^{1}

^{2}

^{*}

^{3}

^{1}

^{2}

^{4}

Conceived and designed the experiments: SAP YDK TJS. Performed the experiments: SAP. Analyzed the data: SAP. Wrote the paper: SAP YDK TJS.

The authors have declared that no competing interests exist.

Transduction of graded synaptic input into trains of all-or-none action potentials (spikes) is a crucial step in neural coding. Hodgkin identified three classes of neurons with qualitatively different analog-to-digital transduction properties. Despite widespread use of this classification scheme, a generalizable explanation of its biophysical basis has not been described. We recorded from spinal sensory neurons representing each class and reproduced their transduction properties in a minimal model. With phase plane and bifurcation analysis, each class of excitability was shown to derive from distinct spike initiating dynamics. Excitability could be converted between all three classes by varying single parameters; moreover, several parameters, when varied one at a time, had functionally equivalent effects on excitability. From this, we conclude that the spike-initiating dynamics associated with each of Hodgkin's classes represent different outcomes in a nonlinear competition between oppositely directed, kinetically mismatched currents. Class 1 excitability occurs through a saddle node on invariant circle bifurcation when net current at perithreshold potentials is inward (depolarizing) at steady state. Class 2 excitability occurs through a Hopf bifurcation when, despite net current being outward (hyperpolarizing) at steady state, spike initiation occurs because inward current activates faster than outward current. Class 3 excitability occurs through a quasi-separatrix crossing when fast-activating inward current overpowers slow-activating outward current during a stimulus transient, although slow-activating outward current dominates during constant stimulation. Experiments confirmed that different classes of spinal lamina I neurons express the subthreshold currents predicted by our simulations and, further, that those currents are necessary for the excitability in each cell class. Thus, our results demonstrate that all three classes of excitability arise from a continuum in the direction and magnitude of subthreshold currents. Through detailed analysis of the spike-initiating process, we have explained a fundamental link between biophysical properties and qualitative differences in how neurons encode sensory input.

Information is transmitted through the nervous system in the form of action potentials or spikes. Contrary to popular belief, a spike is not generated instantaneously when membrane potential crosses some preordained threshold. In fact, different neurons employ different rules to determine when and why they spike. These different rules translate into diverse spiking patterns that have been observed experimentally and replicated time and again in computational models. In this study, our aim was not simply to replicate different spiking patterns; instead, we sought to provide deeper insight into the connection between biophysics and neural coding by relating each to the process of spike initiation. We show that Hodgkin's three classes of excitability result from a nonlinear competition between oppositely directed, kinetically mismatched currents; the outcome of that competition is manifested as dynamically distinct spike-initiating mechanisms. Our results highlight the benefits of forward engineering minimal models capable of reproducing phenomena of interest and then dissecting those models in order to identify general explanations of how those phenomena arise. Furthermore, understanding nonlinear dynamical processes such as spike initiation is crucial for definitively explaining how biophysical properties impact neural coding.

Action potentials, or spikes, are responsible for transmitting information through the nervous system

Hodgkin recognized this and identified three basic classes of neurons distinguished by their frequency-current (

The distinction between class 1 and 2 excitability has proven extremely useful for distinguishing neurons with different coding properties

This study set out to identify the biophysical basis for qualitative differences in neural coding exemplified by Hodgkin's three classes. By relating spike initiating dynamics with transduction properties, and by identifying the biophysical basis for those dynamics, we explain how Hodgkin's three classes of excitability result from a nonlinear, time-dependent competition between oppositely directed currents.

Spinal sensory neurons fall into several categories based on spiking pattern

To explain the differences between cell classes, our first step was to reproduce the behavior of each class in as simple a computational model as possible, and then to analyze that minimal model. We found that a 2D Morris-Lecar-like model could display class 1, 2, or 3 excitability with variation of as few as one parameter (_{w} (see below) was set between values giving class 1 or 3 excitability, but phasic-spiking could not be reproduced in the 2D model. Phasic-spiking was achieved by incorporating adaptation (

In the process of building the model (see _{w} was identified as an important parameter given its capacity to convert the model between all three classes of excitability. The biophysical meaning of _{w} is deferred until _{w} (_{w}>−10 mV, but class 2 and 3 excitability coexisted for all _{w}<−10 mV; in other words, class 2 or 3 excitability was exhibited depending on stimulus intensity _{stim}. This is evident in _{w} = −13 mV, rheobasic stimulation elicited a single spike while stronger stimulation elicited repetitive spiking. This pattern is characteristic of phasic-spiking spinal lamina I neurons (^{+} channel inactivation, but such processes were not included in the models analyzed here in order to keep the model as simple as possible and because such strong stimulation is arguably unphysiological in the first place.

