### abstract ###
The threshold firing frequency of a neuron is a characterizing feature of its dynamical behaviour, in turn determining its role in the oscillatory activity of the brain.
Two main types of dynamics have been identified in brain neurons.
Type 1 dynamics shows a continuous relationship between frequency and stimulation current and, thus, an arbitrarily low frequency at threshold current; Type 2 shows a discontinuous f-I stim relationship and a minimum threshold frequency.
In a previous study of a hippocampal neuron model, we demonstrated that its dynamics could be of both Type 1 and Type 2, depending on ion channel density.
In the present study we analyse the effect of varying channel density on threshold firing frequency on two well-studied axon membranes, namely the frog myelinated axon and the squid giant axon.
Moreover, we analyse the hippocampal neuron model in more detail.
The models are all based on voltage-clamp studies, thus comprising experimentally measurable parameters.
The choice of analysing effects of channel density modifications is due to their physiological and pharmacological relevance.
We show, using bifurcation analysis, that both axon models display exclusively Type 2 dynamics, independently of ion channel density.
Nevertheless, both models have a region in the channel-density plane characterized by an N-shaped steady-state current-voltage relationship.
In summary, our results suggest that the hippocampal soma and the two axon membranes represent two distinct kinds of membranes; membranes with a channel-density dependent switching between Type 1 and 2 dynamics, and membranes with a channel-density independent dynamics.
The difference between the two membrane types suggests functional differences, compatible with a more flexible role of the soma membrane than that of the axon membrane.
### introduction ###
It is now more than 60 years since Alan Hodgkin categorized the firing behaviour in his classical study of isolated axons from the crab Carcinus maenas CITATION.
In many respects his experiments still form the basis for analysis of firing patterns in nervous systems.
Using threshold dynamics and maximum frequency as parameters, he identified two major classes of repetitively firing axons : Class 1 axons start firing with very low frequency at threshold stimulation, yielding a continuous f-I stim relationship, whereas Class 2 axons start firing abruptly with a relatively high frequency at threshold, yielding a discontinuous f-I stim relationship.
On the basis of a similar categorization mammalian cortical neurons have also been separated into main classes CITATION, CITATION, one exhibiting Class 1 characteristics and another Class 2 characteristics.
The former class consists primarily of pyramidal neurons and the latter primarily of interneurons.
This differential classification of excitability has been shown to correlate with a differential bifurcation behaviour of corresponding dynamical models CITATION CITATION and successfully been used in analysing the coding properties of neurons CITATION CITATION.
To avoid confusion, and in accordance with the notation of Tateno and Robinson CITATION, we in the following use the terms Type 1 and Type 2 dynamics when referring to continuous and discontinuous f-I stim relationships, respectively.
This classification takes the threshold dynamics of the regular and fast spiking neurons, and that of the Class 1 and 2 axons, into account, but not all behavioural aspects of these classes CITATION .
The intricate interactions between the many factors involved in the dynamical regulation of neuronal firing are poorly understood CITATION.
The dominant idea is that different combinations of ion channel types explain the different patterns CITATION.
In a previous study we proposed a complementary explanation CITATION, CITATION.
We showed that both Type 1 and Type 2 behaviour can be simulated in a dynamical model of a hippocampal neuron CITATION by varying the membrane density of voltage-gated Na and K channels.
The model used was four-dimensional and based on a detailed experimental voltage-clamp study, thus comprising experimentally estimated parameters.
The choice of ion channel densities as bifurcation parameters was due to their physiological and pharmacological relevance.
Many drugs act by specifically blocking channels and thereby reducing ion channel density both at a somatic and at an axonal level.
Perhaps the most used local anaesthetic drug, lidocaine, acts by blocking sodium channels in axons and sensory nerve endings CITATION.
An increasing number of studies suggest a role for physiological regulation of channel densities, even at a relatively short time scale CITATION CITATION .
Each type of dynamics, i.e., Type 1 and 2, was found to be associated with distinct regions in the channel density plane or with corresponding surface areas of an oscillation volume in the FORMULA FORMULA I stim space.
In regions with high FORMULA and low FORMULA values the model exhibits Type 1 dynamics, whereas in regions with higher FORMULA values the model generates Type 2 dynamics.
A bifurcation analysis showed that the Type 1 dynamics of the model is due to saddle-node on invariant circle bifurcations CITATION, CITATION.
Figure 3A portrays such a bifurcation in a V-I stim plot, calculated for the model using region C1 values.
The Type 2 dynamics was found to be due to either local Andronov-Hopf bifurcations and/or global double limit cycle bifurcations CITATION, CITATION.
The dynamics of the model associated with region B values is due to double limit cycle and subcritical Andronov-Hopf bifurcations, while the dynamics associated with region A2 is exclusively due to double limit cycle bifurcations.
The double limit cycle bifurcation implies an unstable limit cycle, which is part of a separating structure which separates trajectories turning to a central stable point and those approaching a stable limit cycle.
However, preliminary calculations suggested that the bifurcation structure at the border between regions B and C1 is more complex than previously described.
When more bifurcation parameters are changed a more intricate loss of stability occurs CITATION .
Thus, to obtain a better understanding of the processes we reanalysed the hippocampal neuron model in more detail.
Furthermore, we extended the analysis to two other well-described excitable membranes, i.e., the myelinated axon of Xenopus laevis CITATION and the giant axon of Loligo forbesi CITATION.
We found that oscillations associated with a subregion of region C1 of the hippocampal model show Type 2 dynamics, and that the oscillations of both axon models exclusively show Type 2 dynamics.
We investigated the mathematical background to these findings, using techniques from bifurcation theory.
The results suggest that the hippocampal soma and the two studied axon membranes represent two distinct types of membrane with respect to the excitability pattern; membranes with a channel-density dependent switching between Type 1 and 2 dynamics, and membranes with a channel-density independent dynamics.
The difference between the two membrane types suggests functional differences, compatible with a more flexible role of the soma membrane than that of the axon membrane.
