the output rate is effectively kept constant. This effect was noted in Song
and Abbott [79] and demonstrated in Figure 11.3 . An intuitive explanation was
given in Song et al. based on balances of excitation and inhibition. Neuron can
operate in two different modes with distinct spike-train statistics and input-output
correlations [1, 14, 85]. If the total amount of excitation overwhelms the amount of
inhibition, the mean input to the neuron would bring it well above threshold if action
potentials were blocked ( Figure 11.5A) . In this situation, the neuron operates in an
input-averaging or regular-firing mode. The postsynaptic spike sequences produced
in this mode are quite regular ( Figure 11.5C) because the timing of the postsynaptic
spikes is not sensitive to presynaptic spike times. There are roughly equal numbers of
presynaptic action potentials before and after each postsynaptic spike [1, 14] ( Figure
11.6A) . Because the area under the STDP curve is slightly negative ( A − −

A + > 0),

for a flat correlation curve,

∞

K ∞
− ∞

dw i ( )

dt =

t

dt P t ) r out r in

i

t )= dt P t ) , ( ( t ) ( t + ( (11.51)

− ∞

where K is a constant, is still negative. Thus the synapses are weakened.
As the excitatory synapses are weakened by STDP, the postsynaptic neuron en-
ters a balanced mode of operation in which it generates a more irregular sequence
of action potentials, and the timing of the postsynaptic spikes becomes more tightly
correlated with the timing of the presynaptic spikes. The total synaptic input in the
balanced mode is, on average, slightly below threshold ( Figure 11.5B) , so the post-
synaptic neuron fires irregularly, primarily in response to statistical fluctuations in
the total input ( Figure 11.5D) . Because action potentials occur preferentially after a
random positive fluctuation, there tend to be more excitatory presynaptic spikes be-
fore than after a postsynaptic response [1, 14, 85] ( Figure 11.6B) . The small excess
of pr es ynaptic spikes just before a postsynaptic spike is described in the rate model

w i r in
τ

e t / τ

m

as 0 It can be gathered in Figure 11.6 that the excess calculated from

simulations is indeed well approximated by an exponential function. The STDP rule
achieves a steady-state distribution of peak synaptic conductances when the excess of

m

j w j r in 2 if

t <