How do PSP parameters shape JR bifurcation?

In a previous post, I showed the bifurcation of the JR over parameter C. Here, I show the impact of excitatory and inhibitory PSP parameters on that bifurcation. Specifically, I show 3D scatters for the amplitude of the PSP (He/Hi) and the time constants (taue/taui). Those parameters determine the effect of the JR operator that converts firing rates into voltage.

Amplitude

Parameters He and Hi define the peak amplitude of post-synaptic potentials (PSPs) generated by excitatory and inhibitory synapses, respectively. Intuitively, they can be interpreted as the strength of the postsynaptic response to a unitary presynaptic spike. In the standard formulation, He corresponds to the gain A, and Hi to the gain B in the differential equations.

Increasing He enhances the effect of excitatory feedback, which can make the circuit more prone to oscillatory behavior and even promote pathological activity if excessively strong. Conversely, increasing Hi boosts inhibitory control, often leading to dampened oscillations or stabilizing otherwise unstable dynamics.

Time constants

Time constants taue and taui (inverses of parameters a and b) represent the duration of the PSPs. They define how fast excitatory and inhibitory populations react and recover after being activated.

A shorter taue means that excitatory neurons respond and decay quickly, leading to sharper but more transient oscillations. A longer taui implies slower inhibitory feedback, which can delay the suppression of activity and allow oscillations to build up more easily.