Fig 1.
(A) Voltage traces of 3 models for the same input current: integrate-and-fire (IF), quadratic (QIF) and Hodgkin-Huxley (HH). The models differ mainly in their spiking responses. (B) Relationships between different single-compartment models. The IF model is a phenomenological model of spike initiation, while the HH model is a biophysical model. Other models can be obtained by reduction, i.e., as approximations of one of the above models under specific conditions.
Fig 2.
Empirical content of models, from the quadratic model (only spike trains are predicted) to multi-compartmental HH models (Membrane potential and ionic currents are predicted at different locations on the neuron).
Fig 3.
Empirical accuracy of the single-compartment HH model.
(A) Spikes of real neurons have a distinct kink at onset (top: human pyramidal cell, adapted from [18]), which does not appear in single-compartment HH models (bottom: HH models of squid giant axon and rat hippocampal interneuron, adapted from [19]). (B) The peak current in somatic voltage-clamp shows a discontinuity as a function of voltage (red), not seen when axonal Na channels are inactivated (black) (adapted from [20]). (C) Spiking responses of cortical neurons in vitro to somatic current injection (black) are well predicted by integrate-and-fire models with sharp spike initiation (blue, first two traces; percentages are proportion of correctly predicted spikes) but not by the Izhikevich model, a variation of the quadratic model (blue, third trace) (adapted from [22]). (D) Phase locking to high frequencies: cortical neurons fire at preferred phases of a sinusoidal current added on top of noise (left), at frequencies up to several hundred Hz (right, vector strength is a measure of phase locking) (adapted from [23]).
Fig 4.
Cooperativity and backpropagation hypotheses.
(A) The cooperativity hypothesis (adapted from [40]). Collective activation curve of Na channels as a function of coupling strength λ (0: no coupling). When cooperativity is strong enough, collective activation becomes step-like: Na channels open all at once when a threshold voltage is exceeded. (B and C) The active backpropagation hypothesis (adapted from [39]). (B) When a spike propagates forward along an active 2 mm-long cylinder of 1 μm diameter (model), its onset becomes progressively sharper. (C) Onset rapidness as a function of distance from soma where the spike is measured, for three models all including a 50 μm long axon (1 μm diameter) and a soma: soma with the same diameter as the axon (“Uniform axon”), large soma (“Soma+Axon”), and large soma and dendrites (“Dend+Soma+Axon”).
Fig 5.
The compartmentalization hypothesis (B-F adapted from [31]).
(A) Dual-patch recording of an action potential in soma (black) and in the axonal initial segment (red) (digitized from [41]). (B) Electrical model of the soma and initiation site (Vs: somatic voltage; Va: axonal voltage; Ra: axial resistance). (C) A current injected close to the soma produces a linear depolarization between soma and injection point (bottom: voltage as a function of distance to soma). The depolarization is proportional to Ra, and therefore to distance (left: injection at 20 μm; left: at 40 μm). (D) Na current (red curve) and axonal current (black lines) as a function of Va when Na channels are placed at 20 μm (left) and 40 μm (right) away from the soma. The 5 lines correspond to Vs between-60 mV and-47.5 mV. (E) Proportion of open Na channels as a function of Vs, with Na channels at the soma (dark blue), at 20 μm (green), and at 40 μm (red) away from the soma. (F) Current-voltage relationship measured at the soma (same color code as in E).