To examine dependence of a discrete process on the history of prior activity it is common to use Poisson train stimuli (Krausz and Friesen, 1977; Sen et al., 1996; Stern et al., 2009), which by definition include a large range of inherent frequencies. Poisson trains have been used before in Hodgkin-Huxley model axons to characterize spike delay and velocity (George, 1977; Moradmand and Goldfinger, 1995), but not commonly in model axons with more complex excitability. We therefore examined the history-dependence of conduction delay in the PD neuron model axon by stimulating one end of the axon and measuring conduction delay between two positions along its length (Figure 1).

Figure 1

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Experimental data from Poisson stimulation of the PD axon show a slow hyperpolarization that is accompanied by overall slowing of propagation (Ballo et al., 2012). History-dependence of delay is manifest in several ways. At the slow timescale (STS) of hyperpolarization, both the mean value and variability of conduction delay increase over a timescale of minutes following the onset of the stimuli (Figure 2A).

Figure 2

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At a fast timescale (FTS), a nonlinear and non-monotonic relationship between delay and the instantaneous stimulus frequency (F_inst) is observed: conduction delay of the PD axon has a minimum value for F_inst around 40 Hz but higher values for lower or higher F_inst (Figure 2B). Note that the FTS effect compares the conduction delay of each spike only to that of the previous one. The FTS effect, that is, the nonlinear and non-monotonic frequency-dependence, changes with the STS effect, as can be seen in Figure 2B in the changing frequency-dependence at different times of the stimulation. These effects are strongest during pharmacological block of I_h, presumably because the axon hyperpolarizes more in the absence of inward rectification (Ballo et al., 2012, see below). The data shown in Figure 2A and B were obtained under these conditions. To match these conditions, the simulations shown in Figure 2C and D were performed without I_h in the model (${\overline{g}}_h$=0), but with I_pump and I_A in addition to I_Leak, I_Na and I_Kd. We note that both the STS and the FTS effects were present with or without I_h in the model and, later on, we will show how I_h influences these effects in the model and experiments. The model captured, qualitatively and quantitatively, both the STS and the FTS history-dependence of conduction delay observed in the experiments (Figure 2C and D). To compare the experimental and model STS effect, we calculated the mean (D_mean) and coefficient of variation (CV-D) of conduction delay in 20 s time bins. The model provided an excellent match to the changes in D_mean and the increase in variability over the 5 min stimulation period (Figure 2E and F).

To test to which degree the STS and FTS history-dependence are the result of more complex membrane excitability, we applied the same Poisson stimulation to an axon modeled using the classical Hodgkin-Huxley equations (not including I_A, I_h, and I_pump). The results of this simulation indicated that the Hodgkin-Huxley model axon showed no slow history-dependence (Figure 2G). It did, however, show a weak FTS history-dependence for F_inst values larger than 40 Hz (Figure 2H). This effect was qualitatively similar to the FTS effect seen in the PD model axon (Figure 2D).

The STS effect occurred over a timescale of minutes and should therefore be related to a slow activity-dependent process of the PD axon. In many axons, slowing of conduction is thought to be because of hyperpolarization caused by the Na⁺/K⁺ pump (Bucher and Goaillard, 2011), and our model did not produce any STS effect without the pump (not shown).

The activity level of I_pump is determined by the fast sodium current (see Materials and methods). Therefore, both the increase of I_pump with time, and the steady state level it reached, depended on the mean rate of stimulation (Figure 3A). Figure 3B and C show that both D_mean and CV-D also increased depending on the stimulation rate, and with time courses very similar to I_pump. With increased stimulation, I_pump produced a hyperpolarization of the baseline membrane potential (Figure 3D). When I_pump was modeled without dynamics and instead set to different constant values during Poisson stimulation, the baseline membrane potential (trough voltage, V_T) showed a linear dependence on I_pump, while the peak voltage of spikes (V_P) was fairly unaffected (Figure 3E). Both D_mean and CV-D appeared to be determined by the level of I_pump, as both showed a positive and linear dependence (Figure 3F). We therefore concluded that the dominant cause of the STS effect is the hyperpolarization mediated by I_pump.

Figure 3

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Alternatively, slow hyperpolarization could be because of cumulative activation of slow K⁺ currents. We therefore tested if a slow conductance can produce a similar STS effect as I_pump. To address this possibility, we replaced the Na⁺/K⁺ pump with a slowly-activating potassium current (I_Ks) which was tuned to produce a slow increase of conductance on a timescale of several minutes. As before, ${\overline{g}}_h$ was set to zero.

The voltage activity of the model axon with I_Ks showed a slow decrease in V_P but little change in V_T (Figure 4A). This was in contrast to the effect of I_pump, which had a strong effect in hyperpolarizing V_T but little effect on V_P (Figure 3D and E). Although g_Ks increased with a similar time course as I_pump in the model version described in Figure 3, the behavior of I_Ks was quite different. Unlike the slow continuous increase of I_pump (Figure 3A), I_Ks increased with each action potential and was reset back to almost 0 between action potentials (Figure 4B, expanded in 4C) and only the peak value of I_Ks increased with a similar time course as g_Ks. As a result, I_Ks was only effective during each spike (reducing V_P) and had no cumulative effect in between spikes, and therefore could not change the baseline membrane potential. The lack of effect of I_Ks on V_T is because of the small driving force of this current during the inter-spike intervals. Interestingly, D_mean increased with a similar time course as seen in the decrease of V_P (Figure 4D and E), and this increase was only slightly smaller than in the model including I_pump. However, the CV-D stayed relatively constant throughout Poisson stimulation (Figure 4D and F). These results suggest that the STS effect on the variability of delay strongly depends on the baseline membrane potential, while increase in mean delay can be because of baseline hyperpolarization or decreasing peak voltage.

Figure 4

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We conclude that the STS effect on variability of delay in the case of the PD axon could not easily be captured with I_Ks, possibly because the minimal driving force between spikes did not allow for a substantial cumulative hyperpolarization.

In the PD axon, inward rectification by I_h plays an important role during repetitive firing. It is modulated by dopamine (DA), which increases g_h 2–3 fold in the range of normal resting membrane potentials (Ballo et al., 2010). DA prevents hyperpolarization and drastically reduces history-dependence of conduction delay, while pharmacological block of I_h increases both hyperpolarization and history-dependence (Ballo et al., 2012). Here, we wanted to examine whether the experimentally-observed effects of DA and CsCl on I_h levels are sufficient to explain their effects on the history-dependence of conduction delay.

We approximated the effects of DA enhancement or pharmacological block of I_h by adjusting the model parameters governing this current (see Materials and methods and Tables 1 and 2), and we will refer to these respective model parameter sets as DA or I_h block. We examined how the STS and FTS effects were influenced by DA or I_h block. Such a comparison is shown in Figure 5 for the Poisson stimulation with a mean rate of 10 Hz. Blocking I_h increased both D_mean and CV-D in the model, an effect that qualitatively matched the experimental application of CsCl (Figure 5A,B). Consistent with experimental data, DA had the opposite influence on the STS effect.

Figure 5

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To quantify the influence of I_h on the FTS effect, we focused on the data in the fifth minute of the Poisson stimulation (Figure 5C). We fit the nonlinear relationship between delay and F_inst with a quadratic function and determined κ_min, the curvature at the frequency at which delay was smallest, as a measure of nonlinearity. Blocking I_h increased the nonlinearity in the F_inst-delay relationship (Figure 5C2; control: κ_min=0.0059; ${\overline{g}}_h$ = 0: κ_min=0.0091), whereas DA decreased the nonlinearity (2 x control ${\overline{g}}_h$: κ_min=0.0032). These effects mimic the influence of CsCl and DA on the FTS effect in experiments (Figure 5C1; control: κ_min=0.0051; CsCl: κ_min=0.01; DA: κ_min=0.001).

Although the biophysical model qualitatively captured the STS and FTS effects of I_h changes seen in biological experiments, there were small quantitative differences between the model and the experiments. These were most notable for the DA effect when the simulation D_mean values were larger than the experimental values (Figure 5A1–A2), and there was a small increase in D_mean that was not observed in the biological data. Additionally, the nonlinearity of the FTS effect was reduced but not removed as in the experiments (Figure 5C1–C2).

Activity-dependent changes in axon excitability and conduction velocity are often gauged with paired-pulse stimulations (Bucher and Goaillard, 2011), particularly for diagnostic purposes in the context of peripheral neuropathies (Bostock and Rothwell, 1997; Bostock et al., 1998; Burke et al., 2001; Krishnan et al., 2009). Such stimulation protocols consist of a single conditioning pulse followed by a test pulse delivered at different intervals. The change in spike threshold, conduction velocity, or conduction delay between conditioning and test pulses can then be analyzed as a function of inter-stimulus interval (ISI). Typically, excitability changes as a function of ISI in a sequence of relative refractory, supernormal, and sometimes subnormal intervals, collectively referred to as the ‘recovery cycle’. Recovery cycle measurements gauge excitability changes occurring at a similar timescale as the FTS effect seen in Poisson stimulations. We predicted that employing paired-pulse stimulations to test conduction velocity changes is equivalent to analyzing the FTS effect on conduction delay at the beginning of the Poisson stimulation.

Instead of conditioning trains of different duration or frequency, we used simple paired-pulse stimulations and mimicked different excitability states of the axon by setting I_pump to different constant values. These values were obtained from the 10 Hz Poisson stimulation results described in Figure 3A, the relatively low mean value during minute 1 (I_pump,LO), and the relatively high mean value during minute 5 (I_pump,HI). Figure 6A shows recovery cycle measurements for both levels of I_pump. Consistent with the STS effects of I_pump, the conduction velocities of the conditioning (line) and test spikes were higher with I_pump,LO than with I_pump,HI. For both levels of I_pump, test spikes showed reduced velocity at small ISIs compared to the conditioning spike (relative refractory period), and increased velocities at larger ISIs (supernormal period), before converging back to the velocity of the conditioning pulse at even larger ISIs. However, note that with different levels of I_pump, the peak conduction velocity of the test pulse corresponded to different ISIs and that the difference between the velocity of conditioning pulse and the peak velocity of the test pulse was larger with I_pump,HI.

Figure 6

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In order to compare the paired-pulse data with the FTS effect seen in the delay vs. F_inst relationship of the Poisson stimulation (Figure 2D), we plotted the data from Figure 6A as delay vs. F_inst and fit these data with a cubic polynomial function. Figure 6B shows plots of these functions together with the data for the first and the fifth minute of the Poisson stimulation (the latter in 10 Hz bins). This comparison showed that the fits of the paired-pulse data provided very good predictions of the FTS effect seen in the Poisson stimulation (R² = 0.939 for minute 1 and I_pump,LO; R² = 0.937 for minute 5 and I_pump,HI). We can therefore conclude that at the fast timescale, delay is primarily determined by the ISI, that is, the timing of the last preceding spike. The STS effect influences the dependence of velocity or delay on ISI or F_inst, but this effect is slow enough that the FTS effect exposed by Poisson stimulation can still be accurately predicted by simple paired-pulse or train-pulse stimulation methods.

We also compared the effect of paired-pulse and Poisson stimulations in the Hodgkin-Huxley model axon. As shown in Figure 2, this model exhibits no STS effect. We therefore predicted that recovery cycle measurements should perfectly match data from Poisson stimulations, even when repetitive activity is used as a conditioning paradigm. To test this, we used a single pulse as conditioning stimulus. The recovery cycle measurement in Figure 6C shows that relative refractory and supernormal periods were restricted to relatively small ISIs. Figure 6D shows that when plotted as delay vs. F_inst, results from paired-pulse stimulations are a near perfect match to results from Poisson stimulations.

The PD neuron is a member of the pacemaker group of the pyloric network and its natural ongoing activity consists of bursts with a cycle frequency of ~1 Hz and a parabolic interval structure within each burst (Szucs et al., 2003; Ballo and Bucher, 2009; Ballo et al., 2012). During ongoing bursting activity, different spikes of each burst have different conduction delays and there is a highly nonlinear relationship between the conduction delay and the spike number in the burst (Ballo and Bucher, 2009; Ballo et al., 2012). We used the natural activity pattern in our model to address two questions. First, we examined whether the natural activity changes the STS and FTS effects or whether these effects remain the same as during Poisson stimulations. Second, we asked if the FTS effect for the natural bursting activity could be as accurately predicted from paired-pulse stimulations as for Poisson stimulations.

Figure 7A shows traces of the model axon from the start and end of a 300 s stimulation with a burst frequency of 1 Hz. Action potentials were produced by stimulating the model with brief current pulses (see Materials and methods) and it captured the dynamics of voltage trajectories described in the experimental data (Ballo and Bucher, 2009). In particular, over the course of the entire stimulation, the baseline membrane potential between bursts slowly hyperpolarized by several millivolts. Within each burst, both the peak and the trough voltages changed. Peak voltages decreased with increasing F_inst towards the middle of bursts and increased again with decreasing F_inst towards the end, because of inactivation of I_Na. Trough voltages were more depolarized in the middle of bursts, because summation resulting from relatively slow spike repolarization is largest at high F_inst.

