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Weak PING rhythms

Overview

This chapter compares two ways a PING population produces a weak, loosely-periodic rhythm instead of a tightly-locked one: an M-current- mediated recovery in the E cells ("M-current PING"), and a Poisson stream of independent excitatory events driving each E cell ("Poisson PING"). It also derives the phase maps used to interpret temporal sub-clustering within a PING volley.

Core ideas

Poisson PING lets irregular, uncorrelated individual events align into gamma-population episodes through recurrent inhibition. M-current PING instead uses a slow, non-inactivating potassium current so that an E cell's own recent spiking suppresses its next spike, independent of the inhibitory volley. Phase maps psi/phi explain recruitment timing within a cycle rather than merely displaying a period.

Essential model

The E cell adds an M-current $I_M=g_M w(v-E_K)$ with $\dot w=(w_\infty(v)-w)/\tau_w(v)$ to the usual RTM E cell; the I cell is the usual WB cell. Poisson examples replace part of the fixed drive with independent random synaptic events per E cell, so alignment must be judged from population timing rather than an identical raster on every run.

Code examples

All thirteen examples live in one notebook, chapter32.ipynb. simulate_m_current_ping_plain is the shared plain-NumPy M-current network stepper (M_CURRENT_PING_1, and PING_CLUSTERS via g_m=0); simulate_m_current_ping_numba is its numba-accelerated sibling, built on the compiled _m_current_run_loop/_m_current_settle_loop, used by simulate_m_current_ping_closeup (M_CURRENT_PING_1_CLOSEUP, _2_CLOSEUP, _3_CLOSEUP -- identical parameters in the legacy scripts, so one function serves all three) and simulate_m_current_ping_1_from_rest (M_CURRENT_PING_1_FROM_REST, which settles at rest for 200 ms before the drive turns on). simulate_ping_clusters reuses the plain stepper with g_m=0 for PING_CLUSTERS. simulate_plot_phi, simulate_plot_psi, and simulate_plot_psi_phi compute the psi/phi inhibitory phase-reset maps. simulate_poisson_ping is the shared Poisson-PING stepper (no M-current, an independent Poisson synaptic stream per E cell); simulate_poisson_ping_1, _2, _3 are thin parameter wrappers around it, and simulate_poisson_ping_3 doubles as POISSON_PING_3_VOLTAGE_TRACE since it already returns the tracked E cell's voltage trace. Several sub-examples expose a natural scalar parameter (g_m, f_stoch, g_I) through an interact() slider.

What to look for

Contrast irregular Poisson input events with aligned population episodes. In the M-current closeups, follow the slow w recovery before assigning an E-I cycle; M_CURRENT_PING_1_FROM_REST's raster should stay empty for the first ~200 ms. Read the phase plots alongside PING_CLUSTERS: phase structure explains clustered timing.

Suggested order

  1. Run M_CURRENT_PING_1, _FROM_REST, and the first closeup.
  2. Inspect the remaining M-current closeups, the phase plots, and PING_CLUSTERS.
  3. Compare all three Poisson-PING cases, then the voltage-trace closeup.

Prerequisites and related chapters

Chapter 9 provides slow-current context, Chapter 30 PING, Chapter 33 beta-rhythm material, and Chapter 38 Poisson-PING coherence.

Running the examples

Open chapter32.ipynb and run the cells top to bottom. NumPy, SciPy, Matplotlib, numba, and ipywidgets are required. Poisson output is stochastic, so compare qualitative timing across runs rather than exact spike times.