_{1}

A previous study proposed a mathematical model of A-type horizontal cells in the rabbit retina. This model, which was constructed based on the Hodgkin-Huxley model, was described by a system of nonlinear ordinary differential equations. The model contained five types of voltage-dependent ionic conductances: sodium, calcium, delayed rectifier potassium, transient outward potassium, and anomalous rectifier potassium conductances. The previous study indicated that when the delayed rectifier potassium conductance had a small value, depolarizing stimulation could change the dynamic state of the model from a hyperpolarized steady state to a depolarized steady state. However, how this change was affected by variations in the ionic conductance values was not clarified in detail in the previous study. To clarify this issue, in the present study, we performed numerical simulation analysis of the model and revealed the differences among the five types of ionic conductances.

A-type horizontal cells in the rabbit retina are classified into two types: one type can generate repetitive spiking [

The present study performed numerical simulations of a mathematical model of a rabbit A-type retinal horizontal cell that does not generate repetitive spiking (the non-spiking cell model), which was developed previously [_{Na}, h_{Na}, m_{Ca}, m_{Kv}, h_{Kv}, m_{A}, and h_{A}). The time evolution of these state variables is described as follows:

where C_{m} (=0.106 nF) is the membrane capacitance, I_{app} is the externally injected current of constant amplitude, I_{Na} (V, m_{Na}, h_{Na}), I_{Ca} (V, m_{Ca}), I_{Kv} (V, m_{Kv}, h_{Kv}), I_{A} (V, m_{A}, h_{A}), I_{Ka} (V), and I_{L} (V) are the sodium, calcium, delayed rectifier potassium, transient outward potassium, anomalous rectifier potassium, and leakage currents, respectively, which are defined in the Equations (9)-(14) below.

where g_{Na}, g_{Ca}, g_{Kv}, g_{A}, g_{Ka}, and g_{L} (=0.5 nS) are the maximal conductances of I_{Na} (V, m_{Na}, h_{Na}), I_{Ca} (V, m_{Ca}), I_{Kv} (V, m_{Kv}, h_{Kv}), I_{A} (V, m_{A}, h_{A}), I_{Ka} (V), and I_{L} (V), respectively, E_{Na} (=55 mV), E_{Ca} (=12.9log[2000/30] mV), E_{K} (=−80 mV), and E_{L} (=−80 mV) are the reversal potentials of I_{Na} (V, m_{Na}, h_{Na}), I_{Ca} (V, m_{Ca}), three types of potassium currents [i.e., I_{Kv} (V, m_{Kv}, h_{Kv}), I_{A} (V, m_{A}, h_{A}), and I_{Ka}(V)], and I_{L} (V), respectively. Refer to reference [

The free and open-source software Scilab (http://www.scilab.org/) was used to numerically solve the above ODEs (initial conditions: V = −80 mV, m_{Na} = 0.026, h_{Na} = 0.922, m_{Ca} = 0.059, m_{Kv} = 0.139, h_{Kv} = 0.932, m_{A} = 0.030, and h_{A} = 0.998). The total simulation time was 10 s in all the simulations. The values of the following system parameters were varied: I_{app}, g_{Na}, g_{Ca}, g_{Kv}, g_{A}, and g_{Ka}. I_{app} between 0.5 and 10 s was varied from 13 to 19 pA at an interval of 1 pA, while I_{app} between 0.0 and 0.5 s was fixed to be zero. Default values of g_{Na}, g_{Ca}, g_{Kv}, g_{A}, and g_{Ka} were 2.4, 9.0, 4.5, 15.0, and 4.5 nS, respectively. Each ofg_{Na}, g_{Ca}, g_{Kv}, g_{A}, and g_{Ka} was varied to 50 or 150% of each default value.

The previous study investigated the responses of the non-spiking cell model to depolarizing stimulations of different conditions [

The present study next investigated how variations in each ionic conductance value changed the stimulation threshold in order to reveal the relationship between the ionic conductances and the dynamics of the model. _{Na} was 50% of the default value with the other conductance values being default values, the model showed a hyperpolarized steady state in response to stimulations from 13 to 15 pA, but showed a depolarized steady state in response to stimulations from 16 to 19 pA. Therefore, the stimulation threshold for the induction of the positive potential was 16 Pa at 50% g_{Na}. Similarly, the thresholds under different g_{Na} conditions were calculated: the threshold at 100% g_{Na} and 150% g_{Na} was 15 pA. The thresholds under conditions in which the other ionic conductance values were varied were also calculated in a similar manner. The threshold was 19 pA at 50% g_{Ca}, 15 pA at 100% g_{Ca}, and 14 pA at 150% g_{Ca}. The threshold was 15 pA at 50% g_{Kv}, 15 pA at 100% g_{Kv}, and 16 pA at 150% g_{Kv}. The threshold was 15 pA at 50% g_{A}, 15 pA at 100% g_{A}, and 16 pA at 150% g_{A}. The threshold was 15 pA at 50% g_{Ka}, 15 pA at 100% g_{Ka}, and 17 pA at 150% g_{Ka}.

The present study performed numerical simulation of a mathematical model of a non-spiking A-type horizontal

cell in the rabbit retina, and revealed the sensitivity of the stimulation threshold for the transition from a hyperpolarized steady state to a depolarized steady state to variations in ionic conductance values. A previous study investigated the effect of variations in ionic conductance values on the dynamics of the model [_{Na} and g_{Ca} induced a decrease in the stimulation threshold, whereas increases in g_{Kv}, g_{A}, and g_{Ka} induced an increase in the stimulation threshold (_{Ca} > g_{Ka} > g_{Na} = g_{Kv} = g_{A}. In particular, the present study revealed that the influence of g_{Ka} on the stimulation threshold was the largest among the three types of potassium conductances (i.e., g_{Kv}, g_{A}, and g_{Ka}). This is an important finding, which has not been reported previously.

Analyses of mathematical models of other retinal cells based on the Hodgkin-Huxley model have been performed previously (e.g., in a retinal ganglion cell model [

Previous studies of other neuron models have reported the relationship between the stimulation threshold and ionic conductances (e.g., a vibrissa motoneuron model [

The present study performed numerical simulation of a model of a nonspiking A-type horizontal cell in the rabbit retina to reveal the effect of variations in ionic conductances on the dynamics of the model. In particular, the present study revealed that (1) the calcium conductance affected the dynamical states of the model highly nonlinearly, and (2) the anomalous rectifier potassium conductance had the largest influence on the changes in the stimulation threshold among the three types of potassium conductances. Neither of these details has been reported previously. The present study contributes to a more detailed understanding of the characteristics of the ionic conductances of the 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 Dynamics of a Mathematical Model of Non-Spiking A-Type Horizontal Cells in the Rabbit Retina. Applied Mathematics,07,1297-1302. doi: 10.4236/am.2016.712114