Ex18: Goldman-Hodgkin-Katz

GHK current equation for ion channel permeability

Model: GHK Current Equation

The Goldman-Hodgkin-Katz flux equation provides a more physically accurate description of ion channel current than the simple Ohmic (\(g(V-E)\)) model, accounting for ionic concentrations on both sides of the membrane:

\[I = P \cdot z^2 \cdot \frac{F^2 V}{RT} \cdot \frac{[\text{ion}]_i - [\text{ion}]_o \exp(-zFV/RT)}{1 - \exp(-zFV/RT)}\]

The underlying cell is Hodgkin-Huxley with GHK ion currents.

1. Define HH Cell in TVBO

from tvbo import SimulationExperiment

exp = SimulationExperiment.from_string("""
label: "NeuroML Ex18: HH with GHK"
dynamics:
  name: HodgkinHuxley
  parameters:
    C:     { value: 10.0 }
    g_Na:  { value: 1200.0 }
    g_K:   { value: 360.0 }
    g_L:   { value: 3.0 }
    E_Na:  { value: 50.0 }
    E_K:   { value: -77.0 }
    E_L:   { value: -54.3 }
    I_ext: { value: 0.08 }
  derived_variables:
    alpha_m:
      equation:
        rhs: "Piecewise((1.0, Eq(v, -40.0)), (0.1*(v + 40.0)/(1.0 - exp(-(v + 40.0)/10.0)), True))"
    beta_m:
      equation:
        rhs: "4.0*exp(-(v + 65.0)/18.0)"
    alpha_h:
      equation:
        rhs: "0.07*exp(-(v + 65.0)/20.0)"
    beta_h:
      equation:
        rhs: "1.0/(1.0 + exp(-(v + 35.0)/10.0))"
    alpha_n:
      equation:
        rhs: "Piecewise((0.1, Eq(v, -55.0)), (0.01*(v + 55.0)/(1.0 - exp(-(v + 55.0)/10.0)), True))"
    beta_n:
      equation:
        rhs: "0.125*exp(-(v + 65.0)/80.0)"
  state_variables:
    v:
      equation:
        rhs: "(-g_Na*m**3*h*(v - E_Na) - g_K*n**4*(v - E_K) - g_L*(v - E_L) + I_ext*1000) / C"
      initial_value: -65.0
      variable_of_interest: true
    m:
      equation: { rhs: "alpha_m*(1 - m) - beta_m*m" }
      initial_value: 0.05
    h:
      equation: { rhs: "alpha_h*(1 - h) - beta_h*h" }
      initial_value: 0.6
    n:
      equation: { rhs: "alpha_n*(1 - n) - beta_n*n" }
      initial_value: 0.32
network:
  number_of_nodes: 1
integration:
  method: euler
  step_size: 0.01
  duration: 300.0
  time_scale: ms
""")
print(f"Model: {exp.dynamics.name}")
Model: HodgkinHuxley

2. Render LEMS XML

xml = exp.render("lems")
print(xml[:1200])

<Lems>

  <!-- Tell jLEMS/jNeuroML which component is the simulation entry point. -->
  <Target component="sim_NeuroML_Ex18__HH_with_GHK"/>

  <!-- ════════════════════════════════════════════════════════════════
       Dimensions & Units (inline — no external includes needed)
       ════════════════════════════════════════════════════════════════ -->

  <!-- Dimensions -->
  <Dimension name="none"/>
  <Dimension name="time" t="1"/>
  <Dimension name="voltage" m="1" l="2" t="-3" i="-1"/>
  <Dimension name="per_time" t="-1"/>
  <Dimension name="conductance" m="-1" l="-2" t="3" i="2"/>
  <Dimension name="capacitance" m="-1" l="-2" t="4" i="2"/>
  <Dimension name="current" i="1"/>
  <Dimension name="resistance" m="1" l="2" t="-3" i="-2"/>
  <Dimension name="concentration" l="-3" n="1"/>
  <Dimension name="substance" n="1"/>
  <Dimension name="charge" t="1" i="1"/>
  <Dimension name="temperature" k="1"/>

  <!-- Units -->
  <Unit symbol="s" dimension="time" power="0"/>
  <Unit symbol="ms" dimension="time" power="-3"/>
  <Unit symbol="us" dimension="time" power="-6"/>
  <Unit symbol="V" dimension="voltage" power="0"/>
  <Unit symbol="mV" dimension="voltage" power="-3"/>
  <Unit symbol=

3. Run Reference

import sys, os
sys.path.insert(0, os.path.dirname(os.path.abspath(".")))
from _nml_helpers import run_lems_example

ref_outputs = run_lems_example("LEMS_NML2_Ex18_GHK.xml")
for name, arr in ref_outputs.items():
    print(f"  {name}: shape={arr.shape}")
  ex18.dat: shape=(50001, 4)

4. Run TVBO

import numpy as np

result = exp.run("neuroml")
da = result.integration.data
tvbo_arr = np.column_stack([da.coords['time'].values, da.values])
print(f"TVBO: shape={tvbo_arr.shape}")
TVBO: shape=(30001, 5)

5. Plot Reference

import matplotlib.pyplot as plt
import numpy as np

for name, ref_arr in ref_outputs.items():
    t = ref_arr[:, 0] * 1000
    fig, ax = plt.subplots(figsize=(10, 4))
    for i in range(1, min(ref_arr.shape[1], 4)):
        ax.plot(t, ref_arr[:, i] * 1000, alpha=0.8, label=f'Cell {i}')
    ax.set_xlabel("Time (ms)")
    ax.set_ylabel("Voltage (mV)")
    ax.set_title(f"Ex18: GHK — {name}")
    ax.legend(fontsize=7)
    ax.grid(True, alpha=0.3)
    plt.tight_layout()
    plt.show()

GHK in TVBO

The GHK flux equation can be expressed as a derived variable in TVBO. The full GHK current replaces the Ohmic \(g(V-E)\) formulation in the voltage ODE.