Dynamics of a single QIF neuron (<i>N</i> = 1 in Eq (1)) in the bistable regime (a)–(d), and in the tonic regime (e)–(h).

(a): Sketch (not to scale) of the bifurcation diagram of steady states (curve) and periodic solutions (cylinder) of the single QIF neuron subject to a constant input K1 = η1 (ε = 0). A stable quiescent state (down state) coexists with a stable tonic firing solution (up state), separated by an unstable equilibrium (dashed curve). A homoclinic bifurcation is present when K1 = h. In this intrinsically bistable regime (K1 = η1 ∈ (h, 0)), the cell selects the up or down state depending on initial conditions. (b): When 0 ε ≪ 1, K1(t) = η1 + A sin(εt) becomes a slowly varying quantity, oscillating around the value of η1 (ellipses on the K1 axes) with amplitude A, and transitions between the up and the down phases become possible. The onset between phases is determined by a family of canard solutions 1–2 (see text); in the bistable regime they appear in the down-down (green), and down-up (purple) transitions. (c): Time profiles of two solutions for the system with slow input K1(t), displaying a down-down and down-up transition, containing a canard segment (1–2). (d) The solutions in (c) are plotted in the variables (V1, K1), and superimposed on the curve of equilibria of the ε = 0 system (grey parabola), providing evidence of canard behaviour (1–2), and part of the orbits greyed out to enhance visibility. Parameters: ε = 0.01, J = 6, τs = 0.3, η1 = −0.2; A values are reported in the panels. (e): Sketch of the bifurcation diagram of steady states and periodic solutions with constant input (ε = 0) in the tonic regime k1 = η1 > 0). In this regime the cell displays solely the firing solution (up state). (f): When 0 ε ≪ 1 transitions between the up and the down phases become possible, mediated by canard solutions which are possible as up-up and up-down transitions (3–4), but not vice-versa. (g): Time profiles of two solutions in the tonic regime, with slow input K1(t), displaying an up-up and up-down transition, containing a canard segment (3–4). (h): The solutions in (f) are plotted in the variables (V1, K1), and superimposed on the curve of equilibria of the ε = 0 system (grey parabola), providing evidence of canard behaviour (3–4), and part of the orbits greyed out to enhance visibility. Parameters: ε = 0.01 J = 6, τs = 0.3, η1 = 0.5; A values are reported in the panels.