Abstract
Hodgkin–Huxley model describes the action potential phenomenon on the basis of electrochemical properties but does not characterize the anesthetic effects. In this paper, we have proposed a model which reframes Hodgkin–Huxley model to be able to identify the parameters affected by anesthesia. The model comprises of set of partial differential equations that describe how the viscosity of fluid moving along the axon impacts the propagation of action potential. It is observed that with the increase in viscosity of the fluid, there is a reduction in the conduction velocity. The viscosity of the fluid moving along the axon has also been characterized with respect to the temperature, the physical parameter considered in the Hodgkin-Huxley model. The model has been solved using finite difference method and implemented using C++ syntax code in an iterative manner. The results obtained are consistent with the freezing point depression theory for the explanation of anesthesia. The model acts as a framework for drug therapists inducing anesthesia to analyze the target parameters responsible for blocking of action potential propagation and hence for possible therapeutic intervention.
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Abbreviations
- \(c_{m}\) :
-
Membrane capacitance per unit area of membrane
- \(D_{Na}\) :
-
Diffusivity of sodium ions in fluid
- \(D_{K}\) :
-
Diffusivity of potassium ions in fluid
- \(D_{L}\) :
-
Diffusivity of chlorine ions in fluid
- \(F\) :
-
Faraday’s constant
- \(\bar{g}_{K}\) :
-
Conductance of potassium ions per unit area of the membrane
- \(\bar{g}_{L}\) :
-
Conductance of leakage current per unit area of membrane
- \(\bar{g}_{Na}\) :
-
Conductance of sodium ions per unit area of the membrane
- \(K_{B}\) :
-
Boltzman Constant
- \(M_{Na}\) :
-
Molar mass of sodium ions
- \(M_{K}\) :
-
Molar mass of potassium ions
- \(M_{L}\) :
-
Molar mass of chlorine ions
- \(r\) :
-
Axon radius
- \(rad_{i}\) :
-
Radius of different ions i
- \(R_{a}\) :
-
Resistance per unit axial length
- \(\rho\) :
-
Density
- \(R_{u}\) :
-
Universal gas constant
- \(S\) :
-
Source term
- \(T\) :
-
Temperature in kelvin
- \(v_{F}\) :
-
Viscosity of fluid inside the axon
- \(v_{W}\) :
-
Viscosity of water
- \(V\) :
-
Membrane voltage
- \(V_{Na}\) :
-
Equilibrium potential of sodium ions
- \(V_{K}\) :
-
Equilibrium potential of potassium ions
- \(V_{L}\) :
-
Equilibrium potential of chlorine ions
- \({\dot{w}_{Na}}^{{{\prime \prime \prime }}}\) :
-
Rate of addition of mass of sodium ions per unit volume
- \({\dot{w}_{K}}^{{{\prime \prime \prime }}}\) :
-
Rate of addition of mass of potassium ions per unit volume
- \({\dot{w}_{L}}^{{{\prime \prime \prime }}}\)::
-
Rate of addition of mass of chlorine per unit volume
- \(Y_{Na}\) :
-
Mass fraction of sodium ions
- \(Y_{K}\) :
-
Mass fraction of potassium ions
- \(Y_{L}\) :
-
Mass fraction of chlorine ions
- \(\bar{g}_{Na}\) :
-
\(1200\,S/m^{2}\)
- \(\bar{g}_{K}\) :
-
\(360\,S/m^{2}\)
- \(\bar{g}_{L}\) :
-
\(3\,S/m^{2}\)
- \(V_{Na}\) :
-
\(0.050\,V\)
- \(V_{K}\) :
-
\(- 0.077\,V\)
- \(V_{Cl}\) :
-
\(- 0.054\,V\)
- \(c_{m}\) :
-
\(0.01F/m^{2}\)
- \(R_{a}\) :
-
\(resistivity/\varPi r^{2} \,\varOmega /m\)
- \(resistivity\) :
-
\(0.354\,\varOmega - m\)
- \(rad_{Na}\) :
-
\(0.102*10^{ - 9}\) meters
- \(rad_{K}\) :
-
\(0.138*10^{ - 9}\) meters
- \(rad_{Cl}\) :
-
\(0.181*10^{ - 9}\) meters
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Acknowledgements
We thank Prof. Karmeshu, JNU, Delhi and Dr. Pramod Bhatia, The NorthCap University, Gurugram for providing valuable suggestions and guidance in writing this paper.
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Bhatia, S., Singh, P., Sharma, P. (2018). Hodgkin–Huxley Model Revisited to Incorporate the Physical Parameters Affected by Anesthesia. In: Pant, M., Ray, K., Sharma, T., Rawat, S., Bandyopadhyay, A. (eds) Soft Computing: Theories and Applications. Advances in Intelligent Systems and Computing, vol 583. Springer, Singapore. https://doi.org/10.1007/978-981-10-5687-1_47
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