Skip to content

chapter17.md

Latest commit

 

History

History
72 lines (54 loc) · 3.05 KB

chapter17.md

File metadata and controls

72 lines (54 loc) · 3.05 KB
Raw

Frequency-current curves

Overview

These scripts compute firing rate as a function of applied current across LIF, theta, HH, RTM, Erisir, reduced HH, INaP-I$_K$, and self-exciting theta models. Forward and backward scans reveal onset frequency, continuous or discontinuous branches, and bistability.

Core ideas

An f--I curve maps steady firing rate to input current. Type-1 onset rises continuously from zero frequency, while type-2 onset starts at a nonzero frequency. Carrying the terminal state from one current to the next gives a forward branch; reversing the scan can reveal a different branch when resting and spiking attractors coexist.

Essential model

For the LIF example, the interspike interval and firing rate are

$$ T=\tau_m\log!\left(\frac{\tau_m I}{\tau_m I-1}\right),\qquad f=1000/T. $$

Here $I$ is normalized applied current, $\tau_m$ is membrane time constant, $T$ is in milliseconds, and $f$ is in hertz. The conductance models numerically count threshold crossings after transients.

Code examples

All ten primary examples (plus three legacy full-model variants) now live in one notebook, chapter17.ipynb: simulate_lif_f_i_curve and simulate_theta_f_i_curve plot the analytic LIF and theta-neuron f-I curves; simulate_setn_f_i computes the self-exciting theta f-I curve; simulate_hh_reduced_f_i_curve, simulate_erisir_f_i_curve, and simulate_wb_f_i_curve compute forward/backward reduced-HH, Erisir, and Wang-Buzsaki rates; simulate_wb_f_i_curve_at_onset magnifies WB onset behavior; simulate_rtm_with_m_current_f_i measures the adapting RTM model's f-I curve; simulate_inapik_f_i_curve computes INaP-I$_K$ forward/backward branches; simulate_inapik_saddle_cycle_distance plots the distance between an INaP-I$_K$ saddle and cycle. A "Legacy Full-Model F-I Curves" section preserves the original (untested) full HH and RTM RK4 scans as simulate_hh_f_i_curve_legacy, simulate_rtm_f_i_curve_legacy, and simulate_rtm_f_i_curve_at_onset_legacy.

What to look for

Compare LIF_F_I_CURVE and THETA_F_I_CURVE for continuous zero-frequency onset. Use each paired forward/backward scan to find hysteresis. The two _AT_ONSET examples use finer current ranges, and INAPIK_SADDLE_CYCLE_DISTANCE links a geometric distance to the transition.

Suggested order

  1. Run LIF_F_I_CURVE, THETA_F_I_CURVE, and SETN_F_I.
  2. Compare HH_REDUCED_F_I_CURVE, ERISIR_F_I_CURVE, and INAPIK_F_I_CURVE.
  3. Use the RTM, HH, and WB onset scripts for detailed model comparisons.

Prerequisites and related chapters

Chapters 12 and 14 distinguish type-1 and type-2 onset. Chapter 16 provides the INaP-I$_K$ and self-exciting theta systems; Chapter 18 explains the bistability visible in some forward/backward curves.

Running the examples

Open chapter17.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. Most scans are a forward+backward sweep over dozens of currents and take tens of seconds to a few minutes each; the notebook notes the slower ones inline.