### The reduced system

We consider the full system (Eqs. (2.1)-(2.3)) as having one fast variable *V* and two slower variables *n* and *c*. The time-scale separation can be accentuated by decreasing the singular perturbation parameter *C*_{
m
} . This facilitates analysis of the system dynamics [28]. In the limit *C*_{
m
} → 0, the trajectories of the system lie on a 2-D surface called the critical manifold. If we define the right hand side of Eq. (2.1) by

f\left(V,c,n\right)=-\left({I}_{Ca}+{I}_{K}+{I}_{K\left(Ca\right)}+{I}_{BK}\right)

(3.1)

then the critical manifold is the surface S satisfying

S\equiv \left\{\left(V,c,n\right)\in {\mathbb{R}}^{3}:f\left(V,c,n\right)=0\right\}.

(3.2)

The equation *f*(*V*, *c*, *n*) = 0 can be solved in explicit form for *n* as

n=n\left(c,V\right)=-\frac{1}{{g}_{K}}\left[{g}_{Ca}{m}_{\infty}\left(V\right)\frac{\left(V-{V}_{Ca}\right)}{\left(V-{V}_{K}\right)}+{g}_{K\left(Ca\right)}{s}_{\infty}\left(c\right)+{g}_{BK}{b}_{\infty}\left(V\right)\right].

(3.3)

The critical manifold (3.3) is a folded surface (Figure 2) that consists of three sheets separated by two fold curves (L^{-} and L^{+}). The lower and upper sheets are attracting (\frac{\partial f}{\partial V}<0) and the middle sheet is repelling (\frac{\partial f}{\partial V}>0). The lower (L^{-}) and upper (L^{+}) fold curves are given by

{\mathsf{\text{L}}}^{\pm}\equiv \left\{\left(V,c,n\right)\in {\mathbb{R}}^{3}:f\left(V,c,n\right)=0\mathsf{\text{and}}\frac{\partial f}{\partial V}\left(V,c,n\right)=0\right\}.

(3.4)

This yields two constant *V* values and two equations for *n* in the form of *n* = *n*(*c*). Thus, the fold curves (L^{±}) are (*V*^{±}, *c*, *n*^{±} (*c*)) where *V*^{-} and *V*^{+} are constant *V* values. The curve L^{+} is projected vertically (along the fast variable *V*) onto the lower sheet to obtain the projection curve P(L^{+}), and similarly for the (L^{-}) projection onto the upper sheet. Figure 2 shows the critical manifold, the fold curves and the projections of the fold curves.

The reduced flow (when *C*_{
m
} → 0) is described by (3.3), the differential equation for *c* (Eq. (2.3)), and a differential equation for *V* which can be obtained by differentiating *f*(*V*, *c*, *n*) = 0 with respect to time. That is,

-\frac{\partial f}{\partial V}\frac{dV}{dt}=\frac{\partial f}{\partial c}\frac{dc}{dt}+\frac{\partial f}{\partial n}\frac{dn}{dt}

(3.5)

where *n* satisfies Eq. (3.3), and \u1e45, \u010b satisfy Eqs. (2.2), (2.3). The two differential equations for the reduced system are thus

-\frac{\partial f}{\partial V}\frac{dV}{dt}=\left(\right.-{f}_{c}(\alpha {I}_{Ca}+{k}_{c}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{c}})\left)\right.\frac{\partial f}{\partial c}+\left(\frac{\left({n}_{\infty}\left(V\right)-n\right)}{{\tau}_{n}}\right)\frac{\partial f}{\partial n}

(3.6)

\frac{d\mathsf{\text{c}}}{dt}=-{f}_{c}\left(\alpha {I}_{Ca}+{k}_{c}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{c}}\right).

(3.7)

Since \frac{\partial f}{\partial V}=0 on *L*^{±}, the reduced system is singular along the fold curves. The system can be desingularized by rescaling time with \tau =-{\left(\frac{\partial f}{\partial V}\right)}^{-1}t. The desingularized system is then

\frac{dV}{d\tau}=\left(\right.-{f}_{c}(\alpha {I}_{Ca}+{k}_{c}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{c}})\left)\right.\frac{\partial f}{\partial c}+\left(\frac{\left({n}_{\infty}\left(V\right)-n\right)}{{\tau}_{n}}\right)\frac{\partial f}{\partial n}

(3.8)

\frac{d\mathsf{\text{c}}}{d\tau}={f}_{c}\left(\alpha {I}_{Ca}+{k}_{c}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{c}}\right)\frac{\partial f}{\partial V}.

(3.9)

Defining

F\left(V,c,n\right)=\left(\right.-{f}_{c}\left(\alpha {I}_{Ca}+{k}_{c}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{c}}\right)\left)\right.\frac{\partial f}{\partial c}+\left(\frac{\left({n}_{\infty}\left(V\right)-n\right)}{{\tau}_{n}}\right)\frac{\partial f}{\partial n},

(3.10)

we have the desingularized system

\frac{dV}{d\tau}=F\left(V,c,n\right)

(3.11)

\frac{d\mathsf{\text{c}}}{d\tau}={f}_{c}\left(\alpha {I}_{Ca}+{k}_{c}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{c}}\right)\frac{\partial f}{\partial V}.

