# examples/HodgkinHuxley

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Model of action potential in squid neurons (1952)

$$\begin{array}{rcl} C_{ m } \cdot \frac{ d V_{ m } }{ dt } & = & \left(\left(\left(I - g_{ Na } \cdot m^{ 3 } \cdot h \cdot \left(V_{ m } - E_{ Na }\right)\right) - g_{ K } \cdot n^{ 4 } \cdot \left(V_{ m } - E_{ K }\right)\right) - G_{ lk } \cdot \left(V_{ m } - E_{ lk }\right)\right) \\ \frac{ d n }{ dt } & = & \left(\alpha_{ n } - n \cdot \left(\alpha_{ n } + \beta_{ n }\right)\right) \\ \frac{ d m }{ dt } & = & \left(\alpha_{ m } - m \cdot \left(\alpha_{ m } + \beta_{ m }\right)\right) \\ \frac{ d h }{ dt } & = & \left(\alpha_{ h } - h \cdot \left(\alpha_{ h } + \beta_{ h }\right)\right) \\ \alpha_{ n } & = & \frac{ 0.01 \cdot \left(V_{ m } + 10\right) }{ \left(e^{ \frac{ \left(V_{ m } + 10\right) }{ 10 } } - 1\right) } \\ \alpha_{ m } & = & \frac{ 0.1 \cdot \left(V_{ m } + 25\right) }{ \left(e^{ \frac{ \left(V_{ m } + 25\right) }{ 10 } } - 1\right) } \\ \alpha_{ h } & = & 0.07 \cdot e^{ \frac{ V_{ m } }{ 20 } } \\ \beta_{ n } & = & 0.125 \cdot e^{ \frac{ V_{ m } }{ 80 } } \\ \beta_{ m } & = & 4 \cdot e^{ \frac{ V_{ m } }{ 18 } } \\ \beta_{ h } & = & \frac{ 1 }{ \left(e^{ \frac{ \left(V_{ m } + 30\right) }{ 10 } } + 1\right) } \\ \end{array}$$
Variable Type Description Value (default) Datatype
$$C_{ m }$$ parameter membrane capacitance $$1$$ Real
$$g_{ Na }$$ parameter conductance $$120$$ Real
$$g_{ K }$$ parameter conductance $$36$$ Real
$$g_{ L }$$ parameter conductance $$0.3$$ Real
$$V_{ Na }$$ parameter potential $$115$$ Real
$$V_{ K }$$ parameter potential $$-12$$ Real
$$V_{ lk }$$ parameter leak reveral potential $$-49.387$$ Real
$$E_{ Na }$$ parameter equilibrium potential $$-190$$ Real
$$E_{ K }$$ parameter equilibrium potential $$-63$$ Real
$$E_{ lk }$$ parameter equilibrium potential $$-85.613$$ Real
$$n$$ parameter dimensionless; 0 to 1 $$0.31768$$ Real
$$m$$ parameter dimensionless; 0 to 1 $$0.05293$$ Real
$$h$$ parameter dimensionless; 0 to 1 $$0.59612$$ Real
$$V_{ m }$$ free membrane voltage potential  Real
$$I$$ free membrane current $$1$$ Real
$$\alpha_{ n }$$ free  Real
$$\alpha_{ m }$$ free  Real
$$\alpha_{ h }$$ free rate constants  Real
$$\beta_{ n }$$ free  Real
$$\beta_{ m }$$ free  Real
$$\beta_{ h }$$ free rate constants  Real

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

#### 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.

## Modelica Code

model HodgkinHuxley
"Model of action potential in squid neurons (1952)"
parameter Real C_m=1e0  "membrane capacitance";
parameter Real g_Na=120  "conductance";
parameter Real g_K=36  "conductance";
parameter Real g_L=3e-1  "conductance";
parameter Real V_Na=115  "potential";
parameter Real V_K=-12  "potential";
parameter Real V_lk=-4.9387e1  "leak reveral potential";
parameter Real E_Na=-190  "equilibrium potential";
parameter Real E_K=-63  "equilibrium potential";
parameter Real E_lk=-8.5613e1  "equilibrium potential";
parameter Real n=3.1768e-1  "dimensionless; 0 to 1";
parameter Real m=5.293e-2  "dimensionless; 0 to 1";
parameter Real h=5.9612e-1  "dimensionless; 0 to 1";
Real V_m "membrane voltage potential";
Real I=1e0  "membrane current";
Real alpha_n;
Real alpha_m;
Real alpha_h "rate constants";
Real beta_n;
Real beta_m;
Real beta_h "rate constants";
equation
(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 = ((1e-2 * (V_m + 10)) / ((e ^ ((V_m + 10) / 10)) - 1));
alpha_m = ((1e-1 * (V_m + 25)) / ((e ^ ((V_m + 25) / 10)) - 1));
alpha_h = ((7e-2 * e) ^ (V_m / 20));
beta_n = ((1.25e-1 * e) ^ (V_m / 80));
beta_m = ((4 * e) ^ (V_m / 18));
beta_h = (1 / ((e ^ ((V_m + 30) / 10)) + 1));
end HodgkinHuxley;


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