Simulation and dynamical explanation of change in spiking pattern.

<p>(<b>A</b>) Spiking pattern during sustained depolarization was converted from onset-only (normal, β_w = −21 mV) to repetitive (neuropathic, β_w = −13 mV) by varying a single parameter. Onset-only spiking was observed in the neuropathic model but for only a narrow stimulus range. (<b>B</b>) According to bifurcation analysis in which stimulation (<i>I</i>_stim) was systematically varied, repetitive spiking was produced by the neuropathic model when <i>I</i>_stim exceeded a critical value required for a subcritical Hopf bifurcation. In contrast, the normal model did not undergo a bifurcation, which means spiking was limited to single spikes generated through a QS-crossing (see below). Generation of a single spike does not constitute a change in steady-state behavior, consistent with the absence of a bifurcation. (<b>C</b>) Phase planes show the fast activation variable <i>V</i> plotted against the slower recovery variable <i>w</i>. Nullclines (color) indicate where <i>V</i> or <i>w</i> do not change. Excitatory stimulation shifts the <i>V</i>-nullcline upward without affecting the <i>w</i>-nullcline. In the neuropathic model, <i>V</i>- and <i>w</i>-nullclines intersect at a stable (<i>s</i>) fixed point prior to stimulation, but that point becomes unstable (<i>u</i>) during stimulation – this corresponds to a Hopf bifurcation and is responsible for repetitive spiking. In the normal model, the fixed point remains stable during stimulation despite the <i>V</i>-nullcline shifting upward, but a single spike can nonetheless be generated depending on how the system moves to the newly positioned fixed point. The trajectory can be predicted by reference to a quasi-separatrix (QS), which corresponds to a manifold in phase space from which trajectories diverge. Quasi-separatrices were plotted here by integrating with a negative time step with initial values indicated by * on the phase planes (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002524#s4" target="_blank">Methods</a>). Like the <i>V</i>-nullcline, the QS shifts instantaneously with stimulation. If, as shown, the original fixed point ends up below the shifted QS, the trajectory to the newly positioned fixed point must follow an indirect route around the end of the QS (*), thus producing a spike; a more direct, subthreshold route would require the trajectory to cross back over the QS, which is not possible. If the original fixed point remained above the shifted QS, the trajectory would follow a direct route and no spike would be produced (not illustrated).</p

Simulation and dynamical explanation of bursting.

<p>(<b>A</b>) Sample responses at different average membrane potentials in the neuropathic model (β_w = −13 mV) with slow adaptation mediated by <i>I</i>_AHP. Noise was included in all simulations and makes the bursting irregular (and thus more realistic) but noise is not necessary for bursting. Duration and frequency of bursts increased with average depolarization. (<b>B</b>) Bursting depends on hysteresis caused by bistability associated with the subcritical Hopf bifurcation. Inset shows <i>V</i> and <i>z</i> during sample burst, where <i>z</i> controls activation of <i>I</i>_AHP. The same response, with its differently colored burst and interburst phases, was projected onto the bifurcation diagram created by treating <i>z</i> as a bifurcation parameter. The model tracks the stable limit cycle branch, spiking repetitively as <i>z</i> increases until the end of the branch is reached, at which point the burst stops. The model then tracks the stable fixed point as <i>z</i> decreases (during which noise-dependent MPOs wax and wane) until the fixed point becomes unstable, at which point another burst starts. Hysteresis is evident from the bursts starting and stopping at different values of <i>z</i>. This bifurcation diagram is flipped horizontally relative to those shown in other figures because the bifurcation parameter here controls <i>I</i>_AHP, which is an inhibitory current, whereas <i>I</i>_stim (the bifurcation parameter used elsewhere) is excitatory.</p

Simulating the continuum of pathological change.

<p>(<b>A</b>) Summary of <i>I</i>_stim thresholds to elicit onset-only or repetitive spiking for different values of β_w in our standard 2-D model. Reduction in threshold equates with an increase in excitability. Summary of peak MPO power (<b>B</b>) and peak frequency (<b>C</b>) across a range of β_w values. All simulations included noise. For each β_w value, <i>I</i>_stim was chosen relative to the threshold for repetitive spiking: <i>high</i> and <i>low I</i>_stim were 1.3 and 4 µA/cm² below threshold, respectively. Those values were chosen in order to include or exclude, respectively, noise-independent MPOs when a supercritical Hopf bifurcation occurs. Peak MPO amplitude and frequency decreased as β_w was increased. That trend is not attributable to noise-independent MPOs occurring at certain β_w values since noise-independent MPOs were excluded when testing with <i>low I</i>_stim (see above). Moreover, re-setting γ_m from 18 mV to 15 mV prevented the supercritical Hopf bifurcation from occurring at any β_w, but the same trend in MPO power and frequency was observed (data not shown). Dotted curve in C shows minimum sustainable firing rate. * indicates data points that include a noise-independent MPO component.</p

Relating parameter changes in the 2-D model with more biologically meaningful changes in a 3-D model.