Thus, neurons should not strictly be labeled class 2 or 3 but, rather, as exhibiting predominantly class 2 or 3 excitability based on the range of _{stim} over which they exhibit each class. However, phasic-spiking lamina I neurons exhibited class 3 excitability over a negligible stimulus range (i.e., <5% of the range for _{stim} tested as high as 200 pA) and single-spiking neurons never exhibited class 2 excitability for _{stim} as high as 500 pA. So, although a neuron may exhibit both class 2 and class 3 excitability, lamina I neurons exhibit

Having reproduced each class of excitability in a 2D model, our next step was to exploit the simplicity of that model to uncover the spike initiating dynamics associated with each class. Because the model is 2D, its behavior can be explained entirely by the interaction between two variables: a fast activation variable

Right panels of _{stim} exceeded rheobase), two of the intersection points were destroyed and the class 1 model began to spike repetitively. Disappearance of the two fixed points and the qualitative change in behavior that results (i.e., the transition from quiescence to repetitive spiking) is referred to most precisely as a saddle-node on invariant circle (SNIC) bifurcation

(A) Phase planes show the fast activation variable _{stim} is indicated beside each curve); _{stim} is increased. A bifurcation represents the transition from quiescence to repetitive spiking. Type of bifurcation is indicated on each plot. The range of _{stim} over which a QSC occurs is indicated in gray and was determined by separate simulations since a QSC is not revealed by bifurcation analysis.

Center panels of

Left panels of _{stim}>rheobase, meaning spike initiation occurred without a bifurcation. The system moved to the _{stim} changes; therefore, stimulation can move the quasi-separatrix so that a point (

The quasi-separatrix was plotted by integrating backward in time (−0.01 ms time step) from point * indicated on

_{w} = −10 mV, which, on the phase plane, corresponds to the _{w} = −21 mV and _{stim}<80 µA/cm^{2}; stronger stimulation eventually caused a Hopf bifurcation, but such stimulation is unphysiological and is likely to activate other processes not included in the model (see above). The range of _{stim} over which spike initiation occurred through a QSC is indicated with gray shading. Notice that a neuron that generated repetitive spiking through a Hopf bifurcation (_{w} = −13 mV) would, for a narrow range of weaker _{stim}, generate a single spike through a QSC, consistent with data in

Since model parameters were chosen to produce one or another spiking pattern, simply reproducing a given pattern is not necessarily informative—this constitutes an inverse problem akin to circular reasoning. To ensure that the model does not simply mimic spiking pattern, it must predict behaviors separate from those used to choose parameters. The model makes such a prediction: spikes initiated through different dynamical mechanisms are predicted to exhibit different variability in their amplitudes. Specifically, spikes initiated through an SNIC bifurcation should have uniform amplitudes because all suprathreshold trajectories follow the invariant circle formed when the stable manifolds (green curves on

(A) Spikes initiated through a QSC or SNIC bifurcation exhibit different spike amplitude variability. Data are from 2D models stimulated with noisy _{stim} (σ_{noise} = 10 µA/cm^{2}). _{stim} varies continuously during stimulation. Spikes initiated through a QSC exhibit variable amplitudes (yellow shading) because variations in _{stim} affect the _{stim} fluctuations just exceeding rheobase) produce smaller spikes than trajectories starting further from the quasi-separatrix (produced by larger _{stim} 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 (_{noise} = 10 pA.

To test this, we stimulated the model neurons with noisy, near-threshold _{stim} fluctuations. As predicted, the class 1 model produced uniformly sized spikes whereas the class 3 model produced variably sized spikes (

With the model thus validated, our next step was to compare class 1, 2 and 3 models to identify differences in parameters that could be related to differences in the biophysical properties of real neurons. As shown in _{w} converted the model between all three classes of excitability. _{w} controls horizontal positioning of the _{slow} in the 2D model (_{slow} into _{K,dr} and _{sub}, thus transforming the 2D model into a 3D model. Grouping currents with similar kinetics is a method for reducing dimensionality (e.g.,

(A) The _{slow}. Horizontal positioning of that curve is controlled by _{w}. Differences between class 1, 2, and 3 models may thus reflect differences in the voltage-dependency of _{slow}. (B) It is more likely, however, that the components of _{slow} vary between cells of different classes (see _{slow} may comprise multiple currents with similar kinetics. If _{slow} = _{K,dr}+_{sub}, the position of the net _{sub} (see insets) without changing the voltage-dependencies of _{sub} (_{z} = −21 mV, _{z} = 15 mV) or of _{K,dr} (_{y} = −10 mV, _{y} = 10 mV); voltage-dependencies of _{sub} and _{K,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.

We fixed the voltage-dependencies of _{K,dr} and _{sub}, and varied the direction and magnitude of _{sub} in order to represent variable expression levels of a channel carrying inward or outward current. Those changes affected the net slow current (_{sub}+_{K,dr}) in the 3D model the same way that varying _{w} affected _{slow} in the 2D model (compare _{sub} is such that the current activates at subthreshold potentials. These data therefore suggest that spike initiating dynamics may differ between neurons depending on the expression of different slow ionic currents, and, more specifically, that class 3 neurons express a subthreshold outward current and/or class 1 neurons express a subthrehsold inward current, with class 2 neurons expressing intermediate levels of those currents.

Several lines of experimental evidence support this prediction. First, the response to brief, subthreshold depolarizing pulses was amplified and prolonged relative to the equivalent hyperpolarizing response in class 1 neurons, consistent with effects of a subthreshold inward current (^{+} current that was blocked by TTX in the aforementioned experiments is taken into account; this current is expressed exclusively in tonic-spiking neurons ^{+} current in single-spiking lamina I neurons (

(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 ^{+} current (_{Na,p}), which is expressed exclusively in tonic-spiking neurons, was blocked; to correct for this, _{Na,p} measured in separate voltage clamp ramp protocols

Thus, lamina I neurons express the sort of currents predicted by our model, but that result is purely correlative, i.e., based on comparison of simulated and experimental ^{2+} or K^{+} current in class 1 and 3 neurons, respectively. As predicted, spiking was converted to a phasic pattern (

(A) Blocking a subthreshold Ca^{2+} current with Ni^{2+} 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 Ni^{2+} and 4-AP converted class 1 and 3 neurons, respectively, to class 2 neurons according to the

To demonstrate the sufficiency of subthreshold currents for determining excitability, we explicitly incorporated a subthreshold inward or outward current by adding an additional term for _{sub} to the 2D model with _{w} = −10 mV (see Equation 7); recall that the 2D model lies at the interface between class 1 and 2 excitability when _{w} = −10 mV (see _{sub} (which is controlled by the maximal conductance, _{sub}) (_{w} = −13 mV (like the class 2 model in _{sub} affected the _{w} in the 2D model (compare _{sub} = 0 mS/cm^{2}, although that value varied depending on _{w} (see above). For a given value of _{stim}, class 1 and 2 excitability were mutually exclusive whereas class 2 and 3 excitability coexisted. Furthermore, the 3D model exhibited constant or variably sized spikes depending on whether the model was class 1 or 3, respectively (

(A) Responses from 3D model described in _{sub}, the model operated at the interface between class 1 and 2 excitability (see (C)). Adding an outward current (_{sub} = −100 mV) produced class 2 or 3 excitability, with the latter becoming more predominant (i.e. over a wider range of _{stim}) as _{sub} was increased. Adding an inward current (_{sub} = 50 mV) produced class 1 excitability. (B) Bifurcation diagrams show voltage at fixed point and at max/min of limit cycle as _{stim} 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 _{stim} and _{sub}. These data are qualitatively identical to those for the 2D model (see _{sub} are sufficient to explain different classes of excitability. The phasic-spiking that results from adaptation (see _{noise} = 10 µA/cm^{2}.

Thus, expression of distinct subthreshold currents accounts for the different classes of excitability observed amongst spinal lamina I neurons. But can other biophysical properties also account for differences in excitability? And, if so, do those properties confer the same or different spike initiating dynamics than those described above? In other words, can we generalize our biophysical explanation of excitability?

We summarize here how spike initiating dynamics can be inferred from the phase plane geometry of the 2D model. We start by considering _{w} (_{w}>−10 mV). A Hopf bifurcation occurs when the _{w}<−10 mV); although necessary, this is not strictly sufficient for the bifurcation (see below), but it is a close enough approximation for our demonstration. A QSC can occur when the

(A) _{w} controls positioning of the _{slow}). For _{w} = 0 mV, the nullclines intersect tangentially at rheobasic stimulation, which translates into an SNIC bifurcation. For _{w} = −13 mV, the _{w} = −21 mV, the _{m} controls positioning of the _{fast}). Reducing _{m} had the same effect on phase plane geometry as increasing _{w}. The predicted consequences for excitability are confirmed on the bifurcation diagrams. Like _{slow}, _{fast} 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 _{w} = −10 mV, _{w} = 13 mV, and all other parameters are as indicated in _{fast} changed the shape rather than positioning of the _{slow} also changed the shape of the _{fast}, but with the same consequences for excitability. (E) Varying _{w}, which controls the slope of the voltage-dependent activation curve for _{slow}, altered the _{w} = 0 mV.

Moving the _{m}) should have the same effect as moving the _{w}), which indeed it did (_{m} causes a hyperpolarizing shift in the voltage-dependency of _{fast}, causing _{fast} to be more strongly activated by perithreshold depolarization and thus encouraging class 1 excitability. As explained in _{w}, a change in _{m} may reflect the contribution of a second fast current (inward or outward) with different voltage-dependency than the classic Na^{+} current comprising most of _{fast}. Increasing _{fast} in the 2D model without altering its voltage-dependency should also have an effect comparable to reducing _{m}, which indeed it did (_{w}, _{m}, or _{fast} all affect phase plane geometry (i.e., how the nullclines intersect) in essentially the same way and with equivalent consequences for spike initiating dynamics. Although the specific biophysical mechanism is different in each case (voltage-dependency of _{slow}, voltage-dependency of _{fast}, or magnitude of _{fast}, respectively), the common outcome is a change in the balance of fast and slow currents near threshold.

It stands to reason, therefore, that reducing _{slow} (where _{slow} is outward) should have effects comparable to increasing _{fast}, which indeed it did (_{fast} or _{slow} were required to convert excitability from class 1 to class 3, but one must consider that both of those net currents are most strongly activated at suprathreshold potentials. If spike initiating dynamics are dictated by currents at perithreshold potentials (see above), changes in maximal conductance should have small effects if the conductance is only marginally activated near threshold. By comparison, small changes in _{sub} were sufficient to alter spike initiating dynamics in the 3D model (see _{sub} was strongly activated at perithreshold potentials. Accordingly, reducing slope of the _{w}) extended the tail of the activation curve for _{slow} so that _{slow} was more strongly activated at perithreshold potentials; this predictably encouraged class 3 excitability (

Results in

Interpretation of the phase plane geometry can be formalized by doing local stability analysis near the fixed points (_{w}/_{w}. Subthreshold activation of _{slow} produces a steady state _{fast} (_{fast} (e.g., at the onset of an abrupt step in _{stim}), fast-activating inward current can overpower slow-activating outward current—the latter is stronger when fully activated, but can only partially activate (because of its slow kinetics) before a spike is inevitable. Through this mechanism, a single spike can be initiated before negative feedback forces the system back to its stable fixed point, hence class 3 excitability. Speeding up the kinetics of _{slow} predictably allows _{slow} to compete more effectively with _{fast} (see below).

