chapter09.md

Spike-frequency adaptation

Overview

These examples add slow negative feedback to spiking models. They cover a voltage-dependent M-current, a calcium-dependent afterhyperpolarization (AHP) current, a reset-based adaptation variable, and maps that describe how the slow variable changes from one spike to the next.

Core ideas

Adaptation accumulates during activity and reduces subsequent excitability, which can lengthen interspike intervals. The M-current uses a slow gate, whereas the AHP current uses a calcium-like state. Resting variants set the external drive to zero, showing the models' zero-drive resting trajectories alongside sustained spiking regimes. A spike-to-spike map condenses continuous evolution into a one-dimensional update.

Essential model

For the adaptation LIF example, the equations and spike update are

$$ \frac{dV}{dt}=-\frac{V}{\tau_m}+I-wV,\qquad \frac{dw}{dt}=-\frac{w}{\tau_a},\qquad V\ge1\Rightarrow (V,w)\leftarrow(0,w+\Delta). $$

Here $V$ is normalized voltage, $t$ is time, $\tau_m$ is membrane time constant, $I$ is applied input, $w$ is the nonnegative adaptation state, $\tau_a$ is its decay time constant, and $\Delta$ is its increment at a spike. In the RTM M-current model, the added current is $I_M=g_Mw(V-E_K)$, where $g_M$ is maximal M conductance and $E_K$ is the potassium reversal potential. In the AHP model it is $I_{AHP}=g_{AHP}c(V-E_K)$, where $g_{AHP}$ is maximal AHP conductance and $c$ is the calcium-like state.

Code examples

All nine examples now live in one notebook, chapter09.ipynb: M_CURRENT and CALCIUM_RISE plot the underlying steady-state laws; RTM_M/RTM_M_RESTING and RTM_AHP/RTM_AHP_RESTING add each slow current to a driven vs. zero-drive RTM neuron; LIF_ADAPT integrates the reset-based adaptation model; ADAPTATION_MAP computes the spike-to-spike map φ(z); and V_V_TILDE compares two subthreshold voltages with different initial adaptation amplitudes. Each section has an ipywidgets slider to explore its parameters interactively.

What to look for

In RTM_M and RTM_AHP, watch the slow state rise while spiking continues; the resting versions show what remains when drive is removed. LIF_ADAPT makes the discrete spike increment explicit. In ADAPTATION_MAP, intersections of $\phi(z)$ with the diagonal identify fixed spike-to-spike adaptation levels, where $z$ is the adaptation value just after a spike and $\phi(z)$ is its value after the next spike.

Suggested order

1. Run M_CURRENT and CALCIUM_RISE to inspect the slow feedback laws.
2. Compare each driven RTM model with its _RESTING counterpart.
3. Run LIF_ADAPT, V_V_TILDE, and finally ADAPTATION_MAP to connect continuous slow dynamics with a per-spike description.

Prerequisites and related chapters

Chapter 05 introduces the RTM neuron used here, and Chapter 07 supplies the LIF reset convention. The QIF and theta models in Chapter 08 provide other ways to represent spike events and phase.

Running the examples

Open chapter09.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.