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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.

A synapse is a signaling junction between a neuron and another biological cell. The sending side is presynaptic and the receiving side postsynaptic. At a chemical synapse, released neurotransmitter acts on a receptor in the receiving cell membrane. At an electrical synapse, gap junctions provide a direct path for electric current between cells. See 1.1 Elements of Neuronal Systems.

In a common sequence at a chemical synapse, an arriving action potential opens voltage-gated ion channels selective for calcium. Calcium entry triggers fusion of synaptic vesicles with the presynaptic cell membrane, releasing neurotransmitter into the synaptic cleft. Binding to a receptor changes postsynaptic electrical resistance and conductance, directly through an ion channel or through an intracellular signaling pathway. The resulting change in membrane potential is a synaptic potential. See Ion Channels and the Electrical Properties of Membranes.

A useful conductance approximation writes the inward-positive synaptic electric current aswhere $g_{\mathrm{syn}}\ge 0$ describes the activated electrical resistance and conductance, $V$ is the postsynaptic membrane potential, and $E_{\mathrm{syn}}$ is the reversal potential, the voltage at which this current vanishes. Opening these channels tends to pull $V$ toward $E_{\mathrm{syn}}$; it does not prescribe a current of fixed sign independently of $V$.

For a worked example, let $V=-65\mathrm{mV}$ and $g_{\mathrm{syn}}=1\mathrm{nS}$. A reversal potential of $0\mathrm{mV}$ gives $I_{\mathrm{syn}}=65\mathrm{pA}$, an inward electric current that raises membrane potential. A reversal potential of $-75\mathrm{mV}$ instead gives $-10\mathrm{pA}$, which lowers membrane potential. These are illustrative calculations, not measurements. A reversal potential of $-60\mathrm{mV}$ can raise $V$ from $-65\mathrm{mV}$ yet oppose reaching a threshold of $-50\mathrm{mV}$: a depolarizing input need not be excitatory.

An excitatory synapse favors a subsequent action potential, whereas an inhibitory synapse opposes it. The effect depends on the receptor, the permeating ions, their gradients and the reversal potential; a neurotransmitter does not have one fixed excitatory or inhibitory sign in every context. Even with $E_{\mathrm{syn}}=V$, an added conductance can reduce responses to other inputs by lowering effective electrical resistance, an effect called shunting inhibition. See Purves and colleagues: excitatory and inhibitory postsynaptic potentials. This dependence matters when translating a biological synapse into a signed weight or an imposed electric current in computational neuroscience.