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

The Jacobian matrix at a diagonal steady state isIts eigenvectors $(1,1)$ and $(1,-1)$ represent common and opposing perturbations, with respective eigenvalues $(-1+g^{\prime })/\tau$ and $(-1-g^{\prime })/\tau$.

For a nonzero equilibrium, ${\sigma }^2+a^2{z}^2=m{a}^{2}z$ and ${\sigma }^2/a^2=z(m-z)$. Consequently $g^{\prime }(z)=2(m-z)/m$. The lower root has $g^{\prime }>1$ and is a saddle equilibrium; the upper root has $0<g^{\prime }<1$ and is a stable node. At the origin $g^{\prime }(0)=0$, so both eigenvalues are $-1/\tau$.

The supplied parameters give $\sigma /a=4$, and the nonzero roots solve $z^2-10z+16=0$, giving $2$ and $8$. The numerical stability results areBiologically, the origin is a quiescent state and the upper equilibrium a sustained mutually reinforced activity state. The saddle represents a collective activation threshold: a perturbation crossing its basin boundary can switch the network from low to high activity. This persistent-state mechanism can model a memory or decision variable, but the simplified rate equations do not establish that a particular biological circuit uses it.