chapter19.md

Bursting

Overview

These examples create bursts by adding a slow potassium current to Erisir and INaP-I$_K$ neurons. The slow state moves the fast subsystem between silent and spiking attractors, yielding alternation between quiescence and rapid spikes.

Core ideas

Bursting is a slow-fast rhythm: for an almost fixed slow gate, the fast subsystem has either a resting state or a spiking cycle. As the slow potassium gate rises during spiking, it reduces effective drive; as it falls during silence, it restores excitability. Hysteresis between the fast-subsystem transitions closes the loop.

Essential model

The added slow outward current is

$$ I_{K,\rm slow}=g_{K,\rm slow}n_{\rm slow}(v_K-v),\qquad \dot n_{\rm slow}=\frac{n_{\rm slow,\infty}(v)-n_{\rm slow}} {\tau_{n,\rm slow}}. $$

Here $n_{\rm slow}$ is the slow potassium activation, $g_{K,\rm slow}$ is its conductance, $v_K$ is the potassium reversal potential, and $\tau_{n,\rm slow}$ separates its evolution from the fast voltage and gate dynamics.

Code examples

All nine examples now live in one notebook, chapter19.ipynb: simulate_inapik_plus_slow_i_k simulates an INaP-I$K$ burst with the reference slow conductance ($g{K,\rm slow}=5$; an interactive slider lets you sweep it); simulate_inapik_plus_weak_slow_i_k/ simulate_inapik_plus_strong_slow_i_k show the response with weaker/stronger slow potassium feedback; simulate_inapik_show_slow_i_k plots INaP-I$_K$ voltage with the effective drive and transition thresholds; simulate_square_waves marks the quasi-steady (square-wave) portions of an INaP-I$_K$ burst; simulate_inapik_plus_slow_i_k_3d plots the settled INaP-I$K$ cycle in $(v,n,n{\rm slow})$ space; simulate_erisir_plus_slow_i_k simulates an Erisir burst with slow potassium feedback and is reused by plot_erisir_show_slow_i_k (voltage with effective drive and thresholds) and plot_ellipses (local extrema marked on the trace).

What to look for

Compare the weak, reference, and strong INaP-I$_K$ slow conductances (or sweep g_k_slow with the interactive slider) to see how slow feedback changes the rhythm. The _show_slow_i_k plots make the effective current cross thresholds while the voltage alternates. Use simulate_inapik_plus_slow_i_k_3d to connect the trace to a three-dimensional loop.

Suggested order

1. Run simulate_inapik_plus_slow_i_k, its weak and strong variants, and simulate_square_waves.
2. Run simulate_inapik_show_slow_i_k and simulate_inapik_plus_slow_i_k_3d.
3. Compare simulate_erisir_plus_slow_i_k with plot_erisir_show_slow_i_k and plot_ellipses.

Prerequisites and related chapters

Chapter 15 introduces slow-fast canard behavior, Chapter 17 provides the fast-subsystem f--I interpretation, and Chapter 18 shows how slow currents create hysteresis and bistability.

Running the examples

Open chapter19.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom.