Elegant Geometry of Neural Computations

Biophysical neuron models, such as the Hodgkin–Huxley equations, describe neurons not as static input–output units but as dynamical systems evolving over time in a state space defined by membrane voltage and channel variables.

Simplified neuron models with two state variables—voltage and a potassium gating variable—can be analyzed using nullclines and phase portraits to understand excitability and spiking.

Certain neurons exhibit bistability: at the same constant input current they can either sit quietly at rest or fire repetitively, with the actual state determined by recent perturbations and history.

Neurons transition from silent to repetitively spiking regimes via qualitative changes in their phase portraits known as bifurcations, triggered by changes in parameters like input current.

Neurons can be broadly categorized as integrators or resonators based on how their dynamics respond to input amplitude and timing, reflecting different bifurcation structures.