Encyclopedia of Computational Neuroscience
DOI 10.1007/978-1-4614-7320-6_151-1
# Springer Science+Business Media New York 2014
Excitability: Types I, II, and III
Steven A. Prescott*
Neurosciences and Mental Health, The Hospital for Sick Children, Toronto, Canada
Department of Physiology, University of Toronto, Toronto, Canada
Definition
In 1948, Hodgkin distinguished between three classes of excitability on the basis of spiking patterns
observed in crustacean axons. This classification scheme is still applied today because it captures
fundamental differences in excitability that stem from how action potentials, or spikes, are generated. The nonlinear dynamical mechanism responsible for spike initiation in each cell class has been
worked out, and the biophysical implementation of those dynamical mechanisms is reasonably well
understood. Moreover, the differences in spike initiation that define each class confer differences in
metrics like the phase-response curve (PRC) and spike-triggered average (STA), meaning that this
simple classification scheme is predictive of important differences in neural coding.
Detailed Description
Historical Background
Based on single fiber recordings from the crab Carcinus maenas, Hodgkin (1948) identified three
different classes of excitability. Class 1 axons were capable of spiking repetitively across a broad
range of rates (5–150 spikes/s) in proportion to the stimulus intensity. Class 2 axons were also
capable of spiking repetitively but across a narrower range (75–150 spikes/s), implying that they
could not maintain slow repetitive spiking. Class 3 axons were by in large incapable of repetitive
spiking and were also noted to spike at a shorter latency after stimulation than the other two classes.
The differences that Hodgkin described can be readily distinguished from frequency-current (f-I)
curves, as illustrated in Fig. 1: the f-I curve is continuous for class 1 neurons, discontinuous for class
2 neurons, and undefined for class 3 neurons.
Differences in Nonlinear Dynamics
Hodgkin’s classification scheme has stood the test of time because it captures fundamental
differences in neuronal excitability. The distinction between continuous and discontinuous f-I curves
resurfaced most notably when, in 1989, Rinzel and Ermentrout provided dynamical explanations for
the emergence of zero and nonzero frequency oscillations. Using a Morris-Lecar model, they
showed that repetitive spiking in a class 1 neuron arises through a saddle node on invariant circle
(SNIC) bifurcation, whereas repetitive spiking in a class 2 neuron arises through a subcritical Hopf
bifurcation (Rinzel and Ermentrout 1989). These nonlinear dynamical mechanisms are illustrated in
Fig. 2 and are summarized below. See also Izhikevich (2007) for an in-depth discussion.
The Morris-Lecar model they used comprises only two variables, a fast activation variable V and
a slower recovery variable w. Behavior of the model can thus be explained entirely on the basis of
how those two variables interact. That interaction can be visualized by plotting Vagainst w, as shown
*Email: steve.prescott@utoronto.ca
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Encyclopedia of Computational Neuroscience
DOI 10.1007/978-1-4614-7320-6_151-1
# Springer Science+Business Media New York 2014
Fig. 1 Defining differences in frequency-current (f-I) curves. Class 1 neurons can fire repetitively at arbitrarily slow
rates, whereas class 2 neurons cannot, and class 3 neurons typically fire only a single spike even at high stimulus
intensities
in Fig. 2a. How the variables evolve relative to one another is explained by how the V- and
w-nullclines intersect, where nullclines represent everywhere in phase space where V or w remain
constant. Importantly, depolarizing stimulation shifts the V-nullcline upward and can therefore
change if and how that nullcline intersects the w-nullcline.
In class 1 excitability, the V- and w-nullclines intersect at three points prior to stimulation. The left
intersection constitutes a stable fixed point that controls membrane potential. Stimulation shifts the
V-nullcline upward. If it is shifted far enough (i.e., if stimulation exceeds current threshold or
rheobase), two of the intersection points will be destroyed and repetitive spiking ensues – this
constitutes an SNIC bifurcation. In effect, so long as the stable fixed point exists, membrane
potential will stay at that point; but when that point is absent (which occurs during suprathreshold
stimulation), the membrane potential enters a stable limit cycle that corresponds to repetitive
spiking. Importantly, the trajectory around the limit cycle slows down each time it passes close to
where the stable fixed point used to exist, which translates into a reduced rate of repetitive spiking.
Indeed, at the precise stimulus intensity that causes the V-nullcline to shift just far enough that the left
and center fixed points fuse, a homoclinic orbit is formed and firing rate is infinitesimally small.
In class 2 excitability, the V- and w-nullclines intersect at a single, stable point prior to stimulation.
Spiking starts when the vertical shift in the V-nullcline caused by stimulation destabilizes that
point – this constitutes a Hopf bifurcation. To a rough approximation, destabilization occurs when
the V- and w-nullclines intersect to the right of the local minimum in the V-nullcline, whereas the
fixed point is stable when the nullclines intersect to the left of the minimum. Local stability analysis
near the fixed point should be applied to determine stability more accurately.
These changes in nullcline geometry are directly reflected in the bifurcation diagrams shown in
Fig. 2b. In bifurcation analysis, a parameter of interest (such as stimulus intensity) is systematically
varied to determine at what values the system qualitatively changes behavior, e.g., switches from
quiescence to repetitive spiking. The SNIC and Hopf bifurcations described above are labeled on
those diagrams. Note that class 3 excitability is not associated with a bifurcation; the single fixed
point remains stable throughout the entire range of tested stimulus intensities. However, upon an
abrupt increase in stimulus intensity, a single spike can be produced, as shown by gray shading. The
dynamical explanation for this spike initiation mechanism was worked out (Prescott et al. 2008) and
is described below.
In class 3 excitability, the V- and w-nullclines intersect at a single stable point which, as already
mentioned, remains stable even when stimulus intensity is very high. However, upon an
abrupt increase in stimulus intensity, the V-nullcline moves instantaneously, whereas the system
variables V and w lag behind. The system will move to the newly positioned stable fixed
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Encyclopedia of Computational Neuroscience
DOI 10.1007/978-1-4614-7320-6_151-1
# Springer Science+Business Media New York 2014
Fig. 2 Differences in the nonlinear dynamical mechanisms responsible for spike initiation. (a) Phase planes show the
fast activation variable V plotted against the slower recovery variable w. Bottom panels show enlarged views of mauveshaded regions in top panels. Colored lines show V-nullclines (i.e., conditions for which dV/dt ¼ 0) and gray lines show
w-nullclines (i.e., conditions for which dw/dt ¼ 0). Positions at which the V- and w-nullclines intersect represent fixed
points and can be stable (s) or unstable (u) based on whether trajectories move toward or away from that point. On each
phase plane, the V-nullcline is represented twice: once without stimulation (red) and once during excitatory stimulation
(blue). Excitatory stimulation shifts the V-nullcline upward and can qualitatively change how it intersects the
w-nullcline, as summarized in bifurcation diagrams in B. (b) Bifurcation diagrams show the fixed points and limit
cycles (represented by their maximum and minimum V) as stimulus intensity Istim is systematically increased. In class
1 excitability, repetitive spiking begins when the stable fixed point and its neighboring unstable fixed point are destroyed
through a saddle node on invariant circle (SNIC) bifurcation. In class 2 excitability, repetitive spiking begins when the
stable fixed point is destabilized through a subcritical Hopf bifurcation. In class 3 excitability, a single spike is generated
when abrupt stimulation shifts the V-nullcline enough that the start of the trajectory (which corresponds to the initial
stable fixed point labeled with a red s) ends up below the shifted quasi-separatrix (shown as a dotted blue curve).
A quasi-separatrix crossing can also generate single spikes in class 2 neurons for intermediate stimulus intensities. See
text for details (Modified from Prescott et al. (2008))
point, but can do so via different paths depending on where the trajectory starts relative to the quasiseparatrix – a boundary in phase space from which trajectories diverge. Trajectories starting below
the quasi-separatrix make a long excursion around the elbow of the V-nullcline, resulting in a single
spike, whereas those starting above the quasi-separatrix followed a direct, subthreshold route.
