Neuron Modelling: Membrane Potential

Explore the generation and maintenance of membrane potential.

1. Introduction

This post serves as a comprehensive test of the Quarto deployment pipeline. We will verify: 1. \(\LaTeX\) Rendering: For mathematical derivations. 2. Python Execution: Running simulation code during build time. 3. Visualization: Embedding generated figures. 4. Layout: Using callouts and columns.

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2. Mathematical Model

The Leaky Integrate-and-Fire (LIF) neuron describes the evolution of membrane potential \(V(t)\) over time. The subthreshold dynamics are governed by the linear differential equation:

\[ \tau_m \frac{dV(t)}{dt} = -[V(t) - E_L] + R_m I_e(t) \]

Where:

• \(\tau_m\): Membrane time constant (\(R_m C_m\))
• \(E_L\): Resting potential (Leak reversal potential)
• \(R_m\): Membrane resistance
• \(I_e(t)\): Input current

NoteThreshold Condition

When \(V(t)\) reaches the threshold \(V_{th}\), the neuron fires a spike, and \(V(t)\) is instantly reset to \(V_{reset}\).

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3. Simulation Code

Here we implement the Euler method to solve the equation numerically. The time step is \(\Delta t\).

\[ V[t+1] = V[t] + \frac{\Delta t}{\tau_m} (- (V[t] - E_L) + R_m I_e) \]

import numpy as np
import matplotlib.pyplot as plt

# 1. Parameters
T = 100.0        # Total time (ms)
dt = 0.1         # Time step (ms)
time = np.arange(0, T+dt, dt)

tau_m = 20.0     # Time constant (ms)
E_L = -70.0      # Resting potential (mV)
R_m = 10.0       # Resistance (Mohm)
V_th = -55.0     # Threshold (mV)
V_reset = -75.0  # Reset potential (mV)
I_e = 2.0        # Input current (nA)

# 2. Initialization
V = np.zeros(len(time))
V[0] = E_L
spikes = []

# 3. Simulation Loop (Euler Method)
for i in range(len(time)-1):
    # Differential equation
    dV = (-(V[i] - E_L) + R_m * I_e) / tau_m
    V[i+1] = V[i] + dV * dt
    
    # Spike mechanism
    if V[i+1] >= V_th:
        V[i+1] = V_reset
        spikes.append(time[i+1])

# 4. Plotting
plt.figure(figsize=(10, 5))
plt.plot(time, V, label='Membrane Potential', color='#2c3e50', linewidth=1.5)
plt.axhline(V_th, color='#e74c3c', linestyle='--', label='Threshold')
plt.axhline(E_L, color='#95a5a6', linestyle=':', label='Resting Potential')

# Mark spikes
for t_spike in spikes:
    plt.axvline(t_spike, color='#e74c3c', alpha=0.3)

plt.title(f"LIF Neuron Response (I = {I_e} nA)")
plt.xlabel("Time (ms)")
plt.ylabel("Potential (mV)")
plt.legend(loc='upper right')
plt.grid(True, alpha=0.3)

# This confirms Python generated the plot
plt.show()

Figure 1: Simulation of an LIF neuron under constant current injection.

4. System Info

To ensure we are using the correct environment, here is the system information:

import sys
import datetime

print(f"Python Version: {sys.version.split()[0]}")
print(f"Build Date: {datetime.datetime.now().strftime('%Y-%m-%d %H:%M:%S')}")

Python Version: 3.11.11
Build Date: 2026-01-26 17:41:38

5. Conclusion

If you can see the math equations properly rendered and the blue plot above, congratulations! Your Neuro-Engineering Blog stack is fully operational.

Next step: Integrate interactive Streamlit or Gradio modules via Docker.