Evaluation of Kinetic Properties of Dendritic Potassium Current in Ghostbursting Model of Electrosensory Neurons

A ghostbursting model is a mathematical model (a system of coupled nonlinear ordinary differential equations) that is based on the Hodgkin-Huxley formalism. The ghostbursting model describes bursting similar to the in vitro bursting of electrosensory neurons of weakly electric fish. Doiron and coworkers have focused on two system parameters of the model: maximal conductance of the dendritic potassium current ( ) , Dr d g and the current injected into the somatic compartment ( ) s I . They performed bifurcation analysis and revealed that the ( ) , , Dr d s g I -parameter space was divided into three dynamical states: quiescence, periodic tonic spiking, and bursting. The present study focused on a third system parameter: the time constant of dendritic potassium current inactivation ( ) pd τ . A computer simulation of the model revealed how the dynamical states of the ( ) , , Dr d s g I -parameter space changed in response to variations of pd τ .


Introduction
Hodgkin and Huxley [1] proposed a mathematical model that is composed of a system of four-coupled nonlinear ordinary differential equations (page 518 in [1]) and that describes the action potential regeneration of the squid giant axon and the biophysical mechanisms underlying the action potential generation.Various types of mathematical models describing the electrical excitability of neurons and endocrine cells have been developed on the basis of the concepts proposed by Hodgkin and Huxley [1], and analyses of these models, including the RPeD1 neuron model in [2], various bursting models in Chapter 5 of [3], and pituitary lactotroph bursting model in [4], are important research areas in the field of applied mathematics.The concepts proposed by Hodgkin and Huxley [1] are also important in the fields of theoretical physics [5] and mathematical physics [6].The Hodgkin-Huxley model is also used in drug-disease modeling (see Chapter 5.2.2 in [7]).A ghostbursting model [8], which is a mathematical model based on the concepts proposed by Hodgkin and Huxley [1], describes a system of six-coupled nonlinear ordinary differential equations [see Equations (1) to (6) in Section 2].This model exhibits bursting similar to that observed in in vitro recordings of pyramidal cells in the electrosensory lateral line lobe (ELL) of the weakly electric fish Apteronotus leptorhynchus.This model consists of two compartments: the somatic compartment [see Equations ( 1) and ( 2

They performed ( )
, , Dr d s g I -parameter bifurcation analysis of the model (see Figure 6 in [8]).This figure indicates that the organizing center of the ( ) Dr d s g I -parameter bifurcation diagram is a codimension-two bifurcation point and that unfolding the codimension-two bifurcation point yields two types of bifurcation manifolds: a curve for a saddle-node bifurcation of fixed points (SNFP curve) and a curve for a saddle-node bifurcation of limit cycles (SNLC curve).The SNFP and SNLC curves divide the ( ) Dr d s g I -parameter space into three dynamical states: quiescence, periodic tonic spiking, and bursting.When crossing the SNFP curve with an increase in s I under a condition in which value at the codimension-two bifurcation point, the dynamical state changes from quiescence to periodic tonic spiking.The periodic tonic spiking further changes into bursting when the SNLC curve is crossed with an increase in s I .In addition, various bursting patterns are shown in Figure 13 in [8] and Figure 3 in [9].Vo et al. have indicated that it is important to investigate the kinetic properties of ionic conductance for understanding the dynamics of pituitary cell models [10].In other words, variations in the time constant values of ionic conductance can change the dynamical states of the cell model (Figure 4 in [10]).Doiron et al. have also suggested that the appropriate setting of the time constant value in dendritic potassium current inactivation is important for bursting dynamics (see the last paragraph of Section 3.3 in [8]).However, how variations in the time constant values affect the ( ) , , Dr d s g I -parameter space was not revealed in their study.Therefore, to contribute to an in-depth understanding of the kinetic properties of dendritic potassium current inactivation, in the present study, we performed numerical analysis and clarified the influence of time constant variations on the ( ) Dr d s g I -parameter space.

