_{1}

Electrosensory pyramidal neurons in weakly electric fish can generate burst firing. Based on the Hodgkin-Huxley scheme, a previous study has developed a mathematical model that reproduces this burst firing. This model is called the ghostbursting model and is described by a system of non-linear ordinary differential equations. Although the dynamic state of this model is a quiescent state during low levels of electrical stimulation, an increase in the level of electrical stimulation transforms the dynamic state first into a repetitive spiking state and finally into a burst firing state. The present study performed computer simulation analysis of the ghostbursting model to evaluate the sensitivity of the three dynamic states of the model (i.e., the quiescent, repetitive spiking, and burst firing states) to variations in sodium and potassium conductance values of the model. The present numerical simulation analysis revealed the sensitivity of the electrical stimulation threshold required for eliciting the burst firing state to variations in the values of four ionic conductances (i.e., somatic sodium, dendritic sodium, somatic potassium, and dendritic potassium conductances) in the ghostbursting model.

The ghostbursting model is a mathematical model of electrosensory pyramidal neurons in weakly electric fish, which is described by a system of nonlinear Ordinary Differential Equations (ODEs) (see Methods in [

Studies of the characteristics of the potassium conductance of the dendritic compartment have previously been carried out, as described above. However, characteristics of the potassium conductance of the somatic compartment and the sodium conductances of the somatic/dendritic compartments have not been investigated in detail. Doiron and coworkers have implied that variations in values of the sodium conductances of the somatic/ dendritic compartments or the potassium conductance of the somatic compartment may affect electrical stimulation thresholds (see the last sentence of Discussion of [

The ghostbursting model used in this study is described by a system of ODEs, which consists of six state variables: the membrane potential of the somatic compartment [V_{s}(t) (mV)] [t is time (ms)], the activating variable of the potassium conductance of the somatic compartment [n_{s}(t)], the membrane potential of the dendritic compartment [V_{d}(t) (mV)], the inactivating variable of the sodium conductance of the dendritic compartment [h_{d}(t)], the activating variable of the potassium conductance of the dendritic compartment [n_{d}(t)], and the inactivating variable of the potassium conductance of the dendritic compartment [p_{d}(t)]. The dynamic states of the ODEs can change depending on the following system parameters: electrical stimulation of the somatic compartment (I_{s}), the maximal sodium conductance of the somatic compartment (g_{Na}_{,s}), the maximal potassium conductance of the somatic compartment (g_{Dr}_{,s}), the maximal sodium conductance of the dendritic compartment (g_{Na}_{,d}), and the maximal potassium conductance of the dendritic compartment (g_{Dr}_{,d}). The ODEs that describe the ghostbursting model are described as follows:

I_{s} (μA/cm^{2}) is changed from 5.6 to 9.6. The default values of conductances are g_{Na}_{,s} = 55 mS/cm^{2}, g_{Dr}_{,s} = 20 mS/cm^{2}, g_{Na}_{,d} = 5 mS/cm^{2}, and g_{Dr}_{,d} = 15 mS/cm^{2}. Detailed explanations of the ODEs are described in [

The free and open source software Scilab (http://www.scilab.org/) was used to numerically solve the ODEs (initial conditions: V_{s} = −70 mV, n_{s} = 0.00005, V_{d} = −70 mV, h_{d} = 0.973, n_{d} = 0.002, and p_{d} = 0.697). Solving the equations was performed using the lsoda solver implemented in the program Scilab.

Under conditions in which all conductance values were default values [see the columns of 100% conductance in Figures 1(a)-(d)], the dynamic states of the ghostbursting model were a quiescent state (× in _{s}

was 5.6, a repetitive spiking state (○ in _{s} was between 5.8 and 8.4, and a bursting firing state (● in _{s} was between 8.6 and 9.6. The repetitive spiking threshold (I_{s} = 5.8) did not change even if g_{Na}_{,s} decreased to 95% or increased to 105% of the default value [_{s} = 8.6) was sensitive to variations in g_{Na}_{,s}: it decreased to 8.2 when g_{Na}_{,s} decreased to 95% of the default value, whereas it increased to 9.0 when g_{Na}_{,s} increased to 105% of the default value [_{s} = 5.8) increased to 6.0 when g_{Na}_{,d} decreased to 95% but did not change when g_{Na}_{,d} increased to 105% [_{s} = 8.6) was sensitive to variations in g_{Na}_{,d}; it increased to 9.0 when g_{Na}_{,d} decreased to 95%, whereas it decreased to 8.2 when g_{Na}_{,d} increased to 105% [_{s} = 5.8) did not change even if g_{Dr}_{,s} decreased to 90% or increased to 110% [_{s} = 8.6) was sensitive to variations in g_{Dr}_{,s}; it increased to 8.8 when g_{Dr}_{,s} decreased to 90%, whereas it decreased to 8.4 when g_{Dr}_{,s} increased to 110% [_{s} = 5.8) did not change even if g_{Dr}_{,d} decreased to 95% or increased to 105% [_{s} = 8.6) was sensitive to variations in g_{Dr}_{,d}; it decreased to 7.8 when g_{Dr}_{,d} decreased to 95%, whereas it increased to 9.6 when g_{Dr}_{,d} increased to 105% [

The present numerical simulation analysis revealed the sensitivity of the repetitive spiking threshold and the bursting threshold to variations in the values of four ionic conductances (i.e., g_{Na}_{,s}, g_{Na}_{,d}, g_{Dr}_{,s}, and g_{Dr}_{,d}) in the ghostbursting model. A previous study illustrated the sensitivity of the repetitive spiking threshold and the bursting threshold to variations in g_{Dr}_{,d} [_{Na}_{,s}, g_{Na}_{,d}, and g_{Dr}_{,s}. However, this was not shown explicitly [

The importance of the present findings are that they clearly demonstrate the sensitivity of the repetitive spiking threshold and the bursting threshold to variations in g_{Na}_{,s}, g_{Na}_{,d}, and g_{Dr}_{,s}. Specifically, 1) similar to the case of g_{Dr}_{,d}, the repetitive spiking threshold is insensitive to variations in the other ionic conductances, except g_{Na}_{,d}; 2) the bursting threshold is sensitive to variations in the other three ionic conductances; 3) similar to the previous case, in which an increase in g_{Dr}_{,d} increases the bursting threshold [_{Na}_{,s} increases the bursting threshold, whereas an increase in g_{Na}_{,d} and g_{Dr}_{,s} decreases the bursting threshold; and 4) the degree of sensitivity occurs as follows: g_{Dr}_{,d} > g_{Na}_{,s} = g_{Na}_{,d} > g_{Dr}_{,s}.

The effect of variations in ionic conductance values on the behaviors of mathematical models of excitable cells is an important topic of investigation. For example, action potential duration is differentially modulated by variations in the slow-inward calcium conductance (G_{si}) versus the delayed rectifier potassium conductance (G_{K}) in the LR1 model [_{CaS}, G_{h}, G_{Kd}, G_{A}, and G_{KCa}) in regulating maximal gain modulation is revealed in a model of lobster somatogastric neurons [_{Na}_{,s}, g_{Na}_{,d}, g_{Dr}_{,s}, and g_{Dr}_{,d}. Notably, the present study examines conductances of the same ion type in both somatic and dendritic compartments (i.e., the model incorporates differences between g_{Na}_{,s} and g_{Na}_{,d}, and between g_{Dr}_{,s} and g_{Dr}_{,d}). Previous studies described above [

Previous work has reported the sensitivity of the repetitive spiking threshold and the bursting threshold to variations in g_{Dr}_{,d} but has not characterized the sensitivity of these thresholds to variations in three other ionic conductances (g_{Na}_{,s}, g_{Na}_{,d}, and g_{Dr}_{,s}) [_{Na}_{,s}, g_{Na}_{,d}, g_{Dr}_{,s}, and g_{Dr}_{,d}) in the ghostbursting model.

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

Takaaki Shirahata, (2016) The Relationship of Sodium and Potassium Conductances with Dynamic States of a Mathematical Model of Electrosensory Pyramidal Neurons. Applied Mathematics,07,819-823. doi: 10.4236/am.2016.79072