chapter25.md

Phase response curves (PRCs)

Overview

A phase response curve reports how much a single perturbation shifts a neuron's firing phase: deliver it at phase $\varphi$ (fraction of the cycle since the last spike) and measure the resulting shift $g(\varphi)$. The examples move from the theta neuron's closed-form PRC to numerically measured PRCs for RTM, WB, HH, and Erisir neurons, under both synaptic-pulse and instantaneous-voltage-kick perturbations.

Core ideas

For conductance-based neurons, $g$ is measured by splay-initializing a population of independent neurons across every phase, delivering the same perturbation to all of them at time 0, and recording each neuron's next spike time $t_\ast$: $g(\varphi)=1-\varphi-t_\ast/T$, with $T$ the unperturbed period. Weak-perturbation PRCs (small synaptic conductance or small voltage kick) approach a linear-response limit -- normalizing by the perturbation strength collapses curves at different strengths onto one shape.

Essential model

Each PRC example reuses one of the conductance-based neuron models from earlier chapters (RTM, WB, HH, Erisir), perturbed either by a synaptic pulse

$$ I_{\rm syn}=g_{\rm syn},s,(v_{\rm rev}-v),\qquad s(0)=0,\ q(0)=1, $$

or by an instantaneous voltage kick $v\to v+\Delta v$ at $t=0$. The theta neuron instead has a closed-form return map $f(\varphi)=\frac1\pi\arctan!\big(\tan(\pi\varphi-\tfrac\pi2)+\delta v/\sqrt{\tau_mI-\tfrac14}\big)+\tfrac12$ and PRC $g=f-\varphi$.

Code examples

All thirteen examples now live in one notebook, chapter25.ipynb: theta_f/theta_prc/theta_prc_short_weak give the theta neuron's closed-form return map, PRC, and linear-response PRC. rtm_init (shared with Chapter 24) finds the single-cell RTM limit cycle for splay initialization. simulate_synaptic_pulse_prc computes the RTM synaptic-pulse PRC, reused directly for the interaction-function panel ($f=\varphi+g$), the three-weak-pulses comparison (simulate_three_weak_prcs), and the weak-pulse linear-response check (simulate_weak_prc_comparison). simulate_voltage_kick_prc is the voltage-kick analogue, reused for the strong-kick and weak-kick-comparison (simulate_short_weak_prc_comparison) examples. simulate_phase_shift shows the shift directly as two overlaid voltage traces (simulate_baseline and simulate_perturbed) instead of a PRC curve. simulate_wb_prc (shared WB gating and wb_init) covers both the inhibitory-pulse WB example and the WB panel of the model-comparison figure; simulate_hh_prc and simulate_erisir_prc provide the other two panels of that comparison.

What to look for

Compare the theta neuron's smooth, everywhere-positive PRC to the conductance-based neurons' PRCs, which can change sign (a pulse can delay or advance firing depending on phase) or even go negative across most of the cycle for inhibition. Check that halving the perturbation strength collapses the normalized PRC onto the same curve (linear response), and that the phase-shift trace directly shows the same effect the PRC curve predicts.

Suggested order

1. Start with the theta-neuron trio (theta_f, theta_prc, theta_prc_short_weak) for the closed-form baseline.
2. Move to simulate_synaptic_pulse_prc and the interaction-function panel, then the three-weak-pulses and weak-pulse linear-response checks.
3. Compare with simulate_voltage_kick_prc and its weak-kick check, then simulate_phase_shift for the same effect shown as raw traces.
4. Finish with the WB inhibitory-pulse example and the WB/HH/Erisir model-comparison panel.

Prerequisites and related chapters

Chapter 20 develops the synaptic gate kinetics ($\tau_r,\tau_d,\tau_{dq}$) used throughout. Chapter 24's RTM limit-cycle finder is reused directly here. Chapters 26-29 build on PRCs to analyze phase locking and synchrony stability in coupled-oscillator networks.

Running the examples

Open chapter25.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. Several sliders (interact) let you adjust perturbation strength or timing and watch the PRC or trace update.