= Solution
The <Jacobian matrix> at a diagonal <steady state> is
$$
J(z)=\frac1\tau\begin{pmatrix}-1&g'(z)\\g'(z)&-1\end{pmatrix},\qquad
g'(z)=\frac{2ma^2\sigma^2z}{(\sigma^2+a^2z^2)^2}.
$$
Its <eigenvectors> $(1,1)$ and $(1,-1)$ represent common and opposing perturbations, with respective <eigenvalues> $(-1+g')/\tau$ and $(-1-g')/\tau$.
For a nonzero equilibrium, $\sigma^2+a^2z^2=ma^2z$ and $\sigma^2/a^2=z(m-z)$. Consequently $g'(z)=2(m-z)/m$. The lower root has $g'>1$ and is a <saddle equilibrium>; the upper root has $0<g'<1$ and is a <stable node>. At the origin $g'(0)=0$, 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 are
$$
\boxed{\begin{array}{c|cc|c}
(x,y)&\lambda_{(1,1)}&\lambda_{(1,-1)}&\text{type}\\\hline
(0,0)&-0.1&-0.1&\text{stable node}\\
(2,2)&0.06&-0.26&\text{saddle}\\
(8,8)&-0.06&-0.14&\text{stable node}
\end{array}}
$$
Biologically, 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.
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