chapter07.md

Linear integrate-and-fire neurons

Overview

The linear integrate-and-fire (LIF) model keeps the passive accumulation of a membrane but replaces an action potential's ionic mechanism with a threshold and reset rule. The examples also measure the effective subthreshold behavior of the HH model before comparing the two descriptions.

Core ideas

Below threshold, voltage decays toward a driven equilibrium with membrane time constant $\tau_m$. Crossing the chosen threshold is an event: the script records a reset instead of resolving spike-generating sodium and potassium currents. This makes LIF economical, but it cannot reproduce the HH gates or their biophysical refractory dynamics without additional rules.

Essential model

The subthreshold equation and event rule are

$$ \frac{dV}{dt}=-\frac{V}{\tau_m}+I,\qquad V\ge V_{\mathrm{th}}\Rightarrow V\leftarrow V_{\mathrm{reset}}. $$

Here $V$ is the model voltage, $t$ is time, $\tau_m$ is the membrane time constant, $I$ is constant input in the code's normalized units, $V_{\mathrm{th}}$ is threshold, and $V_{\mathrm{reset}}$ is the voltage assigned after a threshold crossing. For the HH comparison, the instantaneous effective time constant is $\tau=C/(g_{\mathrm{K}}n^4+g_{\mathrm{Na}}m^3h+g_{\mathrm{L}})$, where $C$ is capacitance, $g_{\mathrm{K}}$, $g_{\mathrm{Na}}$, and $g_{\mathrm{L}}$ are conductances, and $m$, $n$, and $h$ are HH gates.

Code examples

All five examples now live in one notebook, chapter07.ipynb, with two simulate_* functions covering them: simulate_hh_subthreshold backs LIF_NEURON_WITH_HH, SUBTHR_FOR_HH, and TAU_M_FOR_HH's figures (HH voltage vs. the LIF linear approximation, the current decomposition, and the effective time constant); simulate_lif_neuron(tau_m, i) backs LIF_VOLTAGE_TRACE and LIF_VOLTAGE_TRACE_2. Each has an ipywidgets slider to explore its parameters interactively.

What to look for

In the LIF traces, the discontinuous reset is a declared convention, not a numerically resolved membrane event. Compare it with the smooth HH trace in LIF_NEURON_WITH_HH, then use SUBTHR_FOR_HH and TAU_M_FOR_HH to see why a single fixed passive time constant is only an approximation.

Suggested order

1. Run LIF_VOLTAGE_TRACE and identify threshold and reset.
2. Run LIF_VOLTAGE_TRACE_2 to connect input strength with interspike timing.
3. Read the three HH comparison examples in the order listed above.

Prerequisites and related chapters

Chapters 01--04 provide the HH currents, gates, and trajectories used for comparison. Chapter 08 replaces the linear subthreshold term with a quadratic one; Chapter 09 adds slow adaptation to a LIF-style model.

Running the examples

Open chapter07.ipynb in Jupyter, or via the Colab badge at the top of the notebook. Run all cells top to bottom; each section's static figure reproduces the book's plot, and the interact(...) cell below it lets you adjust that example's parameters with sliders.