chapter26.md

Phase locking of two oscillators

Overview

Two pulse-coupled oscillators can be studied with an event-to-event phase map instead of their full voltage trajectories. The abstract examples build and iterate maps with symmetric or asymmetric PRCs; the RTM example compares the map ingredients with a conductance-based neuron.

Core ideas

A locking relationship is a fixed point of the phase map. Its slope determines local stability: nearby phase differences return only when the map contracts them. Symmetric PRCs can support symmetric phase relationships, whereas asymmetric responses shift or remove them. Directly iterating a pair of pulse oscillators displays the same stable or unstable phase arrangements.

Essential model

If $g$ is a phase-resetting curve, the examples first form the single-pulse reset map $f(\phi)=\phi+g(\phi)$. They then account for the other oscillator's complementary phase with $F(\phi)=f(1-\phi)$, and form the two-event phase-difference map $G(\phi)=F(F(\phi))$. Locking satisfies

$$ G(\phi__)=\phi__. $$

For a one-dimensional map, the fixed point is locally stable when $|G'(\phi_*)|<1$. The examples plot the two-event map and identity line so those intersections can be read directly.

Code examples

All nine examples now live in one notebook, chapter26.ipynb. abstract_phase_grid gives the shared $\varphi\in[0,1]$ grid, and pulse_map_f/pulse_map_bigF/pulse_map_bigG build $f=\varphi+g(\varphi)$, $F=f(1-\cdot)$, and $G=F\circ F$ from any PRC function g; check_f_monotonic prints a warning if $f$ turns out not to be strictly increasing. plot_pulse_coupling_full (a 2x2 figure of $g,f,F,G$) and plot_pulse_coupling_g_and_bigG (just $g$ and $G$) are the two plotting helpers reused across the five abstract PRCs: g_pulse_1 ($\varphi^2(1-\varphi)$, asymmetric positive), g_pulse_2 ($\epsilon\varphi(1-\varphi)^3$), g_pulse_3 (antisymmetric about $\varphi=1/2$), g_pulse_4 (a shifted-locking asymmetric response), and g_pulse_5 (the negative counterpart of g_pulse_1). simulate_f_tilde/ plot_f_tilde plot the transformed map $\tilde f$ used to locate phase-map fixed points geometrically. simulate_two_pulse_coupled_osc iterates two abstract pulse-coupled oscillators event by event from given starting phases (using PRCs g_two_pulse_1 and g_two_pulse_2), and plot_two_pulse_coupled_osc draws the resulting spike raster. rtm_init finds the RTM limit cycle for splay initialization; simulate_rtm_g computes the RTM phase-resetting curve from a single synaptic pulse (the same recipe as Chapter 25's PRC examples), compute_bigG_from_g folds it into the two-event map by linear interpolation, and simulate_rtm_plot_g chains the two for direct comparison with the abstract maps above.

What to look for

For every two-event map plot, find intersections with the diagonal and then compare the nearby slope with one. Iterate the two-oscillator cases to confirm which intersections attract trajectories. Comparing the symmetric and asymmetric abstract cases shows that symmetry of a phase relation is a model property, not a general guarantee. Use simulate_rtm_plot_g to connect the map to a neuron model.

Suggested order

1. Run g_pulse_1 through g_pulse_3 with the shared plotting helpers.
2. Compare g_pulse_4, g_pulse_5, and simulate_f_tilde.
3. Run both simulate_two_pulse_coupled_osc examples, then simulate_rtm_plot_g.

Prerequisites and related chapters

Chapter 25 defines phase responses and interaction functions. Chapter 27 adds delays to the pulse-map construction, and Chapter 28 develops a continuous weak-coupling version of phase-difference dynamics.

Running the examples

Open chapter26.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. The abstract maps are quick NumPy/Matplotlib plots; simulate_rtm_plot_g integrates many RTM phases and takes longer.