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Mathematical relationship between membrane conductance and conduction velocity?

Mathematical relationship between membrane conductance and conduction velocity?



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By what factor would the myelin need to decrease membrane conductance (gm) if you wanted conduction velocity in a 10 μm myelinated mouse axon to be 100 times faster than in a 10 μm unmyelinated mouse axon?

This is a question from a homework set. I understand WHY myelin decreases membrane conductance, and WHY decreasing membrane conductance would increase conduction velocity. I just do not understand how to calculate what factor specifically the myelin would need to decrease membrane conductance by.


The conduction velocity is inversely proportional to the membrane time constant $ au_m$ and proportional to the length constant $lambda$. Membrane time constant is proportional to membrane resistance (inverse of membrane conductance), and the length constant is proportional to the square root of membrane resistance.


University College Dublin and The National Rehabilitation Hospital, School of Electrical, Electronic, & Mechanical Engineering, Dublin, Ireland

University College Dublin and The National Rehabilitation Hospital, School of Electrical, Electronic, & Mechanical Engineering, Dublin, Ireland

Abstract

The speed with which an action potential (AP), or excitatory impulse, travels along the membrane of a skeletal muscle fiber is known as its conduction velocity (CV). Muscle fiber CV may be estimated from the electromyogram (EMG) and can be used as a sensitive measure of the excitability of the muscle fiber membrane in addition to being related to the type, diameter, rate of activation, temperature, and physiological or pathological state of the muscle fiber. Conduction velocity estimates are frequently used as a fatigue index, with applications in ergonomics, kinesiology, rehabilitation, functional electrical stimulation, and, more recently, as a diagnostic tool. Conduction velocity estimation is discussed in the following pages from its genesis to issues regarding its measurement and potential clinical and diagnostic value.


I. Introduction

There are several physiological scenarios in which the electrical interactions between cardiac myocytes and nonmyocytes are important. First, fibroblasts are more numerous than myocytes in the heart and under certain diseased states, can locally proliferate and possibly couple to the myocytes, affecting electrical activity [1]–[3]. Second, the presence of fibroblasts in ischemic or scarred regions represents a possible target for cell therapy to restore normal electrical conduction [4]. Finally, the use of autologous nonmyocytes such as myoblasts or mesenchymal stem cells has been proposed to restore cardiac structure and function [5]. Recent studies have suggested that the success of this cell therapy was affected by the quality of myocyte-nonmyocyte electrical interactions [6]. Because the electrical activity in one cell type can affect the behavior of the other it is critical to understand the mechanism of the potential interactions.

While nonexcitable, these nonmyocytes are not passive. However, when the time constant of nonmyocyte activation is larger than the rising time of the action potential, nonmyocytes are expected to act as a passive load during the depolarization process. Because the electrical activity of a myocyte-nonmyocyte pair is driven by the myocyte action potential [1], this time constant is expected to also depend on the coupling resistance. This paper describes in terms of coupling and nonmyocyte properties the conditions under which general nonmyocytes act as a passive load. The consequences on conduction velocity (CV) are explored in a tissue model incorporating recent detailed fibroblast models [7], [8], reproducing a setup in which the effect on CV has been demonstrated in vitro [3]. The simulations reveal the range of coupling conductances for which passive load approximation is valid and explain how the CV change with coupling is not always monotonic.


Origins of the nervous system

Metazoan electrical communication

danh/MyelinEvolution/images/EvoSpeed_Fig-3.jpg" />
Figure 3 : Conduction velocity for nerve fibers vs fiber diameter. Lines indicate general relations over a range of diameters, many taken from Bullock and Horridge (1965) but adjusted to a standard temperature of 20 C using a Q10 of 1.8 (Chapman and Pankhurst 1967) and an internal and external ionic conductivity of a squid axon (35.4 &Omega cm). Thus vplot = vmeas = 1.8 (20-T)/10 (35.4/Raxoplasm) where T is the temperature in degrees C for the measured velocity vmeas and Raxoplasm is the specific resistance of the axoplasm, if available, or the extracellular medium if otherwise. Specific labeled points or lines from the following sources: Squid: Hartline & Young 1936 cited in Pumphrey and Young (1938) Earthworm: Eccles, Granit & Young (1932) Penaeus and Macrobrachium : Kusano (1966). Crayfish: Govind and Lang (1976) Hydromedusa: Mackie and Meech (1985).


Results

Perinexal width

We previously demonstrated that elevating perfusate Ca 2+ can reduce perinexal width (Wp) in hearts perfused with 145-mM Na + [18, 27]. To confirm this finding when hearts are perfused with 155-mM Na + , we here present representative transmission electron micrographs of hearts perfused with 155-mM Na + and either 1.25- or 2.00-mM Ca 2+ (Fig. 1a). The measurements from 45 to 105 nm were averaged for each heart (Fig. 1b) and compared between Ca 2+ concentrations. Consistent with our previous findings, increasing Ca 2+ to 2.00 mM can decrease Wp compared to 1.25-mM Ca 2+ (Fig. 1c) [18, 27].

Conduction velocity

Representative isochrone maps and summary data of all Na + and Ca 2+ perfusate combinations at 4.6-mM K + are presented in Fig. 2a. Altering Na + and/or Ca 2+ at baseline, in the presence of 4.6-mM K + , does not significantly change CVT or CVL, as can be seen in the summary data within Fig. 2b.

Altering Na + , Ca 2+ , or Na + and Ca 2+ does not change CVT or CVL at 4.6-mM K + . (a) Representative isochrone maps for each Na + /Ca 2+ perfusion combination. The 145-mM Na + /1.25-mM Ca 2+ map is marked with CVT and CVL designations for visualization purposes, (b) Summary of CVT and CVL at 4.6-mM K + . p < 0.05 denoted by *, significance determined by one-way ANOVA with Dunnett’s correction for multiple comparisons. (n = 12, 14, 15, 15 from left to right respectively for both CVT and CVL)

Transverse conduction velocity

As expected, varying K + between 4.6 and 10.0 mM produces a biphasic response in CVT (Fig. 3 Supplemental Figure 2). Representative isochrone maps in Fig. 3a suggest certain ionic combinations increase CV, as evidenced by fewer isochrones and colors, while other maps suggest conduction slowing. Specifically, faster CVT (supernormal conduction) is observed at 6.4-mM K + in hearts perfused with the 145-mM Na + /1.25-mM Ca 2+ , 145-mM Na + /2.00-mM Ca 2+ , and 155-mM Na + /2.00-mM Ca 2+ solutions. The 155-mM Na + /2.00-mM Ca 2+ group also demonstrates faster CVT at 8.0-mM K + (Fig. 3b). CVT slowing with severe hyperkalemia (10-mM K + ) is observed in hearts perfused with three of the four solutions: 145-mM Na + /1.25-mM Ca 2+ , 145-mM Na + /2.00-mM Ca 2+ , and 155-mM Na + /1.25-mM Ca 2+ solutions (Fig. 3a and b). Yet, CVT is not significantly different at 10-mM K + relative to 4.6-mM K + when hearts are perfused with 155-mM Na + /2.00-mM Ca 2+ (Supplemental Figure 1 Supplemental Table 1). These results suggest that the combination of elevated Na + and elevated Ca 2+ can attenuate CVT slowing caused by severe hyperkalemia. We further attempted to compare the CVT-K + relationship across Na + /Ca 2+ variations by comparing the datasets fit with a quadratic model (Supplemental Figure 2). Unfortunately, this approach did not reveal any statistically significant differences among curves.

Simultaneously increasing Na + and Ca 2+ preserves CVT and CVL at 10.0-mM K + . (a) Representative isochrone maps for each Na + /Ca 2+ perfusion combination at 4.6- and 10.0-mM K + , (b) Summary of CVT as a function of K + for all Na + and Ca 2+ perfusion combinations, (c) Summary of CVL as a function of K + for all Na + and Ca 2+ perfusion combinations. p < 0.05 denoted by *, significance determined by one-way ANOVA with Dunnett’s correction for multiple comparisons (n = 12, 14, 15, 15 from left to right, respectively).

Longitudinal conduction velocity

Similar to the analysis of CVT, the relationship of CVL and K + between 4.6 and 10.0 mM is biphasic (Fig. 3c Supplemental Figure 2). Faster CVL is observed at 6.4-mM K + in hearts perfused with the 145-mM Na + /1.25-mM Ca 2+ , 155-mM Na + /1.25-mM Ca 2+ , and 155-mM Na + /2.00-mM Ca 2+ solutions (Fig. 3c). CVL slowing with severe hyperkalemia (10-mM K + ) is observed in hearts perfused with the 145-mM Na + /1.25-mM Ca 2+ , 145-mM Na + /2.00-mM Ca 2+ , and 155-mM Na + /1.25-mM Ca 2+ solutions (Fig. 3c). Consistent with CVT, CVL is not significantly different in hearts perfused with 155-mM Na + /2.00-mM Ca 2+ at 10-mM K + relative to 4.6-mM K + . Taken together, these results further support the finding that the combination of elevated Na + and Ca 2+ can preferentially attenuate CV slowing caused by severe hyperkalemia.

Incidence of asystole

A surprising finding in the study was that incidence of asystole at 10-mM K + varied among the different ionic concentrations. Specifically, all hearts perfused with the 145-mM Na + /1.25-mM Ca + solution develop asystole in the presence of 10-mM K + (Fig. 4a). Elevating Ca 2+ alone does not significantly reduce the incidence of asystole (145-mM Na + /2.00-mM Ca 2+ ). However, asystole is significantly reduced in hearts perfused with 155-mM Na + with or without elevated Ca 2+ .

Incidence of asystole during 10.0-mM K + perfusion. (a) Elevating perfusate Na + significantly increases the preservation of intrinsic rhythm at 10.0-mM K + perfusion (significance determined by Fisher’s exact test * denotes p < 0.05 as compared to 145-mM Na + , 1.25-mM Ca 2+ , 10.0-mM K + perfusate group), (b) Inhibiting GJs with CBX (30 μM) significantly increases the preservation of intrinsic rhythm in the presence of 145-mM Na + when compared to the control condition (significance determined by Fisher’s exact test $ denotes p < 0.05 compared to each perfusates’ respective CBX – group). There were no significant differences in preservation of intrinsic rhythm across groups perfused with CBX

To further probe the mechanism of intrinsic rhythm preservation with elevated Na + , we pharmacologically inhibited GJC with CBX (30 μM). Interestingly, CBX significantly reduces the development of asystole at 10-mM K + in the 145-mM Na + perfusion groups but does not further reduce the incidence of asystole in the 155-mM Na + perfusion groups (Fig. 4b).

Carbenoxolone

We find that CBX reduces CV regardless of Na + , Ca 2+ , or K + concentrations, consistent with previous findings [20]. In summary, CV values in Fig. 5a are significantly reduced at 4.6-mM K + with CBX relative to measured values in Fig. 3 (p < 0.05 for all comparisons), but once again the combinations of Na + and Ca 2+ do not produce significant differences in CVT (Fig. 5b) or CVL (Fig. 5c) at 4.6-mM K + with CBX. Interestingly, the expected CV increase between 4.6- and 8-mM K + is not observed with CBX with any Na + or Ca 2+ combination, and a post hoc comparison of the CV change between 4.6-mM and 6.4-mM K + is also not significantly different for any experimental combination with or without CBX.

Following GJ inhibition with CBX, simultaneously increasing Na + and Ca 2+ preserves CVT and CVL at 10.0-mM K + . (a) Altering Na + , Ca 2+ , or Na + and Ca 2+ does not change CVT or CVL at 4.6-mM K + , (b) Summary of CVT as a function of K + in the presence of CBX (n = 6, 7, 7, 8 from left to right, respectively), (c) Summary of CVL as a function of K + in the presence of CBX (n = 6, 7, 7, 8 from left to right, respectively). Significance determined by ordinary one-way ANOVA with Dunnett’s correction for multiple comparisons (p < 0.05 denoted by *)

Similar to our studies without CBX, CV slowing with severe hyperkalemia (10-mM K + ) is observed in hearts perfused with 145-mM Na + /1.25-mM Ca 2+ (CVT and CVL slowing), 145-mM Na + /2.00-mM Ca 2+ (CVT slowing only), and 155-mM Na + /1.25-mM Ca 2+ (CVT and CVL slowing) solutions (Fig. 5b and c). Furthermore, CV does not significantly decrease at 10-mM K + with 155-mM Na + /2.00-mM Ca 2+ .

Computational model predictions

To explore potential mechanisms that may explain the experimental results above, the CV-K + relationship was investigated in silico. The computational model used in Fig. 6 includes GJC and sodium channel localization at the ends of myocytes facing a shared and restricted extracellular cleft with a variable cleft resistance inversely proportional to perinexal width (WP). The family of curves in each panel of Fig. 6a–d represents narrow (0.5 × WP), nominal (1 × WP), and wide clefts (2 × WP) to demonstrate how modulating EpC alters the CV-K + relationship.

Computational predictions of modulating perinexal width (WP), extracellular sodium concentrations (Na + ), gap junctional coupling (GJC), and the fast sodium channel conductance (gNa). (a) Increasing WP to reduce EpC is associated with increased conduction velocity (CV, black to white points). The positive slope calculated from a linear fit of CV over the range of extracellular potassium (K + ) from 4.56 to 7 mM is reduced as WP decreases. The negative slope associated with sodium channel inactivation over the range of 9- to 10-mM K + decreases to a greater extent with narrow WP, (b) Increasing Na + by 10 mM (+10-mM Na + ) decreases the positive slope of the CV-K + relationship without substantively altering the negative slope or CV overall, (c) Reducing GJC by 50% (0.5 × GJC) slows CV for all values of K + without changing the positive CV-K + slope. However, 0.5 × GJC is associated with enhanced CV slowing during sodium channel inactivation measured between 9- and 10-mM K + relative to the nominal condition in panel b. CV slowing was still the lowest with the narrowest cleft widths (0.5 × WP), (d) Reducing gNa by 50% (0.5 × gNa) not only reduces CV for all K + , but it also reduces both the positive and negative slopes of the CV-K + relationship, without altering predictions that WP associated with the slowest CV attenuates CV-dependent changes on K +

The model predicts that progressively narrowing WP (white to black curves), which enhances EpC, can decrease CV under conditions of robust GJC and EpC (Fig. 6a). Altering WP does not dramatically change the positive slope of the linear portion of the CV-K + curve estimated during supernormal conduction (4.56- through 7-mM K + ). However, the negative CV-K + slope estimated between 9- and 10-mM K + is significantly reduced by narrowing WP. Specifically, the negative slope is −1.1 cm·mM/sec for the widest clefts (2 × WP) and −0.8 cm·mM/sec for the narrowest clefts (0.5 × WP). In short, the model predicts that narrow clefts can attenuate CV slowing during sodium channel inactivation due to hyperkalemia without notably altering supernormal conduction.

Elevating Na + by 10 mM in this model can modestly increase CV, and narrowing WP still slows CV. Interestingly, elevating Na + decreases the positive slope of the linear portions of the CV-K + relationship (Fig. 6b). In contrast, the negative slope associated with sodium channel inactivation at 10-mM K + is not as dramatically affected by elevating Na + , and the values are similar to the negative slopes under the nominal case in Fig. 6a. Taken together with Fig. 6a, the model predicts that elevating Na + can reduce the sensitivity of CV to changes in K + during supernormal CV, but the negative slope is still predominantly determined by intercellular cleft width (WP) and therefore EpC. Importantly, elevating Na + and reducing WP attenuate the sensitivity of CV to K + during supernormal conduction and sodium channel inactivation.

The CV response to reducing GJC by 50% (0.5 × GJC) can be found in Fig. 6c. As expected, reducing GJC slows CV, and narrowing WP still slows CV. The model also suggests that inhibiting GJC does not substantively alter the CV-K + relationship during supernormal conduction between 4.56- and 7-mM K + , but GJC inhibition can increase CV sensitivity to sodium channel inactivation as evidenced by larger negative slopes in Fig. 6c relative to 6a. Yet, the sensitivity of CV slowing during sodium channel inactivation is still the lowest under conditions of the narrowest cleft widths (0.5 × WP) and therefore elevated EpC.

When the peak conductance of the fast sodium current is inhibited by 50% (0.5 × gNa), CV slows dramatically as expected (Fig. 6d). Although narrowing WP still slows CV, reducing WP also reduces the positive and negative slopes of the CV-K + relationship. In summary, the model predicts that enhancing EpC by narrowing WP reduces CV sensitivity to loss of functional sodium channels induced by increasing sodium channel inactivation (10-mM K + ) or reduced peak current (0.5 × gNa). In summary, computational models incorporating both EpC and GJC predict complex CV-K + relationships with a consistent finding that narrowing WP attenuates CV slowing during sodium channel loss of function induced by sodium channel inactivation or reduced peak sodium current.


Results

In this section, we demonstrate how the EMI model may be used to investigate properties of cardiac conduction. First, we consider how a non-uniform distribution of sodium channels affects the conduction velocity, the discontinuous nature of conduction and the time delays across gap junctions of reduced coupling. We also consider how the conduction velocity along a strand of cells is affected by the length of the cells. Finally, we use the EMI model to study the possibility of ephaptic coupling acting as an alternative pathway for conduction between cells and investigate how the sodium channel dynamics are affected by ephaptic effects.

Effect of sodium channel distribution on conduction velocity

As a first example of the application of the EMI model, we will use the model to study how a non-uniform distribution of sodium channels on the cell membrane affects the conduction velocity. In Fig 4, we show the conduction velocity computed in a number of simulations of a strand of 15 cells with an increasing percentages of sodium channels moved to the horizontal parts of the cell ends (see Fig 3B). Here, 0% represents the uniform case, and 100% represents the non-uniform case, when all sodium channels are located close to the cell ends. The total sodium channel conductance of the cell remains the same in each simulation as explained above. We observe that as a larger percentage of the sodium channels are moved to the cell ends, the conduction velocity increases.

The parameters used in the simulations are given in Table 1, and the simulation includes 15 cells. The conduction velocity is computed as the distance between the cell centers of the fifth and tenth cells divided by the time between the two cell centers first reach a membrane potential of v = 0 mV.

Since the largest difference from the uniform case is observed for the case when all sodium channels are moved to the cell ends, we will in the experiments below compare just these two extremes the uniform case with a constant distribution of sodium channels on the entire membrane and the non-uniform case with all sodium channels located near the cell ends.

Discontinuous conduction

It is considered well-established that the electrical conduction in cardiac tissue is discontinuous with significant conduction delays when the wave crosses the gap junctions [7]. This discontinuous conduction is conveniently studied using the EMI model because the boundaries between cells are explicitly represented in the model. In Fig 5, we consider a single strand of cells and show the points in time when each of the x-values along the cell membranes first reach a membrane potential of v = 0 mV. We consider both a uniform and a non-uniform distribution of the sodium channels (see Fig 3A and 3B), and we consider the case of the default value of Rg in addition to three cases of increased Rg. In the figure, we observe that there are clearly visible gaps in time between each part of the gap junctions reach v = 0 mV, and that the size of these gaps increases as the gap junction resistance is increased. In addition, we observe that the gaps in time seem to be longer for the uniform case compared to the non-uniform case, and the overall time spent crossing the five cells is longer for the uniform case for all values of Rg.

The plots show the time at which each of the x-values along the membrane of the five center cells in a simulation of a strand of seven cells first reach a membrane potential of v = 0 mV. The title above each panel indicates the factor with which the default value of Rg = 0.0045 kΩcm 2 is multiplied in the simulation. The remaining parameter values are specified in Table 1, except that the time step is set to Δt = 0.0005 ms.

Activation times for a single cell.

In Fig 6, we focus on the x-values corresponding to a single cell. We observe that the curves for the activation times are not straight lines, but bend along the length of the cell. Moreover, the shape of the curves is clearly different in the uniform and non-uniform cases. In the uniform case, the curves seem to steepen towards the cell end, while for the non-uniform case, the curves seem to flatten out towards the end of the cell. In fact, for the non-uniform case, the activation time is shorter for the far-right part of the cell than at about 80% of the cell length in the cases of increased gap junction resistance.

The figure shows the results from Fig 5 zoomed in on the x-values corresponding to a single cell in the center of the domain.

Furthermore, we observe that for both sodium channel distributions, the time between the start and the end of the cell reaches v = 0 mV is shorter for the case with a high gap junction resistance compared to the case with the default value. This means that wave travels faster over a single cell as the gap junction resistance is increased and, as seen in Fig 5, the time delays across the gap junctions are increased.

Effect of sodium channel distribution on the upstroke velocity

As seen in Fig 4, the conduction velocity is increased for a non-uniform distribution of sodium channels compared to a uniform distribution. To investigate the reason for this effect, we consider the upstroke velocity of the action potential computed for the two sodium channel distributions. In the left panel of Fig 7A, we report how the membrane potential changes with time for a grid point located at the beginning of the seventh cell, at the center of the cell and at the end of the cell in the uniform case and in the non-uniform case with all sodium channels located at the cell ends. In the right panel, we plot the corresponding upstroke velocity (dv/dt) as a function of time. We observe that the upstroke velocity is quite similar in the three points along the cell, but that the upstroke velocity is clearly increased in the non-uniform case compared to the uniform case. This increased upstroke velocity might explain the increased conduction velocity reported in Fig 4.

Comparison of the action potential upstroke and upstroke velocity (A), INa (B), integrated INa (C), Igap (D), and gating dynamics of INa (E) for the U and NU cases (see Fig 3A and 3B). The parameters used in the simulations are given in Table 1, and the simulations include 15 cells. We consider the seventh cell and record the membrane potential, upstroke velocity and currents for the first x-value, the center x-value and the last x-value of the cell.

Effect of the sodium channel distribution on the sodium channel dynamics.

In order to investigate the difference in the upstroke velocity observed between the NU and U cases in Fig 7A, we in Fig 7B report the sodium current density at the same three membrane points for the two sodium channel distributions. In the NU case, the sodium current density is zero in the center of the domain, but has a much larger magnitude than for the U case at the points close to the cell ends.

The total sodium current integrated over the entire membrane is reported in Fig 7C, and we observe that the integrated current is larger for the U case than for the NU case. In addition, in Fig 7E, we report the gating dynamics of the sodium current, and we observe that the open probability of the sodium channels is slightly larger for the U case compared to the NU case. The increased upstroke velocity for the NU case therefore does not seem to be caused by a larger total sodium current over the cell membrane, but rather by the locally increased sodium current density close to the cell ends.

Effect of the sodium channel distribution on the gap junction current.

In Fig 7D, we report the size of the current density through the gap junctions, Igap, for the NU and U cases. We observe that the gap junction current density is larger for the NU case than for the U case.

In order to investigate whether the difference in upstroke velocity between the NU and U cases may be explained by this increased Igap, we report in Fig 8 the upstroke velocity computed for the NU and U cases in a single cell simulation (with no gap junctions present). Again, we observe that the upstroke velocity is considerably higher for the NU case than for the U case. Consequently, the increased upstroke velocity for the NU case observed in Fig 7A seems to be a direct consequence of the locally increased sodium current density at the cell ends and not caused by the increased gap junction currents observed in Fig 7D.

The plot shows the upstroke velocity at the start of the cell, the center of the cell and the end of the cell for the NU and U sodium channel distributions for a simulation including only a single cell. The cell is triggered by applying the initial condition v = −55 mV for all membrane nodes in both the NU and U cases. The parameter values used in the simulation are given in Table 1.

Effect of the sodium channel distribution on the time delays across gap junctions.

In Figs 5 and 6, we observed that both the gap junction delays and the time spent traversing a single cell was decreased for a non-uniform distribution of sodium channels compared to a uniform distribution. Furthermore, we observed that the travelling wave spends the majority of time crossing the gap junctions. Therefore, decreased gap junction delays could be expected to have the most significant effects on the observed difference in conduction velocity between the NU and U cases. In Fig 9 we report the relationship between the time delays across a gap junction, the maximal upstroke velocity, and the maximal gap junction current density for the simulations reported in Fig 4. We observe that an increased maximal upstroke velocity is associated with increased maximum gap junction currents and decreased time delays across the gap junctions.

The time delay is defined as the time between the membrane potential at the last computational node before the gap junctions between the sixth and seventh cells, and the first computational node after the gap junctions reach 0 mV. The upstroke velocity is computed at the first computational node after the gap junctions. The gap junction current density is computed between the sixth and seventh cell at the center of the domain in the y- and z-directions. The results are computed for the simulations displayed in Fig 4.

Time delays across gap junctions of reduced coupling

In Fig 10, we show how the time delay over the gap junctions increases as the resistance of the gap junctions are increased. We consider both the case of a uniform distribution of sodium channels and the case of a non-uniform distribution with all sodium channels located close to the cell ends. We observe that the time delays across the gap junctions are longer for the uniform case than for the non-uniform case for all values of the gap junction resistance, Rg. Furthermore, the difference between time delays associated with each of the two sodium channel distributions increases as the gap junction resistance is increased. In addition, the value of Rg for which the wave is completely blocked is lower for the uniform case than for the non-uniform case.

The simulation includes a row of ten cells and the gap junction resistance between the fifth and sixth cells is increased by a factor of up to 70 from the default value of Rg = 0.0045 kΩcm 2 . The default value of Rg is used for the remaining gap junctions. The remaining parameters used in the simulations are given in Table 1, except that the time step is set to Δt = 0.01 ms. The timings reported in the plot are the time between the end of the fifth cell and the start of the sixth cell reach a membrane potential of v = 0 mV. In the NU case, all the sodium channels are located on the horizontal part of the cell ends (see Fig 3B).

Fig 11 illustrates the propagating wave for the uniform and non-uniform cases when the gap junction resistance is increased by a factor of 10 from the default value of 0.0045 kΩcm 2 to 0.045 kΩcm 2 . We observe that the wave is delayed by about a millisecond when it reaches the gap junctions of reduced coupling, but that it eventually crosses the gap junctions. This happens faster for the case with a non-uniform distribution of sodium channels than for the uniform case.

The figure shows the intracellular potential in the plane at the center of the domain in the z-direction at four points in time for the case with a uniform (U) and a non-uniform (NU) distribution of sodium channels. The default gap junction resistance is increased by a factor of 10 from 0.0045 kΩcm 2 to 0.045 kΩcm 2 between the fifth and sixth cells. The remaining parameter values used in the simulations are given in Table 1, except that the time step is set to Δt = 0.01 ms.

Fig 12 similarly illustrates a case in which the propagating wave is only able to cross the gap junctions of increased resistance for a non-uniform distribution of sodium channels. In this example, the gap junction resistance is increased by a factor of 70 compared to the default value on the gap junctions between the fifth and sixth cells. We observe that the propagating wave is able to cross the gap junctions of increased resistance after a long time delay for the non-uniform case, but is completely blocked in the uniform case. Also, it is worth observing that the repolarization is considerably slower in the NU case compared to the U case. However, we generally observed slower repolarization when a wave is able to propagate across a point of increased resistance, and this also holds when the sodium channels are uniformly distributed.

The figure shows the intracellular potential in the plane at the center of the domain in the z-direction at four points in time for the case with a uniform (U) and a non-uniform (NU) distribution of sodium channels. The default gap junction resistance is increased by a factor of 70 from 0.0045 kΩcm 2 to 0.315 kΩcm 2 between the fifth and sixth cells. The remaining parameter values used in the simulations are given in Table 1, except that the time step is set to Δt = 0.01 ms.

Effect of cell length on the conduction velocity

In this section, we investigate how the conduction velocity depends on the cell length if the number of sodium channels per cell remains constant. Assuming that the number of sodium channels per cell remains constant, the density of sodium channels on the cell membrane will decrease as the cell length is increased. In this respect, it seems reasonable to expect that the conduction velocity would decrease if we increase the length of the cells, because the sodium channels are important for maintaining the membrane excitability necessary for cardiac conduction [57]. On the other hand, as the cell length is increased, the distance between the cell boundaries in the x-direction will increase, and, for a given propagation length, the propagating wave will have to cross less cell boundaries. This contrarily suggests that the conduction velocity would increase as the cell length is increased. As a result of these opposing effects, we might expect that there could be some optimal cell length which maximizes the conduction velocity.

In Fig 13, we investigate this property and report the conduction velocity computed for a number of simulations with different cell lengths. We consider both a uniform and a non-uniform distribution of the sodium channels. The length of ΩO is varied and the cell width and the sizes of ΩW and ΩE are kept constant in each simulation (see the left panel of Fig 2). In order to keep the total number of sodium channels constant for each cell length, we replace the actual cell membrane area, Ac, by the membrane area of the default cell size in Table 1 when computing the sodium channel conductance density by (22) in the NU case. In the U case, we similarly let the sodium channel conductance be scaled by a factor .

In Fig 13, we observe that the conduction velocity indeed reaches a maximum for a given cell length and that the conduction velocity decreases as the cell length is increased or decreased from this value. In particular, for the parameters chosen here (see Table 1), a cell length of approximately 100 μm and 150 μm appears to lead to the maximal conduction velocity in the uniform and non-uniform cases, respectively.

