Figure 1

Perturbation by a square pulse of current switches the regime of activity from silence into bursting. (a) A negative, hyperpolarizing pulse of current $I_{\mathrm{inj}}=-0.05$ nA switches activity from silence to bursting. The minimum magnitude of $I_{\mathrm{inj}}$ that can induce the switch is –0.02131 nA. (b) A positive, depolarizing pulse of current $I_{\mathrm{inj}}$ $=$ 0.05 nA also initiates the bursting activity. To switch the activity, $I_{\mathrm{inj}}$ has to be larger than the value of 0.0175 nA. For either polarity of the pulse, the pulse duration was 0.03 sec. The parameters of the leak current are $E_{\mathrm{leak}}$ $=$ –0.0635 V and $g_{\mathrm{leak}}$ $=$ 10.7 nS.Reuse & Permissions

Figure 2

The strength-duration relationships of the minimum depolarizing (gray) and hyperpolarizing (black) square pulses of current switching the silent regime into the bursting regime. To toggle the switch the pulse strength must be larger than the thresholds $I_{\mathrm{th}}$ determined by the relationships. The two data sets (dots) were obtained numerically and then fitted to the Lapicque formula (curves). The parameters of the leak current are the same as in Fig. 1.Reuse & Permissions

Figure 3

Bifurcation diagram of the stationary states and hyperpolarized subthreshold oscillations of the model. (a) The solid green line represents the stable hyperpolarized equilibrium (EP1). Subcritical Andronov-Hopf bifurcation (AH) occurs at $g_{\mathrm{leak}}=10.67$ nS (marked by the left vertical dashed line, $E_{\mathrm{leak}}=-0.0635$ mV) where the equilibrium loses its stability and gives rise to the unstable periodic orbit. For the values of $g_{\mathrm{leak}}$ smaller than this critical value the equilibrium is unstable (dashed blue line). The dashed blue line located above it depicts the saddle equilibria (EP2). The curve of the depolarized equilibrium has the membrane potential near +0.01 V and is not shown. The unstable periodic orbit corresponds to the unstable subthreshold oscillations (USTO). It is marked by the two dashed red curves and the dashed brown curve locating the minimum, maximum, and average values of the membrane potential of the oscillations correspondingly. The unstable orbit disappears at the homoclinic bifurcation (Hom), marked by the dashed vertical line at the right, $g_{\mathrm{leak}}=10.873$ nS. (b) Red dots graph the values of the period, $P$, of USTO as the $g_{\mathrm{leak}}$ is varied between the two bifurcation values. As the unstable periodic orbit approaches the homoclinic bifurcation the period grows as $-1.198\ln \left(\right|10.873-g_{\mathrm{leak}}\left|\right)$ as shown by the blue curve. (c) Inset shows the part of the graph in (b) taken in the vicinity of the homoclinic bifurcation, where $b$ $=$ 1.198, $g_{\mathrm{leak}}^{\mathrm{Hom}}=10.873$ nS.Reuse & Permissions

Figure 4

Coexistence of bursting and silence. The negative square pulse of $I_{\mathrm{inj}}$ perturbed the bursting activity. Bursting and transient activities are represented by black solid line. (a), (c), (e) Two-dimensional (2D) projection reveals unstable oscillations represented by dashed red (gray) loop along with the two attractors: the bursting and the hyperpolarized equilibrium, green (gray) dot. (a), (b) The pulse with the amplitude of $-0.01044$ nA moved the phase point toward the unstable periodic orbit, allowed recording the subthreshold oscillations for eight periods, but did not switch the activity from bursting into silence. (c), (d) The pulse with the amplitude $-0.01045$ nA again allowed recording the subthreshold oscillations for a few periods, moved the phase point into the basin of attraction of the equilibrium, thus producing the switch from bursting into the silence. The dashed red (gray) lines in plots (b) and (d) mark the minimum and maximum values of the membrane potential of the unstable orbit obtained with the bifurcation analysis. (e), (f) This is the part of the trajectory with the unstable oscillations (solid) revealed by the perturbation plotted together with the unstable periodic orbit (dashed). The parameters of the leak current were the same as in Fig. 1.Reuse & Permissions

Figure 5

Bursting, the saddle orbit and silent regimes at different values of the controlling parameter $g_{\mathrm{leak}}$. The saddle orbit, stable and saddle equilibria on the 2D projection onto $(m_{\mathrm{CaS}},h_{\mathrm{CaS}})$ plane presented for three $g_{\mathrm{leak}}$ values, 10.76 nS (a), 10.83 (b), 10.87 (c); and panels (a1), (b1), (c1) show corresponding bursting activities. On panels (a), (b), and (c), the saddle orbit is marked by the dashed loop; and the stable and the saddle equilibrium are shown by the filled green (gray) and open blue (gray) circles, correspondingly. The homoclinic bifurcation occurs at $g_{\mathrm{leak}}$ $=$ 10.873 nS, where the saddle orbit merges with the saddle equilibrium. Panels (a1) and (b1) show the bursting activities of the models such that in parameter space one is relatively far from the bifurcation [(a1)] and the other is close to the transition from bursting to silence [(b1)]. Both cases exhibit the bistability of bursting and silence. At the parameter value close to the bifurcation [(c), (c1)], bursting no longer exists as a regime; and silence is the only attractor [(c1)]. The value of $E_{\mathrm{leak}}$ was $-0.0635$ V for all the plots.Reuse & Permissions

Figure 6

Bifurcation diagram of the oscillatory and stationary regimes. The green curve is composed of numerically found points where the transition from bursting activity into silence occurs. The subcritical Andronov-Hopf bifurcation (AH) of the hyperpolarized rest state (silent regime) is shown by the blue curve and marks the boundary where the silent regime loses stability giving rise to the unstable subthreshold oscillations. The red circles correspond to the homoclinic bifurcation (Hom) of the unstable subthreshold oscillations. The area between these two curves (the blue and the red dots) marks the parameter regime where unstable subthreshold oscillations exist. The gray curve marks the period-doubling bifurcation (PD) of large amplitude periodic spiking and marks the transition from tonic spiking to bursting activity. The area of bursting activity is located between the PD and green curves.Reuse & Permissions