_{1}

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Sim and Forger have proposed a mathematical model of circadian pacemaker neurons in the suprachiasmatic nucleus (SCN). This model, which has been formulated on the Hodgkin-Huxley mo-del, is described by a system of nonlinear ordinary differential equations. An important feature of the SCN neurons observed in electrophysiological recording is spontaneous repetitive spiking, which is reproduced using this model. In the present study, numerical simulation analysis of this model was performed to evaluate variations in two system parameters of this model: the maximal conductance of calcium current (
*g*
* _{Ca}*) and the maximal conductance of sodium current (

*g*

*). Simulation results revealed the spontaneous repetitive spiking states of the model in the (*

_{Na}*g*

*,*

_{Ca}*g*

*)-pa-rameter space.*

_{Na}Repetitive spiking activity is an important feature of excitable cells such as neurons and muscle cells. In the field of applied mathematics, this activity can be extensively analyzed using various mathematical models, which are described by a system of nonlinear ordinary differential equations (ODEs). One example is the Hodgkin-Huxley model, which can generate repetitive spiking because of an interaction between the sodium conductance and the potassium conductance (page 25 in [

Another type of mathematical model of the repetitive spiking activity is the one in which the repetitive spiking is generated by an interaction among sodium, calcium, and potassium conductances. One example is a mathematical model of circadian pacemaker neurons in the suprachiasmatic nucleus (SCN) [_{Ca}) and the maximal conductance of sodium current (g_{Na}), and we performed numerical simulation analysis.

A mathematical model of the circadian pacemaker neuron, which has been used in the present study, was a model developed by Sim and Forger [

where C is membrane capacitance (5.7 pF); I_{Na}(V, m, h), I_{K}(V, n), I_{L}(V), and I_{Ca}(V, r, f) are a sodium current, a potassium current, a leak current, and a calcium current, respectively, which are defined in Equations (3)-(6) below, in this order; τ_{X}(V) (ms) and X_{∞}(V) are time constants of activation/inactivation and steady-state activation/inactivation functions, respectively, which are defined in Equations (7)-(16) below, in this order; and I_{app} is applied current.

where g_{Na}, g_{K}, g_{L}, and g_{Ca} are the maximal conductances of a sodium current (the g_{Na} value was varied in the present study; the default value was 229 nS), a potassium current (14 nS), a leak current (1/11 nS), and a calcium current (the g_{Ca} value was varied in the present study; the default value was 65 nS), respectively; E_{Na}, E_{K}, E_{L}, and E_{Ca} are the reversal potentials of a sodium current (45 mV), a potassium current (−97 mV), a leak current (−29 mV), and a calcium current (61 mV), respectively.

Detailed explanations of the model equations have been described previously [

The free and open source software Scilab (http://www.scilab.org/) was used to numerically solve ODEs (initial conditions: V = −80 mV, m = 0.34, h = 0.045, n = 0.54, r = 0.01, f = 0.04).

The time courses of the membrane potential of the SCN neuron model under different parameter values are shown in _{Ca} was completely blocked, the model showed the stable steady state instead of the spontaneous spiking state (_{Na} was completely blocked, the model showed a similar pattern, as illustrated in _{Ca} and g_{Na} were set to be values larger than the default values, the model showed the stable steady state (

_{Ca}, g_{Na})-parameter space. The dynamical states of the model were classified into three states: the hyperpolarized steady state (white circle), the spontaneous spiking state (black circle), and the depolarized steady state (double circle). A decrease in g_{Na} induced a decrease in the g_{Ca} range in which repetitive spiking occurred, and finally, the g_{Ca} range disappeared. Therefore, it is concluded that when g_{Na} is zero, repetitive spiking cannot occur, no matter how large the g_{Ca} value is. An increase in g_{Na} induced a decrease in the g_{Ca} threshold required to induce the transition from the hyperpolarized steady state to the spiking state. For example, even when g_{Ca} greatly decreased to 30 nS, repetitive spiking occurred under conditions in which g_{Na} greatly increased to 1603 nS (a seven-fold increase from the default value) (_{Ca} was zero, repetitive spiking occurred under conditions in which g_{Na} was 1603 nS (

The present study revealed the sensitivity of the model dynamics to variations in g_{Ca} and g_{Na} of circadian pacemaker neurons. Although Sim and Forger revealed the difference in time courses between the calcium and sodium currents (_{Ca} and g_{Na} were indispensable to repetitive spiking under the default condition. Interestingly, the present study also revealed the difference between the two conductances. When we considered repetitive spiking under the condition in which the g_{Ca} value was smaller than its default value but the g_{Na} value was much larger than its default value, the sodium conductance was indispensable but the

calcium conductance was dispensable to this type of repetitive spiking (

Previous studies have reported the relationship between ionic conductances and neuronal spiking in various types of mathematical models [_{Na} induced a decrease in the g_{Ca} threshold required to induce the transition from the nonpacemaking state to the pacemaking state and (2) the pacemaking occurred in the absence of the calcium conductance under conditions in which the sodium conductance was set to a large value. The present circadian pacemaker neuron model also exhibited these two characteristics. However, the present results revealed two important differences between these two models: (1) the (g_{Ca}, g_{Na})-parameter space was divided into only two states (the pacemaking and nonpacemaking states) in the midbrain dopaminergic neuron model (_{Ca}, g_{Na})-parameter space was divided into three states (the depolarized steady state, the repetitive spiking state, and the hyperpolarized steady state) in the circadian pacemaker neuron model (

The present investigation focused on a mathematical model of circadian pacemaker neurons, performed numerical simulation analysis, and compared this numerical result with that of the previous studies. The important and novel findings of the present study are as follows: 1) there was a difference in the contribution to repetitive spiking under certain conditions between the sodium and calcium conductances: for the generation of repetitive spiking under certain conditions, the sodium conductance played an indispensable role, whereas the calcium conductance was not necessarily essential, and 2) the (g_{Ca}, g_{Na})-parameter space of the circadian pacemaker neuron model showed a different pattern compared with that of the midbrain dopaminergic neuron model. These findings can contribute to our in-depth understanding of the influence of the sodium and calcium conductances on neuronal repetitive spiking.

The author would like to thank Enago (www.enago.jp) for the English language review.