chapter29.md

Stability of the synchronous state

Overview

Synchrony is stable when an inhibitory perturbation contracts a small timing difference rather than amplifying it. These examples calculate the contraction maps and their condition numbers for LIF and RTM cells, then give a geometric theta-neuron "river" picture of the same sensitivity.

Core ideas

Two nearby phases can be represented by their pre-pulse and post-pulse time separations. An inhibitory pulse may contract that separation, but numerical or parameter sensitivity can make the inferred stability fragile. Condition numbers quantify this sensitivity. LIF reset dynamics and smooth RTM dynamics can therefore react differently even when both receive an inhibitory pulse.

Essential model

Let $P_0$ and $P_1$ denote the two pulse-to-spike timings. The LIF examples define their mean timing and normalized separation as

$$ P=\frac{P_0+P_1}{2},\qquad S=\frac{P_0-P_1}{P}. $$

Here $S$, rather than a derivative of $P$, is the plotted stability quantity. The sensitivity calculations report how changes in parameters alter the mean timing and its inference.

Code examples

All seven examples now live in one notebook, chapter29.ipynb: simulate_p0_and_p1 traces the two timing maps before and after a pulse (plot_p0_and_p1). P0, P1, P, and S are the shared LIF timing maps used by both LIF examples below: simulate_lif_p_and_s plots the LIF mean timing $P$ and normalized separation $S$ while varying synaptic decay, conductance, and drive (plot_lif_p_and_s); compute_lif_condition_numbers returns a dictionary of baseline LIF mean timings and percent changes under parameter perturbations for a weak/slow and a strong/fast synapse. simulate_lif_pulse_panels simulates LIF cells receiving a common inhibitory pulse and records their timing response (plot_lif_pulse_panels). rtm_init and simulate_rtm are shared by the two RTM examples: simulate_rtm_pulse_panels performs the RTM analogue of the LIF inhibitory-pulse experiment (plot_rtm_pulse_panels, reused by the condition-numbers example below); compute_rtm_condition_numbers returns RTM timing sensitivities (and the same three voltage traces) for no, weak/slow, and strong/fast inhibitory conductances. simulate_river and plot_river draw the theta-neuron river geometry that organizes how nearby trajectories are drawn toward or away from a timing relation.

What to look for

In the pulse simulations, compare the timing gap before and after the common inhibition rather than just the voltage deflection. Read the printed LIF and RTM sensitivity results alongside the $P$ and $S$ plots and the RTM voltage traces: a conclusion that changes strongly with parameters deserves more caution. In the river plot, follow neighbouring theta trajectories to connect the geometric flow with phase contraction.

Suggested order

1. Run simulate_p0_and_p1, simulate_lif_p_and_s, and simulate_lif_pulse_panels.
2. Run compute_lif_condition_numbers and compare the sensitivity with the LIF map.
3. Run simulate_rtm_pulse_panels, compute_rtm_condition_numbers, and simulate_river.

Prerequisites and related chapters

Chapter 7 provides LIF reset dynamics, Chapter 25 supplies PRCs, and Chapters 26-28 introduce phase-map, delay, and weak-coupling stability ideas. The theta-neuron geometry builds on Chapter 8.

Running the examples

Open chapter29.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. RTM calculations and condition-number sweeps take a little longer than the LIF and geometric examples but still finish in seconds.