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We propose a theoretical model consisting of coupled differential equation of membrane potential phase and temperature for describing the neuronal signal in mammals cold receptor. Based on the results from previous work by Roper et al., we modified a nonstochastic phase model for cold receptor neuronal signaling dynamics in mammals. We introduce a new set of temperature adjusted functional parameters which allow saturation characteristic at high and low steady temperatures. The modified model also accommodates the transient neuronal signaling process from high to low temperature by introducing a nonlinear differential equation for the “effective temperature” changes which is coupled to the phase differential equation. This simple model can be considered as a candidate for describing qualitatively the physical mechanism of the corresponding transient process.

Mammals complex thermoreceptor systems consisting of free nerve ending fibers are located in the dermis, muscle, skeleton, liver, and hypothalamus [

In this report, we focus our discussion on the dynamics of mammals cold receptor. In a low temperature condition, the corresponding neuronal signals produce periodic bursts with uniform duration and slow oscillation characteristic, but with nonuniform spike frequencies for each burst. When the temperature is raised up by a quasistatic process, the amount of spikes per burst tends to decrease forming a periodic single spike or beating. At a relatively higher temperature, the spike pattern becomes aperiodic; namely, it can also exhibit either double spike or stochastically phase-locked spike (skipping) phenomenon [

Nowadays, many models have been proposed to explain the dynamical characteristics of mammals cold receptor. One of the most profound models is the conductance-based model which relies on the conductance voltage-dependent phenomenon due to the existence of Na^{+} and K^{-} ions. For example, Braun et al. [

In the meantime, there is a certain type of ion channel called transient receptor potential melastatin 8 (TRPM8) that plays an important role in delivering the cold receptor neuronal signal (see [

Apart from those conductance-based models, a fully ionic model has been proposed by Longtin and Hinzer [

Based on this fact, in the present report we discuss a possible modification on the corresponding Roper’s model for the nonstochastic limit by introducing a new functional form of parameters that appeared in the corresponding model. Furthermore, we also discuss an extension of the corresponding modified model to accommodate the dynamical response of neuronal signals during a transition process from high to low temperature condition. This dynamical model is able to explain the phenomenon of sudden increasing amount of spikes per burst due to decreasing temperature, which is followed by a gradual decreasing of the corresponding amount of spikes per burst until the receptor reaches a steady condition at the lower temperature [

We organize the report as follows: Section

The corresponding nonstochastic phase differential equation for steady condition of neuronal signaling at a specific temperature developed previously by Roper et al. [

It is seen that there are two important terms in (

(a) The plot of

Based on the above formulation, it is clearly seen that these linear assumptions will lead to an unrealistic scenario at the high and low temperature conditions, since all those parameters are not saturated at these limits. Therefore, it is reasonable to assume that phenomenologically at those temperatures the neuronal signals become saturated since in that range the receptor becomes less sensitive [

To develop a more realistic model, we consider the modification of

Demonstrated in Figure

Comparison between the previous set of functional parameter forms (dash curve) with the new one (solid curve) for (a)

The corresponding neuronal signals at steady temperatures for

Bursting characteristics resulted from previous model (red curve) [

Mean SB at steady condition resulted from previous model (cross) [

The other important characteristic, namely, the interspike interval histogram (ISIH) of the neuronal signal at the corresponding different temperatures for the tanh functional forms, is given in Figure

(a) From top to bottom panel, interspike interval histogram (ISIH) of the Roper (red curves) and modified (black curves) model at steady conditions for

During a transient transition from high to low temperature, the existence of a peak response with relatively large amount of spikes per burst at a certain time was shown experimentally as the transition process begins as reported in [

Illustration of a Morse-like function.

It should be emphasized that the existence of the abovementioned peak response with large amount of spike per burst during the transient transition is the reason to define the term “effective temperature” as a tuning factor in our formulation based on the following argument: as shown in Figure

It is easy to prove that the function given by (

We expect the parameters

Matching condition between the first bursting and lowest effective temperature from transient transition from

Approximate function of

Using this new model, the simulation results of transition processes from

Bursting characteristics resulting from modified model for a transient transition from (top to bottom panels)

Amount of SB (left panel) and BP (mid panel) resulting from present modified model (left panel) and parametric plot in phase-plane of

An example of the approximate Morse-like function found from (

To validate this modified model with experimental data, we focus on comparing qualitatively the SB and BP characteristics with the results reported by Braun et al. [

The burst phenomenon (top panel) and

(a) SB and (b) BP resulted from present modified model (solid circle) for transition (left to right panel) from

On the other hand, in comparison with Olivares’s model [

Furthermore, although our model is able to describe the existence of peak response at matching condition, however, it should be noted that the model leads to the increasing pause duration or the time distance between two consecutive bursts at the corresponding condition, while in reality this is not the case as reported previously [

To this end, apart from the above mentioned problems, it is also realized that this modified model should be improved further, since the related effective temperature differential equation given by (

We have discussed a modified Roper’s model for describing the characteristics of neuronal signaling in mammals cold receptor, especially for the temperature transition processes. The model consists of coupled phase-temperature nonlinear differential equations equipped with a set of functional parameters that saturate at low and high temperature. It was shown that our modified model is able to describe the experimental fact that the characteristics of neuronal signal in a transient transition process from high to low temperature exhibit the existence of large amounts of spikes per burst right after the process initiated, namely, by introducing the new functional parameter “effective temperature,” which plays a role as a dynamical tuning factor to explain the corresponding phenomenon. We propose that this dynamical tuning factor might be interpreted as a perceived temperature by the mammal brain in which its perception of temperature at

Firman Ahmad Kirana and Husin Alatas are co-first authors.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank Yessie W. Sari from the Department of Physics, Bogor Agricultural University, for a useful discussion.