Importantly, this mechanism requires that the stimulus intensity changes faster than the system
variables V and w; in geometrical terms, this means that stimulation can move the quasi-separatrix
so that a point (V,w) that is above the quasi-separatrix before stimulus onset ends up below the quasiseparatrix after stimulus onset. Accordingly, this mechanism of spike initiation is referred to as
a quasi-separatrix crossing because spike initiation requires that system variables cross from one
side of the quasi-separatrix to the other.
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Encyclopedia of Computational Neuroscience
DOI 10.1007/978-1-4614-7320-6_151-1
# Springer Science+Business Media New York 2014
Fig. 3 Biophysical basis for differences in spike initiation mechanism. Top panels show I-V curves for individual
conductances comprising a modified Morris-Lecar model. Bottom panels show curves from top panels combined to give
instantaneous and steady-state I-V curves. In class 1 excitability, the steady-state I-V curve has a local maximum
(arrowhead), depolarization above which activates slow inward current. This slow positive feedback process acts
cooperatively with activation of the fast sodium current, a fast positive feedback process. Under these conditions, spike
initiation can occur arbitrarily slowly. In class 2 excitability, the steady-state I-V curve has no local maximum; instead,
depolarization above the local maximum of the instantaneous I-V curve causes increasing outward current. Under these
conditions, slow negative feedback competes with fast positive feedback, meaning spike initiation must occur before
slow negative feedback “catches up.” Class 3 excitability resembles class 2 excitability except that the outward current at
perithreshold potentials is even stronger. In this example, voltage dependency of the slow conductance was varied
(indicated by red arrow in top panels) to shift the model between different classes of excitability, but a multitude of other
parameter changes can have equivalent effects that can be understood by the logic described above
Differences in Threshold Mechanism
Spike generation is essentially a two-dimensional phenomenon involving fast activation (positive
feedback) to produce the rapid upstroke and slower recovery (negative feedback) to produce the
subsequent downstroke. The more important issue is to understand what happens at spike threshold.
The three nonlinear dynamical mechanisms described above – the SNIC bifurcation, subcritical
Hopf bifurcation, and quasi-separatrix crossing – provide that explanation, the implication being
that Hodgkin’s three classes of excitability reflect differences in threshold mechanism. The differences are most clearly revealed by considering local stability analysis near the fixed point in
conjunction with the steady-state and instantaneous I-V curves, as illustrated in Fig. 3 and explained
below.
In class 1 excitability, @Iss/@V ¼ 0 at the saddle node (i.e., for just-threshold stimulation). This
means that the steady-state I-V curve is non-monotonic with a local maximum above which
depolarization activates net inward current at steady state; the negative feedback responsible for
spike repolarization only begins to activate at suprathreshold potentials. Under these conditions,
fast positive feedback has no slow negative feedback with which to compete at perithreshold
potentials, and the voltage trajectory can, therefore, pass slowly through threshold. In fact, the net
slow-activating current is inward at threshold, meaning that it cooperates (rather than competes)
with fast-activating current to encourage spike initiation.
In class 2 excitability, C1 @I@Vinst > ftww at the unstable fixed point, meaning positive feedback (whose
rate of activation is described by the left side of the inequality) outpaces negative feedback (whose
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Encyclopedia of Computational Neuroscience
DOI 10.1007/978-1-4614-7320-6_151-1
# Springer Science+Business Media New York 2014
rate of activation is described by the right side of the inequality) and repetitive spiking ensues.
Unlike in class 1 excitability, the steady-state I-V curve is monotonic. This means that net slowactivating current is outward at threshold and competes (rather than cooperates) with fastactivating current to discourage spike initiation. Under these conditions, stimulation must not
only force V high enough to activate fast-activating inward current, it must also do so fast enough
that slower-activating outward current cannot catch up. It stands to reason that spiking below
a certain rate is disallowed since the voltage trajectory between spikes would be too slow to avoid
strongly activating the slow outward current.
