Figure 1.
Simulated effect of antipsychotic drugs in the DA neuron.
The influence of the drugs is modeled as excitation by applied current (lower trace) Tonic firing in the DA neuron (upper trace) is interrupted with excessive excitation. DA neuron remains silent after complete withdrawal of excitation due to hysteresis. Parameters for the DA neuron are from Table 1.
Table 1.
Parameter values for the DA neuron and the HH neuron.
Figure 2.
The Na+ current dynamics in the DA and HH neurons.
A) Activation and B) inactivation functions of the Na+ current in the DA neuron (solid curve) and the HH neuron (dashed curve). C) The time constant function of the Na+ current. Note that the functions are shifted by around 20 mV for a better comparison of the slopes. The ranges for the HH neuron are at the top and to the right.
Figure 3.
The K+ current dynamics in the DA and HH neurons.
A) Activation and B) time constant functions of the K+ current from the DA neuron (solid curve) and the HH neuron (dashed curve). The ranges for the HH neuron are at the top and to the right.
Figure 4.
One-parameter bifurcation diagrams for the DA neuron and the HH neuron.
Hysteresis at the upper boundary of the oscillatory range (where it exists) is indicated by arrows showing direct and reverse transitions. A) Oscillatory solution stays isolated from the equilibrium state in the DA neuron. This is the Class 3 excitability. Parameters for the DA neuron are from Table 1. B) The oscillatory solution connects to the equilibrium state in an Andronov-Hopf bifurcation. Parameters are from A), except that the K+ current half-activation is increased by 4 mV (vnh = −31 mV). C) Hysteresis is not present at the upper boundary of the oscillatory range in the HH neuron. Parameters for the HH neuron are from Table 1. D) The oscillatory solution connects to the equilibrium state in an Andronov-Hopf bifurcation. Parameters are from C), except that the K+ current half-activation is decreased by 5 mV (vnh = −58 mV). Thin curves represent equilibrium states, thick curves - limit cycles. Solid (dashed) curves represent stable (unstable) solutions. HB is the Andronov-Hopf bifurcation, SNLC is the saddle-node of limit cycles bifurcation.
Figure 5.
Two-parameter bifurcation diagrams of the DA neuron and the HH neuron in vnh/vhh/vmh and Iapp planes.
The hysteresis regions are shaded gray. A) Hysteresis is strong in the DA neuron. Parameters are from Table 1. B) Hysteresis is removed and the oscillatory region expands in the DA neuron with the increase in the half-inactivation of the Na+ current. Parameters are from Figure 4B. C) Hysteresis is not reduced with the decrease in the half-activation of the Na+ current. Parameters are from Figure 4B. D) Hysteresis is weak in the HH neuron. Parameters are from Table 1. E) Hysteresis is removed in the HH neuron with an increase of vhh. Parameters are from Figure 4D. F) Hysteresis is weak in the HH neuron with an increase of vmh. Parameters are from Figure 4D. A solid curve represents an Andronov-Hopf bifurcation, a dashed curve – a saddle-node bifurcation of limit cycles. Horizontal dotted lines in A) and D) represent the values of half-activation of the K+ current taken in B), C) and E), F), correspondingly. Horizontal dotted lines in B), C) and E), F) represent the values of half-(in)activations from Table 1 for the DA and HH neurons, correspondingly.
Figure 6.
Two-parameter bifurcation diagrams of the DA and HH neuron for the change in slope factors of (in)activation functions.
The hysteresis regions are shaded gray. A) A gradual voltage dependence of the Na+ current inactivation function removes hysteresis in the DA neuron. B) Steeper voltage dependence of the activation of the Na+ current has almost no effect on hysteresis in the DA neuron. C) More gradual voltage dependence of the K+ current has little effect on hysteresis in the DA neuron. Parameters are from Figure 4B. D) More gradual voltage dependence of the Na+ current reduces and then completely abolishes hysteresis in the HH neuron. E) Gradual voltage dependence of the activation of the Na+ current has little effect on hysteresis at the upper boundary of oscillatory region in the HH neuron. F) Hysteresis range peaks at intermediate values of Sn in the HH neuron. A solid curve represents an Andronov-Hopf bifurcation, a dashed curve – a saddle-node bifurcation of limit cycles. Horizontal dotted lines in A) and D) mark the value of slope parameter from Table 1 for the HH neuron.
Figure 7.
Changing kinetics of gating variables in the DA neuron and the HH neuron.
The hysteresis regions are shaded gray. A), B) Bistability range shortens with accelerating the gating variables kinetics in the DA neuron. fh = 1 and fn = 1 correspond to parameter set from Figure 4B. C) Simultaneous acceleration of both n and h variables decreases the size of bistability range. fn,h = 1 corresponds to parameter set from Figure 4B. D), E) Hysteresis is reduced or eliminated in the HH neuron with slowing the individual current kinetics. fh = 1 and fn = 1 correspond to parameter set from Figure 4D. F) Hysteresis is increased with simultaneous slowing of gating variables n and h. fn,h = 1 corresponds to parameter set from Figure 4D. A solid curve represents an Andronov-Hopf bifurcation, a dashed curve – a saddle-node bifurcation of limit cycles. Horizontal dotted lines (where shown) give the values of fh and fn for which the maximum value of the corresponding time constant function for the DA (HH) neuron matches the maximum value of the time constant for the HH (DA) neuron.
Figure 8.
Two-parameter diagrams for the change in maximal conductances and equilibrium potential.
The hysteresis regions are shaded gray. A) Increase in gK expands the oscillatory region, shortens the instability region and increases hysteresis in the DA neuron. B) Increase in K+ current reversal potential shortens oscillatory region much faster than the instability region, reducing hysteresis in the DA neuron. C) Increase in the maximum conductance of the leak current shortens both instability and oscillatory regions and finally eliminates oscillations in the DA neuron. D) Hysteresis exists in a narrow range of parameter gK in the HH neuron. E) Hysteresis at the upper boundary of the oscillatory region is not affected by the decrease in EK in the HH neuron. F) Hysteresis is slightly reduced with the decrease in the maximum leak conductance in the HH neuron. A–C) Parameter values from Figure 4B. D–F) Parameter values from Figure 4D. A solid curve represents an Andronov-Hopf bifurcation, a dashed curve – a saddle-node bifurcation of limit cycles. Horizontal dotted lines mark parameter values from Table 1 for the DA and the HH neurons, correspondingly.
Table 2.
Normalized parameter contribution to hysteresis in the DA and HH neurons.