The sigmoid to explain JR's oscillations

Don’t miss out these animated plots that are the central components of this post.

The Jansen-Rit model is a neural mass model that captures population-level dynamics of a cortical column. It’s widely used for simulating EEG-like signals and understanding rhythms such as alpha and theta oscillations. At the heart of this model lies a deceptively simple non-linearity: the sigmoid function.

This post offers a visual deep-dive into the sigmoid’s role in shaping brain-like oscillations. With interactive illustrations, I explore how the average membrane potentials of three neural subpopulations — excitatory interneurons, inhibitory interneurons, and pyramidal cells — are dynamically fed into the sigmoid, and how small changes in model parameters can give rise to different regimes: from damped responses to sustained oscillations.

The sigmoid as a thresholding mechanism

In the Jansen-Rit model, the sigmoid transforms average membrane potentials into firing rates. This function captures the saturation and thresholding behavior observed in real neurons. It’s defined as:

\[S(v) = \frac{2 e_0}{1 + \exp{(r (v_0 - v))}}\]

where:

• $ v $ is the input potential,
• $ v_0 $ is the potential at which firing rate is half-maximal,
• $ e_0 $ is the maximum firing rate,
• $ r $ controls the slope (gain) of the sigmoid.

This nonlinearity is essential: it’s where non-trivial dynamics emerge, especially when feedback loops are introduced between subpopulations.

To understand the model’s behavior, it’s useful to track the instantaneous input to the sigmoid function over time. In the Jansen-Rit model, each subpopulation receive inputs from others in the shape of PSP. As these postsynaptic inputs vary, so do their firing rates.

Oscillations emerge from feedback and delay

The rhythm generation in this model comes from a combination of: the nonlinear response of the sigmoid, the delayed feedback via synaptic impulse responses (modeled as second-order linear systems), and the interaction between excitation and inhibition. With specific parameter values the system can enter the limit cycle regimes where self-sustained activity give rise to oscillatory behavior resembling M/EEG rhythms.

I used the default parameters of the JR (see Cabrera-Álvarez, 2023), and varied the average intrinsic input (p), to observe different model operation regimes. Dashed lines in the following bifurcation diagram represent the p values chosen for the sigmoidal plots below.

In the following two animated graphs, I show the dynamics of the two limit cycles in the JR: slow delta/theta (p=0.12) and fast alpha (p=0.24).

Next, I show three fixed point regimes: one in the pre- saddle node bifurcation space (p=0.09), and two in the post- supercritical bifurcation space (p=[0.4, 0.54]). When you click play, you might not perceive the balls moving, but they are in a very small fluctuation range - note the signal movement. You could make zoom into the sigmoid to see the movement. Finally, note that the difference between the last two graphs are just that one is more static (p=0.54) than the other (p=0.4) in its equilibrium.

Some configurations show a tight oscillation near the sigmoid’s steepest region, while others drift into saturation or exhibit more complex dynamics. Finally, a summary of the subpopulation movements on the sigmoidal as a function of $p$.

Final thoughts

Understanding the Jansen-Rit sigmoid in dynamic context is key to building an intuition for how nonlinear population models generate realistic brain rhythms. Before diving into frequency analysis or fitting models to empirical EEG data, it’s worth watching how oscillations arise from the interplay of synaptic input, non-linearity, and feedback.

Let the sigmoid breathe — and you’ll see the rhythm of the cortex begin to emerge.