(A) Top panels show individual currents in 2D model; bottom panels show how they combine to produce the instantaneous (_{inst}) and steady state (_{ss}) _{w} on the voltage-dependency of _{slow}. Class 3 neuron: _{slow} activates at lower _{fast}, meaning slow negative feedback keeps _{slow} calculated as a function of _{stim} = 60 µA/cm^{2}. Class 2 neuron: _{slow} and _{fast} activate at roughly the same _{slow} activates at higher _{fast}, meaning slow negative feedback does not begin activating until after the spike is initiated. This gives a steady state _{ss}/∂_{fast} counterbalances _{leak} and any further depolarization will cause progressive activation of _{fast}. (B) Changing _{fast} in the 2D model had equivalent effects on the shape of the steady state _{fast} 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 _{slow} impacts the onset of class 2 and 3 excitability. Compared with original model (_{w} = 0.15; black), increasing _{w} to 0.25 (red) increased _{stim} required to cause a Hopf bifurcation or a QSC, but did not affect _{stim} required to cause an SNIC bifurcation; reducing _{w} to 0.10 (green) had the opposite effect (summarized in right panel). Increasing _{w} also widened the discontinuity in the class 2 _{stim} because of the faster recovery between spikes; reducing _{w} had the opposite effects.

In class 2 neurons, _{fast} activates more rapidly than _{slow} in order for positive feedback to outrun negative feedback, since the latter would dominate and prohibit spiking if given enough time to fully activate. The difference from class 3 excitability is that fast positive feedback can outrun slow negative feedback with constant stimulation in a class 2 neuron; in the class 3 neuron, positive feedback can outrun negative feedback only during the stimulus transient.

In class 1 neurons, ∂_{ss}/∂_{fast} races subthreshold activation of _{slow} to determine whether a spike is initiated. And although class 2 neurons can spike repetitively, they cannot maintain spiking below a critical frequency lest slow-activating outward current catch up with the fast-activating inward current.

Slope of the steady state _{fast} must compete. Changing the direction and magnitude of _{sub} in the 3D model had the same consequences on the steady state _{w} in the 2D model. Changing other parameters in the 2D model, such as _{Na}, also had similar effects (

The _{slow} by varying _{w} in the 2D model. Consistent with our dynamical explanations of spike initiation, speeding up _{slow} increased the minimum stimulation required to produce class 2 or 3 excitability (especially the latter), whereas slowing down _{slow} had the opposite effects; the minimum stimulation required to produce class 1 excitability was unaffected (_{z} had the same effects in the 3D model (data not shown).

If the balance of fast and slow currents at perithreshold potentials is the crucial determinant of excitability, then perithreshold ^{+} current, _{K,A}) should encourage class 1 excitability the same way perithreshold activation of a slow inward current does. To test this, we incorporated _{K,A} by warping the _{K,A} introduced a region of negative slope into the steady state _{K,A} was explicitly incorporated to produce a 3D model (data not shown). Thus, _{K,A} increased rheobase but its slow inactivation as voltage passed through threshold amounted to a slow positive feedback process that encouraged class 1 excitability. The converse has been shown in medial superior olive neurons, where inactivation of _{Na} encourages coincidence detection associated with class 3 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} = −10 mV, _{w} = 10 mV, _{w*} = −60 mV, _{w*} = = 20 mV, and ξ = 0.1. Under these conditions, the ^{+} current at subthrehsold potentials gave a region of negative slope on the steady state

Our results demonstrate that the three classes of excitability first described by Hodgkin

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.

Our approach for uncovering the biophysical basis for Hodgkin's classification was to forward engineer a simple model in order to help reverse engineer complex neurons. The benefit of such an approach is that the model starts simple and is made only as complex as required to reproduce the phenomena of interest; extraneous details are thus excluded. Building a biologically realistic, high-dimensional model that exhibits one or another firing pattern is reasonably straightforward, but such a model almost certainly contains extraneous detail and may fail to provide greater insight than the experiments upon which it is based. The challenge when forward engineering simple models is that one may reproduce the phenomena of interest through a mechanism that is not the same as that used by real neurons; for example, a QSC produces a single-spiking pattern, but single-spiking neurons may use a completely different mechanism to produce the same result. This constitutes an inverse problem that requires careful consideration in order to validate the forward engineered model, as we demonstrated in

The forward engineering approach gave a model complex enough to reproduce each class of excitability yet simple enough for its spike initiating dynamics to be rigorously characterized using phase plane analysis. By expressing the problem geometrically, we were able to visualize and uncover the functional equivalence of changing different model parameters (

