The Story of the Spiking Threshold
The study of the spiking threshold has a long and interesting history, so I have decided to go into a little bit more detail. Apologies beforehand for the formulas and being somewhat nerdy!
The idea that the neuron works like a slowly charging (integrating) capacitor which generates a spike when a threshold is reached—the integrate-and-fire model—is more than 100 years old. It was first proposed by Louis Lapicque (Fig. Louis Lapicque) in 1907 (Recherches quantitatives sur l’excitation électrique des nerfs traitée comme une polarisation, J. Physiol. Pathol. Gen. 9:620–635; 1907).
Fig. Louis Lapicque and his lab. His wife Marcelle Lapicque visible in the lab was a proficient neuroscientist in her own right. (Left photo and right photo by anonymous, both in public domain).
The precise character of the threshold has been a matter of discussion in the research literature ever since. In his seminal book Biophysics of Computation (Oxford University press, 1999), Koch reaches the conclusion that the threshold should be in terms of voltage or current rather than charge (p. 394):
We conclude that in any cell with a substantial dendritic tree, the occurrence of an action potential is not caused by the charge at the spike initiating zone exceeding some fixed threshold value.
On the other hand, Gerstner, Kistler, Naud and Paninski in their treatise Neuronal Dynamics (MIT Press, 2011) state the following (p. 82):
Despite the fact that [Hodgkin-Huxley] neuronal action potential firing is often treated as a threshold-like behavior, such a threshold is not well defined mathematically. […] We conclude that, for short current pulses, it is not the amplitude of the current that sets the effective firing threshold, but rather the integral of the current pulse or the charge.
Hille’s landmark textbook Ion Channels of Excitable Membranes (Sinauer Associates, 2001) says (p. 59):
[…] For all practical purposes, a healthy axon does show a sharp threshold for firing an action potential.
How can we reconcile these authoritative but seemingly contradictive statements?
Naundorf, Wolf and Volgushev (Unique Features of Action Potential Initiation in Cortical Neurons, Nature, 2006) observed that the sharp “kink” at the onset of the spike observed for biological neurons is inconsistent with the equations of the Hodgkin-Huxley (HH) model. Later, I found by simulation that this problem can be solved by defining the spike initialization zone as a separate compartment (therefore going beyond the HH model, which is single-compartment), but I also discovered that this solution had already been found and published in a reply to Naundorf et al. by Brette (Sharpness of Spike Initiation in Neurons Explained by Compartmentalization, PLoS Comput. Biol., 2013), who should be credited as the person who finally solved the problem.
Here is how I like to describe what happens, using the notation in Fig. The AIS compartment.
Fig. The AIS compartment defines the spiking threshold.
The Na channels in the AIS compartment allow current to pass through depending on the local potential \(V_a\) (Hille 2001, id.):
\[I_{\mathrm{Na}} = I_{\mathrm{Na}_0} ~e^{\alpha V_a} .\]
Here, \(I_{\mathrm{Na}_0} < 0\). You can think of these channels as FET transistors, if you are familiar with these. In fact, one can appropriately say that an Na channel is a kind of FET transistor.
A DC-analysis of the circuit in Fig. The AIS compartment shows that, by Kirchoff’s laws,
\[\frac{V_p - V_a}{R_{pa}} = \frac{V_a}{R_a} + I_{\mathrm{Na}_0} ~e^{\alpha V_a} ,\]
or
\[V_p = V_a \left(1 + \frac{R_{pa}}{R_a}\right) + R_{pa} I_{\mathrm{Na}_0} ~e^{\alpha V_a} .\]
Plotting \(V_p\) as a function of \(V_a\) produces a graph like the one in Fig. Threshold graph.
Fig. Threshold graph.
Given \(V_p\) up to a maximal value, there is at least one solution \(V_a\). However, when \(V_p\) is larger than that value, there is no solution, and \(V_a\) will rise indefinitely. The maximal value of \(V_p\) for which there is a solution is the threshold potential \(V_{th}\). Clearly, this happens when \(dV_p/dV_a = 0\), or
\[\left(1 + \frac{R_{pa}}{R_a}\right) + R_{pa} I_{\mathrm{Na}_0} \alpha ~e^{\alpha V_a} = 0 .\]
Solving this equation for \(V_a\) gives
\[V_a = \frac 1{\alpha}\ln \left(-\frac{R_{pa}+{R_a}}{R_{pa}R_aI_{\mathrm{Na}_0}\alpha} \right)\]
and
\begin{align} V_{th} &= \left(V_a - \frac 1{\alpha} \right) \left(1 + \frac{R_{pa}}{R_a}\right) \\ &= \frac 1{\alpha} \left(1 + \frac{R_{pa}}{R_a}\right) \ln \left(-\frac{R_{pa}+{R_a}}{R_{pa}R_aI_{\mathrm{Na}_0} \alpha e}\right). \end{align}
Now, turning to AC-analysis, we note that because the membrane area of the AIS compartment is small, \(C_a\) is small, and \(V_a\) will rise very quickly when \(V_p > V_{th}\). The neuron will show precisely the behaviour reported by Naundorf et al.
For additional material on this topic, please see Brette’s review in PLoS Computational Biology and his blog entries [1, 2, 3, 4].