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Model neurons of bifurcation type 3

Overview

This chapter presents type-3, or phasic, excitability. Persistent sodium and potassium dynamics generate transient responses, and self-exciting theta neurons show the same geometry in a phase model with a slow feedback state.

Core ideas

A type-3 neuron can answer a step with only a brief spike or a few spikes, rather than tonic firing. Fixed points and phase portraits organize this phasic response. In a self-exciting theta neuron, a slow state jumps or rises around spikes and changes the phase velocity, creating a geometric feedback mechanism.

Essential model

The persistent-sodium/potassium (INaP-I$_K$) reduction uses

$$ C\dot v=g_{\rm Na}m_\infty(v)(v_{\rm Na}-v)+g_Kn(v_K-v) +g_L(v_L-v)+I,\qquad \dot n=(n_\infty(v)-n)/\tau_n. $$

Here $v$ is voltage, $n$ is potassium activation, $m_\infty$ is instantaneous persistent sodium activation, and $I$ is applied current.

Code examples

All five examples now live in one notebook, chapter16.ipynb: simulate_inapik_fixed_points scans current and classifies INaP-I$_K$ equilibria; simulate_inapik_phase_plane plots phase-plane trajectories, fixed points, and cycles at several currents; simulate_self_exciting_theta_neuron simulates a theta neuron with discrete slow-state increments; simulate_self_exciting_theta_smooth uses smooth slow feedback in the theta model; simulate_setn_phase_plane draws theta-slow-state phase planes and threshold markers. The two theta-neuron voltage-trace examples have ipywidgets sliders.

What to look for

Use INAPIK_FIXED_POINTS before the multi-panel INAPIK_PHASE_PLANE so the markers have a clear meaning. Then compare the jump-based and smooth theta feedback in the two SELF_EXCITING examples; SETN_PHASE_PLANE reveals the threshold structure underlying those traces.

Suggested order

  1. Run INAPIK_FIXED_POINTS and INAPIK_PHASE_PLANE.
  2. Run SELF_EXCITING_THETA_NEURON and its _SMOOTH variant.
  3. Use SETN_PHASE_PLANE to connect the time traces to phase geometry.

Prerequisites and related chapters

Chapter 08 introduces theta neurons. Chapters 10--14 supply phase-plane and bifurcation language, and Chapter 17 compares the firing-rate response of the INaP-I$_K$ and theta models.

Running the examples

Open chapter16.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. The SETN_PHASE_PLANE cell is noticeably slower than the others (a couple of minutes).