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The PING model of gamma rhythms

Overview

PING is a pyramidal-interneuron gamma rhythm: E cells recruit I cells, then recurrent inhibition ends the E-cell episode until recovery starts the next cycle. The examples go from a two-cell loop to sparse and random populations, with rasters and LFP-like output.

Core ideas

E-to-I excitation is followed by I-to-E inhibition. Drive strength, E/I heterogeneity, and recurrent E/I connectivity decide which cells participate. Population alignment is read from E and I rasters together with mean-voltage (LFP-like) traces.

Essential model

The E and I cells receive conductance currents such as $I_{\rm IE}=g_{\rm IE}s_I(E_I-v_E)$ and $I_{\rm EI}=g_{\rm EI}s_E(E_E-v_I)$. Random-network weights are normalized by expected in-degree, allowing sparse and dense networks to be compared.

Code examples

All eleven examples live in one notebook, chapter30.ipynb. A shared block of RTM/WB gating functions, tau_peak_function/ tau_d_q_function (synapse rise-time solver), rtm_init_population/ wb_init_population (population splay-state initializers), and make_random_connectivity/make_fixed_degree_connectivity (connectivity builders) is defined once and reused throughout the chapter, along with the numba-accelerated network stepper simulate_ping_network (and its plotting companions plot_ping_drive/plot_ping_panels), which is the shared core of every population example (PING_1-PING_9):

  • simulate_2_cell_ping/plot_2_cell_ping integrate the two-cell E-I mechanism and report the E period (2-Cell PING).
  • run_condition_numbers perturbs drive, inhibition, and decay to report period sensitivity (2-Cell PING Condition Numbers).
  • simulate_ping_population (PING_1-PING_4) runs a random E/I population and returns E/I spike times plus an LFP-like trace; called with different heterogeneity (sigma_e), connectivity (p_XY), and size (num_e, num_i) for each of PING_1 (heterogeneous, 50% connectivity), PING_2 (homogeneous, all-to-all), PING_3 (heterogeneous, sparse), and PING_4 (PING_3 scaled to 800/200 cells).
  • run_ping5_panels (PING_5) and run_connectivity_panels/main (PING_6, a plain-NumPy reference version via simulate_ping_network_plain) each compare dense random, sparse random, and sparse fixed-degree connectivity.
  • simulate_ping_drive (PING_7, PING_8, PING_9) is a generalized wrapper exposing I-cell drive strength, E-E coupling, and I-cell initialization mode as parameters.

Interactive sliders let you sweep the I-E coupling strength in the 2-cell loop, the connectivity density p in the PING_1/PING_3 population, and the mean I-cell drive in the PING_7/PING_8 population.

What to look for

E spikes should precede I spikes, followed by a silent inhibitory interval. Compare raster bands with LFP peaks; connectivity and drive should change participation and timing, not simply voltage amplitude. Use the condition numbers to spot parameter-sensitive periods.

Suggested order

  1. Run the 2-Cell PING and 2-Cell PING Condition Numbers sections.
  2. Compare PING_1, PING_2, and PING_3, then run PING_4.
  3. Compare the plotted drive cases PING_5 through PING_9.

Prerequisites and related chapters

Chapter 20 introduces chemical synapses; Chapters 23-29 introduce phase and synchrony tools. Chapter 31 contrasts all-inhibitory ING, and Chapter 32 studies weak and stochastic PING.

Running the examples

Open chapter30.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom. These are spiking-network simulations and some cells (particularly PING_4 and PING_6's default 2-second run) can take a while to finish -- that is expected.