Figure 7

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Figure 7B1 shows PD axon conduction delays over the course of the burst duration for all 300 bursts in the 5 min stimulation, measured in one experiment. As in Poisson stimulations, D_mean and CV-D increased during the entire stimulation. Note that the delay of the first spikes in each burst, the ones occurring after an inter-burst interval of 650 ms, increases disproportionately. This corresponds to the large delays at small F_inst as seen in Figure 2B for the FTS effect during Poisson stimulation. This increase in delay of the first spike in the burst was even more pronounced in the model (Figure 7C1).

In exploring spike conduction delay under natural bursting conditions, we also considered the effect of neuromodulation on history-dependence. As stated above, conduction delay in the PD axon is affected by the level of I_h, which is modulated by DA (Ballo et al., 2010, 2012). Figure 7B2 and B3 show that both the slow increase in mean delay and the variability of delay are more severe when I_h was blocked by CsCl, and less severe when I_h was increased by application of DA. In the model, these effects were mimicked when I_h was blocked (Figure 7C2) or increased (Figure 7C3). As with Poisson stimulations, the increase in the conduction delay of each subsequent burst was because of the slow increase of I_pump with stimulation (not shown), leading to a more hyperpolarized resting potential. This effect was counteracted to different degrees by different levels of I_h.

As with the FTS effect during Poisson stimulations, we examined whether the nonlinear relationships between conduction delay and time, seen in the burst stimulation of the biophysical model, can be predicted with paired-pulse stimulations at different levels of I_pump. A comparison of the paired-pulse estimates of these relationships for the first and last bursts (1 and 300) is shown in Figure 7D1–3 for the three conditions: control, I_h block and DA. The level of I_pump was set to the mean value of the dynamical I_pump at the time of burst 1 or the time of burst 300. The estimates for the paired-pulse stimulations shown in these panels were obtained from the polynomial fits (as in Figure 6B) and the ISIs of individual spikes. The nonlinear relationship between conduction delay and spike number was captured in all cases.

In addition to exploring the history-dependence of conduction delay at a phenomenological level, we also investigated which ionic current dynamics could account for this history-dependence. In the PD axon, delay is not a simple linear function of spike trajectory measures like peak voltage, trough voltage, or duration (Ballo et al., 2012). A possible reason for this is the dependence of conduction velocity on changes in total membrane conductance, which because of opposing inward and outward currents are not necessarily apparent in voltage traces (Bucher and Goaillard, 2011). A number of theoretical studies have provided equations to describe conduction velocity of a single isolated spike. The Matsumoto-Tasaki equation provides an estimate of velocity as a function of the total conductance level at the peak of the spike (see Materials and methods), and has been verified experimentally for single spikes in the squid giant axon (Matsumoto and Tasaki, 1977; Tasaki and Matsumoto, 2002; Tasaki, 2004). If the history-dependence of conduction delay is because of changes in the total conductance level, the Matsumoto and Tasaki equation should be able to predict the STS and FTS effects seen in our simulations. We therefore used this equation to estimate the delay of each spike in the 300 s, 10 Hz Poisson stimulation of our model (as shown in Figure 2C,D).

Although the Matsumoto and Tasaki equation provided a good first-order estimate of conduction delay, it did not capture the STS effect in the model axon (Figure 8A1). Additionally, for the FTS effect, this equation only showed a decrease of delay at low F_inst and did not capture the variability (Figure 8A2). In general, the overall ability of the Matsumoto and Tasaki equation to predict the conduction delay in our model was poor (Figure 8A3).

Figure 8

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An alternative method for estimating conduction velocity in the Hodgkin-Huxley model axon is provided by the Muratov equation (Muratov, 2000). This estimate uses the values of the I_Na inactivation variable h_Na and the I_Na activation rate ${\alpha }_m=m_{\infty }/{\tau }_m^{Na}$, both evaluated at rest (see Materials and methods). Figure 8B shows a comparison of the delays in the Poisson stimulation of our model and the predicted delays using the Muratov equation. The predictions of delay by the Muratov equation were qualitatively similar to the simulation results for the STS effect, in that they showed an increase of D_mean and CV-D. However, the increases in these two factors were limited and the equation failed to provide a quantitatively accurate prediction (Figure 8B1). Additionally, for the FTS effect, the increase of delay with larger values of F_inst was not captured by this equation (Figure 8B2). Therefore, the overall ability of this equation to predict the conduction delays in the model axon was also poor (Figure 8B3). In conclusion, although these equations provide good predictions for conduction delay of an isolated spike, neither could accurately predict the short- and long-term history-dependence of conduction delay of the model axon.

Because previously published equations failed to predict accurately the history-dependence of conduction delay, we used an empirical approach to explore which model parameters contributed most to this history-dependence. Our model included five distinct ionic currents, each of which depended on a maximal conductance and reversal potential and, with the exception of I_Leak, on parameters that determine the activation kinetics and, in the cases of I_Na and I_A, inactivation kinetics. As a first step in understanding the contribution of these parameters to history-dependence of conduction delay, we examined the sensitivity of STS and FTS effects to changes in these parameters (see Materials and methods). The sensitivity was measured by examining how the different attributes that describe the STS and FTS effects (Figure 9A) depend on small changes in the parameter values. For the STS effect, these attributes were D_mean and CV-D. For the FTS effect, we used quadratic fits to determine the minimum delay (D_min), the frequency (F_min) at which D_min occurred, and the curvature of the fit at this point (κ_min).

Figure 9

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Note that a small sensitivity value does not imply that the model attribute is not dependent on the parameter, but rather that small changes in the parameter do not affect that attribute strongly or monotonically. The large effect of I_pump on conduction delay and history-dependence seen in Figure 3 was because of several-fold changes in pump activation over several minutes. In contrast, the comparatively small (±5% and±10%) changes of I_pump used here had little effect on sensitivity values (not shown). For all other parameters, we performed the sensitivity analysis at two different constant values of I_pump corresponding to the mean I_pump value at minute 1 (red) or 5 (blue) of the Poisson stimulation.

