= Solution
Put $g(z)=s(az)=ma^2z^2/(\sigma^2+a^2z^2)$. The <nullclines> are
$$
x=g(y),\qquad y=g(x).
$$
Any <steady state> has nonnegative coordinates because $g\geq0$. For $a\ne0$, $g$ is strictly increasing on the positive half-line. If a steady state had $x>y$, then $x=g(y)<g(x)=y$, a contradiction. The case $y>x$ is identical, so all steady states lie on the diagonal. Write $x=y=z$. Besides $z=0$, the equation $z=g(z)$ gives
$$
a^2z^2-ma^2z+\sigma^2=0,
$$
and hence
$$
\boxed{z_\pm=\frac{m\pm\sqrt{m^2-4\sigma^2/a^2}}2.}
$$
Thus \b[three distinct steady states exist exactly when $|a|m>2\sigma$], with $a\ne0$ and the positive parameter assumptions stated above. For positive excitatory coupling this is $am>2\sigma$. At equality the two positive states merge; below it only the origin remains. If $a=0$, the origin is again the only steady state.
Both nullclines begin at the origin and approach the limiting activity $m$ along their saturating direction; they are reflections of each other across $x=y$. Flow points to the right where $x<g(y)$ and to the left where $x>g(y)$. It points upwards where $y<g(x)$ and downwards where $y>g(x)$. The plot shows these nullclines and typical trajectories for the parameters of the next part.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-73-neural-phase.png]
{title=Nullclines and flow of the bistable two-neuron rate network}
The origin and upper diagonal state attract nearby trajectories, while the lower positive state is a saddle. Its <stable manifold> separates the two attracting regions: trajectories in each <basin of attraction> approach its stable node, giving <bistability>. The diagonal is invariant, and on it the scalar activity increases between $z_-$ and $z_+$ and decreases between zero and $z_-$ or above $z_+$. Activities remain nonnegative and bounded because $0\leq g\leq m$. The vector field has constant negative divergence $-2/\tau$, so the <Bendixson-Dulac criterion> also rules out periodic orbits in the physical quadrant. The network is a switch between persistent activity levels, rather than an oscillator.
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