Processes 2019, 7,20
and similarly the fast subsystem
dV
dτ fast
=F 1 (V, d, f , x),
dd
dτ fast
= F 2 (V, d, f , x),
d f
dτ fast
= 0,
dx
dτ fast
= 0.
(16)
From (10), using ˙
d ∞ =
∂d ∞
∂V
˙
V we can derive immediately the desingularised slow flow:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
dV
dτ 1
=
∂F 1
∂ f
G 1 (V, d ∞ , f , x)+δ
∂F 1
∂x
G 2 (V, d ∞ , f , x),
d f
dτ 1
= −
∂F 1
∂V
G 1 (V, d ∞ , f , x),
(17)
restricted to C 0 , where τ = −
∂F
∂V
τ 1 . Remember that system (17) is the desingularised version of
−
∂F 1
∂V
dV
dτ 1
=
∂F 1
∂ f
G 1 (V, d ∞ , f , x)+δ
∂F 1
∂x
G 2 (V, d ∞ , f , x),
d f
dτ 1
=G 1 (V, d ∞ , f , x),
(18)
restricted to C 0 . An ordinary singularity of (17) is given if G 1 = G 2 = 0, while a fold point z • =
(V • , d ∞ (V • ), f • , x • ) T ∈ F is a folded singularity of (17)if
∂F 1
∂ f
(z • ) · G 1 (z • )+δ
∂F 1
∂x
(z • ) · G 2 (z • )=0 and
∂F 1
∂V
(z • )=0.
This yields explicit expressions for f • and x • depending on V • , i.e.,
x • = −
¯
G Ca 2+
¯
G K +
d ∞ (V • ) f •
(V • − E Ca 2+ )
(V • − E K + )
and
f • =
1
1 − δ
f ∞ (V • )+
δ
1 − δ
¯
G K + (V • − E K + )
¯
G Ca 2+ (V • − E Ca 2+ )
x ∞ (V • )
d ∞ (V • )
.
At this stage we see that the shape of the critical manifold is not depending on δ or the choice
of τ f and τ x , but the location of the folded singularities and their stability. Moreover, notice that the
Jacobian of (17) has at least one zero eigenvalue. Furthermore, varying the ratio τ f /τ x changes the
desingularised slow flow (17). Notice that for δ → 0 we have an one dimensional slow flow, where f is
determined by the critical manifold and x is constant. Therefore, it does not make sense to consider
both limits ε → 0 and δ → 0 simultaneously. However, varying δ may compensate the effect of an
enhanced calcium current, cf. [15]. Moreover, the critical manifold C 0 as well as the desingularised
slow flow are depending on ¯
G K + and ¯
G Ca 2+ , cf. (14) and (17). Hence, varying ¯
G K + and/or ¯
G Ca 2+ has an
influence on (14) and (17). Furthermore, τ d and C m have only an influence on the time scale separation
argument and after passing to the singular limit ε → 0 our discussion is independent on τ d and C m .
In the following, we consider G Ca 2+ = 0.032
mS
cm 2 . Computing the critical manifold C 0 (14) together
with two fold lines L ± = {(V, d, f , x) ∈ C 0 : F 1V (V, d, f , x)=0, F 1VV (V, d, f , x) = 0} , the folded node
(V, f , x) ≈ (−24.7923, 0.5804, 0.7027) with eigenvalues λ 1 ≈−0.1974 and λ 2 ≈−1.7305, an ordinary
singularity (V, f , x) ≈ (−30.2250, 0.7666, 0.8760) and the singular orbit, we gain Figure 2. Notice that
for τ f and τ x satisfying the ratio δ = τ f /τ x ≡ 4/15 the folded node will be the same—similarly if
G Ca 2+ /G K + = 16/25. The critical manifold is divided into two attracting sheets S ±
a and one repelling
sheet S r , where S r lies between the two fold lines L ± . The fold lines are nondegenerate since ∂F 1 /∂ f = 0
or ∂F 1 /∂x = 0 or both is satisfied. Moreover, we have an ordinary singularity on S r . Notice that spiking,
70
and similarly the fast subsystem
dV
dτ fast
=F 1 (V, d, f , x),
dd
dτ fast
= F 2 (V, d, f , x),
d f
dτ fast
= 0,
dx
dτ fast
= 0.