The primary result of the sensitivity analysis was that the parameters which produced the largest variations in any of the attributes of the STS (Figure 9B and C) or FTS (Figure 9D–9F) effect were those directly involved in spike generation, that is, the parameters of I_Na, I_Kd and I_Leak. Although, as described above, I_h contributes greatly to the history-dependence of conduction delay, the different attributes of history-dependence were not greatly sensitive to the parameters of I_h, nor to that of the transient potassium current I_A.

The value of D_mean (Figure 9B) strongly and negatively correlated with the equilibrium potential of K⁺ (E_K) and the leak reversal potential (E_Leak). This sensitivity was most likely because of the contribution of E_K and E_Leak to the resting membrane potential. D_mean also strongly and negatively depended on the equilibrium potential of sodium (E_Na) and its maximum conductance (${\overline{g}}_{Na}$). This is consistent with the predictions of the Matsumoto-Tasaki equation, because an increase in either of these parameters results in an increase in g_Na, and therefore total conductance. Additionally, D_mean strongly and positively depended on the activation and inactivation time constants of I_Na. The sensitivity of CV-D to model parameters was not as consistent as that observed for D_mean. Several parameters produced different (E_K, E_Leak, ${\overline{g}}_{Na}$, ${\tau }_h^{Na}$) effects on CV-D, when the I_pump value was low or high (Figure 9C).

Of the FTS attributes, F_min was sensitive primarily to E_K (Figure 9D). Although it is difficult to gain a clear intuition on how different parameters affect F_min, this observation indicates that the fastest spikes can be obtained at a higher frequency if E_K is shifted to a more depolarized value. Not surprisingly, the dependence of D_min on the model parameters (Figure 9E) was similar to that of D_mean. The non-monotonic dependence of delay on the instantaneous stimulus frequency was captured primarily by the curvature of the quadratic fit. A larger curvature κ_min implies a larger nonlinearity and therefore a larger difference between conduction delays at different F_inst values. κ_min is negatively dependent on E_Leak, ${\overline{g}}_{Leak}$ and E_Na, but positively on ${\overline{g}}_{Kd}$, ${\overline{g}}_{Na}$ E_K and ${\tau }_h^{Na}$ (the time constant of I_Na inactivation, Figure 9F). These dependencies are consistent, but the reasons underlying their effect on κ_min are not obvious.

The sensitivity analysis included two types of parameters: those with constant values (maximal conductances and reversal potentials), and those whose values depend on the membrane potential and can change with time (ionic current time constants). The latter parameters can change in value for each spike and therefore are the only ones that can potentially contribute to the history-dependence of conduction delay. Within this category of parameters, the FTS and STS effects showed the highest sensitivity to the time constants of I_Na.

The dependence of STS and FTS effects on the gating time constants of I_Na activation and inactivation, ${\tau }_m^{Na}\left(V\right)$ and ${\tau }_h^{Na}\left(V\right)$, led us to explore whether the factors that determine these time constants were responsible for the history-dependence of conduction delay.

${\tau }_m^{Na}\left(V\right)$ and ${\tau }_h^{Na}\left(V\right)$are determined by four rates (Figure 10 inset): the opening and closing rates of activation, α_m(V) and β_m(V), and the opening and closing rate of inactivation, α_h(V) and β_h(V). We therefore examined how these rates changed for each spike during the Poisson stimulation. Because the opening rate is relevant at the onset of the spike and the closing rate during the spike itself, we evaluated α_m and α_h at the spike trough voltage (V_T, the voltage minimum right before the spike), and β_m and β_h at the spike peak voltage (V_P). In order to compare the changes in these parameters directly with those of conduction delay, we plotted the reciprocal values of the parameters, which have units of time.

Figure 10

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All four parameters changed substantially during the Poisson stimulation and showed both STS and FTS effects (Figure 10A and B), but no single parameter followed exactly the STS and FTS effects on conduction delay. However, both 1/α_m(V_T) and 1/α_h(V_T), the (reciprocals of the) rates of activation and inactivation gates opening at the spike onset, showed an STS effect that was qualitatively similar to that of conduction delay (Figure 10A1 and A2); yet the STS effect for these parameters was decreasing rather than increasing, and neither parameter captured the non-monotonic FTS effect (Figure 10B1 and B2). In contrast, neither 1/β_m(V_P) nor 1/β_h(V_P), the (reciprocal) rates of activation and inactivation gates closing at the spike peak, qualitatively captured the increase in variance over time of the STS effect very well (Figure 10A3 and A4), and both these parameters showed a monotonic FTS effect (Figure 10B3 and B4). In summary, no single parameter could predict the overall time course of changes in conduction delay.

We next examined whether any combination of two parameters would better capture the STS and FTS effects. There are six possible combinations of two parameters out of these four. We fit the model STS and FTS effects with all six combinations. That is, for any combination of two parameters x and y out of the four, we fit the Poisson stimulation data of Figure 2C–D with the equation

The coefficients c_i (i = 1,2,3) were determined with a Powell’s optimization method and the estimated delay was compared with the simulation results. We found that five of the combinations produced a fit to the model delay values that had a coefficient of determination R² of >0.9, but only two of these qualitatively captured both the STS and FTS effects seen in Figure 2C–F (data not shown). These combinations were 1/α_h(V_T), 1/β_h(V_P), and 1/α_m(V_T), 1/β_h(V_P). Of these, the better fit was obtained from the combination that included both the activation and inactivation gating variables (R² = 0.99 compared to 0.97). We therefore used the following empirical equation as the best estimate for the model conduction delay:

Figure 10C1 and C2 show that this equation captures both STS and FTS effects very well. Figure 10C3 shows the high accuracy with which this equation predicts the conduction delays in the model (R² = 0.99). Although, for brevity, we do not show the dependence of conduction delay on α_m(V) and β_h(V), scaling these functions had a direct effect on the conduction delay. We therefore conclude that the opening rate of the activation gate of the sodium channel right before the spike, and the closing rate of the inactivation gate of the sodium channel at the peak of the spike are good proxies for, and likely important determinants of, the spike conduction velocity in axons.

