From c08fefa5e30e680e348acf7817201193b1a9634f Mon Sep 17 00:00:00 2001 From: bnewbold Date: Mon, 16 Jan 2017 15:45:26 -0800 Subject: clean up hodgkin-huxley --- examples/hodgkin_huxley/model.modelica | 46 ++++++++++-------- examples/hodgkin_huxley/page.md | 86 ++++++++++++++++++++++++++++++++++ 2 files changed, 113 insertions(+), 19 deletions(-) diff --git a/examples/hodgkin_huxley/model.modelica b/examples/hodgkin_huxley/model.modelica index 782cb20..7e44547 100644 --- a/examples/hodgkin_huxley/model.modelica +++ b/examples/hodgkin_huxley/model.modelica @@ -1,24 +1,32 @@ model HodgkinHuxley - "Predator-Prey Model" - parameter Real G_Na =120 "conductance"; - parameter Real G_K =36 "conductance"; - parameter Real G_lk =0.3 "conductance"; - parameter Real C =1.0 "membrane capacitance"; + "Model of action potential in squid neurons (1952)" + parameter Real C_m =1.0 "membrane capacitance"; + parameter Real g_Na =120 "conductance"; + parameter Real g_K =36 "conductance"; + parameter Real g_L =0.3 "conductance"; parameter Real V_Na =115 "potential"; parameter Real V_K =-12 "potential"; - parameter Real V_lk =-49.387 "leakage"; - Real n; - Real alpha_n; - Real beta_n; - Real m; - Real alpha_m; - Real beta_m; - Real h; - Real alpha_h; - Real beta_h; + parameter Real V_lk =-49.387 "leak reveral potential"; + parameter Real E_Na =-190 "equilibrium potential"; + parameter Real E_K =-63 "equilibrium potential"; + parameter Real E_lk =-85.613 "equilibrium potential"; + parameter Real n =0.31768 "dimensionless; 0 to 1"; + parameter Real m =0.05293 "dimensionless; 0 to 1"; + parameter Real h =0.59612 "dimensionless; 0 to 1"; + Real V_m "membrane voltage potential"; + Real I =1.0 "membrane current"; + Real alpha_n, alpha_m, alpha_h "rate constants"; + Real beta_n, beta_m, beta_h "rate constants"; equation - C * der(V_m) = - G_Na * (Vm - E_Na) - G_K * (V_m - E_K) - G_lk * (V_m - V_lk); - der(n) = - (alpha_n + beta_n) * n + alpha_n; - der(m) = - (alpha_m + beta_m) * m + alpha_m; - der(h) = - (alpha_h + beta_h) * h + alpha_h; + C_m * der(V_m) = I - g_Na * m^3 * h * (V_m - E_Na) - g_K * n^4 * (V_m - E_K) - G_lk * (V_m - E_lk); + der(n) = alpha_n - n * (alpha_n + beta_n); + der(m) = alpha_m - m * (alpha_m + beta_m); + der(h) = alpha_h - h * (alpha_h + beta_h); + + alpha_n = 0.01 * (V_m + 10) / (e^((V_m + 10)/10) - 1); + alpha_m = 0.1 * (V_m + 25) / (e^((V_m + 25)/10) - 1); + alpha_h = 0.07 * e^(V_m / 20); + beta_n = 0.125 * e^(V_m / 80); + beta_m = 4*e^(V_m/18); + beta_h = 1 / (e^((V_m + 30)/10) + 1); end HodgkinHuxley; diff --git a/examples/hodgkin_huxley/page.md b/examples/hodgkin_huxley/page.md index e69de29..4598c97 100644 --- a/examples/hodgkin_huxley/page.md +++ b/examples/hodgkin_huxley/page.md @@ -0,0 +1,86 @@ + +The Hodgkin–Huxley model, or conductance-based model, is a mathematical model +that describes how action potentials in neurons are initiated and propagated. +It is a set of nonlinear differential equations that approximates the +electrical characteristics of excitable cells such as neurons and cardiac +myocytes, and hence it is a continuous time model, unlike the Rulkov map for +example. + +Alan Lloyd Hodgkin and Andrew Fielding Huxley described the model in 1952 to +explain the ionic mechanisms underlying the initiation and propagation of +action potentials in the squid giant axon. They received the 1963 Nobel Prize +in Physiology or Medicine for this work. + +## Mathematical