Table 1 Variables

\(V_{i}\)

Membrane potential for population i

\(h_{i}\)

Deinactivation of persistent sodium current for population i

\(s_{i}\)

Slow synaptic gating variable for population i

\(I_{\mathit {Na}P}\)

Persistent sodium current

\(I_{\mathrm {syn}}\)

Synaptic input from the network

\(I_{\mathrm {ext}}\)

External synaptic input

\(F(V_{i}, h_{i}, s_{i})\)

Right hand side of the voltage differential equation

\(g_{i}(V_{i},h_{i})\)

Right hand side of the persistent sodium differential equation

\(g_{\mathrm {syn}}^{i,j}\)

Synaptic weight of the synapse from population j to population i

\(i_{i}^{\mathrm {ext}}\)

Weight of external drive to population i

s

Vector of all synaptic variables in the network

\(V_{i,X}(h,\mathbf{s})\)

Left (X = L), middle (X = M) or right (X = R) branch of the cubic voltage nullcline for population i

\(p_{i,X}(\mathbf{s})\)

Fixed point located on the X∈{L,M,R} branch of the voltage nullcline for population i

\((V_{i}^{\mathrm {JU}}(\mathbf{s}),h_{i}^{\mathrm {JU}}(\mathbf{s}))\)

Jump up curve, curve in slow phase space from which population i may enter the active phase

\((V_{i}^{\mathrm {JD}}(\mathbf{s}),h_{i}^{\mathrm {JD}}(\mathbf{s}))\)

Jump down curve, curve in slow phase space from which population i may enter the silent phase

\(s_{\max}\)

Maximum value achieved by synaptic gating variable

\(s_{\mathrm {dynamic}}\)

Synaptic gating variable evolving in time for a given portion of the rhythm, while the other synaptic gating variable is fixed

\(I=\{i_{\mathit {IP}}^{\mathrm {ext}}, i_{\mathit {EP}}^{\mathrm {ext}}, i_{\mathit {ER}}^{\mathrm {ext}}, i_{\mathit {IR}}^{\mathrm {ext}} \}\)

Set of external drives to populations of interneurons

\(T_{\mathrm {active}}^{j}(I)\)

Length of time population j is active for a given I

\(s_{\mathrm {SN}}\)

Value of s at which saddle-node bifurcation occurs

\(s_{\mathit {ER}}^{\min}(I)\)

Minimum value achieved by \(s_{\mathit {ER}}\) for a given I

\(I_{s}= [ s_{\mathit {ER}}^{\min}(I),s_{\mathrm {SN}} ]\)

Values of \(s_{\mathit {ER}}\) from which KE can enter the active phase

\(h_{\max}\)

\(h_{\mathit {EP}}^{\mathrm {JD}}(s_{\max}) = h_{\mathit {ER}}^{\mathrm {JD}}(s_{\max})\), the largest \(h_{\mathit {KE}}\) value at which KE can enter the silent phase

\(h_{\min}(I)\)

\(h_{\mathit {ER}}^{\mathrm {JD}}(s_{\mathit {ER}}^{\min}(I))\), the value of \(h_{\mathit {KE}}\) on the ER curve of jump down knees corresponding to \(s_{\mathit {ER}}^{\min}(I)\)

\(I_{h}= [ h_{\min}(I),h_{\max} ]\)

Values of \(h_{\mathit {KE}}\) at which KE can enter the silent phase

\(\mathit {LK}_{I_{s}}\)

The part of the curve of jump up knees corresponding to \(s \in I_{s}\)

T(I)

Time for s to decay from \(s_{\mathrm {SN}}\) to \(s_{\mathit {ER}}^{\min}(I)\)

h(a;b,c)

\(h_{\mathit {KE}}\) value at time a for a trajectory that started at time 0 with initial condition \((h_{\mathit {KE}},s)=(b,c)\)

\(h_{\mathit {ER}}^{\min}(I)\)

h value on the ER jump up curve given by \(s_{\mathit {ER}}^{\min}(I)\)

\(h_{\mathrm {SN}}^{+}\)

Forward flow of \((h_{\mathrm {SN}}, s_{\mathrm {SN}})\) for time T(I)

\(h_{\mathrm {SN}}^{-}\)

Backward flow of \((h_{\mathrm {SN}}, s_{\mathrm {SN}})\) to the line \(s=s_{\max}\)

\(h_{s_{\min}}^{-}\)

Backward flow of \((h_{\mathit {ER}}^{\mathrm {JU}}(s_{\mathit {ER}}^{\min}(I)), s_{\mathit {ER}}^{\min}(I))\) to the line \(s=s_{\max} \)

\(t^{*}\)

Minimal time spent in the silent phase by KE