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M-current PING and PINB

Overview

Chapter 33 covers the book's beta-rhythm material. The examples examine M-current PING period skipping and PING-with-inhibition/burst (PINB), including gap junctions and cell-assembly timing.

Core ideas

Slow M-current recovery can make E cells skip gamma opportunities, yielding beta-scale population timing. PINB organizes which E assemblies participate and when; gap junctions can change inhibitory coordination. A slower rhythm therefore need not mean that every cell fires slowly.

Essential model

The PING E-I loop is supplemented by M-current-controlled E-cell availability:

$$ I_M = \hat g_M, w,(v_K - v), \qquad \dot w = \frac{w_\infty(v) - w}{\tau_w(v)}. $$

Population period and skipped cycles are useful timing measures. Gap-junction currents are proportional to $v_j-v_i$, providing electrical coupling without chemical-synapse delay. PINB networks additionally split the I-cell output into two independently-gated synapse pools -- a fast one onto other I-cells and a (typically slower or stronger) one onto E-cells -- both driven by the same I-cell spikes.

Code examples

All nine examples live in one notebook, chapter33.ipynb. The network sims integrate 40-250 coupled cells over tens of thousands of time steps, so their per-timestep update loops are @njit-compiled (numba), while population initializers and the tau_d_q bisection search stay plain NumPy.

simulate_m_current_beta_with_gj/plot_beta_gj_raster cover M_CURRENT_BETA_WITH_GJ: a gap-junction-coupled population of M-current E cells with no synaptic coupling, exposing g_hat_gap through an interact() slider.

simulate_m_current_ping is the shared M-current PING network integrator (RTM E cells with M-current, WB I cells, configurable E-E/E-I/I-E/I-I connectivity). simulate_m_current_ping_4 through simulate_m_current_ping_8 are thin wrappers around it, each exposing one differentiating parameter through interact(): g_m (M_CURRENT_PING_4), g_hat_ee (M_CURRENT_PING_5 and M_CURRENT_PING_6, the latter with recurrent excitation confined to the E_P sub-assembly), i_ext_i_mean (M_CURRENT_PING_7, with E_S-to-I synapses halved), and the E_P drive-boost factor (M_CURRENT_PING_8). All are plotted with plot_m_current_ping_raster.

simulate_pinb is the shared PINB network integrator (plain RTM/WB cells, no M-current, with independent I-I and I-E synapse pools driven by the same I-cell spikes). simulate_pinb_1 (slider: tau_d_ie), simulate_pinb_2 (slider: g_hat_ie), and simulate_pinb_3 (slider: i_ext_e_mean) are thin wrappers, plotted with plot_pinb (raster plus the E-cell-averaged LFP-like trace).

What to look for

Count skipped gamma opportunities before calling the population rhythm beta. In the gap-junction case, inspect cluster timing, not only voltage. In PINB rasters, identify participating E assemblies and their order relative to inhibition.

Suggested order

  1. Run M_CURRENT_PING_4 through M_CURRENT_PING_8.
  2. Compare M_CURRENT_BETA_WITH_GJ.
  3. Run PINB_1, PINB_2, and PINB_3.

Prerequisites and related chapters

Chapter 9 introduces M-current context, Chapter 21 gap junctions, Chapters 30 and 32 PING, and Chapter 34 nested slow/fast rhythms.

Running the examples

Open chapter33.ipynb and run the cells top to bottom. The network sims (M_CURRENT_PING_4-M_CURRENT_PING_8 and PINB_1-PINB_3) integrate 40-250 coupled cells over 200-500ms with a fixed 0.01ms step, so each can take from several seconds to about a minute once numba has compiled the step loop.