Figure 1.
Step responses of the HH model without (left) and with a calcium current (right).
(A) Time-evolution of the applied excitatory current (bottom) and of the corresponding membrane potential (top) in HH model (the reduced model leads to almost the same behavior (Fig. S1)). (B) Phase portraits of the reduced Hodgkin-Huxley model in resting (top) and spiking states (bottom). The - and
-nullclines are drawn as a full and a dashed line, respectively. Trajectories are drawn as solid oriented red lines. Black circles denote stable fixed points, white circles unstable fixed points, and cross saddle points. The presence of calcium channels strongly affects the phase-portrait and the corresponding electrophysiological time-response of the neuron to excitatory inputs.
Figure 2.
Total ionic currents for fixed as a function of
without (left) and with calcium channels (right).
Blank portions corresponds to the values of where
, shaded portions corresponds to the values of
where
, and the dashed lines correspond to the values of
where
. The hyperpolarizing calcium pump current
accounts for the vertical shift of the curve in the right figure. Note that the total ionic currents monotonically increase only in the absence of calcium channels (left).
Figure 3.
One parameter bifurcation diagram of the reduced Hodgkin-Huxley model without (left) and with calcium channels (right).
Thin solid lines represents stable fixed points, while dashed lines unstable fixed points or saddle points. The thick lines labeled and
represent the minimum and the maximum voltage of stable limit cycles, respectively.
, with
, denotes the value of the input current for which the system undergoes the bifurcation
.
Figure 4.
Transcritical bifurcation as the main ruler of neuronal excitability.
(A) Cartoon of the V-nullcline transition through a singularly perturbed transcritical bifurcation. Black circles denote stable fixed points, white circles unstable fixed points. (B) Continuation of the stable (in green) and the unstable
(in red) manifolds of the saddle away from the singular limit (i.e.
). They dictate the transition from the resting state (
) to the the spiking limit cycle (
) via a saddle-homoclinic bifurcation (
).
Figure 5.
Unfolding of the transcritical bifurcation in the global reduced Hodgkin-Huxley phase portrait.
(A and B) Phase portraits or the original reduced HH model without (a) and with a calcium current (b). The unshaded area represents the area of physiological relevance. A constant inhibitory current of increasing amplitude (from left to right) is applied to the model. The transcritical bifurcation (green circle) is non physiological in the classical reduced HH model (A) but plays an important physiological role in the presence of calcium channels (B).
Figure 6.
Schematic phase-portraits of the transcritical hybrid model for different values of and
. The
- and
-nullclines are drawn as full and dashed lines, respectively.
The trajectories are drawn as red oriented lines. Many different phase-portraits derive from the transcritical hybrid model, including the one of the fold hybrid model, which only captures the shaded area.
Figure 7.
Step responses of a conductance-based TC neuron model (adapted from [46]) in low (left) and high calcium conductance modes (right).
(A and B): Membrane potential variations of the simulated TC neuron over time in both conditions. The model reproduces the firing patterns exhibited by TC relay cells, namely tonic and burst firing.
Figure 8.
Phase-portraits of the reduced conductance based TC model in low (A) and high calcium conductance modes (B).
The - and
-nullclines are drawn as full and dashed lines, respectively, and the hyperpolarized state as a filled circle
. The trajectories are drawn as red oriented lines. In high calcium conductance mode, the phase portrait shows two
-nullclines branches which derives from a transcritical singularity.
Figure 9.
Comparison of the experimental step response of a TC neuron in vitro [41] (top) and the step response of the proposed transcritical hybrid model (bottom) in low (left) and high calcium conductance modes (right).
(A and B): Membrane potential variations of the recorded TC neuron over time in both conditions. (C and D) Membrane potential variations of the modeled TC neuron over time in both conditions. A variation of , which is an image of the calcium conductance, is sufficient to generate the switch of firing pattern physiologically observed in TC cells.
Figure 10.
Phase-portrait of the transcritical hybrid model of a TC cell in the low (, left) and high calcium conductance modes (
, right).
Trajectories are depicted as solid oriented red lines. The reset point is depicted as a square , while the (instantaneous) hyperpolarized state as a filled circle
. The
-nullcline is depicted as a dashed line. In (A), the gray (black) thin solid line is the
-nullcline when the current step in off (on). In (B), the
-nullcline is depicted as gray thin lines of different darkness. As sketched in the figure, light gray correspond to large values of
, whereas dark gray to small.
Figure 11.
Nominal step response (left) and step response in the presence of small current pulses in the 200 compartments TC neuron model (A), the transcritical hybrid model (B), and the fold hybrid model of TC neuron ([11]) (C).
Figure 12.
Comparison of the ADP generation mechanisms in the fold (left) and in the transcritical hybrid models (right).
The stable manifold of the saddle () is depicted in green. In the fold hybrid model, ADPs are generated by sliding near the stable manifold of the saddle and crossing the
-nullcline from below. In the transcritical hybrid model, ADPs are robustly generated along the attractor
.