chapter27.md

Phase locking with delays

Overview

Propagation delays change the phase at which an oscillator receives a pulse, so they change the locking map itself. These examples simulate delayed pulse coupling for two and three oscillators and realize delayed locking with theta neurons.

Core ideas

A delayed pulse arrives after its source phase has advanced, and this timing must be included before applying a PRC. Consequently a delay can stabilize a phase difference that was not locked without delay, or destabilize synchronous timing. Three oscillators add consistent pairwise timing constraints; their locked pattern need not be a simple two-cell extension.

Essential model

For natural period $T$ and transmission delay $d$, an event map applies a phase shift at a phase displaced by $d/T$. In a theta-neuron realization, the continuous angle evolves between discrete arrivals, so the delay is held as an event-time condition rather than being folded into a static voltage term.

Code examples

All three examples now live in one notebook, chapter27.ipynb: simulate_two_delayed_pulse_coupled_osc iterates a two-oscillator phase map with delayed pulses, running a short-delay ($\delta=0.1$) and a long-delay ($\delta=0.7$) case with the shared simulate_two_delayed_pulse_pair engine (two_pulse_g/two_pulse_f give the pulse and interaction-function maps). simulate_three_delayed_pulse_coupled_osc extends delayed pulse coupling to three all-to-all oscillators using simulate_three_delayed_pulse_pair (three_pulse_g for the pulse map), comparing $\delta=0.45$ against $\delta=0.55$ and returning each run's event times for plot_three_delayed_pulse_coupled_osc. simulate_two_theta_neurons_grid classifies a $9\times9$ grid of delayed, pulse-coupled theta-neuron simulations (simulate_two_theta_neurons_pair, theta_neuron_inc, sync_measure) as synchronized or unsynchronized and plot_two_theta_neurons_grid draws that region in the $(\epsilon,\delta)$ parameter plane.

What to look for

Vary the initial conditions mentally while following the event sequence: a locked state repeats the same pulse-arrival phases. In the three-cell plot, distinguish a repeating collective order from exact simultaneous spikes. For the theta pair, read the red and blue grid points as synchronized and unsynchronized outcomes, respectively, and compare them with the plotted boundary in the $(\epsilon,\delta)$ plane.

Suggested order

1. Run simulate_two_delayed_pulse_coupled_osc.
2. Run simulate_three_delayed_pulse_coupled_osc and identify its repeating order.
3. Run simulate_two_theta_neurons_grid to connect the phase-map result to continuous neuron dynamics.

Prerequisites and related chapters

Chapter 26 establishes undelayed pulse-coupled phase maps, and Chapter 8 introduces theta neurons. Chapter 28 treats weak coupling, while Chapter 29 focuses on the stability of a synchronous phase relation.

Running the examples

Open chapter27.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. The theta-neuron grid cell is the slowest (about a minute and a half): it runs 81 sampled $(\epsilon,\delta)$ pairs, each a long fine-step theta-neuron simulation.