The cell length here refers to the total length of ΩO, ΩW and ΩE (see Fig 2), but only the length of ΩO is varied in the simulations. The remaining parameter values are given in Table 1, except that for cell lengths shorter than 20 μm, the length of ΩN and ΩS in the x-direction is reduced to the cell length minus 6 μm so that they fit on ΩO. For the simulations of cell lengths of up to 104 μm, the simulation includes 20 cells, the first four cells are stimulated and the conduction velocity is calculated between cells number seven and thirteen. For the remaining cell lengths, the simulation includes ten cells, the first two cells are stimulated and the conduction velocity is computed between cells number three and seven.

Ephaptic coupling of cardiomyocytes

Action potential propagation from cardiomyocyte to cardiomyocyte is primarily believed to be enabled by current through the gap junctions connecting individual cells [59]. However, ephaptic coupling has been proposed as an alternative or supporting mechanism for conduction between cells [19]. The EMI model is well-suited for studying this mechanism because the extracellular space is explicitly represented in the model as a geometrical subdomain.

Ephaptic coupling for closed gap junctions.

In order to investigate the possibility of extracellular potentials alone being an alternative pathway of conduction between neighboring cells, we consider two cells with a gap junction conductance, , set to zero on the intercalated disc between the cells. We stimulate the first half of the first cell and investigate whether the resulting propagating wave is able to pass to the second cell despite the fact that there is no current through the gap junctions between the cells.

In the upper panel of Fig 14, we report the intracellular potential, the extracellular potential and the membrane potential in a grid point located on the membrane of the second cell, just after the gap junctions with zero conductance. This point is illustrated by a red circle in the domain description in the panel above the plots. We consider a number of different distances d between the cells, and observe that as d is decreased, the magnitude of the minimum extracellular potential increases considerably for the non-uniform case. Indeed, for a cell distance of d = 5 nm, the extracellular potential reaches a value of approximately −30 mV. For the uniform case, however, the magnitude of the extracellular potential does not increase considerably, even for a cell distance of d = 5 nm.

The figure shows the intracellular potential, the extracellular potential and the membrane potential observed after blocked gap junctions for different values of the cell distance, d. The distance d is the combined length of ΩW and ΩE (see Fig 2 and the upper panel of this figure). In the NU case, we distribute all sodium channels on the vertical part of the cell ends (see Fig 3C). We consider two cells with gap junction conductance and stimulate the first half of the first cell. The potentials are recorded in a grid point located just after the blocked gap junctions, on the membrane of the second cell, illustrated by a red circle in the upper panel. The parameter values used in the simulations are given in Table 1, except that we use Δt = 0.01 ms, Δz = 1 μm and a slightly reduced cell size. We let ΩO be of size 100 μm × 12 μm × 12 μm, ΩW and ΩE be of size d/2 × 4 μm × 4 μm and ΩS and ΩN be of size 4 μm × 2 μm × 4 μm. Furthermore, we use Δx = d/4. The center panel shows the intracellular potential, the extracellular potential and the membrane potential as functions of time. The lower panel shows the maximum intracellular potential, the minimum extracellular potential and the maximum membrane potential as functions of 1/d.

In the lower panel of Fig 14, we report the maximum intracellular potential, the minimal extracellular potential and the maximum value of the membrane potential for the same grid point as a function of 1/d. We observe that the minimum value of ue seems to be almost proportional to 1/d for the NU case. In the plot, we illustrate this proportionality by comparing the computed minimal extracellular potentials for the NU case to the linear approximation min(ee) ≈ a/d, where a = −0.15 mV μm.

The size of the intracellular potential does not change much for the considered values of d, and the increased membrane potential observed in the rightmost panel of Fig 14 is therefore entirely caused by the decrease in the extracellular potential (recall that v = uiue). We observe that for a cell distance of d = 5 nm, the membrane potential just after the blocked gap junction increases to about −52 mV. This is, however, not enough to initiate an action potential in the second cell, so we do not achieve successful propagation through ephaptic coupling in this case.

Ephaptic coupling for a decreased extracellular conductivity.

As observed in Fig 14, the extracellular potential reaches a value of almost −30 mV for a cell distance of 5 nm, but this is not enough to support propagation of the action potential for the case of closed gap junctions. However, this conclusion is expected to depend on the choice of parameter values used in the simulation. For example, if we assume that the extracellular conductivity is 10 mS/cm instead of the default value of 20 mS/cm, the magnitude of the extracellular potential is large enough to enable propagation though ephaptic coupling alone, as illustrated in Fig 15. Here, we show the intracellular potential, the extracellular potential and the membrane potential in the point of the membrane of the second cell illustrated by a red circle in the upper panel of Fig 14. In this case, the magnitude of the extracellular potential seems to be large enough to bring the membrane potential up to a value that triggers the activation of the sodium channels on the membrane of the second cell, and thereby to a significantly increased intracellular potential in the second cell.

The figure shows the intracellular potential, the extracellular potential and the membrane potential observed after an intercalated disc with blocked gap junctions for a simulation with the same setup as in Fig 14, except that the value of σe is reduced from 20 mS/cm to 10 mS/cm. We consider a cell distance of d = 5 nm and a non-uniform distribution of sodium channels.

Ephaptic effects on the INa dynamics.

In Fig 16, we investigate how ephaptic coupling affects the conduction properties when there is intracellular current through the gap junctions. We consider a case with two cells like in Fig 14, but where the gap junction resistance between the two cells is set to the default value given in Table 1. In particular, we investigate how the distribution of sodium channels and the cell distance affect the dynamics of the sodium channels. The figure shows that, both at the end of the first cell (A) and the start of the second cell (B), NU channel localization accelerates the rate of INa activation with respect to time, v, and both ui and ue. These effects are exaggerated at short cell distances, particularly in the second cell (B), to which the AP is being transmitted. In panel C, the ephaptic effects on the sodium channels in the NU case are illustrated further, by considering INa, ui and ue at the beginning of the second cell as functions of time. We observe that as the cell distance is decreased, there is a significant increase in the magnitude of the extracellular potential, activating the sodium channels at an earlier point in time and for a lower intracellular potential. Moreover, in panel D, we integrate the total INa influx over the entire membrane of the second cell. We observe that the charge movement required for the AP upstroke is reduced substantially for the NU case, especially at short cell distances. Together, these results suggest that NU localization and short cells distances may interact to potentiate sodium channel activation during the AP upstroke, thus reducing both gap junctional delay and the net charge movement required for AP propagation.

(A) Raw (top panels) and normalized (fraction of peak, bottom panels) INa currents for the distal cell-end membrane in the first cell of a two-cell strand. INa in this membrane region is shown at each of three different cell distances (d = 5 nm, 10 nm, 160 nm) for the U and NU cases. (B) Similar results for the membrane at the beginning of the second cell. (C) Time development of INa, ui and ue for the beginning of the second cell. (D) Integrated whole-cell INa influx in the second cell over the entire upstroke for the NU and U cases. The simulation set up and parameters are the same as in Fig 14, except that the gap junction resistance is set to the default value of Table 1.

Ephaptic effects for large cell distances.

In Figs 14–16, we observed that for small cell distances, the magnitude of the extracellular potential increases considerably in the small extracellular junctions between the cells for a non-uniform distribution of sodium channels, enabling ephaptic effects between the cells. In the remaining simulations of this paper, however, the cell distance is much larger than in Figs 14–16, and we therefore expect that the results in Figs 4–13 are not caused by ephaptic effects between the cells.

For example, Fig 17 shows the conduction velocity for 0%, 50% and 100% of the sodium channels moved to the cell ends for an increasing value of σe. The left panel shows the maximum absolute value of ue during the simulation for each of these values of σe, and the right panel shows the corresponding conduction velocities. We observe that as the value of σe increases, the magnitude of the extracellular potential decreases, but the conduction velocity seems to remain constant. In other words, the effect that the conduction velocity is increased for a non-uniform distribution of sodium channels seems to be unaffected by a decreased magnitude of the extracellular potential, and the results of Fig 4 therefore do not seem to be caused by ephaptic effects.

The left panel shows the maximum absolute value of the extracellular potential as the value of σe is increased in simulations using the same setup as in Fig 4, with 0% (U), 50% and 100% of the sodium channels moved to the cell ends. The right panel shows the conduction velocity computed in each of the simulations.


Discussion

These investigations assume that the four test species inhabit different thermal environments. It is therefore appropriate to comment on the magnitude of this temperature range. At the outset, it should be stressed that the precise temperature range for these species is not known, a common problem for marine organisms. Geographically distinct subpopulations probably experience temperature variation as well. For the present purpose, we intend only to establish that each species' temperature range is different and that collectively they span approximately 20°C.

L. opalescens is commonly found off the Pacific coast of North America and is particularly abundant off California. In the winter,populations are found in southern California, whereas for the rest of the year they are found off northern and central California. In both cases, surface temperatures rarely exceed 13°C and are often several degrees lower. These squid have also been observed below the thermocline at temperatures as low as 6-7°C (Neumeister et al.,2000). Specimens of L. pealei are commonly encountered off the Atlantic coast of North America from the Gulf of Maine to the Gulf of Mexico. Their temperature range has been discussed in greater detail elsewhere(Boyle, 1983 Rosenthal and Bezanilla,2000b). In Woods Hole, this squid is most commonly encountered between approximately 10 and 20°C, although their abundance is greatest when water temperatures are at the low end of this range. They are reported actively to avoid temperatures below 8°C(Summers, 1969). Specimens of L. plei are commonly found in the Gulf of Mexico and in the Caribbean Sea off northern South America. In Mochima, Venezuela, where the specimens for this study were collected, water temperatures at the capture site (and depth)were between 18 and 20.5°C. In the Gulf of Mexico, these squid are often encountered in the same areas as L. pealei however, the abundance of each species is stratified with depth. L. plei prefer warmer surface waters (<50 m), whereas L. pealei prefer cooler, deeper waters(Hixon et al., 1980 Whitaker, 1978, 1980). S. sepioideaare normally found adjacent to shallow reefs in the Caribbean Sea and the Gulf of Mexico (Voss, 1956). In Mochima, where specimens were collected, data loggers indicated that daily temperatures fluctuated between 23.5 and 27°C in April and May. In July and August, temperatures were significantly higher (26-29°C). It should be noted that both L. plei and S. sepioidea were maintained at 29-30°C in flowing seawater tanks for several days before use with no obvious detrimental effects.

The data presented in this report identify three properties of the propagated action potential that vary. First, the duration of the action potential's falling phase differs among all species tested and correlates with the thermal environment of the species when measured at equivalent temperatures, action potentials from species that inhabit colder environments are shorter. Second, the action potential's rising phase is relatively slow in S. sepioidea, but equivalent in all three species of Loligo. Third, the same relationship holds true for the conduction velocity.

The absolute levels of gK and gNa are consistent with all three changes in the propagated action potential. It should be stressed that species-dependent conductances were measured using two independent sets of experiments (voltage-clamped axons and action potential interruptions) and both yielded comparable results. Clearly, an increase in gK will decrease the duration of the action potential's falling phase, an assertion supported by the Hodgkin and Huxley model (data not shown). In fact, for a number of systems, the action potential's duration is regulated by changing the level of gK (Chen et al.,1996 Kaab et al.,1996 Thuringer et al.,1996 Zhang and McBain,1995). The relatively slow action potential rise time and conduction velocity in S. sepioidea axons are consistent with its low levels of gNa. On the basis of the Hodgkin and Huxley model axon, the level of gNa correlates well with conduction velocity and rise time(Adrian, 1975 Hodgkin, 1975 Huxley, 1959). However, more quantitative extrapolations of conduction velocity and rise time, using our measured conductance values, were not attempted. Years of scrutiny have indicated that several aspects of the Hodgkin and Huxley model require further refinement to predict properties of conduction with sufficient accuracy for the present purposes (Armstrong and Bezanilla, 1977 Armstrong et al., 1973 Armstrong and Hille,1998 Vandenberg and Bezanilla, 1991 Bezanilla,2000 Bezanilla and Armstrong,1977). For example, as seen in Fig. 6, at the action potential's peak, there is virtually no gK, a result not predicted by the Hodgkin and Huxley equations using the original parameters(Hodgkin and Huxley,1952).

There are many plausible mechanisms for regulating ionic conductances in the giant axon. The most straightforward would be to modify the surface expression of Na + or K + channels by transcriptional mechanisms, by translational mechanisms or by changing turnover rates. Conceivably, the unitary conductance of Na + and K + channels could also be regulated. No evidence supports or refutes any of these possibilities. Species-dependent differences in slow inactivation could also influence the available levels of ionic conductances. However, at the holding potentials used for these studies (≤-65 mV), no significant slow inactivation was apparent for any species. For gK, regulation may also be at the level of subunit assembly. Recent evidence suggests that SqKv1.1A, a K + channel mRNA that is thought to underlie some or all of delayed rectifier gK in the giant axon(Rosenthal et al., 1996),contains several anomalous amino acid residues in a channel domain (T1)responsible for subunit tetramerization(Liu et al., 2001). The amino acids at these positions, which are regulated by RNA editing(Rosenthal and Bezanilla,2000a), strongly influence the functional expression of heterologously expressed channels in Xenopus oocytes. Further evidence suggests that the pattern of RNA editing in the T1 domain of SqKv1.1A varies among different species of squid(Rosenthal and Bezanilla,1998).

The conductance level changes identified in this study are a plausible regulatory mechanism for the changes observed in the propagated action potentials. However, other possibilities exist. For example, inactivation and deactivation of gNa could contribute to the action potential's rate of decline. The axon's cable properties (e.g. resistance or capacitance) could also vary among species. Previous work has demonstrated that, within a species, seasonal changes in the giant axon's internal resistance can influence conduction velocity (Rosenthal and Bezanilla, 2000b).

Do these data describe temperature adaptations? To answer this question,several factors should first be considered. Does the giant axon system mediate an equivalent function in all the species tested? On the basis of similarities in anatomy and behavior, it is assumed that for each species the giant axon system regulates the jet-propelled escape response(Otis and Gilly, 1990 Young, 1938) and plays a role in feeding behavior (Preuss and Gilly,2000). Are these species sufficiently close on a phylogenetic level to permit a meaningful comparison? Although there are no established standards, all four species are members of the family Loliginidae. On the basis of recent phylogenetic studies using mitochondrial DNA sequences, the three Loligo species are more closely related to each other than to other members of the genus. The species S. sepioidea is the outlier(Anderson, 2000). Therefore,when considering our data, more weight should be given to physiological differences that varied within the genus Loligo (e.g. the level of gK). All three species of Loligo are active pelagic predators. Although S. sepioidea are more domercile, it is probable that the giant axon system mediates the same basic function for all the squid species tested.

It is also important to consider why a physiological difference would be adaptive. First, the action potential's duration will be considered in terms of simple rate compensation. Is it necessary for species that inhabit different thermal environments to maintain a similar action potential duration at their native temperatures? If so, modifications of the underlying ionic conductances are required. Data from this study only partially support this view. Action potentials from L. opalescens giant axons are relatively fast and those from S. sepioidea are slow. However, the rate compensation is not complete. On average, there is an approximately 5 °C shift in the fall times between L. opalescens and S. sepioidea and an overall shift of approximately 7.5 °C in the total action potential duration (fall time plus rise time). The temperature difference between these two species' environment is approximately double this. Clearly, there is not perfect rate compensation. When considering the purpose of the giant axon's action potential, to stimulate an escape jet, the necessity to compensate the duration in the first place is not immediately apparent. A single action potential in the giant axon produces an all-or-none contraction of the mantle musculature(Prosser and Young, 1937 Young, 1938). Does the duration of this action potential matter? A better understanding of the relationship between temperature, action potential duration and mantle contraction could help shed light on this issue.

An alternative hypothesis is that the action potential broadening, seen in the species from warmer habitats (L. plei and S. sepioidea),is merely a consequence of an adaptation to avoid action potential failure at high temperatures. As discussed in the Results section, action potentials in the two species from colder habitats fail at approximately 29 °C. The temperature range for both L. plei and S. sepioidea can approach this level. In fact, in our hands, both species survive in tanks maintained at this temperature. Using the Hodgkin and Huxley paradigm,decreases in gK lead to increases in the failure temperature because less gK competes with gNa following stimulation. Action potentials from both `warm' species exhibited a reduced gK and elevated failure temperature. Other factors, however, could also contribute to differences in failure temperature (e.g. differences in Na + channel inactivation). Clearly, avoiding action potential failure would be an adaptational advantage.

The reported changes in normalized conduction velocity can also be viewed in terms of compensatory adaptations. The conduction velocity of the giant axon of S. sepioidea, at all temperatures, is slower than that of Loligo species. In fact, the shift of approximately 10 °C between the conduction velocity versus temperature relationships is in reasonable agreement with the temperature ranges for these two species. Thus,to make the escape response as fast as possible, Loligo species have more gNa in their axons to compensate for a relatively cold environment. It is important to note, however, that there is no apparent rate compensation within the genus Loligo, even though the species utilized in this study span a considerable range of habitat temperature. A previous study, using two Loligo species, hypothesized that a change in the slope of the conduction velocity versus temperature relationship was a temperature adaptation(Easton and Swenberg, 1975). In our study, at temperatures above 7 °C, there is no evidence for a change in the slope of the conduction velocity versus temperature realtionship. Members of the genus Loligo are not commonly found below this temperature.

At present, we consider the physiological properties identified in this study as reasonable candidates for temperature adaptations. Similar investigations utilizing more representatives within the family Loliginidae would help clarify the issue. These findings are interesting in light of the considerable attention that has been paid to the theory of homeoviscous adaptation (Macdonald, 1988 Sinensky, 1974). No comparative data exist for membrane lipid viscosity in the giant axonhowever, homeoviscous adaptation would be expected to affect the rates of ion channel gating. These data suggest that absolute conductance levels are more likely to be targets of adaptation. However, Na + channel inactivation and deactivation kinetics need to be measured. For these and other studies, the squid giant axon remains an excellent system in which to study temperature adaptation in nerve.


A mathematical model of the nerve impulse at the molecular level

A model of the axonal membrane that has as one of its guiding principles the integration and extension of the ideas of the Hodgkin-Huxley school and the Nachmansohn school is constructed, based upon the Bass-Moore alkalosis theory [3] and the pH dependence of the activities of suitably selected enzymes in the membrane medium. The underlying principles, basic postulates, and governing equations are described and their application to the cases of the membrane potassium conductance g K and sodium conductance g N a is discussed. The same set of equations is used to define the enzyme systems controlling g K and g N a by simply letting the time constant with which the membrane pH relaxes back to its resting value following a sudden displacement of the membrane potential difference be finite in the region responsible for the control of g N a and be effectively infinite in the region responsible for the control of g K , it is shown how the model offers an explanation for the observed transient increase of g N a and the observed maintained increase of g K under a maintained voltage-clamped depolarization.


Materials and Methods

Model of Action Potential Conduction on the Surface of an Axonal Membrane

An action potential is mainly produced in neurons by the influx of sodium ions from the extracellular fluid to the intracellular fluid through voltage-gated sodium (NaV) channels (Lai and Jan, 2006 Hu et al., 2009). Differing from a closed electric circuit with a metal lead, in which the bearer of the electric charge is free electrons, the bearer of the electric charge in the intracellular fluid is multiple electrolyte ions (e.g., sodium, potassium, or chloride ions).

The resistivity of the axonal membrane (lipid bilayers) is far larger than that of the intracellular fluid thus, the conduction of some kinds of electric charges within lipid bilayers is unrealistic (Naumowicz et al., 2003). It is known that the membrane interior between lipid bilayers is positively charged, because of the molecular polarity of phospholipids (Berkovich et al., 2012). However, this space, less than 1 [nm] width, is too narrow to allow the electrolytes (ionic radius around 100� [pm]) to travel freely (Gurtovenko and Anwar, 2009). In the first place, hydrophobic molecules are facing close to each other in this space thus, electrolytes are unlikely to exist at sufficient level within this space. Considering these facts together, in this report, lipid bilayers are regarded as insulators and are unable to transfer electric currency within them.

Another concern is that whether the inflowing sodium ion particles could be directly transmitted as carriers of electric charges from one NaV channel to the next or not. The ionic migration velocity within a solution is known to be very slow, usually far less than 1.0 [mm/s]. As described in the next section, the allowed time lapse at each NaV channel in nerve conduction is much less than 10 𢄦 [s] (= 1.0 [μs]). The velocity of ionic migration could be accelerated to some extent by Coulomb force from the inflowing sodium ions, but it would not be enough to make the ion particles to travel all the way to the next NaV channel within such a short time period. Thus, the transmission of membrane potential from one NaV channel to the next would be realized not by the migration of the inflowing sodium ions all the way to the next channel, but by the transmission of some remote forces like Coulomb force along the surface of an axon membrane.

Allowed Time Lapse in One NaV Channel

As described in a later section, the density of NaV channels on the axon membrane of unmyelinated nerves is known to be 5� [channels/μm 2 ](Hu and Jonas, 2014) suggesting that the distance between each of the adjacent NaV channels is less than 0.50 [μm]. Now, we will consider an unmyelinated nerve with an axonal length of 10 [cm] (= 10 5 [μm]). The estimated number of NaV channels on a longitudinal straight line from the neuron body to a synapse is at least 2.0 × 10 5 [channels / longitudinal line]. Here, the conduction velocity of unmyelinated nerves of an average size of diameter is known to be around 1.0 [m/s] (= 10 6 [μm/s]) which means only 0.1 [s] (= 10 2 [ms]) would be required to travel across an axon 10 [cm] in length. Based on these premises, the elapsed time at each site of a NaV channel on the longitudinal line is calculated to be less than 10 𢄦 [s] (= 10 𢄣 [ms] = 1.0 [μs]).

As we know, it takes more than 10 𢄡 [ms] for an axon membrane to generate an action potential from the rise of the membrane potential (Platkiewicz and Brette, 2010). Based on these facts, the most advanced part of some kind of nerve conductive stimulant is already a longitudinal distance of more than 100 [channels] away from one specific NaV channel when a sufficient amount of ionic conductance and action potential are created around the channel. This means that the conduction in some form, which could trigger the opening of the NaV channels, passes much earlier than the generation of an action potential at each site of a NaV channel. This unexpected finding suggests that propagation of the nerve conduction happens without the generation of an action potential.

Based on these findings, in this report, we assume a new conductive model focusing on ionic migration based on Coulomb force inside the axon from a viewpoint different from the conventional equivalent circuit model.

Generation of Action Potential

In the conventional cable theory, membrane potential at a specific coordinate and time on the membrane can be calculated by the Equation (1). However, as we described above, there is no actual metal leads spreading across the internal and external surfaces of the axonal membranes. The equivalent circuit model and cable theory are truly wonderful and splendid theories as a model to explain the phenomenon of action potential and nerve conduction. But in the actual sites of the internal surface of axons, there is no closed metal circuit in which the Ohm's law and Kirchhoff's law are realized. In the actual site, under the condition of steady state with no net charge transfer through the membrane, the membrane potential can be calculated by an equation using gradient of ionic concentrations along the axonal membrane as below, known as Goldman-Hodgkin-Katz (GHK) voltage equation (Goldman, 1943 Delamere and Duncan, 1977).

(F: Faraday constant, R: the ideal gas constant, T: the temperature in kelvins, Pion: the permeability for each ion, [ion]in: the intracellular concentration of each ion, [ion]out: the extracellular concentration of each ion).

Though this equation does not stand when the steady state is broken, like the state of action potential, transfer of ions like sodium and potassium ions takes place in producing action potential. After the inflow of sodium ions at a specific NaV channel, actual change of ionic concentration around the next NaV channel is needed to make the membrane potential around the next NaV channel surpass the threshold of membrane potential to trigger the opening of the next NaV channel (Platkiewicz and Brette, 2010 Carter et al., 2012). This is one of the most basic and invariable principles in the generation of action potential, but could not be fully reflected in conventional theories based on Ohm's law and Kirchhoff's law.

As described before, in actual sites on an axonal membrane, the transmission of electric charges from one NaV channel to the next is realized not by the transmission of ion particles or free electrons all the way to the next NaV channel, but by the transmission of remote forces like Coulomb force from the inflowing mass of sodium ions to the ions around the next NaV channel. Without the change of ionic concentrations gradient across the membrane at the next NaV channel, at least only around the NaV channel, the generation of potential change around the next NaV channel or the opening of the next NaV channel would not be realized.

This relation is an important and fundamental principle, which was not fully considered in the conventional conduction models. In this paper, on the basis of this relation, we will focus on what the Coulomb force from a mass of inflowing sodium ions at one NaV channel will do to remote ion particles around the next NaV channel.

Coulomb Force and Ionic Migration Around the Next NaV Channel

The initial velocity of ionic inflow from a NaV channel within very short period less than the first 1.0 [μs], an allowed limit of time period at one NaV channel as shown in the Equation (2), can be regarded as approximately constant, because the electrostatic repulsion from the formerly flown sodium ions to the later flowing sodium ions through a NaV channel can be disregarded within such a very short time period (Ohshima and Kondo, 1987 Chung and Kuyucak, 2002). Thus, we define the amount of electric charge of inflowing sodium ion at a specific NaV channel within a time period of 1.0 [ns] (= 10 𢄣 [μs]) in the first 1.0 [μs] as a constant of +Q1 [C]. Then, the total amount of sodium ions flown through a NaV channel within the first t1 [ns] can be described as below.

Next, we define the elementary charges of monovalent cations and anions as a constant of ±e [C] (𢒁.602 × 10 � [C]). Because the ionic migration within fluid is quite slow and almost ignorable within a short period of nerve conduction, we can disregard the size of the inflowing mass of sodium ion from one NaV channel, compared to the distance from one NaV channel to the next. Then, the absolute values of the Coulomb forces from the mass of inflowing sodium ions (flown from 0 [ns] to t [ns]) to each of the cations and anions around the next NaV channel, which is r [m] away from the previous NaV channel, can be described as below.

Here, there is an equation as below showing the ionic migration velocity and ionic flux (J) of a specific area in a specific time period within a fluid (Teorell, 1935). Ratio of ionic particles and water molecules is reflected in the value of the molar mobility (ω) and ionic concentration (C).

(ω: molar mobility, C: ionic concentration, F: driving force per gram-ion)

Based on (6) and (7), Coulomb force to each ion (F [N]) at a specific distance from the mass of inflowing sodium ion particles is proportionate to t [ns] thus, combined with (9), the following relationship is true.

Ionic Movement along the Axon Membrane

Ion concentration in the intracellular fluid is roughly regarded as homogeneous throughout the axoplasm. Charge balance between cations and anions is preserved, even in the state of action potential, and the axoplasm in total is not strongly electrically-charged (Kurbel, 2008).

But, strictly speaking, the cation-to-anion ratio within axoplasm is different based on the distance from the internal surface of axon membrane. The anion-densed layer on the internal surface of axon membrane is known to be restricted only within 1𠄲 [nm] from the internal surface, known as �ye length” (Quinn et al., 1998 Bulai et al., 2012) (Figure 2). These anions within the Debye length from the internal surface of the membrane will be attracted by the Coulomb force from the inflowing sodium ion. The high density of water molecules, some of which hydrates the anions, and the ionic bands with cations would decelerate the ionic migration of the anions to some extent. However, the anions-densed horizontal layer within Debye length from membrane surface in total would be attracted by the Coulomb force and create a fluid flow toward the mass of inflowing sodium ions around the preceding NaV channel.

Figure 2. Ions and water molecules along the internal surface of an axon membrane. Within the Debye length (1𠄲 [nm]) from the internal surface of an axon membrane, anions are densely distributed thus, this layer is negatively charged in total, only with the exception that cations crowded around the NaV channels which have negative charges on their surface. The axoplasm out of the Debye length from the surface are consisted of almost equal amount of anions and cations thus, the axoplasm out of the Debye length is not electrically charged in total. Abbreviation: NaV, voltage-gated sodium ion channel.

Here, there is an exception about the anion-crowded circumstance on the internal surface of axon membrane. Usually, cation channels have negative charges on their surfaces, which should produce an excess of cations near the internal surface of the axon membrane around NaV channels (Dani, 1986 Cukierman et al., 1988 Miedema, 2002).

Now, we consider about the situation when a mass of sodium ion particles flowing through a NaV channel from extracellular fluid to axoplasm (Figure 3). As shown in the figure, the bulk of inflowing sodium ion affect charged ions in axoplasm cations will receive repulsive forces and anions will receive attractive forces. Because of this inverse direction of Coulomb force between anions and cations, cation-to-anion ratio within the Debye length in the limited space around the next NaV channel will be dramatically changed by the Coulomb force from inflowing sodium ion. If the accumulated sum of the Coulomb force surpass the minimum threshold and the ionic milieu around the next NaV channel has been sufficiently changed, it causes conformational change on the structure of next NaV channel and triggers its opening. Propagation of the forefront line of such minimum threshold in Coulomb force can be illustrated as shown in Figure 4. As described later, threshold Coulomb force to trigger opening of the next NaV channel differs based on the channel density, in other words, based on the distance of two-adjacent NaV channels.

