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A Wilson--Cowan model of an oscillatory E--I network

Overview

Wilson--Cowan equations replace individual membrane voltages with firing-rate variables for excitatory and inhibitory populations. These examples show how recurrent excitation and delayed inhibitory feedback make a population rhythm, and how that rhythm appears in rate traces, a phase plane, and a rastergram.

Core ideas

The variables $E$ and $I$ are population activities filtered on their own time scales. Nonlinear response functions turn recurrent input into rates. Excitation raises the E population, inhibition subsequently suppresses it, and the cycle can repeat. Lowering recurrent excitation changes whether this loop can sustain a rhythm.

Essential model

A representative system is

$$ \tau_E\dot E=-E+f(w_{EE}E-w_{IE}I+P_E),\qquad \tau_I\dot I=-I+g(w_{EI}E-w_{II}I+P_I). $$

Nullclines are the points where one derivative is zero; their geometry helps explain the direction of a trajectory and any oscillatory feedback loop.

Code examples

All four examples now live in one notebook, chapter22.ipynb: simulate_wilson_cowan_e_and_i plots the excitatory and inhibitory rate traces over several oscillation cycles. simulate_wilson_cowan_lowering_w_ee lowers recurrent excitation and shows the altered long-time activity. plot_wilson_cowan_phase_plane draws the E--I nullclines and the flow field in the rate phase plane. simulate_wilson_cowan_rastergram samples the rates into a population-style raster representation.

What to look for

In simulate_wilson_cowan_e_and_i, compare the timing of E and I peaks rather than only their heights. In the phase plane, follow the trajectory relative to both nullclines. Then compare the lowered-$w_{EE}$ trace and the raster: a rate oscillation is summarized differently by continuous activity and by events.

Suggested order

  1. Run simulate_wilson_cowan_e_and_i.
  2. Use plot_wilson_cowan_phase_plane to interpret the rate cycle.
  3. Compare simulate_wilson_cowan_lowering_w_ee and simulate_wilson_cowan_rastergram.

Prerequisites and related chapters

The earlier phase-plane and bifurcation chapters provide the geometric tools for rate nullclines. Chapter 24 studies synchronization in spiking excitatory networks, whereas this chapter describes the population-level E--I mechanism.

Running the examples

Open chapter22.ipynb in Jupyter, or via the Colab badge at the top of the notebook, and run all cells top to bottom.