In class 3 excitability, C1 @I@Vinst < ftww at the stable fixed point. This means that at steady state, positive
feedback activates too slowly to overcome negative feedback. Subthreshold activation of Islow
produces a steady-state I-V curve that is monotonic and sufficiently steep near the apex of the
instantaneous I-V curve that V is prohibited from rising high enough to strongly activate fast
inward current (Ifast). However, because the two feedback processes have different kinetics,
a spike can be initiated if the system is perturbed from steady state: If V escapes high enough to
activate Ifast (e.g., at the abrupt onset of stimulation), 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. In this way, a single spike is
initiated before negative feedback forces the system back to its stable fixed point. Thus, like in
class 2 excitability, slow-activating outward current competes with fast-activating inward current,
but in class 3 excitability, the fast-activating current will only win that competition during
stimulus transients. In other words, constant stimulation is never strong enough to sufficiently
bias the competition in favor of fast-activating current, and constant stimulation thus never
succeeds in driving repetitive spiking in class 3 neurons.
Interestingly, class 2 neurons exhibit a narrow stimulus range in which an abrupt increase in
stimulus intensity to a level that is insufficient to produce a Hopf bifurcation can nonetheless drive
a single spike generated through a quasi-separatrix crossing (See Fig. 1). This has been demonstrated
experimentally and implies that spike generation through two distinct mechanisms – a subcritical
Hopf bifurcation and quasi-separatrix crossing – can occur in the same neuron based on different
stimulus conditions. By comparison, the SNIC bifurcation cannot coexist with the other two spike
initiation mechanisms. Nonetheless, a given neuron can shift between different classes of excitability (i.e., initiate spikes through different mechanisms) based on changes in its intrinsic membrane
properties. In fact, different regions of the same neuron may generate spikes through different
mechanisms based on subcellular variations in the densities of various ion channels.
To summarize, Hodgkin’s three classes of excitability reflect differences in spike initiation
dynamics. Fast-activating current is necessarily inward (depolarizing) at spike threshold. In class
1 excitability, net slow-activating current is also inward at perithreshold voltages and thus
cooperates with fast-activating current during spike initiation. In class 2 and 3 excitabilities, net
slow-activating current is outward (hyperpolarizing) at perithreshold voltages and thus competes
with fast-activating current during spike initiation. Class 2 excitability exists if fast-activating
inward current overpowers slow-activating outward current when constant stimulation exceeds
threshold. Class 3 excitability exists if fast-activating inward current overpowers slow-activating
outward current only during a stimulus transient, which precludes repetitive spiking during
sustained stimulation.
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Encyclopedia of Computational Neuroscience
DOI 10.1007/978-1-4614-7320-6_151-1
# Springer Science+Business Media New York 2014
Biophysical Basis
The fact that Hodgkin’s three classes of excitability can be reproduced in a two-dimensional model
demonstrates that nonlinear interaction between as few as two variables can account for the different
threshold mechanisms that are the basis for each class. In other words, explaining the differences that
Hodgkin described does not require complicated models with dozens of different ion channels. That
said, this does not mean that dozens of different ion channels are not contributing in one way or
another to differences in excitability. Whereas currents with dissimilar kinetics interact nonlinearly,
currents with similar kinetics sum linearly. Thus, net slow-activating current may, in a real neuron,
comprise many different ion channels with similarly slow activation kinetics, and, likewise, net fastactivating current may comprise many different ion channels with similarly fast activation kinetics.
The nonlinear interaction between the net-fast current and net-slow current could be qualitatively
altered by alteration of any one of the contributing currents, meaning the class of excitability could
be switched by any one of those changes. And of course ion channel inactivation is just as important
as activation in determining the contribution of a given ion channel to net current. Furthermore,
ultraslow processes like adaptation through slow activation of potassium currents or slow inactivation of sodium currents will modulate the fast-slow interaction; in other words, the ultraslow process
can be approximated as a constant bias current over the time scale of a single fast-slow interaction
(i.e., an individual spike), but that bias current will differ from one spike to the next. It stands to
reason that spike initiation dynamics (ascertained in the two-dimensional subsystem) and the class of
excitability that they convey will be dynamically modulated.
