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Thresholding in PING

Overview

PING can have sharp boundaries between suppression and participation. This chapter's notebook isolates a non-reset threshold mechanism, runs a baseline PING network to locate the boundary, magnifies the transition where a small timing or drive change flips a cell's participation, and computes the boundary directly for a single cell driven by a periodic inhibitory conductance.

Core ideas

Thresholding is a network property: excitation must arrive during the interval left open by recurrent inhibition. A non-reset reference cell separates continuous voltage crossing from an imposed reset. Near a boundary, a cell can miss a cycle or get recruited into the next one from what looks like a negligible change in drive or timing.

Essential model

The E-to-I and I-to-E PING conductance loop evolves continuously, while a voltage threshold selects spike events. In the non-reset comparison, crossing does not reset the voltage. In the direct threshold construction, a single RTM cell is driven by a fixed periodic inhibitory conductance trace $g(t)=e^{\cos^4(\pi t/25)}-1$ (scaled to a target mean $\bar g$); the onset drive $I_L$ (first spike) and the drive $I_R$ at which the cell locks onto the full 39 Hz rhythm bracket a thresholding window $w=I_R-I_L$ that narrows as $\bar g$ grows.

Code examples

All four examples live in one notebook, chapter37.ipynb.

  • no_reset_time_constants/simulate_no_reset/plot_no_reset (NO_RESET) build the piecewise-exponential, never-reset voltage trace and its two effective time constants (tau_m, tau_m_hat).
  • simulate_ping_thr is the shared, @njit-compiled RTM-E/WB-I PING network stepper (200 E cells with a linearly ramping drive, 50 I cells, all-to-all connectivity) reused by both PING examples below. simulate_ping_thr_1/plot_ping_thr_raster (PING_THR_1) run it for 200 ms and raster the whole population, with the E-cell index doubling as a drive axis. simulate_ping_thr_1_zoom/plot_ping_thr_zoom (PING_THR_1_ZOOM) run it for 500 ms and zoom into E-cells 72-78 to inspect the suppression/participation boundary cycle by cycle.
  • firing_rate, bisect_threshold, and threshold_width_sweep (THRESHOLDING) compute the onset drive I_L, the full-rate-locking drive I_R, and the resulting window w = I_R - I_L for five values of g_bar; the per-drive RTM simulation is @njit-compiled since dozens of bisection evaluations are needed per g_bar.

interact() sliders are provided for g_bar (NO_RESET) and g_hat_ie (PING_THR_1), the two single-parameter knobs each example is built around.

What to look for

Use NO_RESET to distinguish a threshold event from reset dynamics. In PING_THR_1_ZOOM, inspect whether an E cell crosses when inhibition relaxes enough to recruit it, or remains suppressed cycle after cycle. A boundary represents changed cycle participation, not merely a small voltage difference. In THRESHOLDING, note how w shrinks by roughly a factor of 2.5-3 for each 0.05 increment of g_bar.

Suggested order

  1. Run NO_RESET and THRESHOLDING.
  2. Run PING_THR_1.
  3. Use PING_THR_1_ZOOM to inspect its boundary.

Prerequisites and related chapters

Chapter 7 discusses reset dynamics, Chapter 30 PING, Chapter 35 inhibitory windows, and Chapter 38 gamma coherence.

Running the examples

Open chapter37.ipynb and run the cells top to bottom. The PING network cells (PING_THR_1, PING_THR_1_ZOOM) integrate 250 cells over 200-500 ms at a fixed 0.01 ms step, and THRESHOLDING runs dozens of 5000 ms bisection evaluations; each of these can take from several seconds to about a minute.