The mechanistic explanation of excitability afforded by quantitative analysis (i.e., phase plane analysis, local stability analysis, and bifurcation analysis) is precisely what is needed to make sense of the ever accumulating mass of experimental data. It provides an organizing framework for understanding which parameters are important and why, for instance, by explaining the functional equivalence of different biophysical changes (see

According to our results, the direction, magnitude, and _{slow} is absent or inward at perithreshold potentials, positive feedback mediated by _{fast} faces no competition as it drives voltage slowly through threshold; a slow voltage trajectory between spikes means that the neuron can fire repetitively at low rates, thus producing the continuous _{slow} is outward at perithreshold potentials, then _{fast} must compete with slow negative feedback. To compete successfully, _{fast} must exploit its fast kinetics, which means driving voltage through threshold with sufficient rapidity that _{slow} cannot catch up; a rapid voltage trajectory between spikes means that the neuron cannot fire repetitively at low rates, thus producing the discontinuous _{slow} is “strong enough” to prevent repetitive spiking altogether (thus producing class 3 excitability) depends on _{stim}, hence the diagonal border between class 2 and 3 excitability on _{stim}, hence the vertical border between class 1 and 2 excitability on the same plots.

The adaptation observed in phasic-spiking neurons is also interesting insofar as it indicates that shifting the balance of fast and slow currents has important consequences for coding, and that a given neuron is not restricted to a unique spike initiation mechanism. Effects of activating an outward current or inactivating an inward current on ultra-slow time scales (across several ISIs) can be predicted from plots like

Although the dynamical bases for class 1 and 2 excitability have been established for some time

Experimental study of class 3 excitability has been neglected at least partly because of the mistaken assumption that all neurons displaying a single-spiking pattern are unhealthy (i.e., that the quality of the recording is poor). Indeed, an unhealthy neuron will often fail to spike repetitively, but many other indices of neuronal health (e.g., resting membrane potential) have proven that a single-spiking pattern is not synonymous with dysfunction. Indeed, “healthy” single-spiking neurons have been described not only in the superficial dorsal horn of the spinal cord

Class 3 excitability has been most extensively studied in the auditory system where the single- (or onset-) spiking pattern has been shown to result from a low-threshold K^{+} current ^{+} current factors into the coding properties of those neurons, arguing that well-timed spikes are generated by rapid depolarizing input that minimizes activation of _{K,lt} and inactivation of _{Na}; ideally, that rapid depolarization is preceded by hyperpolarizing input that primes the neuron by deactivating _{K,lt} and deinactivating _{Na}. That biophysical explanation is consistent with our data. Single-spiking cells in the auditory system also exhibit variably sized spikes (e.g., _{w} (see Equation 3) and is liable to vary with temperature

As explained in the Introduction, Hodgkin identified three classes of neurons based on phenomenological differences in their spiking pattern

To conclude, Hodgkin's three classes of excitability result from different outcomes in a competition between fast and slow currents. The kinetic mismatch between currents is crucial for allowing single-spiking (class 3 excitability) or repetitive spiking faster than a critical frequency (class 2 excitability) despite the net steady state current being outward at threshold. Moreover, reproduction of qualitatively different spiking patterns in a 2D model emphasizes that rich dynamics are possible in simple systems based on their nonlinearities. Identifying functionally important nonlinearities and then determining how they are biologically implemented represents a powerful way of deciphering the functional significance of biophysical properties.

All experiments were performed in accordance with regulations of the Canadian Council on Animal Care. Adult male Sprague Dawley rats were anesthetized with intraperitoneal injection of sodium pentobarbital (30 mg/kg) and perfused intracardially with ice-cold oxygenated (95% O_{2} and 5% CO_{2}) sucrose-substituted artificial cerebrospinal fluid (S-ACSF) containing (in mM) 252 sucrose, 2.5 KCl, 2 CaCl_{2}, 2 MgCl_{2}, 10 glucose, 26 NaHCO_{3}, 1.25 NaH_{2}PO_{4}, and 5 kynurenic acid; pH 7.35; 340–350 mOsm. The spinal cord was removed by hydraulic extrusion and sliced in the parasagittal plane as previously described

