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ING rhythms

Overview

Interneuron gamma (ING) is an inhibitory network rhythm in which recovery from shared inhibition determines the next spike window. These examples move from a one-cell timing reference and abstract inhibitory pulse-coupling maps, through full spiking-network simulations of an inhibitory population (with optional electrical coupling), to ING entraining a population of excitatory cells.

Core ideas

ING synchrony is organized by inhibitory recovery rather than recurrent excitation. Inhibitory phase-reset maps predict stable timing; gap junctions can alter clustering. The population outputs show whether an inhibitory rhythm creates repeatable windows for excitatory spikes.

Essential model

Recurrent inhibition has the form $I_{{\rm II},i}=g_{\rm II}\sum_j s_j(E_I-v_i)$, optionally supplemented by gap-junction voltage-difference currents. A pulse map $\phi\mapsto F(\phi)$ reduces inhibitory timing to fixed points and their slopes.

Code examples

All sixteen examples live in one notebook, chapter31.ipynb. The ING_1-ING_10 and entrainment population sims share @njit-compiled (numba) per-timestep update loops, and most sub-examples expose one natural parameter (synaptic/gap-junction strength, heterogeneity, connectivity, drive, ...) through an interact() slider so you can sweep it and re-plot without re-running the whole notebook. simulate_1_cell_ing/plot_1_cell_ing give the single self-inhibited-cell timing reference. compute_condition_numbers measures the one-cell period's sensitivity to i_ext, g_ii, and tau_d. simulate_abstract_pulse_coupling_inh/plot_abstract_pulse_coupling_inh and simulate_abstract_pulse_coupling_inh_2/plot_abstract_pulse_coupling_inh_2 plot the two inhibitory pulse-coupling maps. simulate_ing_population is the shared WB inhibitory-population network integrator (random or fixed-indegree synaptic wiring, optional sparse gap junctions); simulate_ing_1 through simulate_ing_10 are thin wrappers around it with different heterogeneity/connectivity/gap-junction parameters, plotted with plot_ing_raster, plot_ing_raster_scalebar (ING_7), or plot_ing_raster_zoom (ING_8-ING_10). build_entrainment_network and simulate_entrainment are the shared RTM-E/WB-I network for the entrainment examples: simulate_ing_entraining_e_cells runs the reference configuration, and run_drive_panels/main sweep three E-cell drive levels for the second entrainment figure.

What to look for

Cells should fire after inhibitory conductance decays enough to open a recovery window. Compare abstract-map fixed points with persistent raster timing. In the entrainment examples, identify E spikes relative to the inhibitory window rather than expecting every E cell to fire every cycle.

Suggested order

  1. Run the one-cell and abstract-map examples (simulate_1_cell_ing, compute_condition_numbers, the two pulse-coupling maps).
  2. Compare simulate_ing_1 through simulate_ing_10.
  3. Finish with simulate_ing_entraining_e_cells and run_drive_panels/main.

Prerequisites and related chapters

Chapter 21 introduces gap junctions, Chapters 25-29 phase maps, Chapter 30 PING, and Chapter 35 periodic inhibition.

Running the examples

Open chapter31.ipynb and run the cells top to bottom. The population network sims (ING_1-ING_10 and the two entrainment examples) integrate 100-500 coupled cells over 500ms with a fixed 0.01ms step, so each can take from several seconds to about a minute.