The equation to predict conduction delay from activation and inactivation variables served to gain insight into underlying mechanisms. However, it is of limited value for experimental work, because activation and inactivation variables cannot be measured during ongoing spiking. We therefore asked if there are good enough correlates of activation and inactivation variables to be found in the voltage trajectory of spikes that would allow the prediction of delay. Single spike measures like peak or trough voltages are poor predictors of delay in the PD axon (Ballo et al., 2012), but the dependence of delay on channel gating at both trough and peak shown here suggests that more than one measure has to be considered. We noticed that although α_m and β_h and are nonlinear functions, in the range of membrane potentials restricted to the trough or peak of the spike (V_T and V_P), both α_m(V_T) and β_h(V_P) were almost linear functions of their respective variables. We therefore treated V_T and V_P as linear approximations of 1/α_m(V_T) and 1/β_h(V_P), respectively, and used them instead in our equation:

With the same optimization process as before, the coefficients were determined for the Poisson stimulation data of the model axon. The resulting prediction of conduction delay captured the history-dependence of delays with 99% accuracy (Figure 11A1–A3). Furthermore, the equation with the same coefficient values predicted the delays in a novel set of Poisson stimulation data with the same accuracy (not shown).

Figure 11

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We also determined the contribution of each variable to the fit. To this end, we examined how well the simulated delays can be fit using only 1/V_T or only 1/V_P as the variable. In these fits, the other variable (V_P or V_T) was set to its mean value measured during the 300 s Poisson stimulation. The results of these fits are shown in Figure 11B and C. Neither variable perfectly captured the STS effect although this effect was better captured with fits using only 1/V_T (Figure 11B1). This result is consistent with the sensitivity of D_mean to ${\tau }_m^{Na}$ (see Figure 9B), because a larger ${\tau }_m^{Na}$ results in a slower opening rate of I_Na activation variable at V_T. Similarly, for the FTS effect, neither variable alone captured the non-monotonic relationship of delay with F_inst; however, the decrease of delay with F_inst was best approximated by 1/V_T (Figure 11B2) whereas its increase was best approximated by 1/V_P (Figure 11C2). Therefore, although neither 1/V_T nor 1/V_P can accurately predict conduction delay on their own, the multivariate linear regression using both1/V_T and 1/V_P provides an accurate prediction of history-dependence of conduction delay in the biophysical model.

Our description of delay changes suggests that they are dominated by the I_Na activation state before the spike and the inactivation state at the peak, and well reflected in the voltage values at these time points (Figure 10). We therefore predicted that, while other conductances can influence the gating state of I_Na through their effect on membrane potential, the recovery cycle should be accurately predicted by our equation describing the dependence of delay on trough and peak voltages. To test this prediction, we used the estimated delays from our equation to predict the results of the paired-pulse stimulations shown in Figure 6A (with I_pump,HI). As described above, the coefficients c_i (i = 1,2,3) were independent of the stimulation pattern once the model axon was fixed. We therefore used coefficients obtained by a 1 min Poisson stimulation protocol to predict the conduction velocities of the spikes in the paired-pulse stimulation. With these coefficients and the V_T and V_P values, the conduction velocities were accurately predicted for both levels of I_pump (Figure 11A4).

The extent to which either V_T or V_P can separately predict delays during Poisson stimulation was discussed above (for Figure 11B and C). We also tested the extent to which V_T and V_P contributed to the different periods of the recovery cycle. We restricted this analysis to the paired-pulse simulation results for high levels of I_pump (Figure 11B4 and C4). In order to understand the contribution of V_T to changed excitability at different intervals, we set V_P to a constant value equal to the trough voltage of the conditioning spike in the paired-pulse stimulation. Using the coefficients from the Poisson stimulation and V_T from the paired-pulse stimulation, the predicted conduction velocity decreased with increasing ISIs in the range of relative refractory and supernormal periods (Figure 11B4). Therefore, changes in V_T captured the supernormal period. The decrease of velocity was consistent with the results for Poisson stimulations shown in Figure 11B2: predicted conduction delay decreased with F_inst if only V_T was used.

To capture the contribution of V_P, we fixed the value of V_T to a constant equal to the peak voltage of the conditioning spike in the paired-pulse stimulation. Again, using the same coefficients and the V_P values from the paired-pulse stimulation, we found that the predicted conduction velocity monotonically increased with ISI (Figure 11C4). Thus the changes in V_P captured the relative refractory period, which was consistent with the results shown in Figure 11C2: predicted conduction delay increased with F_inst if only V_p was used.

Because both V_T and V_P are readily measured from experimental intracellular recordings (Ballo et al., 2012), we asked whether our equation can be used to predict the history-dependence of conduction delays in the PD axon. We used conduction delays measured with a Poisson stimulation at 10 Hz and fit the delay values with our equation. We then used this equation to predict the delay values of a novel dataset produced with the same stimulation rate. The results showed that we could predict both the STS and the FTS effects of the experimental data relatively accurately (Figure 12A–B). The fit produced a coefficient of determination of 0.87, indicating that 87% of the variance of the experimental delays can be predicted by this equation (Figure 12C). The history-dependence of conduction delay in experiments can therefore be predicted from the spike voltage trajectories without any need for computational modeling.

Figure 12

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We describe the history-dependence of conduction delay in a biophysically realistic model axon designed to match properties of the PD motor axon of the crustacean stomatogastric nervous system. The model captures well how delay changes as a function of prior spiking in this axon, at two different timescales (Ballo et al., 2012)(Figure 2).

At the slow timescale (STS effect), repetitive activity leads to cumulative hyperpolarization and conduction slowing. Over the course of minutes, mean delay and variability of delay increases. We show that the STS effect is due to the Na⁺/K⁺ pump (Figure 3). It has been proposed for many axons that slow hyperpolarization and slowing of conduction is caused by this pump, which is activated by the massive Na⁺ influx during repetitive spiking and causes a net deficit of positive charge (Van Essen, 1973; Gordon et al., 1990; Bostock and Bergmans, 1994; Robert and Jirounek, 1994; Vagg et al., 1998; Baker, 2000; Kiernan et al., 2004; Moldovan and Krarup, 2006; Scuri et al., 2007). Direct experimental evidence of the involvement of the pump in activity-dependent dynamics for the PD axon is missing because of the difficulty that pharmacological block of the pump also interferes with its overall role in maintaining a functional membrane potential (Ballo and Bucher, 2009). However, other slow outward currents have not been identified in the PD axon (Bucher, unpublished results).