Materials and Methods
The ghostbursting model [Equations ( 1)-( 6)] contains the following six state variables: the somatic membrane potential ( ) ( ) where the definitions and values of the above-mentioned parameters are listed in Table 1.Equation ( 1) indicates that the time evolution of the somatic membrane potential ( ) V is regulated by the fast inward sodium current Table 1.Values of the parameters in Equations ( 1)-( 6) from [8]. (the 2 nd term), outward delayed-rectifying potassium current (the 3 rd term), leak current (the 4 th term), and electrotonic diffusive current between the somatic and dendritic compartments (the 5 th term).Similarly, Equation (3) indicates that the time evolution of the dendritic membrane potential ( ) V is regulated by the fast inward so- dium current (the 1 st term), outward delayed-rectifying potassium current (the 2 nd term), leak current (the 3 rd term), and electrotonic diffusive current between the somatic and dendritic compartments (the 4 th term).Equations ( 2), ( 4), (5), and (6) indicate that the activating or inactivating variables approach the steady-state function , , , .x s d y ns hd nd pd = = at a rate that depends on the time constant y τ ( ) , , , .y ns hd nd pd = .
For detailed explanations of the model, see [8].

Reproduction of Previous Results
The ghostbursting model can show the three dynamical states: quiescence (Figure 1(a)), periodic tonic spiking (Figure 1(b)), and bursting (Figure 1(c)).The present study shows that the regions of these dynamical states

Effects of Changes in pd
τ on the ( ) , ,

Dr d s g I -Parameter Space
The simulation results under conditions in which the pd τ value was decreased and increased are shown in

Discussion
In the field of dynamical systems, it is important to investigate the dependence of the solutions of ordinary differential equations on system parameters.The present study illustrates the dependence of the qualitative nature of the solutions of ordinary differential equations on the following system parameters:  study, which was not reported in the previous study [8], is that there was a nonlinear relationship between pd  was divided into stable equilibria and unstable equilibria regions by a Hopf bifurcation curve and that an increase in I x shifted the Hopf bifurcation curve upward, resulting in an expansion of the stable equilibria re- gion.
Study [13] proposed an algorithm for the visualization of the bifurcation manifolds in the three-dimensional parameter space.In the three-dimensional parameter space, the parameter sets at which codimension one-bifurcation occurs are visualized as bifurcation surfaces.Higher codimension bifurcations are located at intersections of the bifurcation surfaces.For example, analyses of a socioeconomic model have revealed codimension-one bifurcation surfaces: a Hopf bifurcation surface and a saddle-node bifurcation surface (Figure 5 in [13]).In addition, the following codimension-two bifurcation curves were visualized: a Gavrilov-Guckenheimer bifurcation curve and a Takens-Bogdanov bifurcation curve.In contrast to the findings of the previous study [13], in the present study, we did not visualize bifurcation manifolds in the three-dimensional ( ) , , , The parameter sets at which codimension-two bifurcation occurs are thought to form a bifurcation curve at the intersection of the surfaces of SNFP and SNLC.

Conclusion
In conclusion, the novelty of this paper is that it reveals in detail the influence of pd τ variations on the dynam- ical states in the ( ) Dr d s g I -parameter space of the ghostbursting model.
space was divided into three dynamical states: quiescence, periodic tonic spiking, and bursting.The present study focused on a third system parameter: the time constant of dendritic potassium current inactivation ( ) pd τ .A computer simulation of the model revealed how the dynamical states of the ( ) ) in Section 2] and the dendritic compartment [see Equations (3) to (6) in Section 2].Doiron et al. have focused on two system parameters of the model: maximal conductance of the delayed-rectifying potassium current in the dendritic compartment (3)] and the current injected into the somatic compartment ( ) s I [see Equation (1)].
activating variable of the somatic delayed-rectifying potassium current ( ) variable of the dendritic sodium current ( ) d h , activating variable of the dendritic delayed-rectifying potassium current ( ) d n , and inactivating variable of the dendritic delayed- rectifying potassium current ( ) d p .The time evolution of these variables is described with the following equa- tions: under different values of pd τ .The total simulation time was 1.2 s, and the constant depolarizing current pulse ( ) s I was injected between 0.1 s and 1.1 s.Otherwise, the injected current was zero.

(Figure 2 (
space change in response to pd τ variations (Figure 2).The results at low pd τ are shown in Figure 2(a), those at intermediate pd τ are shown in Figure 2(b), and those at high pd τ are shown in Figure 2(c).First, in the present study, we performed a simulation of the model with ( ) b)), which was the same condition as that used in Figure 6 in [8].At a low s I value (5.6 μA/cm 2 ), the dynamical state of the model was that of quiescence, irrespective of the , Dr d g value (× in Figure 2(b)).An example of the time course of the somatic membrane potential during the quiescent state is shown in Figure 1(a).At high s I values (≥5.8 μA/cm 2 ), the dynamical state was that of pe- riodic tonic spiking (○ in Figure 2(b)) or bursting (• in Figure 2(b)).In other words, when the , Dr d g value was small (≤12.0 mS/cm 2 ), the dynamical state was that of bursting.In contrast, when the , Dr d g value was large (≥12.2mS/cm 2 ), the dynamical state was that of periodic tonic spiking at smaller s I values and that of bursting at larger s I values, and the s I threshold between periodic tonic spiking and bursting increased as the , Dr d g value was increased.Examples of the time courses of the somatic membrane potential during periodic tonic spiking and bursting are shown in Figure 1(b) and Figure 1(c), respectively.When the above-mentioned results were compared with previous findings (Figure 6 in [8]), the present numerical analysis could reproduce the previous results.Based on the previous results (Figure 6 in [8]), SNFP was thought to occur at certain s I values between × and • in Figure 2(b).In addition, SNFP was thought to occur at certain s I values between × and ○ in Figure 2(b).SNLC was thought to occur at certain s I values between ○ and • in Figure 2(b).Codimension-two bi- furcation was thought to occur at a certain ( ) surrounded by ×, ○, and • in Figure 2(b).

Figure 2 (
Figure 2(a) and Figure 2(c), respectively.At a low I s value (5.6 μA/cm 2 ), the dynamical state was that of quiescence, irrespective of the , Dr d g value (× in Figure 2(a) or Figure 2(c)), which is the same as that shown in Figure 2(b).The s I threshold between quiescence and bursting, which is the boundary between × and • in Figure 2(a) and Figure 2(c), is the same as that shown in Figure 2(b).The s I threshold between quiescence and periodic tonic spiking, which is the boundary between × and ○ in Figure 2(a) and Figure 2(c), is also the same as that shown in Figure 2(b).These results suggested that changes in the pd τ values did not affect SNFP.At high s I values (≥5.8 μA/cm 2 ), patterns similar to Figure 2(b) were observed.In other words, when the , Dr d g value was small (≤12.8 mS/cm 2 in Figure 2(a) and ≤11.6 mS/cm 2 in Figure 2(c)), the dynamical state was that of bursting only (• in Figure 2(a) and Figure 2(c)).In contrast, when the , Dr d g value was large (≥13.0mS/cm 2 in Figure 2(a) and ≥11.8 mS/cm 2 in Figure 2(c)), the dynamical state was that of periodic tonic

Figure 1 .
Figure 1.Examples of the time courses of the simulated somatic membrane potential ( ) s V at different , Dr d g and I s values at 5.0 pd τ = model, there were three qualitatively different dynamical states: quiescence, spiking, and bursting.In particular, the present results revealed how the dynamical states of the two-dimensional ( ) space changed in response to variations in the third parameter pd τ .These results are important in that they imply a relationship between pd τ and bifurcation manifolds in the ( ) , , Dr d s g I -parameter space.In other words, these findings suggested that an increase in the pd τ value did not shift the SNFP curve in the ( ) , , Dr d s g I -parameter space but rather shifted the SNLC curve upward.A very interesting finding in the present

Figure 2 ..
Figure 2. The effects of variations in the pd τ value on the dynamical states in the two-dimensional

τ
and the area of the bursting state.In other words, although the amount of pd τ decrease was the same (−0.8ms) between the changes from Figure 2(c) to Figure 2(b) and the changes from Figure 2(b) to Figure 2(a), the amount of increase in the area of the bursting state in the latter case was much larger than that in the former case.Other examples that illustrate how the dynamical states of two-dimensional parameter space change in response to variations in the third parameter are (1) a model of CA1 pyramidal neuron spiking dynamics (Figure12in[11]) and (2) a compartmental model of Cheyne-Stokes respiration (Figure5in[12]).In analysis of the CA1 model, Bianchi et al. have focused on the following three parameters: the injected current ( ) inj I , half-activation voltage of the transient sodium current of the Cheyne-Stokes respiration model, Atamanyk and Langford focused on the following three parameters: the partial pressure of CO 2 in the inspired air ( ) I x , ventilation-perfusion ratio ( ) A V F , and slope of the Hill function ( ) µ .Their findings revealed that the two-dimensional ( ) , A V F µ -parameter space , when considering the changes in the dynamical states of the two-dimensional ( ) meter space in response to variations in pd τ (Figure2), one can roughly imagine the bifurcation manifold in the three-dimensional () .In other words, in the three-dimensional parameter space that is defined as a three-dimensional orthogonal coordinate system with axis lines , at which SNFP occurs are thought to form the surface of SNFP that is orthogonal to the the parameter sets at which SNLC occurs are thought to form the surface of SNLC that is not orthogonal to the ( )