Text S2

Abstract

Derivation of the nullcline of the model.
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Text S2: Derivation of the nullcline of the model
We follow the approach of Van Ooyen and Van Pelt [1] and assume:
dst
ij
dt = 0,∀i, j ∈N(1)
and
dξt
i
dt =ξ0−ξt
i
τξ
+κ
N
X
j=1
st
ij Θ(ξt
j−%t
j) = 0,∀i, j ∈N(2)
Averaging over all neurons gets us:
0 = 1
N
N
X
i=1


ξ0−ξt
i
τξ
+κ
N
X
j=1
st
ij Θ(ξt
j−%t
j)
.(3)
Thus
0 = ξ0
τξ
−Ξt
τξ
+κ
N
X
j=1
Θ(ξt
j−%t
j)1
N
N
X
i=1
st
ij .(4)
Now we replace the Heaviside function with a sigmoid as we can assume:
Θ(ξt
j−%t
j) = lim
1/α→0
1
1 + exp(α(%t
j−ξt
j))
⇒Θ(ξt
j−%t
j)≈1
1 + exp(α(%t
j−ξt
j)) =G(ξt
j)
On average we can set %t
jconstant and equal to the mean value, which is 0.5 getting:
0≈ξ0
τξ
−Ξ
τξ
+κ
N
X
j=1
F(ξj)1
N
N
X
i=1
sij .(5)
Expanding Ginto a Taylor series we have:
G(ξt
j)T
=G(Ξt) + G0(Ξt)(ξt
j−Ξt) + O2(Ξt)
and get:
0≈ξ0
τξ
−Ξt
τξ
+κ
N
X
j=1 G(Ξt) + G0(Ξt)(ξt
j−Ξt)1
N
N
X
i=1
st
ij .(6)
0≈ξ0
τξ
−Ξt
τξ
+κ
N
X
j=1
G(Ξt)1
N
N
X
i=1
st
ij (7)
≈ξ0
τξ
−Ξt
τξ
+κG(Ξt)1
N
N
X
j=1
N
X
i=1
st
ij (8)
≈ξ0
τξ
−Ξt
τξ
+κG(Ξt)St(9)
where we have used that one can write Stas the average connectivity across all neurons, hence St=
1
NPN
j=1 PN
i=1 st
ij , and get finally for the nullcline:
St=Ξt−ξ0
τξκ G(Ξt)(10)
1

References
[1] Van Ooyen A, Van Pelt J (1994) Activity-dependent outgrowth of neurons and overshoot phenomena
in developing neural networks. J Theor Biol 167: 27-43.
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