<p>(<b>A</b>) Changing β_w from −21 mv to −13 mV shifts the <i>I</i>_slow-<i>V</i> curve to the right, which corresponds to a rightward shift in the <i>w</i>-nullcline on the <i>V-w</i> phase plane (inset) and switches the spike initiation mechanism (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002524#pcbi-1002524-g002" target="_blank"><b>Fig. 2</b></a>). In the 2-D model, β_w represents the voltage-dependency of <i>I</i>_slow which, in reality, comprises multiple currents with slow kinetics. The biological realism of the model can be increased by “ungrouping” <i>I</i>_slow into two (or more) components, one representing the delayed-rectifier potassium current <i>I</i>_K,dr and one representing a subthreshold current <i>I</i>_sub that can be inward or outward depending on the reversal potential. (<b>B</b>) Voltage-dependent activation curve for <i>I</i>_sub. Parameter values (indicated on the figure) were determined as explained below. In the 3-D model, the (<i>I</i>_K,dr+<i>I</i>_sub)−<i>V</i> curve was shifted the same as the <i>I</i>_sub-<i>V</i> curve in A by increasing inward <i>I</i>_sub (<b>C</b>) or by decreasing outward <i>I</i>_sub (<b>D</b>) on the basis of varying <i>g</i>_sub. Bifurcation diagrams demonstrate the change in spike initiation mechanism. With <i>I</i>_K,dr properties fixed, maximal conductance and voltage-sensitivity of <i>I</i>_sub were adjusted to recreate the shift shown in A; derived parameters illustrate the importance of the modulated conductance activating at subthreshold potentials. Adding or removing the same subthreshold currents to a Hodgkin-Huxley model (rather than to our starting 2-D Morris-Lecar model) produces equivalent changes in excitability (data not shown). By comparison, modulating current that activates only at suprathreshold potentials (β_y = 0 mV) had no effect on the (<i>I</i>_K,dr+<i>I</i>_supra)−<i>V</i> curve in the perithreshold voltage range, and thus the spike initiation dynamics were unchanged (see <b><a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002524#pcbi.1002524.s004" target="_blank">Fig. S4</a></b>).</p

Neuropathic changes in primary afferent excitability.

<p>(<b>A</b>) Sample responses from large diameter acutely isolated dorsal root ganglion (DRG) neurons under control conditions (<i>normal</i>) and two days after L5 spinal nerve transection (<i>neuropathic</i>). Spiking pattern switches from onset-only to repetitive. (<b>B</b>) Sample responses, with the same average membrane potential of −36 mV, showing development of membrane potential oscillations (MPOs) after nerve injury. (<b>C</b>) Sample response showing bursting after nerve injury. (<b>D</b>) Venn diagram distinguishing classes of molecular changes and their relationship to primary afferent hyperexcitability and neuropathic pain. In this study, we sought to define the red circle. Parts A–C were modified from reference 4.</p

Summary of relationships between molecular and cellular changes.

<p>(<b>Scenario 1</b>) A molecular change of interest outside the red-shaded region is neither necessary nor sufficient to cause hyperexcitability, but may nevertheless be correlated with it. (<b>Scenario 2</b>) In the absence of other changes, a molecular change of interest inside the red-shaded region is both necessary and sufficient to cause hyperexcitability. (<b>Scenario 3</b>) If only one molecular change occurs inside the red-shaded region, that change will be necessary for hyperexcitability but may or may not be sufficient depending on how the change interacts with other changes outside the red-shaded region. (<b>Scenario 4</b>) If multiple molecular changes occur inside the red-shaded region, then the change of interest will not be necessary for hyperexcitability and may or may not be sufficient depending on how that change interacts with other changes. This last scenario (hightlighted in yellow) is the most likely given that nerve injury triggers multiple molecular changes and given the degenerate manner by which spike initiation can be altered, as shown in this study. Degeneracy implies that the red circle is large and thus likely to significantly overlap the gray circle.</p

Effects of multiple co-occurring molecular changes.

<p>(<b>A</b>) A parameter change affecting a subthreshold current (in this case in the original 2-D model) may fail to cause hyperexcitability if that change is offset by a change in a second parameter. In this example, varying β_w shifts the <i>w</i>-nullcline whereas varying γ_m re-shapes the <i>V</i>-nullcline, but the combination of changes results in no change in the geometry of the nullcline intersection. Parameter values are indicated on the bifurcation diagrams; the color of each label corresponds to the color of nullclines shown on the <i>V-w</i> phase plane. (<b>B</b>) Changes in excitability (quantified as the <i>I</i>_stim threshold for repetitive spiking) caused by co-varying β_w and γ_m. The increase in excitability caused by varying only β_w (arrow <i>a</i>) could be produced by a much smaller change in γ_m (arrow <i>b</i>) or by small combined changes in β_w and γ_m (arrow <i>c</i>). Arrow <i>d</i> shows conditions in A. Note that the reduction in γ_m required to offset a “neuropathic” change in β_w (arrow <i>e</i>) is larger than the increase in γ_m required to produce neuropathic excitability (arrow <i>b</i>). Systematic testing of all parameter combinations is beyond the scope of the current study, but this example highlights the importance of parameter co-variation.</p

Dynamically equivalent effects of varying other model parameters in the 2-D model.

<p>Effects of changing β_w (<b>A</b>), <i>g</i>_fast (<b>B</b>), <i>g</i>_slow (<b>C</b>), and β_m (<b>D</b>) in the original 2-D model. All parameters were at their default values (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002524#s4" target="_blank">Methods</a>) except for the parameter of interest, which was varied from its default value (green) to a value causing neuropathic excitability (red) as indicated on each panel. In each case, the shape and/or positioning of the <i>V</i>- or <i>w</i>-nullcline (shown on phase planes; left panels) was affected in a distinct way, but the geometry of the nullcline intersection showed the equivalent “neuropathic” change, as evidenced by the bifurcations diagrams (right panels); specifically, all “neuropathic” bifurcation diagrams exhibit a Hopf bifurcation. For A–D, the “normal” bifurcation diagrams are equivalent and correspond to that shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002524#pcbi-1002524-g002" target="_blank"><b>Fig. 2B</b></a><b> top</b>.</p