Axonal model for temperature stimulation

Abstract

Recent studies indicate that a rapid increase in local temperature plays an important role in nerve stimulation by laser. To analyze the temperature effect, our study modified the classical HH axonal model by incorporating a membrane capacitance-temperature relationship. The modified model successfully simulated the generation and propagation of action potentials induced by a rapid increase in local temperature when the Curie temperature of membrane capacitance is below 40 °C, while the classical model failed to simulate the axonal excitation by temperature stimulation. The new model predicts that a rapid increase in local temperature produces a rapid increase in membrane capacitance, which causes an inward membrane current across the membrane capacitor strong enough to depolarize the membrane and generate an action potential. If the Curie temperature of membrane capacitance is 31 °C, a temperature increase of 6.6-11.2 °C within 0.1-2.6 ms is required for axonal excitation and the required increase is smaller for a faster increase. The model also predicts that: (1) the temperature increase could be smaller if the global axon temperature is higher; (2) axons of small diameter require a smaller temperature increase than axons of large diameter. Our study indicates that the axonal membrane capacitance-temperature relationship plays a critical role in inducing the transient membrane depolarization by a rapidly increasing temperature, while the effects of temperature on ion channel kinetics cannot induce depolarization. The axonal model developed in this study will be very useful for analyzing the axonal response to local heating induced by pulsed infrared laser.

Keywords: Axon; Laser; Model; Stimulation; Temperature.

Fig. 1

Unmyelinated axon model to simulate action potential generation by temperature stimulation. A: The unmyelinated axon is segmented into many small cylinders of length Δx, each of which is modeled by a resistance-capacitance circuit based on the Hodgkin Huxley model. R_a: Axoplasm resistance. R_m: Membrane resistance. C_m: Membrane capacitance. V_a: Intracellular potential. V_e: Extracellular potential.

Fig. 2

Action potential is induced by a rapid increase in local temperature at the 4.5 mm location along the axon and propagates in both directions. The local temperature rapidly increased by 8 °C from 18.5 °C within 1 ms. The length of heated axon is 2 mm with a Gaussian distribution of the temperature centered at 4.5 mm. Axon diameter: 2 μm.

Fig. 3

A rapid increase in local temperature (A) caused an increase in membrane capacitance (B) that generated inward current across the membrane capacitor (C) and produced membrane depolarization and a propagating action potential (D). The solid lines in A–D were taken from Fig. 2 at the center of temperature stimulation (4.5 mm location). Legends in D indicate that an 8 °C increase from 18.5 °C within 1 ms was the threshold temperature for generating a propagating action potential. Axon diameter: 2 μm.

Fig. 4

The rise time (t_r) of local temperature determines the minimal temperature and capacitance increases required for generating an action potential. A. Temperature increase. B. Rate of temperature increase. C. Capacitance increase. Axon diameter: 2 μm. Global axon temperature: 18.5 °C. Length of heated axon: 2 mm.

Fig. 5

Influence of global axon temperature on the minimal temperature and capacitance increases required for generating an action potential. A. Temperature increase. B. Threshold temperature. C. Capacitance increase. Legends in A indicate the global axon temperature. Axon diameter: 2 μm. Length of heated axon: 2 mm.

Fig. 6

Influence of axon diameter on the minimal temperature and capacitance increases required for generating an action potential. A. Temperature increase. B. Threshold temperature. C. Capacitance increase. Legends in A indicate the axon diameter. Global axon temperature: 18.5 °C. Length of heated axon: 2 mm.

Fig. 7

The minimal temperature increase required for generating an action potential is increased as the global heat block temperature (T_b) increases from 31 °C to 40 °C. For a longer length of heated axon (2 mm), the temperature must quickly increase within a short time period (t_r = 0.5 ms) in order to generate an action potential when the global heat block temperature is high (40 °C). Global axon temperature: 18.5 °C. Axon diameter: 2 μm.

Fig. 8

The minimal temperature increase required for generating an action potential is not changed significantly by different decay time of the temperature (A) or by reducing the length of axon segment (B). Global axon temperature: 18.5 °C. Axon diameter: 2 μm. Length of heated axon: 2 mm.