Figure 1.
Hodgkin's three classes of neuronal excitability. (A) Sample responses from spinal lamina I neurons representing each of Hodgkin's three classes. Hodgkin's classification is based on the f–I curve which is continuous (class 1), discontinuous (class 2), or undefined because measurement of firing rate requires at least two spikes (class 3). Data points comprising a single spike (ss) are indicated with open symbols in (A) or gray shading in (B–D). (B) Each cell class could be reproduced in a Morris-Lecar model by varying a single parameter, in this case βw. Like in (A), rheobasic stimulation (minimum Istim eliciting ≥1 spike) elicited a single spike at short latency in class 2 and 3 neurons compared with slow repetitive spiking in class 1 neurons. Despite reproducing the discontinuous f–I curve, the 2D model could not reproduce the phasic-spiking pattern. (C) Phasic-spiking was generated by adding slow adaptation, thus giving a 3D model described by C dV/dt = Istim−g̅fast m∞(V)(V−ENa)−g̅sloww(V−EK)−gleak(V−Eleak)−gadapta(V−EK) and where a controls activation of adaptation and g̅adapt = 0.5 mS/cm2, φa = 0.05 ms−1, βa = −40 mV, and γa = 10 mV. Bottom traces show single-spike elicited by second stimulus applied shortly after the end of first stimulus, which suggests that adaptation slowly shifts the neuron from class 2 towards class 3 excitability. (D) Firing rate (color) is plotted against Istim and βw. Separable regions of the graph correspond to different classes of excitability. Neuronal classification is based on which class of excitability is predominant (i.e., exhibited over the broadest range of Istim) and is indicated above the graph.
Figure 2.
Each class of excitability is derived from a distinct dynamical mechanism of spike initiation.
(A) Phase planes show the fast activation variable V plotted against the slower recovery variable w. Nullclines represent all points in phase space where V or w remain constant. V-nullclines (colored) were calculated at rest (red) and at the onset of stimulation (blue) (Istim is indicated beside each curve); w-nullclines do not change upon stimulation and are plotted only once (gray). Black curves show response of model with direction of trajectory indicated by arrows. Class 1 neuron: Red and gray nullclines intersect at three points (red arrowheads) representing stable (s) or unstable (u) fixed points. Stimulation shifts that V-nullcline upward and destroys two of those points, thereby allowing the system to enter a limit cycle and spike repetitively. The trajectory slows as it passes through constriction between blue and gray nullclines (yellow shading) thereby allowing the neuron to spike slowly, hence the continuous f–I curve. Class 2 neuron: Red and gray curves intersect at a single, stable fixed point. Spiking begins when stimulation destabilizes (rather than destroys) that point. The f–I curve is discontinuous because slow spiking is not possible without the constriction (compare with class 1 neuron). Class 3 neuron: Stimulation displaces but does not destroy or destabilize the fixed point. System variables V,w can follow different paths to the newly positioned fixed point: a single spike is initiated when stimulation instantaneously displaces the quasi-separatrix (dotted curves) so that the system, which existed above the (red) quasi-separatrix prior to stimulation, finds itself below the (blue) quasi-separatrix once stimulation begins; the trajectory must go around the head of the quasi-separatrix (*) to get to the new fixed point – we refer to this mechanism of spike initiation as a quasi-separatrix-crossing or QSC. Dashed black curve shows alternative, subthreshold path that would be followed if trajectory remained above the (blue) quasi-separatrix. (B) Bifurcation diagrams show voltage at fixed point and at max/min of limit cycle as Istim is increased. A bifurcation represents the transition from quiescence to repetitive spiking. Type of bifurcation is indicated on each plot. The range of Istim over which a QSC occurs is indicated in gray and was determined by separate simulations since a QSC is not revealed by bifurcation analysis.
Figure 3.
Comparison of spikes initiated through different dynamical mechanisms.