Figure 3. Ionic migration by Coulomb force from the inflowing sodium ions. When a mass of sodium ions are created around a NaV channel in the process of the generation of an action potential, ions in the axoplasm would receive an attractive or repulsive Coulomb force from the mass. Within the Debye length, electric charge at each site would not be changed, except for the minute space around the next NaV, where the cations crowded around the negatively-charged surface of NaV. Abbreviations: dQ, small change in the amount of inflowing electric charges dt, very short time period NaV, voltage-gated sodium ion channel.

Figure 4. Propagation of the threshold line of Coulomb force to trigger opening of the next NaV. The closer the electric charges are, the stronger the Coulomb force between them. Thus, the ion particles close to the mass of inflowing sodium ions receive stronger Coulomb force than the ion particles far away from the mass. The threshold forefront line of Coulomb force to trigger opening of the next NaV expands as the inflowing sodium ions increases by time. Abbreviation: Cd, channel density of voltage-gated sodium ion channel on the membrane.

Size of Axons

The diameters of axons in unmyelinated nerves are generally known to be smaller than those in myelinated nerves (Ikeda and Oka, 2012). Usually, the diameter of an axon in unmyelinated nerves is 0.2𠄱.2 [μm] (Ritchie, 1982). In this study, the diameter of the axon is described as “D” [μm], which is between 0.2 and 1.2. By using this variable, the cross-sectional periphery and area of an axon can be described as below.

Based on (11) and (12), if the axon is cut with the length of L [μm], the cell membrane area and the intracellular volume of the cut axonal section can be described as below.

Ratio of Intracellular Ions and Inflowing Sodium Ion

The membrane capacitance is known to be around the below-mentioned value (Gentet et al., 2000 Golowasch et al., 2009).

Thus, to produce an electric potential of 100 [mV] (= 0.1 [V]) on the axon membrane with an area of πDL [μm 2 ] (axon with L [μm] in length, D [μm] in diameter), a total amount of πDL × 10 � [C] of electric charge must be transferred from the extracellular to intracellular fluid. The electric charge of one particle of monovalent cation is 1.602 × 10 � [C] thus, to produce an action potential, a minimum of the following number of sodium ion particles must be transferred.

Here, the usual concentrations of intracellular sodium and potassium ions are around 15 [mEq/l] and 140 [mEq/l], respectively. The numbers of intracellular cation particles (sodium and potassium) in the same length of an axon before the inflow of sodium ions can be estimated with the following equations, using the Avogadro constant of 6.02 × 10 23 [mol 𢄡 ]:

Thus, the ratio of inflowing sodium ion particles to the intracellular cation particles before the inflow is as follows:

This ratio shows that the number of intracellular cation particles originally present before the sodium influx is far greater than the number of sodium ion particles producing the action potential, more than 1,000 times higher than sodium influx this suggests that sodium influx through the NaV channel would not significantly change the intracellular concentration of cations. Based on this result, in this study, we ignore the effect of Coulomb force from the inflowing sodium ions on different sides of the axon membrane, and only focus on each pair of two longitudinally-adjacent NaV channels.

Density and Distance of Sodium Ion Channels on an Axonal Membrane

The density of NaV channels in unmyelinated nerves is known to be 5-50 [channels/μm 2 ], and mainly differs depending on the distance from the axon hillock, the root of an axon (Hu and Jonas, 2014). For convenience, in this study, channel density in one axon is regarded as being constant. Here, we define the channel-density of NaV channels on an axon membrane as Cd.

To mathematically describe the distance from one NaV channel to the next NaV channel on distal (peripheral) side, we apply a model in which the NaV channels evenly align circularly on one cross section of an axon, expressed as “NaV channel line” with a ring formation. Then, distance from one channel line to the next adjacent channel line can be described as below (Figure 5).

Figure 5. Schema of the concept of NaV channel density and distribution on the axonal membrane. When the density of NaV channels is Cd [channels/μm 2 ], the distance of the adjacent NaV channel line from one channel line can be described as 1/ C d [μm]. In this report, for convenience, we adapted a model of “NaV channel line”, parallel to the cross-section of axons, and regarded an axon as a pile of such channel lines. Cd, NaV channel density [channels/μm 2 ] D, diameter of axon [μm] NaV, voltage-gated sodium channel rad, radian.

The theoretical relationship between the axonal diameter (D) and NaV channel density (Cd) will be discussed later.

On a cross-section of one NaV channel line of an axon with D [μm] (0.2 ≤ D ≤ 1.2) in diameter, the numbers of NaV channels and the central angle comprised by the two adjacent NaV channels can be described as below.

Conduction Velocity

The nerve conduction in unmyelinated nerves can be theoretically described as the integration of continuous transmission of an action potential, realized by changes in ionic concentrations around NaV channels, from one NaV channel to the next. Thus, the average time period to take for nerve conduction along a specific distance can be calculated by the average time lapse from opening of one NaV channel to opening of the next adjacent NaV channel (tadjacant) multiplied by the number of channel lines existing along the specific distance.

In this equation, angle brackets (< >) means the average on time of the content.

Theoretical Relationship between the Axonal Diameter and Channel Density

Until now, there has been no study showing the relationship between the axonal diameter (D) and channel density (Cd). In this report, we will theoretically estimate the relationship based on the following conditions.

(1) Because there is no organelle inside axons, NaV protein is translated in the neuronal body and then transported by kinesin protein on the rail of microtubules in axon.

(2) As previously known, the numbers of microtubule in axons are proportionate to the axonal cross-sectional area [μm 2 ]. (Hoffman et al., 1985 Iturriaga, 1985).

(3) Sufficient amounts of NaV channels are expressed in the neuronal body in normal condition, so that the rate of gene expression does not affect or control the rate of NaV-channel transportation to the efferent side.

In a state of equilibrium of the channel density, the rate of NaV channel importation and NaV channel breakdown are equal in a specific area of the axon membrane [μm 2 ]. Based on the third condition, the total number of imported NaV channels inside the axon can be proportionate to the number of microtubules inside the axon. Based on these premises, we will consider the theoretical relationship between an axonal diameter and a channel density.

Other Assumptions

In the model of this report, which mainly focuses on the relationship between the axonal diameter and conduction velocity, factors other than the axonal diameter (D) and channel density (Cd), like temperature, density of sodium ion channels on cell membrane, resistivity of the fiber interior, membrane capacitance per unit area, or the unit area resistance of the membrane in its excited state) were regarded as constant, regardless of the axonal diameter (Matsumoto and Tasaki, 1977).


Discussion

We investigated ionic mechanisms in the DCMD axon that allow for faithful high𠄏requency transmission and demonstrate that a current exists that shortens the AHP duration. Computational modeling suggests that shortening the AHP duration could increase high𠄏requency fidelity and conduction velocity.

Divalent cations effects on the transient sodium current

The divalent cations effects on the rise time and max rise slope were indicative of modulation of the transient sodium current. This can arise from the surface charge screening effect where divalent cations bind negative surface charges on the plasma membrane or ion channel, distorting the local electric field (Frankenhaeuser and Hodgkin 1957 Hille etਊl. 1975). This can reduce the voltage difference across the channel, slowing gating properties and decreasing the current conducted by the ion channel. Divalent cations can also directly interact with the activation gate of the transient sodium channel, slowing its activation as well (Gilly and Armstrong 1982b). Surface charge screening and direct slowing of the activation gate would explain the increased rise time we observed in the DCMD. They may also explain why conduction failed over longer exposures as the accumulation of divalent cations further impeded the activation gate eventually causing a conduction block. Modulation of the transient sodium current was likely the primary cause of the divalent cation decreased CV as CV is heavily dependent on active transient sodium channels (Carp etਊl. 2003 De Col etਊl. 2008). Our model showed a significant decline in CV following a 10 mV screening of the sodium current that supports this theory, and provides caution for axonal CV studies that involve large changes (

mmol/L) to extracellular divalent cation concentrations. However, our model also suggests a role for persistent or resurgent sodium currents that can fine tune high𠄏requency CV.

The differences between cadmium's effects at room and high temperature may be an amplification of the screening effect by the temperature scaling of the channel kinetics. Channel kinetics scale nonlinearly with temperature (for a review see Robertson and Money 2012) and can increase channel rates by a factor of 2.5𠄴.2 times over the temperature range we used (assuming Q10 of 2𠄳, respectively). Shifts in the activation curves brought on by the divalent cation screening are then amplified by the temperature scaling, resulting in a larger effect at higher temperatures. This may also explain why the room temperature experiments required longer exposure, to accumulate sufficient divalent cations for the screening effect to be observed.

Increased CV in calcium𠄏ree saline

There are several explanations for how removal and addition of extracellular calcium could increase and decrease CV, respectively. As a divalent cation, calcium may be screening transient sodium channels, and their removal may increase CV. Although we attempted to control for this effect using magnesium as a substitute ion, magnesium tends to be less potent than other divalent cations, including calcium, at binding negative surface charges and screening (Hille etਊl. 1975 Hanck and Sheets 1992). Given how the magnitude of the screening effect can vary across the different divalent cations it would be difficult to accurately balance screening effects.

Calcium's effect may have also been through a calcium�pendent potassium channel that hyperpolarizes the resting membrane potential thereby reducing CV. During heat stress, the resting membrane potential of the DCMD hyperpolarizes (Money etਊl. 2005) that may be caused by a calcium�pendent potassium channel and could explain why calcium's effect are absent at room temperature. Increase in intracellular calcium concentration is also known to occur with higher temperatures and plays an important role in conferring tolerance to heat stress at synapses in Drosophila (Barclay and Robertson 2003 Klose etਊl. 2008).