We nevertheless considered a slowly activating outward current as an alternative mechanism because many axons express slow K⁺ channels, e.g. K_v7 channels that produce M-type currents (Dubois, 1981; Baker et al., 1987; Eng et al., 1988; Devaux et al., 2004; Schwarz et al., 2006; Vervaeke et al., 2006; Buniel et al., 2008). Such currents are thought to contribute to hyperpolarization and reduced excitability and may have cumulative effects during repetitive activity (Baker et al., 1987; Taylor et al., 1992; Stys and Waxman, 1994; Miller et al., 1995; Lin et al., 2000; Burke et al., 2001; Schwarz et al., 2006). An interesting difference between the pump current and currents due to ion channel conductances is that the pump current does not depend on a driving force. Equivalently, an ionic current would change the input conductance of the neuron, whereas the pump current does not. We found that even with a low activation threshold and slow kinetics, the slow voltage-gated K⁺ current cannot cause substantial activity-dependent hyperpolarization because an E_K relatively close to the resting membrane potential rendered the driving force in between spikes too small for substantial summation of current (Figure 4). The same would be true for Cl^- conductances and E_Cl. Yet, this slow K⁺ current reduced the peak voltage of action potentials and reproduced the effect of the pump current in increasing the mean delay D_mean, but not the variability CV-D.

It should be noted, however, that we used an E_K value (−70 mV) in our axon model that is about 10 mV less negative than usually assumed for soma recordings of STG neurons (Golowasch and Marder, 1992). A sufficiently slow voltage-gated K⁺ with E_K substantially lower than the resting potential could possibly reproduce the effect of the pump current in producing the STS effect. We chose the E_K value near the resting potential because spikes in PD axon recordings show an absence of fast after-hyperpolarization (undershoot), and we could only replicate this in the model with a less negative E_K. An alternative approach would have been to model E_K as a dynamic variable and allow it to transiently change to more depolarized values over the course of a single spike. This type of dynamic E_K was used in a model of the squid giant axon, which assumed a significant temporary extracellular K⁺ accumulation, and was successful in rendering a more realistic voltage trajectory of repolarization and undershoot (Clay, 1998). As the PD axon does not actually express slow K⁺ currents, we did not pursue this further. However, our results demonstrate that the increase of variability of delay was dependent on hyperpolarization, as a mere increase in conductance only increased the mean delay.

The dependence of the STS effect on hyperpolarization is further highlighted by the role of I_h (Figure 5), which is thought to play an important role in preventing spike failures in many axons because its inwardly rectifying properties can balance activity-dependent hyperpolarization (Baker et al., 1987; Grafe et al., 1997; Soleng et al., 2003; Kiernan et al., 2004; Baginskas et al., 2009; Tomlinson et al., 2010). In the PD axon, I_h is increased several-fold by dopamine (Ballo et al., 2010), which substantially reduces hyperpolarization and history-dependence of conduction delay (Ballo et al., 2012). Our model provides a mechanistic explanation of our previous experimental observation that the balance of I_h and pump current plays a dynamic role in setting how sensitive conduction is to the history of activity. This represents a level of importance in controlling temporal fidelity that goes beyond just preventing spike failures.

Previous studies have also used a modeling approach to explore the role of slow accumulation of ionic currents and concentrations in axons. The topics explored included explorations of Na⁺ accumulation and the effects of the Na⁺/K⁺ pump in the coupled left-shift of the voltage-dependence of Na⁺ channels due to axonal injury (Boucher et al., 2012; Yu et al., 2012; Lachance et al., 2014), the influence of Na⁺ channel clustering on spike conduction (Freeman et al., 2016) and activity-dependent slowing of spike conduction (Tigerholm et al., 2014).

The model also captures well the PD axon’s nonlinear and non-monotonic dependence of delay on instantaneous frequency (FTS effect), the manifestation of which is influenced by the STS effect. It has long been known that, with increasing spike intervals even well beyond the refractory period, excitability and conduction velocity go through an oscillation of decreases and increases, (Bullock, 1951; Raymond, 1979), collectively termed the recovery cycle. Such activity-dependent changes are often gauged with paired-pulse stimulations (Bucher and Goaillard, 2011), particularly for diagnostic purposes in the context of peripheral neuropathies (Bostock and Rothwell, 1997; Bostock et al., 1998; Burke et al., 2001; Krishnan et al., 2009). The dependence on spike interval changes dramatically when axons are stimulated with trains of conditioning pulses instead of simple paired pulses. In our model, prior activity can be mimicked by setting the pump current to different values (Figure 6). This suggests that FTS and STS effects are completely separated in their time dependence, meaning that delay depends on the last interval and the mean rate of prior activity, but not on the exact temporal pattern of the last few preceding spikes. This independence of exact prior pattern is further demonstrated by our finding that paired-pulse stimulation results predict delays from both Poisson (Figure 6) and burst (Figure 7) stimulations. A previous modeling study on primary afferent C-fibers has also demonstrated an activity-dependent alteration of recovery cycles (Tigerholm et al., 2014).

Our results show that the dynamics of conduction delay can be accurately predicted from two parameters describing the state of the Na⁺ conductance at two time points, the opening rate of the activation gate at the onset of the spike, and the closing rate of the inactivation gate at the peak (Figure 10). In comparison, two equations that accurately describe conduction delay of single spikes fail to capture the dynamics. The Matsumoto-Tasaki equation predicts conduction delay dependent on the change in total membrane conductance elicited by a spike (Matsumoto and Tasaki, 1977; Tasaki and Matsumoto, 2002; Tasaki, 2004). The Muratov equation includes Na⁺ channel gating variables, but only at the rest state (Muratov, 2000). Both equations aim to predict absolute values of delay and include constant biophysical parameters such as diameter and membrane capacitance. In contrast, our empirical equation describes only how conduction delay changes as a function of activity. The magnitude of the delay changes and the absolute range of values are found by scaling the contribution of the two Na⁺ gating rates with two constants and adding a third. These three constants have to be found empirically and may represent stand-ins for actual biophysical parameters.

Interestingly, the gating parameters at the trough potential before a spike and the peak voltage of the spike correlate linearly with these voltages, so that the voltages can be treated as stand-ins for the gating parameters in the empirical fit, which allows us to predict delay solely from voltage trajectories (Figure 11). This illustrates the important point that, while both our sensitivity analysis (Figure 9) and the empirical prediction of delays seem to suggest that Na⁺ channel gating is the key factor, the rates used to predict delay are voltage-dependent and are therefore sensitive to the effect other ionic mechanisms have on the membrane potential trajectory.

It should be noted that the small level of sensitivity to the various currents is because sensitivity analysis only examines small changes in the parameters of these currents which would not greatly alter the membrane potential. However, some of these currents, as shown for the pump current and I_h, undergo large changes that have substantial effects on delay. Again, these mechanisms act at two different timescales. At the slow timescale, the balance of pump current and I_h determines the overall level of hyperpolarization and thus affects Na⁺ channel gating through the steady-state voltage-dependencies of the opening and closing rates. It should be noted that changes in baseline membrane potential overall can have ambiguous effects on conduction velocity. For example, slow hyperpolarization is often associated with slowing of conduction but can also increase removal of Na⁺ channel inactivation and therefore increase velocity (Vervaeke et al., 2006; De Col et al., 2008).

At the fast timescale, K⁺ currents play an important role in shaping the repolarization speed, after-potentials, and re-excitability, the extent of which critically depends on the gating time constants and temporal overlap with the Na⁺ current (Erisir et al., 1999; Baranauskas, 2007; Sengupta et al., 2010). K⁺ currents with different gating time constants may differentially affect both Na⁺ channel activation and its inactivation. Of note is the common presence of inactivating and low-threshold K⁺ currents in axons (Krishnan et al., 2009; Bucher and Goaillard, 2011; Debanne et al., 2011; Bucher, 2015), because their own activation and inactivation states will be critically dependent on slow timescale changes in baseline membrane potential and therefore change their effect on voltage trajectories and Na⁺ channel gating. In the PD axon, relatively slow spike repolarization leads to summation even at moderate frequencies, and spike duration depends on I_A activation and inactivation, which vary substantially with the baseline membrane potential (Ballo and Bucher, 2009).

The model presented here aimed to mimic, as faithfully as possible, the behavior of a well described unmyelinated invertebrate axon. However, it should be noted that the ionic mechanisms considered, like pump-mediated hyperpolarization, inward rectification through I_h, and the presence of both transient and sustained K⁺ currents, are found in many axons across different phyla (Bucher and Goaillard, 2011). In contrast, diversity of axon properties within a single species or even a single part of the nervous system can be very large, which makes the notion of a ‘typical’ axon appear not useful (Bucher, 2015). Nevertheless, spikes in the overwhelming majority of axons are driven by fast transient Na⁺ currents responsible for their regenerative nature. It is plausible as a general mechanism that the gating state of these channels at the onset and peak of the spike acts as a critical determinant of conduction velocity. Different complements of additional voltage-gated or electrogenic transporter currents across axon types may lead to very different dependence of conduction velocity on preceding activity. Yet, we expect that their influence on the Na⁺ channel gating variables, or trough and peak voltages as their proxies, would still serve to determine conduction velocity in each case. Consequently, once the peak and trough voltages of the action potential have been measured, the empirical equation introduced in this study would serve to accurately predict the conduction velocity, independent of the combinations of ionic mechanisms involved.

We constructed a conductance-based biophysical axon model to examine the role of different ionic currents in shaping the history-dependence of conduction delay. All simulations were done in NEURON (Carnevale and Hines, 2006). The model was based on standard cable equations for an unmyelinated axon, standard Hodgkin-Huxley type ionic conductances with different voltage- and time-dependencies, and a pump current representing the Na⁺/K⁺-ATPase:

The parameter a is the radius (=5 μm) of the model axon, R_i (=80 Ωcm) is the specific intracellular resistivity, V is voltage, x is the position along the axon, and C_m (=1 μF/cm²) is the membrane capacitance per unit area. The ionic currents (ΣI_ion), the Na⁺/K⁺ pump current (I_pump), and the stimulus current ($I_{app}^0\left(t\right)$) are described below. A specific membrane resistivity R_m value of 8 KΩcm² was used to calculate the leak conductance, resulting in a passive length constant λ = 1581 μm. Both a and λ were similar to the values found in the PD axon (Ballo and Bucher, 2009). The length of the model axon was set to 1 cm, and divided into 101 identical compartments for simulation. In order to apply the finite difference method to solve the model equations numerically, all compartments were assumed to be isopotential during the simulation process. All axon parameters are listed in Table 1.

Ionic currents used in the model are shown in Figure 1A. Different versions of the model contained different combinations of ionic currents (currents not included in all versions are shown in brackets):

In the simplest version, it included only the standard Hodgkin-Huxley-like leak (I_Leak), fast sodium (I_Na), and delayed-rectifier potassium (I_Kd) currents.

The PD axon expresses two additional voltage-gated ionic currents, a transient potassium current I_A, and a hyperpolarization-activated inward current I_h (Ballo and Bucher, 2009; Ballo et al., 2010). I_h is increased by dopamine (DA, Figure 1A) through a D₁-type receptor mechanism that increases cAMP levels (Ballo et al., 2010). We mimicked DA effects on I_h by using different values of voltage half-activation and slope (Table 2) that were described experimentally (Ballo et al., 2010). However, these changes alone were not sufficient to produce the experimentally-observed effect of I_h increase by DA on axonal delay, presumably because of differences in resting membrane potential between our model and the biological neuron. To match the biological results, we also assumed that DA increased ${\overline{g}}_h$ (Table 1). To mimic block of I_h, ${\overline{g}}_h$was set to 0.

We also tested the effect of a slow potassium current (I_Ks). To this end, we introduced a conductance with the same voltage dependence as I_Kd, but a 5000 times slower time constant of activation.

All ionic currents were Hodgkin-Huxley type (Hodgkin and Huxley, 1952b) and were represented by equations of the following form:

where $\overline{g}$ is the maximum conductance, E_rev is the reversal potential, m and n are the activation and inactivation variables, respectively, and p and q are non-negative integers. As stated above, gating parameters and E_rev for I_h were based on actual measurements in the PD axon (Ballo et al., 2010). I_Na and potassium current parameters were based on common values and tuned to produce spike shapes that matched those in intracellular PD axon recordings (Ballo and Bucher, 2009). Of note in these recordings is the absence of an undershoot following a spike. We only managed to replicate this by setting the K⁺ equilibrium potential E_K to a fairly depolarized value of −70 mV (see Discussion). All gating parameters are listed in Table 2.

The Na⁺/K⁺ pump current was modeled in a simplified form as described previously (Angstadt and Friesen, 1991), and was governed by the following equation:

where I_max,pump is the maximum current, [Na⁺]_in the intracellular sodium concentration, [Na⁺]_1/2 the concentration at which the pump is half active, and 1/[Na⁺]_S describes the sensitivity of the pump to alterations of intracellular sodium concentrations. The rate of change for [Na⁺]_in was governed by:

where I_Na is the level of sodium current, F is Faraday’s constant, Vol is the volume of one compartment of the model axon and α is a scale factor. According to this equation, at steady state, I_pump is equal to one third of I_Na. Because of their comparatively small magnitude, the contributions of I_Leak and I_h to [Na⁺]_in were ignored and changes in [Na⁺]_in were modeled to depend solely on I_Na. The equilibrium potential of Na⁺ in the model axon was calculated according to the change in intracellular sodium concentration from the Nernst equation:

In our model, we made the simplifying assumption that the values of [Na⁺]_out and the K⁺ concentrations inside and outside the axon were not changed by activity.