(3.12)

The desingularized system describes the flow on the critical manifold. Because of the time rescaling, the flow on the middle sheet, where \frac{\partial f}{\partial V}>0, must be reversed to obtain the equivalent reduced flow.

### FoLded singuLarities and canards

Equilibria of the desingularized system are classified as ordinary singularities and folded singularities. An ordinary singularity is an equilibrium of Eqs. (2.1)-(2.3) and satisfies

f\left(V,c,n\right)=0

(3.13)

n={n}_{\infty}\left(V\right)

(3.14)

c=-\frac{\alpha {I}_{Ca}}{{k}_{c}}.

(3.15)

A folded singularity lies on a fold curve (L^{+} or L^{-}), and satisfies:

f\left(V,c,n\right)=0

(3.16)

F\left(V,c,n\right)=0

(3.17)

\frac{\partial f}{\partial V}=0.

(3.18)

A folded singularity is classified as a folded node if the eigenvalues are real and have the same sign, a folded saddle if the eigenvalues are real and have opposite signs, or a folded focus if the eigenvalues are complex [22, 23, 25, 29]. For parameter values used in Figure 2, the system has a folded node (with negative eigenvalues) on L^{+} (FN, blue point, in Figure 2), and a folded focus on L^{-}(not shown).

There are an infinite number of singular trajectories on the top sheet that pass through the folded node (FN). These are called singular canards [22]. The singular canard that enters the FN in the direction of the strong eigenvector is called the strong canard (SC, green curve, in Figure 2). This curve and the fold curve L^{+} delimit the singular funnel that consists of all initial conditions whose trajectories for the reduced system pass through the folded node. The singular funnel and key curves are projected onto the (c, V)-plane in Figure 3. The different panels are obtained with different values of the parameter *g*_{
K
} .

### Singular periodic orbits, relaxation oscillations, and mixed mode oscillations

A singular periodic orbit (Figure 2, black curve with arrows) can be constructed by solving the desingularized system for the flow on the top and bottom sheets of the critical manifold, and then projecting the trajectory from one sheet to the other along fast fibers when the trajectory reaches a fold curve. The singular periodic orbit is the closed curve constructed in this way. This process was discussed in detail in [22, 28, 30]. Briefly, the trajectory moves along the bottom sheet until L^{-} is reached. At this point the reduced flow is singular (\frac{\partial f}{\partial V}=0). The quasi-steady state assumption *f*(*V*, *c*, *n*) = 0 is no longer valid and there is a rapid motion away from the fold curve L^{-}. This rapid motion is seen as vertical movement to the top sheet (the dynamics are governed by the layer problem, see [22, 28]). The trajectory moves to a point on P(L^{-}) and from there is once again governed by the desingularized equations, moving along the top sheet until L^{+} is reached. The fast vertical downward motion along fast fibers returns the trajectory to a point on P(L^{+}) on the bottom sheet, completing the cycle.

When the singular periodic orbit reaches L^{-} it jumps up to a point on P(L^{-}). If this point on P(L^{-}) is in the singular funnel, then the orbit will move through the FN. Otherwise it will not. Let *δ* denote the distance measured along P(L^{-}) from the phase point on P(L^{-}) of the singular periodic orbit to the strong canard (SC in Figure 3). When the phase point is on the strong canard, *δ* = 0. Let *δ* > 0 when the phase point is in the singular funnel and *δ* < 0 when the phase point is outside the singular funnel. Singular canards are produced when *δ* > 0.

In Figure 3(a) the singular periodic orbit jumps to a point on P(L^{-}) outside of the singular funnel (*δ* < 0), so it does not enter the FN. This orbit is a relaxation oscillation [31]. In Figure 3(b)*δ* > 0, so the orbit is a singular canard. Away from the singular limit, this singular canard perturbs to an actual canard that is characterized by small oscillations about L^{+}[22]. The combination of these small oscillations with the large oscillations that occur due to jumps between upper and lower sheets yields mixed mode oscillations [24, 32]. The small oscillations have zero amplitude in the singular case, which grows as \sqrt{{C}_{m}} for *C*_{
m
} sufficiently small [23]. A discriminating condition between relaxation and mixed mode oscillations is *δ* = 0, where the singular periodic orbit jumps to P(L^{-}) on the SC curve.

When *C*_{
m
} > 0 the full system (Eqs.(2.1)-(2.3)) produces spiking for *δ* < 0 and mixed mode oscillations for *δ* > 0. Figure 4 shows these two different cases for *g*_{
BK
} = 0.4 nS. For *g*_{
K
} = 5.1 nS (*δ* < 0 in Figure 3(a)), the nearly-singular periodic orbit produced when *C*_{
m
} = 0.001 pF (Figure 4(a)) perturbs to continuous spiking when *C*_{
m
} = 10 pF (Figure 4(e)). When *g*_{
K
} = 4 nS the singular periodic orbit enters the singular funnel (Figure 3(b)), so when *C*_{
m
} is increased the singular orbit transforms to mixed mode oscillations. For *C*_{
m
} = 0.5 pF mixed mode oscillations with small spikes are produced (Figure 4(d)). As *C*_{
m
} is increased to 10 pF, mixed mode oscillations with larger spikes are produced. This is the genesis of pseudo-plateau bursting (Figure 4(f)).