properties + +The Hodgkin–Huxley model can be thought of as a differential equation with four +state variables, v(t), m(t), n(t), and h(t), that change with respect to time +t. The system is difficult to study because it is a nonlinear system and cannot +be solved analytically. However, there are many numeric methods available to +analyze the system. Certain properties and general behaviors, such as limit +cycles, can be proven to exist. + +## Alternative Models + +The Hodgkin–Huxley model is regarded as one of the great achievements of 20th-century biophysics. Nevertheless, modern Hodgkin–Huxley-type models have been extended in several important ways: + +* Additional ion channel populations have been incorporated based on experimental data. + +* The Hodgkin–Huxley model has been modified to incorporate transition state + theory and produce thermodynamic Hodgkin–Huxley models. + +* Models often incorporate highly complex geometries of dendrites and axons, + often based on microscopy data. + +* Stochastic models of ion-channel behavior, leading to stochastic hybrid + systems + +Several simplified neuronal models have also been developed (such as the +FitzHugh–Nagumo model), facilitating efficient large-scale simulation of groups +of neurons, as well as mathematical insight into dynamics of action potential +generation. + + +## References + +The body of this page is from Wikipedia (see below). + +#### Papers + +"The dual effect of membrane potential on sodium conductance in the giant axon +of Loligo". *The Journal of Physiology*. **116** (4): 497–506. April 1952. +doi:10.1113/jphysiol.1952.sp004719. + +"Currents carried by sodium and potassium ions through the membrane of the +giant axon of Loligo". *The Journal of Physiology*. **116** (4): 449–72. April 1952. +doi:10.1113/jphysiol.1952.sp004717. + +"The components of membrane conductance in the giant axon of Loligo". *The +Journal of Physiology*. **116** (4): 473–96. April 1952. +doi:10.1113/jphysiol.1952.sp004718. + +"The dual effect of membrane potential on sodium conductance in the giant axon +of Loligo". *The Journal of Physiology*. **116** (4): 497–506. April 1952. +doi:10.1113/jphysiol.1952.sp004719. + +"A quantitative description of membrane current and its application to +conduction and excitation in nerve". *The Journal of Physiology*. **117** (4): +500–44. August 1952. doi:10.1113/jphysiol.1952.sp004764. + +#### Interactive Models on the Web + +* ModelDB: [Squid axon (Hodgkin, Huxley 1952)](https://senselab.med.yale.edu/ModelDB/ShowModel.cshtml?model=5426) +* Wolfram Demonstrations: + [Interactive Hodgkin-Huxley](http://demonstrations.wolfram.com/HodgkinHuxleyActionPotentialModel/) + by Shimon Marom and + [Neural Impulses: The Action Potential in Action](http://www.demonstrations.wolfram.com/NeuralImpulsesTheActionPotentialInAction/) + by Garrett Neske +* [Hodgkin-Huxley Simulation with Javascript](http://myselph.de/hodgkinHuxley.html) + by Hubert Eichner, which creates static plots in the browser. +* BioModels database: [](http://www.ebi.ac.uk/biomodels-main/BIOMD0000000020) + +#### Other Links + +* Wikipedia: [Hodgkin–Huxley model](https://en.wikipedia.org/wiki/Hodgkin%E2%80%93Huxley_model) +* [Summary of the Hodgkin-Huxley model](http://ecee.colorado.edu/~ecen4831/HHsumWWW/HHsum.html) +* [Hodgkin-Huxley model in R](http://www.magesblog.com/2012/06/hodgkin-huxley-model-in-r.html) -- cgit v1.2.3