One point that should be stressed is that there is no unique ion channel or set of ion channels that is
responsible for a given class of excitability. On the contrary, there are countless different ion channel
combinations that can implement equivalent spike initiation mechanisms. In that respect, the
biophysical basis for the spike initiation mechanism (and equivalently for Hodgkin’s three classes
of excitability) is highly degenerate. What is important is whether the fast and slow ion channels
compete or cooperate during the spike initiation process.
Computational Consequences
The distinct dynamics associated with spike initiation in each of Hodgkin’s three classes endows
neurons with tuning to different stimulus features. The cooperative spike initiation dynamics
associated with class 1 excitability implements a low-pass filter. Spikes in class 1 neurons are thus
preferentially elicited by constant or slowly fluctuating stimuli, and, as a result, those neurons tend to
behave as good integrators. By comparison, the competitive spiking initiation dynamics associated
with class 2 and 3 excitabilities implements a high-pass filter, although once combined with the
low-pass filter implemented by membrane capacitance, the end result is band-pass filtering. Spikes
in class 2 and 3 neurons are thus preferentially elicited by more rapidly fluctuating stimuli, and, as
a result, those neurons tend to act as good coincidence detectors. Consistent with that description,
class 1 neurons are primarily sensitive to stimulus intensity, whereas class 2 and 3 neurons are
relatively more sensitive to the rate of change of stimulus intensity (Ratté et al. 2013).
Differences in tuning are directly reflected in measures such as the spike-triggered average
stimulus (STA): class 1 neurons have broad, monophasic STAs, whereas class 2 and 3 neurons
have biphasic STAs whose positive phase tends to be quite narrow. This is directly related to the
shape of the phase-response curve (PRC), which is monophasic for class 1 neurons and biphasic for
class 2 neurons (Ermentrout et al. 2007). Differences in the shape of the STA and PRC have a variety
of interesting consequences (e.g., on synchronization), but that discussion exceeds the scope of the
current chapter.
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Encyclopedia of Computational Neuroscience
DOI 10.1007/978-1-4614-7320-6_151-1
# Springer Science+Business Media New York 2014
Summary
Hodgkin’s classification of excitability is based on spiking patterns discerned over a range of
stimulus intensities. This classification scheme is still widely used today, 65 years after it was
initially proposed, because it captures fundamental differences in how neurons generate spikes.
Spikes can arise through cooperation or competition between fast- and slow-activating currents. The
biophysical basis for each spike initiation mechanism is degenerate insofar as many different ion
channels contribute to the process and several different ion channel combinations can produce
equivalent excitability. What is most important is how those various ion channels interact during
spike initiation. Different spike initiation mechanisms confer differences in tuning that are computationally important, thus further enhancing the utility of Hodgkin’s classification scheme.
References
Ermentrout GB, Galan RF, Urban NN (2007) Relating neural dynamics to neural coding. Phys Rev
Lett 99:248103
Hodgkin AL (1948) The local electric changes associated with repetitive action in a non-medullated
axon. J Physiol 107:165–181
Izhikevich EM (2007) Dynamical systems in neuroscience. MIT Press, Cambridge, MA
Prescott SA, De Koninck Y, Sejnowski TJ (2008) Biophysical basis for three distinct dynamical
mechanisms of action potential initiation. PLoS Comput Biol 4:e1000198
Ratté S, Hong S, De Schutter E, Prescott SA (2013) Impact of neuronal properties on network
coding: roles of spike initiation dynamics and robust synchrony transfer. Neuron 78:758–772
Rinzel J, Ermentrout B (1989) Analysis of neural excitability and oscillations. In: Koch C, Segev
I (eds) Methods in neuronal modeling. MIT Press, Cambridge, MA
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