We also tried other, more descriptive, models of the Na⁺/K⁺ pump (Cressman et al., 2009; Yao et al., 2011) in our simulations. These provided somewhat different dynamics but produced very similar qualitative and quantitative results, which for brevity are not shown.

Spikes were generated in the axon by applying a stimulus current in the first compartment:

where t_i and dt (1 ms) are the stimulus time and duration, respectively. Spikes propagated along the length of the 1 cm axon and were recorded at two sites (0.3 and 0.7 cm; Figure 1B). The conduction delay was measured in NEURON using the time that the spike crossed the voltage threshold of −40 mV at the two recording sites. (Delays measured using action potential peak times produce practically identical results.) Delays produced along this 0.4 cm of axon were multiplied by a factor of 9.5 to mimic the conduction delays and velocities of the 4–5 cm biological axon shown in our results for comparison. Different conduction velocities of consecutive spikes lead to gradually changing intervals over distance. Because axonal excitability changes between spikes depend on interval, conduction delays during repetitive spiking do not scale linearly with distance (Kocsis et al., 1979; Bucher and Goaillard, 2011; Bucher, 2015). However, at initial spike frequencies smaller than about 100 Hz, these nonlinearities are relatively subtle and only differentially affect delays over distances of tens of centimeters (Moradmand and Goldfinger, 1995; Bucher and Goaillard, 2011). We confirmed that the conduction delay for a longer model axon, measured at the same relative points, scaled reasonably linearly with the length of the axon.

We characterized the history-dependence of conduction delay with three different stimulus regimes that have been used before in the PD axon (Ballo et al., 2012). First, to describe delay as a function of time and stimulus frequency, we used 300 s Poisson stimulation protocols with different mean rates: 5, 10 and 19 Hz. The last value was chosen to match mean spike frequency in burst stimulation protocols (see below). It should be noted that the Poisson stimulation method is useful in producing a variety of inter-stimulus intervals, but this can be also achieved with other stimulation paradigms that incorporate such a variety, for example through rate adaptation.

Second, we used a paired-pulse protocol to obtain ‘recovery cycle’ measurements (Krishnan et al., 2009; Bucher and Goaillard, 2011). After a single conditioning stimulus, test stimuli were delivered at different intervals (between 10 ms and 10 s). The change of delay or conduction velocity was then analyzed as a function of stimulus interval.

Third, we used a burst stimulation protocol that mimicked normal ongoing rhythmic firing in the PD neuron (Ballo and Bucher, 2009; Ballo et al., 2012). It consisted of 300 bursts at 1 Hz burst frequency, and each burst consisted of 19 spikes (360 ms burst duration) in a parabolic instantaneous frequency structure, increasing from 32 Hz to 63 Hz and decreasing back to 32 Hz.

All simulations were run beginning with a 100 s interval of no stimulation to remove any transient and initial condition effects.

Two equations have been shown to provide good approximations for conduction velocity of a single spike in the squid giant axon and Hodgkin-Huxley model axon. We tested if these equations yielded good estimates of conduction velocities during repetitive spiking. The first is derived based on boundary matching principles which we will refer to as the Matsumoto-Tasaki equation (Matsumoto and Tasaki, 1977; Tasaki and Matsumoto, 2002; Tasaki, 2004):

In this equation, v is the conduction velocity, d is the diameter of the axon, R_total is the total resistance of the membrane of unit area in the excited state, R_i is the axial resistivity of the axon interior, C_m is the membrane capacitance per unit area and κ is defined as the ratio of R_total to R_rest, which is the resistance of the membrane of unit area at rest. In our evaluations, as in those done by Matsumoto and Tasaki, the value of κ was consistently close to 0 and did not make any demonstrable difference in the estimations. We therefore followed Matsumoto and Tasaki’s approach and used the simplified equation

The second equation is an explicit analytical expression developed for calculating conduction velocity in the Hodgkin-Huxley model (Muratov, 2000), which we will refer to here as the Muratov equation:

where R_i and C_m are as above, r is the radius of the axon, ${\overline{g}}_{Na}$ is the maximum fast sodium conductance, h₀ is the value of the inactivation variable h of g_Na at the rest state and

where E_Na is the sodium equilibrium potential and V_rest is the resting membrane potential. To use this equation for repetitive spiking, we used the trough voltage before each spike (the local minimum voltage immediately before the spike) as a proxy for the resting membrane potential.

The sensitivity of conduction delay in the model to parameters was measured by examining how the different attributes that describe the slow- (STS) and fast-timescale (FTS) effects depend on small changes in the parameter values. For the STS effect, these attributes were the mean conduction delay and its coefficient of variation, measured for the final 20 s interval of simulation. For the FTS effect, we used quadratic fits to determine the minimum delay and the frequency at which it occurred, and the curvature of the quadratic fit at this point. To measure sensitivity, each parameter in the (reference) model was decreased or increased by 5% and 10% of its original value while other parameters were kept unchanged. It should be noted that E_Na is a dynamic parameter, which is determined by the intracellular and extracellular concentration of Na⁺. Therefore, the scaling factors were applied to the Nernst equation for Na⁺ in order to rescale E_Na.

The 5% and 10% changes in each direction produced four new models for each parameter. Each new model was subjected to the same Poisson stimulation as the original model and the attributes of the fast and slow timescale effects were measured. We then normalized these measures by their values in the reference model. The resulting four data points, together with the reference value (at the origin) were fit with a line. The sensitivity of an attribute to the parameter was defined as the slope of the linear fit. For example, a sensitivity value of 1 meant that a ± 5% and±10% change in the parameter resulted in a ± 5% or±10% change in the attribute.

For comparison of model results with experimental results, as well as for testing the empirical method of predicting conduction delay, we used experimental delay data from the PD axon obtained in our previous study (Ballo et al., 2012).

All optimization was performed using Powell’s conjugate direction method (Powell's method; Powell, 1994), implemented in Matlab.

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Supported by NIH grants NS083319, MH60605 and NS058825.

1. Received: January 23, 2017
2. Accepted: July 6, 2017
3. Accepted Manuscript published: July 10, 2017 (version 1)
4. Version of Record published: July 20, 2017 (version 2)

© 2017, Zhang et al.

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