Leaky Integrate-and-Fire (LIF) Neuron Model

A simulation of the Leaky Integrate-and-Fire (LIF) neuron — the simplest widely-used spiking neuron model, capturing how a neuron charges up, leaks, and fires without modeling the biophysical machinery of ion channels.

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What it does

• Simulates a noisy LIF neuron via Euler integration with a refractory period
• Plots voltage traces under sub-threshold, supra-threshold, and strong current drive
• Measures the membrane time constant (τ_m) directly from simulated decay/rise curves and compares to theory
• Builds an f-I curve (firing rate vs. injected current) and compares simulation against the exact analytical solution
• Computes inter-spike interval (ISI) statistics, including the coefficient of variation as a measure of spike-timing irregularity

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Background

Unlike the biophysically detailed Hodgkin-Huxley model, the LIF neuron treats the membrane as a simple leaky capacitor: current flows in, voltage rises, and it "leaks" back toward rest. When voltage crosses a threshold, the neuron fires a spike and resets — with no explicit spike-generating currents.

Concept	What it captures
Rheobase (I_rheo)	Minimum constant current needed to ever fire
Membrane time constant (τ_m)	How fast voltage responds to input — sets the "leak" speed
Refractory period (t_ref)	Enforces a maximum possible firing rate
ISI coefficient of variation	Spike-timing regularity — 0 for noiseless/regular firing, ~1 for Poisson-like biological spiking

Despite its simplicity, the LIF model reproduces key neural input-output properties (like the f-I curve) almost exactly, which is why it's the workhorse neuron model in large-scale spiking network simulations.

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Installation

pip install numpy matplotlib

Usage

python lif_neuron.py

Runs all three simulations, saves 3 figures to the working directory, and prints summary spike statistics.

Core function

V, spike_times = simulate_lif(
    I_ext, t, dt, tau_m, V_rest, V_thresh, V_reset, R_m, t_ref,
    sigma=1.0   # noise strength (mV); set to 0 for deterministic dynamics
)

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Outputs

Figure	Shows
fig1_voltage_traces.png	Voltage traces at 0.8×, 1.5×, and 3.0× rheobase current, with spike counts and firing rates
fig2_subthreshold.png	(A) Exponential decay after a brief pulse, used to measure τ_m; (B) exponential rise to steady state under a sub-threshold step current
fig3_fi_curve.png	Simulated firing rate vs. current, overlaid on the exact analytical F-I curve

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Model parameters (defaults)

Parameter	Value	Meaning
tau_m	20 ms	Membrane time constant
V_rest	−70 mV	Resting potential
V_thresh	−55 mV	Spike threshold
V_reset	−75 mV	Post-spike reset potential
R_m	10 MΩ	Membrane resistance
t_ref	2 ms	Refractory period
sigma	1.0 mV	Noise strength

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Math, briefly

Subthreshold dynamics: dV/dt = [−(V − V_rest) + R_m·I_ext] / τ_m (+ noise)

Spike rule: if V ≥ V_thresh → record spike, set V = V_reset, hold for t_ref

Rheobase: I_rheo = (V_thresh − V_rest) / R_m

Analytical F-I curve (noiseless):

V_ss = V_rest + R_m·I
T_ISI = τ_m·ln[(V_ss − V_reset)/(V_ss − V_thresh)] + t_ref     (if V_ss > V_thresh)
f = 1000 / T_ISI  (Hz)

Max firing rate: f_max = 1000 / t_ref

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Roadmap

• Adaptive threshold / adaptation currents (AdEx-style extensions)
• Synaptic input (conductance-based) instead of pure current injection
• Network-level simulations with LIF populations
• Alternative noise models (Ornstein-Uhlenbeck colored noise)

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License

MIT — see LICENSE.

References

• Lapicque, L. (1907) — Recherches quantitatives sur l'excitation électrique des nerfs
• Gerstner, W., Kistler, W. M., Naud, R., & Paninski, L. — Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition
• Dayan, P., & Abbott, L. F. — Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems