_{1}

A previous study has proposed a mathematical model of type-A medial vestibular nucleus neurons (mVNn). This model is described by a system of nonlinear ordinary differential equations, which is based on the Hodgkin-Huxley formalism. The type-A mVNn model contains several ionic conductances, such as the sodium conductance, calcium conductance, delayed-rectifier potassium conductance, transient potassium conductance, and calcium-dependent potassium conductance. The previous study revealed that spontaneous repetitive spiking in the type-A mVNn model can be suppressed by hyperpolarizing stimulation. However, how this suppression is affected by the ionic conductances has not been clarified in the previous study. The present study performed numerical simulation analysis of the type-A mVNn model to clarify how variations in the different ionic conductance values affect the suppression of repetitive spiking. The present study revealed that the threshold for the transition from a repetitive spiking state to a quiescent state is differentially sensitive to variations in the ionic conductances among the different types of ionic conductance.

Type-A medial vestibular nucleus neurons (mVNn) can show spontaneous repetitive spiking without stimulation in vitro. A mathematical model, which reproduces the spiking, has been proposed based on the Hodgkin-Huxley formulation [

Several studies have investigated the characteristics of the ionic conductances of various excitable cell models. For example, various studies have investigated the effect of co-variation of ionic conductances on the dynamics of excitable cell models [

The present study performed numerical simulations of a mathematical model of type-A mVNn, which was developed in a previous study [

where C_{m}(1 mF/cm^{2}) is the membrane capacitance; I_{app} is the externally injected current of constant amplitude; I_{Na}(V, n), I_{Ca}(V, x, [Ca]), I_{K}(V, n), I_{KCa}(V, [Ca]), I_{A}(V, b), and I_{L}(V) are the sodium current, calcium current, delayed-rectifier potassium current, calcium-dependent potassium current, transient potassium current, and leak current, respectively, which are defined in Equations (6)-(11) below;t_{X}(V) (ms) (X = n, x, b) and X_{∞}(V) (X = n, x, b) are the time constants of activation/inactivation and the steady-state activation/inactivation functions, respectively, which are defined in Equations (12)-(17) below; K_{p} (=1) is the calcium influx parameter; and R (=5) is the parameter that regulates calcium removal through calcium transport and mitochondrial uptake.

where g_{Na}, g_{Ca}, g_{K}, g_{KCa}, g_{A}, and g_{L} (=0.3 mS/cm^{2}) are the maximal conductances of I_{Na}(V, n), I_{Ca}(V, x, [Ca]), I_{K}(V, n), I_{KCa}(V, [Ca]), I_{A}(V, b), and I_{L}(V), respectively, and V_{Na}(=55 mV), V_{Ca}(=124 mV), V_{K}(=−80 mV), and V_{L}(=−50 mV) are the reversal potentials of I_{Na}(V, n), I_{Ca}(V, x, [Ca]), three types of potassium currents {I_{K}(V, n), I_{KCa}(V, [Ca]), and I_{A}(V, b)}, and I_{L}(V), respectively. Refer to [

The free and open source software, Scilab (http://www.scilab.org/), was used to numerically solve the above ODEs (initial conditions: V = −60 mV, n = 0.1, x = 0.1, b = 0.9, and [Ca] = 0.1 mM). The total simulation time was 0.6 s in all the simulations. System parameters whose values were varied were I_{app}, g_{Na}, g_{Ca}, g_{K}, g_{KCa}, and g_{A}. I_{app} between 0.2 s and 0.6 s was varied from −2.0 mA/cm^{2} to 2.5 mA/cm^{2} at an interval of 0.5 mA/cm^{2}, whereas I_{app} between 0.0 s and 0.2 s was set to zero. Default values of g_{Na}, g_{Ca}, g_{K}, g_{KCa}, and g_{A} were 20 mS/cm^{2}, 1 mS/cm^{2}, 2 mS/cm^{2}, 1 mS/cm^{2}, and 4 mS/cm^{2}, respectively. Each of g_{Na}, g_{Ca}, g_{K}, g_{KCa}, and g_{A} was varied to 50% or 150% of each default value.

