S2 Appendix: Calculating average weight modification for the ... - PLOS

S2 Appendix: Calculating average weight modification for the suppression model Equation (9) of Methods section enables us to calculate the probability density of the intervals between pre- and post synaptic spikes ( t = tpost tpre ). Assuming that the presynaptic spike train is Poisson, the probability density of

t is

P ( t) = rpre rpost ( t) exp( t rpre ) 8 h t/⌧s ) i > > rpre r¯post + w exp( exp( > > < (Vth Vr )⌧m = > > > > : rpre r¯post exp( t rpre )

(S4) t rpre ) if

if

t

0

t < 0.

When the neuron fires almost regularly, the interval between two consecutive postsynaptic spikes can be considered to be 1/¯ rpost . However, if a presynaptic spike arrives before a postsynaptic spike ( t > 0), it shortens the interval between the upcoming postsynaptic spike and the preceding one. In this case, the interval between consecutive postsynaptic spikes ( tpost ) can expressed as a function of t from equation (S1). Taken together, 8 ⌧s 1 exp( t/⌧s ) i 1 h > > > 1 w t 0 > > ⌧m (Vth Vr ) < r¯post tpost = (S5) > > > 1 > > t < 0. : r¯post

We now have all the components required to calculate the average weight modification under the suppression model (see equation 13). In this model, each nearest neighboring prepost pair of spikes with pairing interval

t induces potentiation or depression depending on

the ordering of the pair. In addition, the previous pre- and postsynaptic spikes participate in plasticity depending on their temporal distance from the spikes in the pair ( tpre and respectively) . Of the three intervals participating in suppression model, stochastic variables, while

tpost is simply a function of

the weight modification over all possibles values of

tpre and

tpost t are

t. Therefore, we should average

t and

tpre . Given the assumption

that the postsynaptic spike fires almost regularly, | t| cannot be longer than the length of a

typical postsynaptic ISI (1/¯ rpost ). Also, in the case where the postsynaptic spike precedes the presynaptic one ( t < 0), there is a lower limit on

tpre : it cannot be shorter than

the pre-post interval | t|. By these considerations, the average weight change can be calculated as dhwi = dt

Z

+1/¯ rpost

d t P ( t) 1/¯ rpost

Z

1 max(0,

d tpre P ( tpre ) Fsupp ( t, tpre , tpost ) t)

where Fsupp is the weight modification (equation 13). Because the presynaptic spike train is assumed to be Poisson, P ( tpre ) is the waiting time of the Poisson process, namely rpre exp(

tpre rpre ). Substituting equations (S4) and (S5) in the above equation and keep-

ing only terms up to first order in hwi results in " A ⌧˜ E(˜ ⌧+ ) dhwi = E(⌧post ) r¯post rpre + + dt 1 + rpre ⌧pre

+ hwi

A ⌧˜ E(˜ ⌧ ) + A ⌧ˆ E(ˆ ⌧ ) ⇣

"

⌧ˆ+ E(ˆ ⌧+ ) (⌧post ⌧s )E(⌧post ) + ⌧s A+ rpre ⌧m ⌧post (Vth Vr ) 1 + rpre ⌧pre ⇣

⌧s ⌧˜+ E(˜ ⌧+ ) 1 with E(⌧ ) = 1

E(⌧post )

exp( 1/(rpost ⌧ )) and time constants defined as ⌧˜+ =

⌧+ 1 + rpre ⌧+

⌧ˆ+ =

⌧ s ⌧+ ⌧ s + ⌧ + + ⌧ s ⌧+

⌧˜

⌧ 1 + rpre ⌧

⌧ˆ =

⌧pre ⌧ . ⌧pre + ⌧ + ⌧pre ⌧

=

Equation (S6) is numerically evaluated in figure 5.

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(S6)