chapter14.md

Model neurons of bifurcation type 2

Overview

These examples study type-2 excitability, where firing begins at a nonzero frequency through Hopf-related geometry. Reduced HH and Erisir models expose fixed points, eigenvalues, attracting cycles, and repelling cycle boundaries.

Core ideas

Unlike type-1 onset, type-2 onset has a finite firing frequency at threshold. The reduced HH construction takes sodium activation to equilibrium and ties inactivation to potassium activation. Eigenvalues show when an equilibrium changes stability, while a repelling cycle can divide resting and spiking basins.

Essential model

The reduced HH equations are

$$ C\dot v=I_{\rm Na}(v,n)+I_{\rm K}(v,n)+I_L(v)+I,\qquad \dot n=\alpha_n(v)(1-n)-\beta_n(v)n, $$

with $m=m_\infty(v)$ and $h=0.83-n$. The Erisir reduction has the same form but uses its rodent-neuron conductances and $h=0.36-n$. Here $v$ is voltage, $n$ is a recovery gate, and $I$ is applied current.

Code examples

All seven examples now live in one notebook, chapter14.ipynb: simulate_erisir_reduced compares three- and two-dimensional Erisir voltage traces; simulate_erisir_2d_fp classifies Erisir reduced fixed points across current; simulate_hh_reduced_count_fp reports the minimum and maximum number of reduced HH fixed points over its current scan; simulate_hh_reduced_fixed_points plots stable and unstable reduced HH fixed-point branches; simulate_hh_reduced_fp_evs plots the real and imaginary parts of fixed-point eigenvalues; simulate_hh_reduced_repelling_cycle traces attracting and backward-integrated repelling cycles; simulate_hh_reduced_cycle_distance zooms the distance between attracting and repelling cycles for several currents. The reduced-HH gating and fixed-point/Jacobian helpers shared by the last five live in one cell (hh_reduced_*).

What to look for

First check how well ERISIR_REDUCED follows its full counterpart. For HH, read HH_REDUCED_FIXED_POINTS together with HH_REDUCED_FP_EVS; a real-part crossing signals the stability change. The two cycle examples then show the nearby attracting and repelling invariant curves.

Suggested order

1. Run ERISIR_REDUCED and ERISIR_2D_FP.
2. Run the three HH fixed-point and eigenvalue examples.
3. Finish with HH_REDUCED_REPELLING_CYCLE and HH_REDUCED_CYCLE_DISTANCE.

Prerequisites and related chapters

Chapter 10 provides reduced HH geometry and Chapter 13 provides Hopf normal forms. Chapter 17 compares the finite-onset frequency of these models with other excitability types.

Running the examples

Open chapter14.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom.