Journal of Biophysics

Volume 2016 (2016), Article ID 2754249, 11 pages

http://dx.doi.org/10.1155/2016/2754249

## A Coupled Phase-Temperature Model for Dynamics of Transient Neuronal Signal in Mammals Cold Receptor

^{1}Theoretical Physics Division, Department of Physics, Bogor Agricultural University, Kampus IPB Darmaga, Jl. Meranti, Bogor 16680, Indonesia^{2}Research Cluster for Dynamics and Modeling of Complex Systems, Faculty of Mathematics and Natural Sciences, Bogor Agricultural University, Kampus IPB Darmaga, Jl. Meranti, Bogor 16680, Indonesia^{3}Applied Physics Division, Department of Physics, Bogor Agricultural University, Kampus IPB Darmaga, Jl. Meranti, Bogor 16680, Indonesia

Received 1 June 2016; Revised 17 August 2016; Accepted 29 August 2016

Academic Editor: Jianwei Shuai

Copyright © 2016 Firman Ahmad Kirana et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

Mammals complex thermoreceptor systems consisting of free nerve ending fibers are located in the dermis, muscle, skeleton, liver, and hypothalamus [1]. It is a phasic receptor which is active when there is a change in environmental temperature and rapidly becomes steady when reaching the stable temperature. Based on its characteristics with respect to the temperature level, it can be classified into warm or cold receptor [2, 3], which is, respectively, sensitive to high or low temperature relative to the normal body temperature, characterized by its way in delivering the neuronal signals. The corresponding neuronal signals are delivered in the form of bursting, that is, rhythmic of action potential consisting of spikes and punctuated by periods of inactivity [4, 5]. Their characteristics depend strongly on the associated temperature levels.

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 [3]. An experimental study on the static and dynamic discharge of a specific mammals cold receptor, that is, cat’s lingual nerve, has been comprehensively conducted by Braun et al. [4]. In particular, they showed that the dynamical response of the associated cold receptor is different for various temperature transitions between 10°C and 40°C.

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. [6] in their report had discussed a Hodgkin-Huxley voltage-conductance type equation in their attempt to understand the role of nonlinearity and noise on the dynamics of nerve cell membrane through mammals cold receptor data. Another conductance-based cold receptor model was also discussed in different reports [7, 8].

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 [9] for review). The role of TRPM8 has been shown experimentally in thermosensation mechanism in mice as discussed in [10–12]. Very recently, a conductance-based model which includes the role of TRPM8 ion channel has been proposed by Olivares et al. and showed a good agreement with the experimental data found from the cold receptor of mice [13]. The corresponding Olivares model successfully resembled the experimental data of increasing the firing rate for quasistatically increasing exposed temperature protocol.

Apart from those conductance-based models, a fully ionic model has been proposed by Longtin and Hinzer [5], which discussed the stochastic action potential phase model specifically for a cat’s lingual cold receptor. This model was further simplified by Roper et al. [14], namely, by introducing a simplified phase differential equation. It was demonstrated that the corresponding model was able to approximate the Longtin-Hinzer model for temperature interval 17.8°C to 40°C. Compared to the conductance-based model, Roper’s model [14] offered a relatively simple mathematical description. However, we discovered that this model did not lead to a realistic description on phenomena that occurred in higher or lower temperature conditions.

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 [4, 15]. We explain this phenomenon by considering an additional differential equation to describe the temperature dynamics, which is coupled to the associated phase differential equation.

We organize the report as follows: Section 2 discusses the phase model for the case of the steady temperature condition. The modified models for static temperature and dynamic transient process from high to low temperature are given in Section 3, namely, by defining a new set of functional parameters in the corresponding phase differential equation and introducing a new differential equation of temperature coupled to the phase differential equation and we focus our discussion on the characteristics of spike per burst, burst period, and interspike interval. We end this report with a conclusion in Section 4. Comprehensive discussions regarding the biological and chemical related properties of the corresponding cold receptor have been given in detail previously [4, 5, 14], such that in this report we only focus on the modified mathematical model.

#### 2. Model and Method

The corresponding nonstochastic phase differential equation for steady condition of neuronal signaling at a specific temperature developed previously by Roper et al. [14] is given as follows:withwhereHere, the symbol represents the phase of membrane potential in the trigger region, in which its full rotation describes the generation of an action potential [14]. The parameter is related to the modulation of the mean potential of the cell, while the term is a zero mean periodic term that oscillates with the frequency , with as the corresponding magnitude. The function , with an inverse time unit, describes the dynamics of the corresponding neuronal signal bursting [14].

It is seen that there are two important terms in (2), namely, and functions, as given by (3) and (4), respectively. The burst occurs when , where the average amount of spikes in each burst is proportional to the maximum width of the overlap area of both curves, denoted by , as exemplified in Figure 1(a) along with the corresponding phase of membrane potential , which is found by solving (1), and neuronal signal bursting functions as shown in Figures 1(b) and 1(c), respectively. It is obvious that the amount of spikes per burst can be controlled by changing the value of and as well as which also determine the period between two consecutive bursts. It was assumed previously [14] that these parameters are of linear functional forms of temperature as follows:with , , , , , and being constants to be determined. This assumption was aimed at yielding a decreasing period between two consecutive bursts when the temperature increases through a quasistatic process. In their work, Roper et al. defined the value of each parameter as follows: , , , , , and . We used all these parameters at = 35°C to depict the example shown in Figure 1.