Ex15: Calcium Dynamics

Intracellular calcium concentration pool and calcium-dependent K channels

Model: Calcium Dynamics

Demonstrates decayingPoolConcentrationModel — a first-order calcium pool:

\[\frac{d[\text{Ca}^{2+}]}{dt} = -\frac{I_{Ca}}{2 F \cdot \text{shell}} - \frac{[\text{Ca}^{2+}] - [\text{Ca}^{2+}]_{\text{rest}}}{\tau}\]

This is a granule cell with 7 channel types (Na, KDr, H, CaHVA, KA, KCa, leak) plus the Ca concentration pool. All are expressible as TVBO state variables and derived variables.

---

1. Define HH Cell in TVBO

from tvbo import SimulationExperiment

# Standard HH representing the core membrane dynamics
exp = SimulationExperiment.from_string("""
label: "NeuroML Ex15: HH Cell (Ca Dynamics)"
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: 55.0 }
    E_K:   { value: -90.0 }
    E_L:   { value: -65.0 }
    I_ext: { value: 0.01 }
  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: 700.0
  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_Ex15__HH_Cell__Ca_Dynamics_"/>

  <!-- ════════════════════════════════════════════════════════════════
       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"/>
  <Un

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_Ex15_CaDynamics.xml")
for name, arr in ref_outputs.items():
    print(f"  {name}: shape={arr.shape}")

  ex15_cond_dens.dat: shape=(700001, 6)
  ex15_curr_dens.dat: shape=(700001, 6)
  ex15_v.dat: shape=(700001, 2)

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=(70001, 5)

5. Plot Reference (Full Granule Cell)

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], alpha=0.8, label=f'col {i}')
    ax.set_xlabel("Time (ms)")
    ax.set_title(f"Ex15: Ca Dynamics — {name}")
    ax.legend(fontsize=7)
    ax.grid(True, alpha=0.3)
    plt.tight_layout()
    plt.show()

Ca Dynamics in TVBO

The calcium pool ODE and Ca-dependent channel rates are expressible as TVBO state variables and derived variables. The full model (~17 state variables) could be defined as a single TVBO Dynamics.