Leaky Integrate-and-Fire (LIF)

The LIF model treats the heart's sinoatrial node as a biological pacemaker neuron. Each heartbeat corresponds to a membrane voltage threshold crossing, producing an inter-beat interval (IBI) directly in physiological time (milliseconds).

Theory

The sinoatrial node drives cardiac rhythm by spontaneously depolarizing and then resetting after each beat. The leaky integrate-and-fire model captures this with a single first-order ODE:

\[\tau \frac{dV}{dt} = -(V - V_{rest}) + R \cdot I\]

The membrane potential $V$ rises toward the fixed point $V^* = V_{rest} + R \cdot I$. When $V$ crosses $V_{threshold}$ (upward), a spike (heartbeat) is recorded and $V$ is immediately reset to $V_{reset}$.

Analytical Period Formula

For constant current $I$ with $V^* > V_{threshold}$ (a necessary condition for repetitive firing), the inter-spike period has a closed form:

\[T = \tau \cdot \ln\!\left(\frac{R \cdot I}{R \cdot I - \Delta V}\right), \quad \Delta V = V_{threshold} - V_{rest}\]

Inverting for the current that produces a target period $T$:

\[I(T) = \frac{\Delta V}{R \cdot \left(1 - e^{-T/\tau}\right)}\]

This inverse formula is the basis of the :analytical fitting method, which applies it to each measured IBI individually.

Parameters

Fixed Physiological Parameters

These are set at construction and are not fitted. They reflect biologically plausible values for the human sinoatrial node.

Fitted Parameter

The bounds on I are set in parameter_space(::LIF). Higher I values drive faster firing; lower values produce slower rhythms. Use fit to determine the I that best matches a measured IBI series.

Fitting Methods

Examples

Create and Simulate

using HeartRateLab

# Create model with physiological defaults
lif = LIF()

# Simulate 500 inter-beat intervals
params = (I = 1.52,)
ibis = simulate(lif, params, 500)

using Statistics
println("Mean IBI: ", round(mean(ibis); digits=1), " ms")
println("Heart rate: ", round(60000 / mean(ibis)), " bpm")

Fit with Analytical Method

The analytical method is the fastest option. It returns a per-beat current series capturing how autonomic drive varies across the recording.

using HeartRateLab

lif = LIF()
ibis = read_txt("data.txt")

result = fit(lif, ibis; method=:analytical)

println("Mean I:  ", round(result.params.I; digits=4))
println("Std(I):  ", round(result.diagnostics["I_std"]; digits=4))

# Per-beat current series: same length as ibis
I_series = parameter_series(result, :I)
println("I range: ", round.(extrema(I_series); digits=4))

# Simulate from the mean fitted I
synthetic = simulate(result.model, result.params, length(ibis))

Fit with Gradient Method

using HeartRateLab

lif = LIF()
ibis = read_txt("data.txt")

result = fit(lif, ibis; method=:gradient)

println("Fitted I:    ", round(result.params.I; digits=4))
println("Converged:   ", result.diagnostics["converged"])
println("Final RMSE:  ", round(result.diagnostics["loss_final"]; digits=2), " ms")

# Generate synthetic IBIs
synthetic = simulate(result.model, result.params, length(ibis))

API Reference

LIF <: AbstractHRVModel

parameter_space(model::LIF) -> NamedTuple

simulate(model::LIF, params::NamedTuple, n_beats::Int) -> Vector{Float64}

fit(model::LIF, data::Vector{Float64}; method=:bayesian, kwargs...) -> ModelFitResult