Although calcium saline manipulation affected the DCMD's performance, it did not mimic the effects of the divalent cations on CV. We could be reasonably certain that calcium𠄏ree saline had removed calcium from the DCMD's extracellular space, as high‐temperature effects occurred within 20 min of exposure and were stable up to 50 min past exposure. This is consistent with T‐type calcium channels and calcium�tivated nonselective cation channels (Haj�hmane and Andrade 1997) not being responsible. Furthermore, it remains unlikely that cadmium and nickel were inhibiting a transient receptor potential (TRP) current, which can also shorten AHP duration, as most channel isoforms that carry this current can conduct divalent cations (Bouron etਊl. 2015) and those isoforms that cannot are activated by cytoplasmic calcium increases (Launay etਊl. 2002 Hofmann etਊl. 2003 Lei etਊl. 2014). Future experiments should focus on rigorously ruling out calcium currents involved in AHP shortening via highly selectively calcium channel blockers.

It's also unlikely the AHP shortening was due to reduction in potassium current brought on by the divalent cations. Divalent cations typically modulate the activation time constant of the delayed‐rectifying potassium channel without a large effect on deactivation (Gilly and Armstrong 1982a Armstrong and Matteson 1986) which does not explain the AHP increase. Likewise, with A‐type potassium currents, divalent cations shift the inactivation curves in a depolarized manner (Erdelyi 1994) resulting in shortening the AHP, not increasing it. Therefore, we believe the most likely cause of the divalent cations were through a persistent or resurgent sodium current as have been described in several axons (Stys etਊl. 1993 Crill 1996 Astman etਊl. 2006 Kim etਊl. 2010).

A role for the persistent/resurgent sodium currents

Persistent sodium currents arise from transient sodium channels that open during an AP and remain open past the hyperpolarizing phase of the AP as they fail to inactivate (Crill 1996). By remaining active at hyperpolarized potentials, they provide a depolarizing potential for high𠄏requency and repetitive firing (Harvey etਊl. 2006). Resurgent sodium currents occur when transient sodium channels reactivate late in the AP and remain active into the hyperpolarizing phase until inactivated and are also involved with high frequency and repetitive firing (Khaliq etਊl. 2003).

In axons, persistent sodium currents increase excitability (McIntyre etਊl. 2002), and resurgent sodium currents secure high𠄏requency fidelity in calyx of Held (Kim etਊl. 2010). Our model supports both functions for the persistent and resurgent sodium current and suggests a possible mechanism to decrease expensive, high𠄏requency signaling that occurs in the DCMD after metabolic stress. After anoxic stress, the DCMD axon conducts fewer and slower high𠄏requency APs (Money etਊl. 2014). However, under hypoxic conditions, persistent sodium currents increase in hippocampal (Hammarstrom and Gage 2000) and cardiac tissue (Ju etਊl. 1996) that should increase excitability, which conflicts with the DCMD's decreased excitability during hypoxia (Money etਊl. 2014). It is possible the persistent sodium currents are more tightly regulated in the DCMD given that it can fully recover from anoxia, whereas hippocampal and cardiac tissue experience cell injury and death. Also, elevated cAMP levels can increase persistent sodium currents (Nikitin etਊl. 2006) and may account for the DCMD's recovery of CV from hypoxia after application of adenylate cyclase activator (Money etਊl. 2014).

A characteristic of neurons with persistent and resurgent sodium currents is an ability to cause bursting activity. In neocortex (van Drongelen etਊl. 2006), the pre𠄋ӧtzinger complex (Thoby𠄋risson and Ramirez 2001 Del Negro etਊl. 2002), cerebral snail (Nikitin etਊl. 2006) and leech heart interneurons (Opdyke and Calabrese 1994), persistent sodium currents underlie intrinsic and network bursting, while resurgent sodium currents are involved in bursting activity of cerebellar Purkinje cells (Raman and Bean 1997 Khaliq etਊl. 2003). In the DCMD, rigorous quantification of its activity in response to a looming visual stimulus reveals robust high𠄏requency bursting that increases as the visual stimulus approaches (McMillan and Gray 2015). It is possible that a persistent or resurgent sodium current allows faithful transmission of high𠄏requency bursts by the DCMD from the LGMD to its postsynaptic partners. Burst generation is commonly observed in sensory pathways as a means of enhancing the signal‐to‐noise ratio at synapses, causing a greater release of neurotransmitter (Krahe and Gabbiani 2004), which is exemplified in the DCMD axon, as several high𠄏requency APs are required to elicit activity in a post‐synaptic interneuron (Santer etਊl. 2006). For bursting neurons that require high𠄏idelity transmission of high𠄏requency bursts, including the thalamic relay interneurons (Steriade etਊl. 1993) and cricket auditory neurons (Marsat and Pollack 2006), our data suggest that AHP shortening can increase CV for high𠄏requency APs which may assist in maintaining fidelity during high𠄏requency bursting. Also, we are the first to demonstrate, to the best of our knowledge, that a persistent or resurgent sodium current can reduce the subnormal conduction of high frequency APs in a computational model, which would maintain the temporal fidelity within a burst of APs during transmission. These computational results may shed light on the functional role of persistent or resurgent sodium currents in other axonal models including optic nerve axons (Stys etਊl. 1993) and at the calyx of Held (Kim etਊl. 2010).

ADP and heat stress

Although we did not show that the current responsible for AHP reduction is involved in the ADP formation at high temperature in the DCMD, our computational modeling suggests this is possible. So what is the functional role of an ADP�using current at high temperature in the DCMD? In the DCMD axon, elevated temperatures result in a reduction in excitability that could be explained by the accompanying hyperpolarization of the membrane, as the magnitude to reach the threshold for AP initiation has increased (Money etਊl. 2005). In heat‐shocked animals hyperpolarization at high temperatures is larger than control animals, however, they exhibit an increase in membrane excitability (Money etਊl. 2005). Hyperpolarization may be a protective mechanism from thermal stress as heat‐shocked locusts experience heat‐induced conduction failure in the DCMD at higher temperatures (Money etਊl. 2009). Heat‐induced conduction failure occurs with an abrupt increase in extracellular potassium (Money etਊl. 2009) and may be due to temperature mismatch between neural activity and potassium clearance mechanisms (Robertson and Money 2012). Hou etਊl. (2014) found locust thoracic ganglia increase the number of Na + /K + ATPase in neuronal plasma membranes after heat‐shock, presumably to improve potassium clearance. This could account for the increased hyperpolarization in the DCMD after heat‐shock if the axon increased the number of Na + /K + ATPase in its plasma membrane. The ADP, which is enhanced in heat‐shocked animals (Money etਊl. 2005), then provides a temporary depolarization to increase excitability of high𠄏requency APs, as predicted by our model (Fig.  5 E), which trigger escape responses (Santer etਊl. 2006 Fotowat etਊl. 2011) without compromising the protective effect of hyperpolarization. Future experiments should determine if the ADP modulates high𠄏requency CV of the DCMD axon and if so, in what frequency band.

Model limitations

Like all computational models, ours have several limitations, including the absence of electrophysiological data characterizing the sodium, potassium, or leakage currents in the DCMD. Channel kinetic data for large axons are distorted by incomplete space clamps while fitting channel kinetics by AP waveform can produce endless combinations (Sengupta etਊl. 2010). Also, the model assumes a fixed radial geometry for the DCMD, which is incorrect, as tapering and branch points occur throughout the axon (O'Shea etਊl. 1974). However, we feel the effects of both limitations are minimal as we were more interested in the AHP's role in conduction rather than an accurate description of the DCMD axon. Also, our results for the persistent sodium current extended to two different models with similar results. Another limitation in the model is the ADP produced. Though similar in magnitude to the ADP described by Money etਊl. (2005), it decays quicker, which could be attributed to the short time constant for the persistent and resurgent sodium current. By increasing the time constant, we could increase the duration the persistent or resurgent sodium current is active and therefore increase the ADP's time course. However, this was not undertaken in our study.

Our model also assumes the presence of a single potassium current even though many potassium channel subtypes have been identified in axons (Debanne 2004). However, we feel a single potassium current is sufficient to capture high𠄏requency firing in the DCMD neuron. We know activity�pendent hyperpolarization of the resting membrane potential of the axon is primarily contributed by the Na + /K + ATPase (Money etਊl. 2014), which suggests that BK and SK channels are absent. Also, to test for a potential interaction of slow�tivating potassium currents with the persistent or resurgent sodium current, the Connor–Steven's model with an A‐type potassium current was included and this still shows similar effects of the persistent and resurgent sodium current on CV profile. Furthermore, given the fast firing frequencies we were interested in (𾄀 Hz primarily) potassium channels with short time constants would have the largest effects in shaping the CV. Kv3 potassium channels have short time constants and contribute to high firing frequency in other neurons (Rudy etਊl. 1999 Rudy and McBain 2001), however, they are TEA‐sensitive and our unpublished observations with TEA in the DCMD found no effect. Although this does not rule out all possible potassium channels and dynamics, our use of a single potassium current sufficiently captures the frequency�pendent effects on CV. Furthermore, our results with the persistent or resurgent sodium current matches well with other models that include more complex potassium dynamics (D'Angelo etਊl. 2001 McIntyre etਊl. 2002 Kim etਊl. 2010). Future models of the DCMD may partition the single potassium current into several to observe their interactions with the persistent and resurgent sodium current which could be insightful for interpreting heat shock in locusts as potassium currents are known to be downregulated by this process (Ramirez etਊl. 1999).

Our models also focused exclusively on the persistent or resurgent sodium current even though many different mechanisms could shorten the AHP. However, currently the multicompartment models are computationally intensive, so we focused on the candidate mechanisms we felt were the most plausible: the persistent and resurgent sodium currents. Future simulations should focus on whether similar effects could be achieved with functionally similar channels including a T‐type calcium current.


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