FitzHugh–Nagumo network

Every neuron is an identical FitzHugh–Nagumo oscillator with fast voltage v and slow recovery w. A neuron sends input whenever its voltage is above zero. The vertical Voltage threshold line at v = 0 separates the shaded inactive (v ≤ 0) and active (v > 0) regions. It is the threshold for sending synaptic input, separate from the cubic nullclines. All quantities, including time, are dimensionless.

For N > 1, I_syn,i = ±g Σ_{j ≠ i} H(v_j)/(N − 1), where H(v) = 1 if v > 0, and 0 otherwise. The plus sign is excitation; the minus sign is inhibition. For one neuron the synaptic input is zero. Dividing by N − 1 keeps the maximum total input at ±g as the network grows.

• Excitatory example: I = 0 and default g = 1. The cubic rises when partners fire. Recovered neurons can fire sooner, closing a firing gap.
• Inhibitory example: I = 1 and default g = 2. Inhibition lowers the cubic and can hold a neuron silent while it recovers. When the active partner stops, the cubic rises and the silent partner can fire. With two neurons this gives alternation.
• All nullclines: the voltage-nullcline family includes every possible count of active partners, from zero to N − 1, at the configured coupling strength. The free cubic has no input. The excited or inhibited cubic has every partner active. Intermediate curves remain visible even when no neuron currently occupies them. With one neuron or zero strength, they coincide with the free cubic. With coupling switched off, the family remains a reference and every neuron's actual input is zero.
• Following neurons: “None” turns off colored nullcline highlighting while keeping every reference nullcline and neuron trajectory visible. “All neurons,” the default, shows every trajectory and highlights all currently occupied cubics in neuron colors. Overlapping current cubics share colored segments. Choosing one neuron highlights its trajectory and current cubic while keeping the entire family visible.
• Nullcline equations: a neuron's instantaneous voltage nullcline is w = v − v³/3 + I + I_syn,i. Its recovery nullcline is w = 2(v + 0.4). A nullcline is a zero-derivative curve, not a trajectory or a fixed threshold.
• More neurons: inhibition can produce staggered firing, clusters, or more irregular rhythms. More than two neurons cannot all be exactly half a cycle apart from one another. Changing strength or timing can change the outcome.
• Starting phases: neurons start at different points on their isolated limit cycle. “Nearby phases” uses a band of 0.30 cycles; “Around the cycle” distributes them around one cycle. The initial states are reproducible, with a small phase offset for larger networks to avoid exact symmetries.
• Reading the traces: the phase plane retains a six-unit trajectory tail. The trace and raster show the chosen time window, with up to 640 units of history retained. Colored cubics show current inputs; older trajectory segments may have evolved under earlier inputs.
• Timing measurement: for two neurons, separation uses the last completed cycle of neuron 1 containing one spike from neuron 2. Zero degrees means coincident firing; 180° means alternation. A result is withheld until there is enough suitable spike history. The readout is a finite observation, not a stability proof.

The equations use threshold-gated additive currents. They illustrate timing mechanisms rather than a conductance-based synapse or a fit to experimental data. Integration uses fourth-order Runge–Kutta with a fixed step of 0.005, independently of playback speed. “Step +1” and “Advance +100” pause playback and integrate by the indicated amount.

Model background: FitzHugh (1961), Impulses and physiological states in theoretical models of nerve membrane. For inhibitory switching mechanisms, see Ausborn et al. (2018), State-dependent rhythmogenesis and frequency control in a half-center locomotor CPG.