(16)
From (10), using ˙
d ∞ =
∂d ∞
∂V
˙
V we can derive immediately the desingularised slow flow:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
dV
dτ 1
=
∂F 1
∂ f
G 1 (V, d ∞ , f , x)+δ
∂F 1
∂x
G 2 (V, d ∞ , f , x),
d f
dτ 1
= −
∂F 1
∂V
G 1 (V, d ∞ , f , x),
(17)
restricted to C 0 , where τ = −
∂F
∂V
τ 1 . Remember that system (17) is the desingularised version of
−
∂F 1
∂V
dV
dτ 1
=
∂F 1
∂ f
G 1 (V, d ∞ , f , x)+δ
∂F 1
∂x
G 2 (V, d ∞ , f , x),
d f
dτ 1
=G 1 (V, d ∞ , f , x),
(18)
restricted to C 0 . An ordinary singularity of (17) is given if G 1 = G 2 = 0, while a fold point z • =
(V • , d ∞ (V • ), f • , x • ) T ∈ F is a folded singularity of (17)if
∂F 1
∂ f
(z • ) · G 1 (z • )+δ
∂F 1
∂x
(z • ) · G 2 (z • )=0 and
∂F 1
∂V
(z • )=0.
This yields explicit expressions for f • and x • depending on V • , i.e.,
x • = −
¯
G Ca 2+
¯
G K +
d ∞ (V • ) f •
(V • − E Ca 2+ )
(V • − E K + )
and
f • =
1
1 − δ
f ∞ (V • )+
δ
1 − δ
¯
G K + (V • − E K + )
¯
G Ca 2+ (V • − E Ca 2+ )
x ∞ (V • )
d ∞ (V • )
.
At this stage we see that the shape of the critical manifold is not depending on δ or the choice
of τ f and τ x , but the location of the folded singularities and their stability. Moreover, notice that the
Jacobian of (17) has at least one zero eigenvalue. Furthermore, varying the ratio τ f /τ x changes the
desingularised slow flow (17). Notice that for δ → 0 we have an one dimensional slow flow, where f is
determined by the critical manifold and x is constant. Therefore, it does not make sense to consider
both limits ε → 0 and δ → 0 simultaneously. However, varying δ may compensate the effect of an
enhanced calcium current, cf. [15]. Moreover, the critical manifold C 0 as well as the desingularised
slow flow are depending on ¯
G K + and ¯
G Ca 2+ , cf. (14) and (17). Hence, varying ¯
G K + and/or ¯
G Ca 2+ has an
influence on (14) and (17). Furthermore, τ d and C m have only an influence on the time scale separation
argument and after passing to the singular limit ε → 0 our discussion is independent on τ d and C m .
In the following, we consider G Ca 2+ = 0.032
mS
cm 2 . Computing the critical manifold C 0 (14) together
with two fold lines L ± = {(V, d, f , x) ∈ C 0 : F 1V (V, d, f , x)=0, F 1VV (V, d, f , x) = 0} , the folded node
(V, f , x) ≈ (−24.7923, 0.5804, 0.7027) with eigenvalues λ 1 ≈−0.1974 and λ 2 ≈−1.7305, an ordinary
singularity (V, f , x) ≈ (−30.2250, 0.7666, 0.8760) and the singular orbit, we gain Figure 2. Notice that
for τ f and τ x satisfying the ratio δ = τ f /τ x ≡ 4/15 the folded node will be the same—similarly if
G Ca 2+ /G K + = 16/25. The critical manifold is divided into two attracting sheets S ±
a and one repelling
sheet S r , where S r lies between the two fold lines L ± . The fold lines are nondegenerate since ∂F 1 /∂ f = 0
or ∂F 1 /∂x = 0 or both is satisfied. Moreover, we have an ordinary singularity on S r . Notice that spiking,
70