Hebbian Learning and Spike-Timing-Dependent Plasticity - Computational Neuroscience: A Comprehensive Approach - page 313

Therefore,

dw i ( t )

dt
= G ∞ 0 dt ∞ − ∞

(11.26)

dt K t ) t ) ∑

j w j r i j (

t ) r in

i

t ) − θ

( P ( t − ( t + r in ( A + − A − ) .

For simplicity, let us assume that the scale of STDP is slow compared to the spiking
rate of the neurons, we can therefore replace r i j (

t ) r in

i

t )

t − ( t + with its average

r i j ( t ) r in

i

t ) > t ,or Q ij ( − t − t ) t − t ) < is stationary

for all subsequent calculations. We can now calculate the rate of change in synaptic
strengths for correlated inputs. From Section 11.1,

t − ( t + . We assume that Q ij ( −

C ij r in 2 e −| t |/ τ r in 2

( )= + + δ ( ) ,

Q ij t

c

t r in (11.27)

ij

and

1

e − t / τ K ( t )= . (11.28)

m

m

Upon collecting terms,

dw i

r in w i A + τ + + τ Hr in 2 r in 2 ) )

dt =

G (

j C ij w j − (

A − − A + )

j w j +

m

τ + τ

(11.29)

m

− θ r in ( A + − A − ) ,

where

∞ 0

1 A +

τ + [

e − t / τ dt e t / τ + e − ( t + t ) / τ dt

= {

H

m c

− t 0 m

− t ∞

A −
τ −

dt e t / τ + e t + t ) / τ dt e − t / τ − e − ( t + t ) / τ

+ ] − }

c c

− ∞ 0

c

A + τ A + τ τ + A + τ τ +

c c

=

) − ) + ( τ − τ + )( τ + τ ( τ − τ + )( τ + + τ ( τ + τ + )( τ + + τ )

. (11.30) c c m c m c m

c

A − τ

−

( τ

c + τ − )( τ c + τ m )

c

A + ( τ −
τ

A − ( τ +
τ

=

+ τ − ) [

A + − A − +

c ) − c ) ( τ + τ )( τ + τ + )( τ

c m c c

2 A + τ τ + ( τ + τ − )

m c

+ ] .

c

( τ + + τ )

m

The first term in the brackets in Equation (11.29) corresponds to the effect of corre-
lations in the inputs, the second term corresponds to the effect of the average input,
and the last term takes into account the correlation introduced by spiking. It is inter-
esting that STDP can switch from a Hebbian to an anti-Hebbian rule depending on
the time scale of the input correlation. In order for the rule to be Hebbian, H has to
be greater than zero. Otherwise, it is anti-Hebbian. Solving H

>

0 for c yields

− x 2

x − 4 ( A − − A + ) y

< , (11.31)

c

2 ( A − − A + )

where

2 A + τ

τ +

m

x = A + τ − − A − τ + +

m ,

(11.32)

τ + + τ