Theory of axo-axonic inhibition

Abstract

The axon initial segment of principal cells of the cortex and hippocampus is contacted by GABAergic interneurons called chandelier cells. The anatomy, as well as alterations in neurological diseases such as epilepsy, suggest that chandelier cells exert an important inhibitory control on action potential initiation. However, their functional role remains unclear, including whether their effect is indeed inhibitory or excitatory. One reason is that there is a relative gap in electrophysiological theory about the electrical effect of axo-axonic synapses. This contribution uses resistive coupling theory, a simplification of cable theory based on the observation that the small initial segment is resistively coupled to the large cell body acting as a current sink, to fill this gap. The main theoretical finding is that a synaptic input at the proximal axon shifts the action potential threshold by an amount equal to the product of synaptic conductance, driving force at threshold, and axial axonal resistance between the soma and either the synapse or of the middle of the initial segment, whichever is closer. The theory produces quantitative estimates useful to interpret experimental observations, and supports the idea that axo-axonic cells can potentially exert powerful inhibitory control on action potential initiation.

Fig 1. Resistive coupling between soma and proximal axon, in a simple biophysical model. A, A current I injected in the proximal soma (orange, 20 µm; red, 40 µm) flows mostly resistively towards the cell body. B, The somatic membrane potential in response to a current (top) injected in the soma (bottom, black) is essentially the same as when the current is injected in the proximal axon (grey, 40 µm). C, This results in a linear (ohmic) change in membrane potential between soma and injection site, with slope proportional to I (here shown at t = 5 ms).

Fig 2. Resistive coupling in simultaneous patch-clamp recordings of soma and axonal bleb of pyramidal cortical neurons (data from [

34]). A, A negative current injected at the soma (top) results in the same hyperpolarization at the soma (middle, black) and proximal axon (red, 75 µm). The bottom trace shows the difference (green). B, The same current injected at the proximal axon (top), hyperpolarizes the axon (middle, red) more than the soma (black), with an additional ohmic component (green) due to the resistance between the soma and axonal injection site.

Fig 3. Modulation of action potential initiation by axonal inhibition.

A, Phase plots of somatic APs triggered by current pulses at the soma, when a negative current I_GABA (0-200 pA) is injected in the middle of the AIS. B, Same as A, when the current is injected at the soma. C, Somatic AP threshold as a function of axonal current. D, Threshold as a function of the mean voltage gradient between soma and AIS, when the axonal current is varied between 0 and 200 pA (dashed: line of slope 1). E, With a synaptic current I_GABA = g(E_GABA-V) (g = 5 nS), threshold as a function of synaptic reversal potential E_GABA. Dashed line: threshold with no synaptic current at the AIS. F, Threshold as a function of synaptic conductance g (E_GABA = -70 mV). Dashed line: theoretical prediction.

Fig 4. Effect of synapse and AIS position on threshold modulation.

A, Threshold as a function of synapse position, when the AIS extends from 5 to 35 µm from the soma (I _GABA  = -100 pA). Dashed line: theoretical prediction with a point AIS. B, Same as A, but the AIS extending from 25 to 55 µm from the soma. C, Change in threshold (relative to no axonal synaptic current) as a function of AIS center position, when synapses are distributed over the AIS and move with it (E _GABA  = -90 mV, g = 5 nS). D, Same as C, but synapses are fixed between 15 and 45 µm from the soma (shaded area).

Fig 5. Transient modulation of AP threshold by axo-axonic inhibition.

A, A 50 pA hyperpolarizing current is applied at the middle of the AIS (decay time ${\tau }_s=$ 20 ms). B, Somatic membrane potential. C, Voltage gradient between soma and current injection site. D, Somatic threshold of AP initiation, measured by applying a 0.7 nA depolarizing current at the soma at different times (in steps of 1 ms).