Supplementary material: INa, IK, IL and IK2 currents
INa, IK and IL are the sodium (Na+), K+ and leakage currents (μA/cm2), respectively; and IK2 is an additional current: the fast transient K+ current. INa, IK, IL[1], and IK2[2] are given by:
, ,, , (1)
where m, n and h are the gating variables,A and B are factors having the same functional significance as factors m and h; VNa, VK, VL and VK2 are the reversal potentials for the Na+, K+, leakage and fast transient K+ currents (all in mV), respectively, and gNa, gK, gL and gA are the maximum ionic conductances of Na+, K+, leakage and the fast transient K+ currents (all inmS/cm2), respectively.The conductance of the ionic current can be regulated by the voltage dependent activation and inactivation variables (gating variables) of the conduction, given as [1]:
or , / (2)where can be any one of the three gating variables, m, n, or h, and , αxand βx being the rate constants (in s-1); is a steady-state voltage dependent (in)activation function of x, is a voltage-dependent time constant, and xfac is a scaling factor introduced to allow greater flexibility in adjusting the firing frequency. Here, and can be calculated by [3]:
, / (3), / (4)
where the steady-state value is a sigmoid function, with half of the activation (or inactivation) occurring at and a slope proportional to , is the time constant and has a bell-shaped curve with its maximum at and half-width determined by σx, andxfac is the scaling factor. Hence, each of the gating variables is described by three parameters (m, n, h). From voltage clamp experiments, αx and βx can be approximated as [1]:
, / (5), / (6)
, / (7)
, / (8)
, / (9)
. / (10)
The factors A and B are determined by [4]:
, , / (11), / (12)
, , / (13)
. / (14)
To the authors’ best knowledge, no data on intradental nociceptors have been reported in the literature; therefore, the parameters for a squid axon [4-6] were used in the present model: Cmem = 2.8 μF/cm2, gA = 47.7 mS/cm2,gNa = 120 mS/cm2, gK = 36 mS/cm2, gL = 0.3 mS/cm2, Afac = Bfac = 7.0, mfac = hfac = 0.263, and nfac = 2.62. Other parameters used in the model are: ENa = 57.19 mV,EK = -78.78 mV [7], and the reversal potential of the leakage current EL = -63.79 mV, which is obtained by adjusting EL until the equilibrium membrane potential is achieved [2, 8].
References
[1] Connor,J.A. and Stevens,C.F., Prediction of repetitive firing behaviour from voltage clamp data on an isolated neurone soma. The Journal of Physiology, 1971, 213(1), 31.
[2] Connor,J.A., Walter,D.A.V.I.D.and McKowN,R.U.S.S.E.L.L., Neural repetitive firing: modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons. Biophysical Journal, 1977, 18(1), 81.
[3]Hille,B., Ion channels of excitable membranes. Sunderland, MA: Sinauer, 2001
[4] Hodgkin,A.L., The conduction of the nervous impulses. Livepool: Liverpool University Press,1964.
[5] Hodgkin,A.L. and Huxley,A.F.,A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of physiology, 1952, 117(4): 500.
[6] Wechselberger,M., Wright,C.L., Bishop,G.A., et al., Ionic channels and conductance-based models for hypothalamic neuronal thermosensitivity. American Journal of Physiology-Regulatory, Integrative and Comparative Physiology, 2006, 291(3), 518-529.
[7] Xu,F., Lu,T.J. and Seffen,K.A., Skin thermal pain modeling—A holistic method. Journal of Thermal Biology, 2008, 33(4), 223-237.
[8] Xu,F., Wen,T., Lu,T.J., et al., Modeling of nociceptor transduction in skin thermal pain sensation. Journal of biomechanical engineering, 2008, 130(4), 041013.