How small should dt be?

Too large (dt = 1): you overshoot and miss dynamics
Too small (dt = 0.00001): you're wasting computation for no extra accuracy

Rule of thumb: dt << tau (the neuron's time constant). For a neuron with tau_m = 10 ms, use dt ≤ 1 ms.

import numpy as np
import matplotlib.pyplot as plt

# Simple exponential decay: dV/dt = -V / tau
V_rest = 0
tau = 10  # ms
V_init = 1

# Test different time steps
for dt in [0.1, 1, 5]:  # too big, good, too big
    t_max = 50
    t = np.arange(0, t_max, dt)
    V = np.zeros_like(t)
    V[0] = V_init
    
    for i in range(len(t)-1):
        dV_dt = -V[i] / tau
        V[i+1] = V[i] + dV_dt * dt
    
    # Plot
    plt.plot(t, V, 'o-', label=f'dt={dt}', markersize=3)

# Analytical solution (true answer)
t_exact = np.linspace(0, 50, 1000)
V_exact = V_init * np.exp(-t_exact / tau)
plt.plot(t_exact, V_exact, 'k--', linewidth=2, label='Analytical')

plt.legend()
plt.xlabel('Time (ms)')
plt.ylabel('Voltage (mV)')
plt.title('Euler Method: Effect of Time Step Size')
plt.show()