Why Two Identical Neurons Behave Differently

Concise summary

Two visually identical neurons can respond very differently to the same input because biological neurons have internal state (memory): their recent activity changes their dynamics. This arises from the neuron’s intrinsic biophysics and is best understood with dynamical-systems / phase-plane reasoning rather than treating neurons as simple input→output units.

---

Key scientific concepts, phenomena and results

Hodgkin–Huxley framework

• Membrane voltage V plus gating variables (m, n, h) form a 4D dynamical system that governs spiking (the Hodgkin–Huxley model).

Time-scale separation and model reduction

• m (sodium activation) is very fast → can be approximated by its steady-state value m∞(V).
• h (sodium inactivation) can often be dropped for qualitative spiking behavior.
• These reductions typically yield a 2D system in (V, n), where the n equation is written as relaxation toward n∞(V) with time constant τn(V).

Phase space / phase portrait intuition

• State-space axes: membrane voltage (V) and potassium gating (n).
• Nullclines:
  • V-nullcline and n-nullcline are curves where one derivative is zero.
  • Their intersections are equilibria.
• Equilibria types: stable node (rest), unstable equilibria, and saddles.
• Limit cycle: a closed trajectory corresponding to repetitive spiking.
• Separatrix: the stable manifold of a saddle that separates basins of attraction — functions as a threshold boundary.

How identical inputs can produce different outcomes

• The same constant input current shifts the V-nullcline. Depending on the neuron’s internal state (its current position in phase space), the same input can lead either to rest or to sustained spiking.
• Transient perturbations appear as horizontal pushes in the phase plane; whether they cross the separatrix determines a single spike versus a transition to sustained firing.

Bifurcations and excitability classes

• Saddle-node bifurcation: a node and a saddle collide and annihilate; can produce bistability and a sudden onset of repetitive firing.
• SNIC (saddle-node on invariant circle): a saddle-node occurring on the spiking orbit; typically associated with monostable integrator behavior.
• Andronov–Hopf bifurcation:
  • Supercritical Hopf: small stable oscillations appear smoothly as an equilibrium loses stability → monostable resonator (spike amplitude/shape depend on perturbation).
  • Subcritical Hopf: a large-amplitude stable limit cycle exists before the equilibrium becomes unstable → coexistence of rest and spiking (bistable resonator) and hysteresis.

Functional classifications

• Monostable vs bistable neurons (single vs coexisting stable attractors).
• Integrators vs resonators:
  • Integrators (saddle-node / SNIC): accumulate inputs; responses depend on cumulative input strength and are less sensitive to precise timing.
  • Resonators (Hopf): show subthreshold oscillations; timing and phase of inputs strongly affect responses (sensitive to rhythm/frequency).

Computational implications

• Hysteresis / bistability provides cellular memory (state maintenance).
• Integrators are well suited to encode stimulus strength (e.g., in some sensory circuits).
• Resonators are suited to temporal pattern processing and rhythm generation (e.g., in motor circuits).
• Spike amplitude and stereotypy differ by the bifurcation mechanism (all-or-none vs graded spike-like detours).

---

Methodology / analysis steps

1. Start from the Hodgkin–Huxley equations (4D).
2. Use time-scale separation:
   • Set the fast gating variable m = m∞(V).
   • Remove h if it is not essential for the qualitative behavior.
3. Reduce the system to two variables (V, n).
4. Plot the V-nullcline and n-nullcline in the (V, n) phase plane.
5. Identify equilibria and assess their stability by inspecting nullcline intersections and local flow.
6. Visualize flow arrows, limit cycles, and separatrices to understand typical trajectories.
7. Model inputs as:
   • Persistent current (a parameter that shifts the V-nullcline).
   • Transient pulses (instant horizontal displacements in phase plane).
8. Vary the input parameter to observe bifurcations (saddle-node, SNIC, Hopf) and classify neuron behavior (monostable/bistable, integrator/resonator).

---

Researchers / sources featured

• Hodgkin & Huxley — Hodgkin–Huxley model
• Eugene Izhikevich — Dynamical Systems in Neuroscience (referenced text)
• Andronov and Hopf — namesakes for Hopf bifurcation
• Brilliant.org — course platform / sponsor mentioned