Ex6: NMDA Synapse

Voltage-dependent NMDA synapse with Mg²⁺ block

Model: NMDA Synapse

NMDA receptor-mediated synapse with voltage-dependent Mg²⁺ block. The block factor is:

\[B(v) = \frac{1}{1 + [\text{Mg}^{2+}]/3.57 \cdot \exp(-0.062 \cdot v)}\]

The synapse conductance follows \(dg/dt = -g/\tau_{\text{decay}}\) with the post-synaptic current \(I_{\text{syn}} = g \cdot B(v) \cdot (v - E_{\text{rev}})\).

TVBO models the HH pre-cell dynamics (same as Ex1/Ex3).

1. Define Pre-Cell in TVBO

from tvbo import SimulationExperiment

# Same HH pre-cell as Ex3, I_ext = 0.065 nA
exp = SimulationExperiment.from_string("""
label: "NeuroML Ex6: HH Pre-Cell (NMDA Network)"
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: 0.065 }
  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: 150.0
  time_scale: ms
""")
print(f"Model: {exp.dynamics.name}")
Model: HodgkinHuxley

2. Render LEMS XML

xml = exp.render("lems")
print(xml[:1500])

<Lems>

  <!-- Tell jLEMS/jNeuroML which component is the simulation entry point. -->
  <Target component="sim_NeuroML_Ex6__HH_Pre_Cell__NMDA_Network_"/>

  <!-- ════════════════════════════════════════════════════════════════
       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="A" dimension="current" power="0"/>
  <Unit symbol="mA" dimension="current" power="-3"/>
  <Unit symbol="nA" dimension="current" power="-9"/>
  <Unit symbol="pA" dimension="current" power="-12"/>
  <Unit symbol="S" dimension="conductance" power="0"/>
  <Unit symbol="mS" dimension="cond

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_Ex6_NMDA.xml")
for name, arr in ref_outputs.items():
    print(f"  {name}: shape={arr.shape}")
  ex6_block.dat: shape=(40001, 2)
  ex6_g.dat: shape=(40001, 2)
  ex6_v.dat: shape=(40001, 2)

4. Run TVBO (Pre-Cell)

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=(15001, 5)

5. Plot Reference

import matplotlib.pyplot as plt
import numpy as np

ref_arr = list(ref_outputs.values())[0]
t = ref_arr[:, 0] * 1000

fig, ax = plt.subplots(figsize=(10, 4))
for i in range(1, min(ref_arr.shape[1], 5)):
    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("Ex6: NMDA Synapse Network — Voltage Traces (NeuroML reference)")
ax.legend()
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

NMDA Dynamics

The NMDA synapse ODE \(dg/dt = -g/\tau_d\) with Mg²⁺ block \(B(v)\) is expressible as a TVBO derived variable + state variable. The network topology (pre→post projections) is NeuroML-native.