chapter28.md

Weakly coupled oscillators

Overview

When coupling is weak, the full neuron equations can be reduced to slowly changing phases. The examples plot interaction and difference functions and simulate identical and heterogeneous oscillator pairs to relate zeros of the reduced equation to stable phase differences.

Core ideas

Weak coupling preserves each oscillator's basic cycle while shifting its phase slightly each period. The interaction function summarizes that shift. Subtracting the two phase equations gives a difference function: its zeros are locked phase differences, and its local sign or derivative determines which are stable. Intrinsic frequency mismatch adds a drift term and can move or remove a locked state.

Essential model

For phases $\theta_1,\theta_2$, a weak-coupling reduction has the form

$$ \dot\theta_i=\omega_i+\varepsilon H(\theta_j-\theta_i). $$

With $\psi=\theta_2-\theta_1$, the difference equation is $\dot\psi=\omega_2-\omega_1+\varepsilon D(\psi)$. A zero with restoring flow on both sides is a stable phase difference.

Code examples

All four examples now live in one notebook, chapter28.ipynb: find_two_fixed_points builds the difference function D_two_fixed_points (from the single-pulse-like PRC g_0) and locates its unstable and stable zeros above $\psi=0.5$ by bisection; plot_d_two_fixed_points draws it. simulate_weakly_coupled_event_driven and simulate_weakly_coupled_de are the shared machinery for an identical weakly coupled pair -- the first advances each oscillator exactly to its next spike (ceiling handles the "just spiked" edge case) and applies the weak-coupling kick from a given g at each spike, the second integrates the reduced difference equation with Heun's method. simulate_weakly_coupled_1 (interaction function wc1_g, $\varphi^2(1-\varphi)$) and simulate_weakly_coupled_2 (interaction function wc2_g, $\varphi(1-\varphi)^3$) each call both for a chosen epsilon, and plot_weakly_coupled overlays the event-driven and reduced-equation traces. simulate_weakly_coupled_heterogeneous_1 adds a period mismatch (B's period is $T_B=1+\varepsilon c$) via simulate_heterogeneous_event_driven and simulate_heterogeneous_de, and plot_weakly_coupled_heterogeneous compares two detunings c.

What to look for

In find_two_fixed_points, identify each zero and inspect the direction of the flow on either side before naming it stable. Compare the long-time phase difference in the two identical-pair simulations (simulate_weakly_coupled_1, simulate_weakly_coupled_2) with those predictions. In the heterogeneous simulation, separate a genuine locked offset from ongoing phase drift caused by frequency mismatch.

Suggested order

1. Run find_two_fixed_points / plot_d_two_fixed_points.
2. Run simulate_weakly_coupled_1 and simulate_weakly_coupled_2.
3. Run simulate_weakly_coupled_heterogeneous_1 and compare it with the identical cases.

Prerequisites and related chapters

Chapter 25 derives PRCs and interaction functions, and Chapters 26-27 use event maps with pulse coupling and delay. Chapter 29 gives a more detailed view of synchronous-state stability under inhibitory perturbations.

Running the examples

Open chapter28.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. These are small NumPy/Matplotlib simulations that display their figures inline.