(A) Spikes initiated through a QSC or SNIC bifurcation exhibit different spike amplitude variability. Data are from 2D models stimulated with noisy Istim (σnoise = 10 µA/cm2). V-nullclines are shown for rest (red) and for one stimulus intensity (blue) although Istim varies continuously during stimulation. Spikes initiated through a QSC exhibit variable amplitudes (yellow shading) because variations in Istim affect the V-w trajectory: trajectories starting close to the quasi-separatrix (produced by Istim fluctuations just exceeding rheobase) produce smaller spikes than trajectories starting further from the quasi-separatrix (produced by larger Istim fluctuations). Spikes initiated through an SNIC bifurcation exhibit little variability (pink shading) because all trajectories follow the invariant circle once the heteroclinic trajectories (green curves) fuse at the moment of the SNIC bifurcation to form a single homoclinic orbit. Histogram shows distribution of voltage maxima; maxima above cutoff (*) are considered spikes. Distributions differed significantly between cell classes after normalizing by maximum or by average spike amplitude (p<0.005 and p<0.001, respectively; Kolmogorov-Smirnov test). (B) As predicted, class 3 (single-spiking) neurons showed significantly greater variability in spike amplitude than class 1 (tonic-spiking) neurons (p<0.001 regardless of normalization by peak or average; Kolmogorov-Smirnov test). σnoise = 10 pA.
Figure 4.
Biophysical correlate of differences in βw.
(A) The w-nullcline (inset) corresponds to the voltage-dependent activation curve for Islow. Horizontal positioning of that curve is controlled by βw. Differences between class 1, 2, and 3 models may thus reflect differences in the voltage-dependency of Islow. (B) It is more likely, however, that the components of Islow vary between cells of different classes (see Results). Islow may comprise multiple currents with similar kinetics. If Islow = IK,dr+Isub, the position of the net I–V curve can be changed in qualitatively the same way as in (A) by changing the direction and magnitude of Isub (see insets) without changing the voltage-dependencies of Isub (βz = −21 mV, γz = 15 mV) or of IK,dr (βy = −10 mV, γy = 10 mV); voltage-dependencies of Isub and IK,dr are different, however, with the former being more strongly activated at subthreshold potentials. These results predict that tonic-spiking neurons express a subthreshold inward current and/or that single-spiking neurons express a subthreshold outward current.
Figure 5.
Different classes of spinal lamina I neurons express different subthreshold currents.
(A) Traces show responses to 60 pA, 20-ms-long depolarizing pulses (black) and to equivalent hyperpolarizing pulses (gray); the latter are inverted to facilitate comparison with former. In class 1 (tonic-spiking) neurons, depolarization was amplified and prolonged relative to hyperpolarization, consistent with effects of an inward current activated by perithreshold depolarization. Class 3 (single-spiking) neurons exhibited the opposite pattern, consistent with effects of a subthreshold outward current, which is also evident from outward rectification (arrow) in the I–V curve. Depolarizing and hyperpolarizing responses were symmetrical in class 2 (phasic-spiking) neurons, consistent with negligible net subthreshold current. (B) Membrane current activated by voltage-clamp steps from −70 mV to −60, −50, −40, and −30 mV. For a given command potential, class 3 neurons exhibited the largest persistent outward current and class 1 neurons exhibited the smallest outward current. Red line highlights difference in current activated by step to −40 mV. (C) Steady-state I–V curves for voltage clamp protocols in (B). Because recordings were performed in TTX to prevent unclamped spiking, the persistent Na+ current (INa,p), which is expressed exclusively in tonic-spiking neurons, was blocked; to correct for this, INa,p measured in separate voltage clamp ramp protocols [26] was added to give a corrected I–V curve (dotted line). Compare with Figure 4. (D) 4-AP-sensitive current determined by subtraction of response before and during application of 5 mM 4-AP to a single-spiking neuron. Protocol included prepulse to −100 mV, which revealed a small persistent outward current active at −70 mV that was deactivated by hyperpolarization to −100 mV (*). Although depolarization also activates a large transient outward current, we are concerned with the persistent component (arrowhead); effects of the transient outward current are beyond the scope of this study and were minimized by adjusting pre-stimulus membrane potential to −60 mV for all current clamp protocols. Gray line shows baseline current.
Figure 6.
Necessity of oppositely directed subthreshold currents to explain excitability in spinal lamina I neurons.
(A) Blocking a subthreshold Ca2+ current with Ni2+ converted tonic-spiking neurons to phasic-spiking (right). Blocking a subthreshold K+ current with 4-AP converted single-spiking neurons to phasic-spiking (left). Compare with naturally occurring phasic-spiking pattern (center). (B) Application of Ni2+ and 4-AP converted class 1 and 3 neurons, respectively, to class 2 neurons according to the f–I curves. Firing rate was determined from the reciprocal of first interspike interval. According to these data, a subthreshold inward current is necessary for class 1 excitability, a subthreshold outward current is necessary for class 3 excitability, and class 2 excitability occurs when neither current is present.
Figure 7.
Sufficiency of oppositely directed subthrehsold currents to explain excitability.
(A) Responses from 3D model described in Figure 4B. Without Isub, the model operated at the interface between class 1 and 2 excitability (see (C)). Adding an outward current (Esub = −100 mV) produced class 2 or 3 excitability, with the latter becoming more predominant (i.e. over a wider range of Istim) as ḡsub was increased. Adding an inward current (Esub = 50 mV) produced class 1 excitability. (B) Bifurcation diagrams show voltage at fixed point and at max/min of limit cycle as Istim was increased. Class 1, 2, and 3 versions of the 3D models exhibited exactly the same spike initiating dynamics seen in class 1, 2 and 3 versions of the 2D models (compare with Figure 2B). (C) Firing rate (color) is plotted against Istim and ḡsub. These data are qualitatively identical to those for the 2D model (see Figure 1D) and indicate that direction and magnitude of Isub are sufficient to explain different classes of excitability. The phasic-spiking that results from adaptation (see Figure 1C) can be understood in terms of slowly activating outward current (or inactivating inward current) causing a shift from class 2 to class 3 excitability. (D) As with the 2D model (Figure 3A), the class 3 version of the 3D model exhibited significantly greater spike amplitude variability than the class 1 version when driven by noisy stimulation (p<0.001, respectively; Kolmogorov-Smirnov test). σnoise = 10 µA/cm2.
Figure 8.
Common phase plane geometries associated with different parameter changes.
(A) βw controls positioning of the w-nullcline (i.e. voltage-dependency of Islow). For βw = 0 mV, the nullclines intersect tangentially at rheobasic stimulation, which translates into an SNIC bifurcation. For βw = −13 mV, the w-nullcline crosses the V-nullcline on its middle arm, which translates into a Hopf bifurcation. For βw = −21 mV, the w-nullcline crosses the V-nullcline on its left arm, meaning spike initiation is limited to a QSC. See Figure 2B for corresponding bifurcation diagrams. Thus, spike initiating dynamics (and the resulting pattern of excitability) are directly related to phase plane geometry (i.e. how the nullclines intersect). (B) βm controls positioning of the V-nullcline (i.e., voltage-dependency of Ifast). Reducing βm had the same effect on phase plane geometry as increasing βw. The predicted consequences for excitability are confirmed on the bifurcation diagrams. Like Islow, Ifast may comprise more than one current; therefore, differences in the voltage-dependency of the net fast current may reflect the expression of different fast currents rather than variation in the voltage-dependency of any one current (see Figure 4). For (B–E), βw = −10 mV, γw = 13 mV, and all other parameters are as indicated in Methods unless otherwise stated. (C) Varying ḡfast changed the shape rather than positioning of the V-nullcline, but both had equivalent consequences for excitability. (D) Varying ḡslow also changed the shape of the V-nullcline, in a slightly different manner than ḡfast, but with the same consequences for excitability. (E) Varying γw, which controls the slope of the voltage-dependent activation curve for Islow, altered the w-nullcline, again, with predictable consequences for excitability. βw = 0 mV.
Figure 9.
Competition between kinetically mismatched currents.