Slices were transferred to a recording chamber constantly perfused at ∼2 ml/min with oxygenated (95% O_{2} and 5% CO_{2}), room temperature ACSF. Lamina I neurons were visualized with gradient-contrast optics on a modified Zeiss Axioplan2 microscope (Oberkochen, Germany) and were patched on with pipettes filled with (in mM) 135 KMeSO_{4}, 5 KCl, 10 HEPES, and 2 MgCl_{2}, 4 ATP (Sigma, St Louis, MO), 0.4 GTP (Sigma); pH was adjusted to 7.2 with KOH and osmolarity ranged from 270–290 mOsm. Whole cell current clamp recordings were performed using an Axopatch 200B amplifier (Molecular Devices, Palo Alto, CA). Functional classification was determined from responses to 900 ms-long current steps _{noise}) and filtering (_{noise}) are reported in the text. DC offset (_{avg}) was adjusted to give roughly equivalent firing rates across different neurons.

Traces were low-passed filtered at 3–10 KHz and stored on videotape using a digital data recorder (VR-10B, Instrutech, Port Washington, NY). Recordings were later sampled at 10–20 KHz on a computer using Strathclyde Electrophysiology software (J. Dempster, Department of Physiology and Pharmacology, University of Strathclyde, Glasgow, UK).

Our starting model was derived from the Morris-Lecar model _{fast}); _{slow}). Both _{fast} and _{slow} may comprise more than one current (see below); currents with similar kinetics were bundled together in order to create a low-dimensional model. The system is described by_{Na} = 50 mV, _{K} = −100 mV, _{leak} = −70 mV, _{fast} = 20 mS/cm^{2}, _{slow} = 20 mS/cm^{2}, _{leak} = 2 mS/cm^{2}, _{w} = 0.15, ^{2}, _{m} = −1.2 mV, _{m} = 18 mV, _{w} = 10 mV, and _{w} was varied as explained below.

This simple 2D model displayed each of Hodgkin's three classes of excitability but excluded details unnecessary for explaining the response properties of interest. Within that minimalist framework, we sought to isolate parameters sufficient to distinguish one class of excitability from another. Parameter values were found by manually varying them to produce a tonic- or single-spiking pattern. Once a set of parameters was found for each pattern, parameters were compared and adjusted to isolate those sufficient to explain each pattern. Varying _{w} was found to be sufficient to convert the model between tonic-spiking (class 1 excitability) and single-spiking (class 3 excitability); varying other parameters including _{fast}, _{slow}, _{m} or _{w} also affected excitability through the same geometrical changes associated with varying _{w} (see

To make the model more biophysically realistic, we converted the 2D model into a 3D model (see _{slow} into its component parts which include the delayed rectifier K^{+} current _{K,dr} and a subthreshold current _{sub} that is either inward or outward depending on _{sub}. Activation of _{K,dr} and _{sub} was controlled by _{sub} was either inward (_{sub} = _{Na} = 50 mV) or outward (_{sub} = _{K} = −100 mV) and _{sub} was varied. Kinetics of _{sub} were adjusted to match experimental data so that _{z} = 0.5 for inward current and _{z} = 0.15 for outward current. The steady-state activation curve for _{z} = −21 mV and _{z} = 15 mV. _{y} = 0.15, _{y} = −10 mV, _{y} = 10 mV, and all other parameters were the same as in the 2D model.

To stimulate the model, _{stim} (in µA/cm^{2}) was varied to produce steps, noise, or ramps comparable to stimuli used in experiments. Equations were integrated numerically in XPP

Nullclines were calculated in XPP

In bifurcation analysis, _{stim} was systematically varied to determine at what point the system qualitatively changes behavior (i.e., starts or stops spiking), which corresponds to a bifurcation. Whereas repetitive spiking is generated through a bifurcation, single-spiking generated through a QSC is not evident on a bifurcation diagram since the system's steady state has not changed. The stimulus range over which a QSC occurs is therefore indicated based on independent simulations.

^{+}channels in sustained high-frequency firing of fast-spiking neocortical interneurons.

^{+}conductances with complementary functions in postsynaptic integration at a central auditory synapse.

^{+}current.