Why Two Identical Neurons Behave Differently

Two neurons that look identical can behave very differently because their electrical activity depends on where their state sits in an internal “history” of dynamical trajectories. In a simplified two-variable model of a neuron—membrane voltage plus potassium-channel activation—small differences in prior perturbations can place the system on opposite sides of a threshold in phase space. That threshold determines whether the neuron returns to rest after a spike attempt or locks into sustained repetitive firing.

The core mechanism is geometric. Starting from the Hodgkin–Huxley framework, the model reduces fast sodium activation to an instantaneous equilibrium and often omits the inactivation gate, leaving coupled equations for voltage and potassium gating. Plotting these variables as axes produces a phase portrait: arrows show how the neuron’s state moves over time. Where arrows vanish are equilibria—resting states and other fixed points. Some equilibria are stable (nearby trajectories flow back), while others are unstable (trajectories run away). Intersections of nullclines—curves where one variable stays constant—create these equilibria, and their stability shapes what the neuron does after perturbations.

When the neuron receives sustained input current, the phase portrait can support a limit cycle: a closed loop that the system repeatedly traverses, corresponding to rhythmic spiking. Crucially, the model often contains both a stable resting equilibrium and an unstable saddle point whose separatrix acts like a watershed. Perturbations that push the state into the limit cycle’s basin of attraction produce persistent firing; perturbations that fall short return the state to rest. If the push is too strong and overshoots the basin, the neuron can also return to rest after a spike detour. This “sweet spot” produces bistability and hysteresis: the same input current can yield either rest or repetitive firing depending on recent history.

Different neurons achieve different computational behaviors through different bifurcations—qualitative changes in the phase portrait as input current (or biophysical parameters) shift. For bistable integrators, increasing current can trigger a saddle-node bifurcation where a stable node and an unstable saddle collide and annihilate, leaving only the limit cycle; the neuron then spikes repetitively. A closely related case, SNIC (saddle-node on an invariant circle), yields monostable integrators: the resting equilibrium blocks spiking until it disappears, after which repetitive firing emerges.

For resonator behavior, the story shifts to Andronov–Hopf bifurcations. In the supercritical Hopf case, a stable equilibrium loses stability and a small limit cycle emerges gradually, producing monostable resonators with sub-threshold oscillations whose timing matters. In the sub-critical Hopf case, a large stable cycle can coexist with a stable equilibrium, separated by an unstable cyclic threshold; this creates bistable resonators where the neuron can either rest or sustain oscillations at the same input level.

Putting it together yields two key dichotomies—monostable vs bistable, and integrator vs resonator—that map onto four bifurcation types. Integrators accumulate perturbations until crossing a threshold, while resonators respond preferentially to inputs at specific phases of oscillations. The result is a dynamical-systems explanation for why neurons can function as memory-like switches or timing-sensitive oscillators even when their outward structure appears identical.