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f--I curves with pulsed excitation

Overview

This chapter compares firing-rate curves driven by periodic excitation with steady-current f--I curves. Square pulses provide an idealized input, and RTM simulations show how pulse timing and amplitude select the spikes that determine rate.

Core ideas

A pulsed f--I curve counts response opportunities per pulse period. It can have steps or saturation because a cell may fire once, skip, or fire more than once per pulse. The square-pulse model isolates this temporal effect before conductance-based RTM dynamics are added.

Essential model

For pulse period $T$, the observed rate is $f=N_{\rm spikes}/T_{\rm observation}$. Unlike steady injected current, drive is nonzero only during a pulse interval; sweeping pulse amplitude produces the pulsed f--I curve.

Code examples

All four examples now live in one notebook, chapter36.ipynb: plot_idealized_f_i_curve draws the schematic type-1-onset $f=\sqrt{I-I_c}$ construction; plot_square_pulses draws the idealized square-pulse forcing signal. The two RTM examples share a single simulation kernel (_rtm_step, _rtm_f_i_curve_constant_python, _rtm_f_i_curve_pulsed_python, rtm_pulse_shape) driven through _compute_rtm_f_i_curves; compute_rtm_f_i_curves_pulsed_excitation (g_l=0.1) and compute_rtm_f_i_curves_pulsed_excitation_2 (g_l=0.2) each return (f_vec_constant, f_vec_pulsed, i_ext_vec), plotted with plot_rtm_f_i_curves_pulsed_excitation.

What to look for

Read rate changes with the pulse schedule: a jump can be an extra spike per pulse and a plateau can be a locked one-spike response. Compare the idealized curve with both RTM outputs before attributing all shape to intrinsic f--I nonlinearity.

Suggested order

  1. Run the "Square Pulses" and "Idealized F-I Curve" sections.
  2. Run "RTM F-I Curve, Pulsed Excitation".
  3. Compare it with "RTM F-I Curve, Pulsed Excitation 2".

Prerequisites and related chapters

Chapter 17 provides steady-current f--I curves, Chapter 23 periodic excitation and entrainment, and Chapter 35 periodic inhibition.

Running the examples

Open chapter36.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. The schematic sections are instant; each RTM F-I sweep integrates 1000 ms across 201 drive values and takes roughly ten seconds with numba.