_{1}

Stern et al. have developed a mathematical model describing pseudo-plateau bursting of pituitary cells. This model is formulated based on the Hodgkin-Huxley scheme and described by a system of nonlinear ordinary differential equations. In the present study, computer simulation analysis of this model was performed to evaluate the correlation between the dynamic states of the model and two system parameters: long-lasting external stimulation (Iapp) and the time constant of delayed-rectifier potassium conductance activation (τn). Computer simulation results revealed that the model showed four different dynamic states: a hyperpolarized steady state, a depolarized steady state, a repetitive spiking state, and a bursting state. An increase in Iapp changed the dynamic states from the hyperpolarized steady state to bursting state to depolarized steady state when τn was fixed at smaller values, whereas it changed the dynamic states from the hyperpolarized steady state to bursting state to repetitive spiking state when τn was fixed at larger values. An increase in τn 1) did not change the dynamic states when Iapp was fixed at a very small value, 2) changed the dynamic states from the depolarized steady state to repetitive spiking state when Iapp was fixed at a very large value, and 3) changed the dynamic states from the depolarized steady state to bursting state to repetitive spiking state when Iapp was fixed at an intermediate value.

Bursting is a dynamic state that appears in nonlinear dynamic systems such as neurons [

Dynamic states of mathematical models that show bursting dynamics are affected by other system parameters, such as the time constant of potassium conductance. For example, changes in the time constant of dendritic delayed-rectifier potassium conductance inactivation induce a nonlinear effect on the dynamic states of the ghostbursting model [

The Stern model analyzed in the present study is formulated based on the Hodgkin-Huxley formalism, and is described by a system of nonlinear ODEs. The model contains four state variables: the membrane potential of corticotroph cells [V(t) (mV)] [t is time (s)], the activating variable of L-type calcium current [m_{L}(t)], the activating variable of delayed-rectifier potassium current [n(t)], and the concentration of calcium ions [Ca(t) (μM)]. Time evolutions of these state variables are described:

in which C_{m} is the membrane capacitance (0.00314 nF), I_{app} (pA) is the applied current corresponding to long- lasting external stimulation (this parameter was varied from −1.8 to 2.0 pA at an interval of 0.2 pA), τ_{n} (s) is the time constant of activating variable of delayed-rectifier potassium current (this parameter was varied from 0.017 to 0.027 s at an interval of 0.001 s), f is the fraction of free calcium (0.01), b is the ratio of cell surface area to cell volume (0.6 μm^{−1}), and I_{CaL}(V(t), m_{L}(t)), I_{CaT}(V(t)), I_{K}(V(t), n(t)), I_{KCa}(V(t), Ca(t)), I_{L}(V(t)), τ_{mL}(V(t)), m_{L}_{∞}(V(t)), n_{∞}(V(t)), J_{exchange}(Ca(t)), J_{influx}(V(t), m_{L}(t)), and J_{efflux}(Ca(t)) are the L-type calcium current, the T-type calcium current, the delayed-rectifier potassium current, the calcium-activated potassium current, the leak current, the time constant of activating variable of L-type calcium current, the steady state activation function of L-type calcium current, the steady state activation function of delayed-rectifier potassium current, the term for exchange between the cytosol and the internal store, the term for influx through calcium channels, and the term for pumping calcium out of the cell, respectively, each of which is calculated:

Detailed explanations of the Stern model are provided elsewhere [

The open-source software Scilab (http://www.scilab.org/) was used to numerically solve ODEs. Initial conditions were V(0) = −57.31515986286935 mV, m_{L}(0) = 0.06191856353928273, n(0) = 0.0003852853926905176, and Ca(0) = 0.4861280925831973 μM.

By systematically varying I_{app} and τ_{n}, we found that the Stern model had four different dynamic states: a hyperpolarized steady state, a bursting state, a depolarized steady state, and a repetitive spiking state. Examples of time courses of the membrane potential are shown in _{n} was 0.020 s, the hyperpolarized steady state appeared in response to current injection of I_{app}= −1.8 pA (_{app} was −1.0 pA, the dynamic state was the bursting state (_{app} was 1.8 pA, the dynamic state was the depolarized steady state (_{n} was 0.027 s under conditions in which I_{app} was 1.8 pA, the dynamic state was the repetitive spiking state (

_{app}, τ_{n})-parameter space. We characterized the dynamic states of the Stern model by slicing the (I_{app}, τ_{n})-parameter space in a horizontal or a vertical direction. First, we sliced the parameter space in a horizontal direction between τ_{n} = 0.022 s and 0.023 s. When τ_{n} was ≤0.022 s, an increase in I_{app} with a value of τ_{n} being fixed changed the dynamic states from the hyperpolarized steady state to

bursting state to depolarized steady state. A decrease in τ_{n} from 0.022 did not change the I_{app} threshold required for inducing the bursting state, whereas it decreased the I_{app} threshold required for inducing the depolarized steady state. Therefore, a decrease in τ_{n} induced a decrease in the I_{app} range in which the model showed the bursting state, finally resulting in a direct transition from the hyperpolarized steady state to depolarized steady state without undergoing the bursting state. When τ_{n} was ≥0.023 s, an increase in I_{app} with a value of τ_{n} being fixed changed the dynamic states from the hyperpolarized steady state to bursting state to repetitive spiking state. An increase in τ_{n} from 0.023 did not change the I_{app} threshold required for inducing the bursting state, whereas it decreased the I_{app} threshold required for inducing the repetitive spiking state. Therefore, an increase in τ_{n} induced a decrease in the I_{app} range in which the model showed the bursting state, finally resulting in a direct transition from the hyperpolarized steady state to the repetitive spiking state without undergoing the bursting state.

Second, we sliced the parameter space in a vertical direction in two ways: 1) between I_{app} = −1.8 pA and I_{app} = −1.6 pA, and 2) between I_{app} = 1.8 pA and I_{app} = 2.0 pA. Changes in the dynamic states can be characterized in three ways. First, when I_{app} was −1.8 pA, an increase in τ_{n} did not change the dynamic state: the state was the hyperpolarized steady state irrespective of the τ_{n} value. Second, when I_{app} was 2.0 pA, an increase in τ_{n} changed

the dynamic state from the depolarized steady state to repetitive spiking state. Third, when I_{app} was between −1.6 and 1.8 pA, an increase in τ_{n} changed the dynamic state from the depolarized steady state to bursting state to repetitive spiking state. In addition, an increase in I_{app} increased the τ_{n} threshold required for inducing the bursting state. In contrast, it decreased the τ_{n} threshold required for inducing the repetitive spiking state, and finally this threshold was constant as long as I_{app} was a positive value. An increase in I_{app} induced a decrease in the τ_{n} range in which the model showed the bursting state.

In the present study, numerical simulation of the Stern model was performed [

Previous studies have revealed that the time constant of potassium conductance plays an important role in the transition between the dynamic states in mathematical models of excitable cells. For example, variations in the time constant of dendritic delayed-rectifier potassium conductance inactivation induces the transition between the repetitive spiking and bursting states [

The present study clarified four dynamic states of the Stern model [

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

Takaaki Shirahata, (2016) Dynamics of a Pituitary Cell Model: Dependence on Long-Lasting External Stimulation and Potassium Conductance Kinetics. Applied Mathematics,07,861-866. doi: 10.4236/am.2016.79077