New curator needed
I see Harold Lecar is listed as curator for this article, meaning that someone else should adopt this article...although I have no idea what procedure that involves. Very sadly for everyone who knew Professor Lecar, he is dead.
<review>A little mistake: Eq.5 should be\[T_w(V)=T_0sech[(V-V_3)/2V_4]\], but not\[T_w(V)=T_0sech[(V-V_3)/2V|4]\].</review>
This is a well-written article on a popular model for neural excitability. I enjoyed reading about the history of the model and its applications. I do have one concern.
The authors state that the transition from single-shot firing to stable limit cycles will always arise from a saddle-node bifurcation. I don’t agree with this since another possibility is a Hopf bifurcation, when a conjugate pair of complex eigenvalues crosses the imaginary axis. Later they discuss the two types of oscillations described by Hodgkin. My understanding is that Class I is characterized by a sharp threshold for firing, long latencies to firing are possible and there may be arbitrarily low frequencies. Class II is characterized by a variable threshold, short latencies and a positive minimal frequency. As pointed out by Rinzel and Ermentrout, Class I corresponds to a saddle-node bifurcation and Class II corresponds to the Hopf bifurcation.
Modifications suggested by reviewer A
I agree with reviewer A's comments about the two classes of oscillator. In fact an earlier version the manuscript had a discussion similar to that suggested by the reviewer, but it was in the the context of a longer discussion about the classes of oscillator. That section was taken out of the paper because it overlapped with another article entirely about the classes of oscillator. I hope the present statement that I inserted accurately describes the two oscillator classes.
Similarly, an earlier version tried to summarize all the singular-point/eigenvalue possibilities, and would have included the Hopf bifurcation to a stable limit cycle. However, the discussion was too cumbersome, and I was losing the thread of the correspondence to real excitable cells.
I thank the reviewer, and have incorporated the changes in the last paragraph of the paper. If I still don't have it quite right, I'll be happy to do it over again.
Possible variant on the Morris-Lecar model
I'm currently working on a calcium-channel based system that may translate to a similar model, but with some interesting differences. A deeper understanding of this system will help clarify our understanding of the processing in the auditory brainstem.
In Sivaramakrishnan S. and D L Oliver (2001) "Distinct K Currents Result in Physiologically Distinct Cell Types in the Inferior Colliculus of Rat," Journal of Neuroscience, 21(8):2861-2877, April 15, 2001, the authors describe some experiments investigating calcium currents in disc cells of the inferior colliculus (IC). After an anode-break spike, they see broad rebound depolarisations in some cells. In 2 micromole TTX, the anode break spikes disappear, showing they are sodium spikes, but the rebound depolarisation remains, with characteristics suggesting it is due to calcium currents. In a combination of 2 micromole TTX, 10 millimole TEA-Cl, and 2 millimole 4-AP, they get regenerative spiking with depolarising current injection, with the spikes typically about 100 msec in duration. The spikes seem to be doublets, perhaps associated with different locations in the neurone.
Since voltage-sensitive sodium and potassium channels are (for the most part) blocked in the last experiment, the system appears to involve just calcium and leak conductances and so is simpler than the Nagumo-Fitzhugh and Morris-Lecar models. The most likely calcium currents involved inactivate more as a function of calcium concentration than voltage, but it is not clear that the resulting system has the necessary properties to produce regenerative spiking.
"Thus V_1 and V_3 set the steepness of the conductance curves". Should-it not be "Thus V_2 and V_4 set the steepness of the conductance curves"?