(A) Top panels show individual currents in 2D model; bottom panels show how they combine to produce the instantaneous (Iinst) and steady state (Iss) I–V curves. Double-headed arrows highlight effect of βw on the voltage-dependency of Islow. Class 3 neuron: Islow activates at lower V than Ifast, meaning slow negative feedback keeps V from increasing high enough to initiate fast positive feedback at steady state. Fast positive feedback (that results in a spike) can be initiated only if the system is perturbed from steady state. Quasi-separatrix (blue) has a region of negative slope (*) indicating where net positive feedback occurs given the kinetic difference between fast and slow currents: positive feedback that activates rapidly can compete effectively with stronger negative feedback whose full activation is delayed by its slower kinetics. If V is forced rapidly past the blue arrowhead, fast positive feedback initiates a single spike before slow negative feedback catches up and forces the system back to its stable fixed point. Quasi-separatrix is plotted as the sum of all currents but with Islow calculated as a function of w at the quasi-separatrix (see phase plane in Figure 2A) rather than at steady state and is shown here for Istim = 60 µA/cm2. Class 2 neuron: Islow and Ifast activate at roughly the same V. A Hopf bifurcation occurs at the point indicated by the arrow, where (see Results). This means that fast positive feedback exceeds slow negative feedback at steady state; as for class 3 neurons, this relies on positive feedback having fast kinetics since the net perithreshold current is still outward (i.e., steady state I–V curve is monotonic). Note that the slope of the steady-state I–V curve is less steep in the class 2 model than in the class 3 model. Class 1 neuron: Islow activates at higher V than Ifast, meaning slow negative feedback does not begin activating until after the spike is initiated. This gives a steady state I–V curve that is non-monotonic with a region of negative slope (*) near the apex of the instantaneous I–V curve. The SNIC bifurcation occurs when ∂Iss/∂t = 0 (arrowhead) because, at this voltage, Ifast counterbalances Ileak and any further depolarization will cause progressive activation of Ifast. (B) Changing ḡfast in the 2D model had equivalent effects on the shape of the steady state I–V curves. Unlike in (A), voltage at the apex of the instantaneous I–V curve (purple arrows) changes as ḡfast is varied; in other words, the net current at perithreshold potentials can be modulated by changing fast currents (which directly impact voltage threshold) rather than by changing the amplitude or voltage-dependency of slow currents. This is consistent with results in Figure 8. (C) Speeding up the kinetics of Islow impacts the onset of class 2 and 3 excitability. Compared with original model (φw = 0.15; black), increasing φw to 0.25 (red) increased Istim required to cause a Hopf bifurcation or a QSC, but did not affect Istim required to cause an SNIC bifurcation; reducing φw to 0.10 (green) had the opposite effect (summarized in right panel). Increasing φw also widened the discontinuity in the class 2 f–I curve and allowed class 2 and 3 neurons to achieve higher spiking rates with strong Istim because of the faster recovery between spikes; reducing φw had the opposite effects.
Figure 10.
Depolarization-induced inactivation of a subthreshold outward current can also produce class 1 excitability.
(A) Inactivation of an A-type K+ current by subthreshold depolarization should shift the balance of inward and outward currents the same way that depolarization-induced activation of an inward current does, and is therefore predicted to encourage class 1 excitability. To test this, we warped the w-nullcline to give it a region of negative slope at subthreshold potentials (see [55]); this was done by changing Equation 5 so that where βw = −10 mV, γw = 10 mV, βw* = −60 mV, γw* = = 20 mV, and ξ = 0.1. Under these conditions, the V- and w-nullclines intersected tangentially at rheobasic stimulation. (B) This phase plane geometry resulted in an SNIC bifurcation and class 1 excitability, as predicted. (C) Inactivation of the A-type K+ current at subthrehsold potentials gave a region of negative slope on the steady state I–V curve that overlapped the apex of the instantaneous I–V curve.
Figure 11.
Summary of phase plane geometry and local stability analysis.
Class 1 excitability results when slow-activating outward current is absent at voltages below threshold; inward current faces no competition and can drive arbitrarily slow spiking. Class 2 excitability results when outward current is activated at subthreshold voltages, but although net current is outward at steady state, fast-activating inward current ensures repetitive spiking above a critical frequency; spiking cannot be sustained below a rate that would allow enough time for slow-activating outward current to activate sufficiently that net current becomes outward during the interspike interval. Class 3 excitability results when outward current is sufficiently strong that repetitive spiking is prohibited despite fast-activating inward current; spike generation is only possible when the system is perturbed from steady state, as during a stimulus transient, during which fast-activating inward current initiates a spike before slow-activating outward current has an opportunity to counteract the positive feedback process.