Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 73 / 3 / a / Solution

Interpret $x$ and $y$ as nonnegative firing-rate activities, rather than individual action potentials. Take $\tau ,m,\sigma >0$. The relaxation time $\tau$ controls how quickly activity changes. The decay terms $-x$ and $-y$ make activity fall in the absence of recurrent drive. Each neuron receives a weighted input from the other, with coupling $a$, and the static input-output function converts that input into a target activity.

The nonlinear response tends to $m$ at large input magnitude and equals $m/2$ at $|p|=\sigma$, so $m$ is its maximum activity and $\sigma$ its half-saturation input scale. At small inputs it is approximately $m{p}^2/{\sigma }^2$. For $a>0$ this is a mutually excitatory saturating rate network: increasing one neuron's activity increases the other's drive, permitting positive feedback. The response is even in $p$, so this particular equation does not model a negative $a$ as ordinary inhibition; its dynamics depend on $a^2$.

The equations combine passive relaxation with instantaneous nonlinear recurrent drive. They omit explicit spike timing, delays, channel dynamics and stochasticity, so they are an activity-level model for computational neuroscience.