The previous study had indicated that a hyperpolarizing stimulation of −3 mA/cm^{2} suppressed spontaneous repetitive spiking [^{2} was applied to the model, it showed a repetitive spiking behavior, but the frequency of spiking became smaller than that before the stimulation (^{2} was applied to the model, it did not show a repetitive spiking behavior, but showed a quiescent state (^{2} as the stimulation threshold for suppressing repetitive spiking (we hereafter referred to this threshold as the “suppression threshold”). To reveal the relationship between the ionic conductances and the dynamics of the model, the present study next investigated how the suppression threshold changed in response to variations of each ionic conductance.

The effects of variations in the ionic conductance values on the dynamics of the model are shown in _{Na} was 50% of the default value with other conductance values being set at the de-

fault values, the model showed a quiescent state in response to stimulation from −2.0 mA/cm^{2} to 2.0 mA/cm^{2}, whereas it showed a repetitive spiking state in response to stimulation of 2.5 mA/cm^{2} (_{Na} was 2.0 mA/cm^{2}. _{Na} and 150% g_{Na} were −1.0 mA/cm^{2} and −2.0 mA/cm^{2}, respectively. Similarly, the suppression thresholds under conditions in which other ionic conductance values were varied were calculated. The suppression threshold at 50% g_{Ca}, 100% g_{Ca}, and 150% g_{Ca} was −1.5 mA/cm^{2}, −1.0 mA/cm^{2}, and −0.5 mA/cm^{2}, respectively (_{A}, 100% g_{A}, and 150% g_{A} was −1.5 mA/cm^{2}, −1.0 mA/cm^{2}, and −0.5 mA/cm^{2}, respectively (_{KCa}, 100% g_{KCa}, and 150% g_{KCa} was −1.5 mA/cm^{2}, −1.0 mA/cm^{2}, and 0.0 mA/cm^{2}, respectively (_{K} was −1.0 mA/cm^{2} for all three conditions (

The present study performed numerical simulation of the type-A mVNn model and revealed the differential sensitivity of the threshold for the transition from a repetitive spiking state to a quiescent state to variations in the different ionic conductance values. The present study revealed the difference in modulation of the threshold among the ionic conductances. An increase in g_{Na} lowered the threshold, whereas an increase in g_{Ca}, g_{A}, or g_{KCa}raised the threshold. Finally, an increase in g_{K} had no effect on the threshold. The sensitivity of the threshold to variations in the ionic conductance values decreased in the following order: g_{Na} > g_{KCa} > g_{Ca} = g_{A} > g_{K}. Although the previous study had investigated the response of the type-A mVNn model to hyperpolarizing and depolarizing stimulation [

It is important to determine whether or not variations in system parameters affect mathematical models of neurons linearly. For example, analysis of the ghostbursting model has reported that changes in the dendritic potassium conductance kinetics induce anonlinear effect on the dynamical states in a two-dimensional parameter space of the ghostbursting model [_{Ca} and g_{A}, whereas the threshold changed highly nonlinearly in response to variations in g_{Na} (the threshold changed slightly nonlinearly in response to variations in g_{KCa}). In particular, the present study clarified that the dynamics of the type-A mVNn model was affected linearly in response to variations in the calcium conductance. However, because neither the ghostbursting model nor the vibrissa motoneuron model contains the calcium conductance variable [

The present study performed numerical simulation of the type-A mVNn model to reveal that the threshold for the transition from a repetitive spiking state to a quiescent state was differentially sensitive to variations in the ionic conductance values among the different types of the ionic conductances, which had not been reported in the previous study of the type-A mVNn model. The present study contributes to a more detailed understanding of the difference among the ionic conductances of the type-A mVNn model.

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

Takaaki Shirahata, (2016) The Effect of Variations in Ionic Conductance Values on the Suppression of Repetitive Spiking in a Mathematical Model of Type-A Medial Vestibular Nucleus Neurons. Applied Mathematics,07,1134-1139. doi: 10.4236/am.2016.710101