Vertebrate neurons interact primarily with action potentials or "spikes", which is why a basic neurophysiological question is to understand precisely what makes neurons spike. What makes it a particularly good subject for a theoretician is that there are biophysical models that allow precise quantitative predictions to be tested. For the empirical work, we are now doing some patch-clamp experiments on slices in the lab (using some modern automation tools). We also collaborate with experimental neurophysiologists, in particular Dominique Debanne in Marseille, Maarten Kole in Amsterdam.

Spike initiation is a surprisingly rich subject, even today, 70 years after Hodgkin and Huxley’s pioneering work. In neurons, spikes are produced by a highly organized system of ionic channels that is spatially distributed on the soma and the start of the axon (called the axonal initial segment or AIS). The channels have different properties depending on their position, and those properties can change with activity. What is the functional logic of this organization, how does it get built, how does it adapt to changing conditions? My recent work on the subject has focused on the development of a theory I called *resistive coupling theory* (see below).

A few things I have found:

- Neurons spike not when a given voltage is reached, but rather when the voltage fluctuates a few mV above its recent average value (6, 8, 12).
- Because of the geometrical discontinuity between the soma and the axon, spikes are produced with a sharp threshold condition as in an integrate-and fire model (11). The surprising consequence for modelers is that the integrate-and-fire model is more realistic than the isopotential Hodgkin-Huxley model (13).
- The geometry of the AIS directly regulates not only the threshold for spiking (11, 19), but also the amount of current transmitted to the soma at spike initiation, which should be matched to the size of the soma (14, 17).

I have also started to write a book on the theory of action potentials (currently in a very sketchy state).

__Resistive coupling theory__

Many people do not realize that the model of Hodgkin and Huxley is not a model of spike initiation in neurons. It is a model of a space-clamped axon (or homogeneous axon), which stems from a syncytium (many cells merging their axons into a giant one); it was intended as a model of the biophysics of excitability. In most vertebrate neurons, spikes initiate in the axonal initial segment, a small structure near the much larger soma, packed with ionic channels and other proteins. The consequences of this heterogeneity are described by *resistive coupling theory,* very briefly reviewed in (17), which I first introduced with the name “compartmentalization hypothesis” (11) (a possibly confusing name). The resistive coupling model consists of a large soma and a small initiation site that are coupled by a resistance modeling the axon. The main point is that, while in an isopotential model the Na+ current is opposed by the leak current, a transmembrane current, in the resistive coupling model the Na+ current is opposed the axial current flowing towards the soma, an intracellular current.

This has a number of important consequences. First, spikes initiate through a dipole consisting of the soma (and proximal dendrites) and the AIS (15, 16), and not a soma-dendrite dipole. Second and most importantly, when the coupling resistance gets large enough, spike initiation suddenly transitions from a one-step phenomenon (soma and AIS spike at the same time) to a two-step phenomenon: the AIS spikes, then that spike charges the soma which then spikes (11, 15). This means that Na channels open abruptly as a function of somatic voltage, which in effect makes the integrate-and-fire model more realistic than the single-compartment Hodgkin-Huxley model (13). It explains why integrate-and-fire models are so good at predicting the responses of neurons to somatically injected currents, which we (and other labs) found out using optimization techniques (5,7). The theory also explains a number of experimental findings thanks to a quantitative formula for the threshold as a function of biophysical parameters (11). For example, spike onset is predicted to be about 6-8 mV higher in the AIS than in the soma. It also predicts that the coupling resistance and therefore AIS position directly regulates excitability. In the supplementary of (11), I also showed that a hyperpolarizing current injected at the AIS should raise the somatic threshold, proportionally to the distance of the AIS.

The mathematical modeling used an exponential approximation of the Na+ current (although it can be done with a more accurate Boltzmann approximation). I previously showed (with Wulfram Gerstner) that, when augmented with adaptive currents, this leads to a simple model that is a good approximation of (single-compartment) Hodgkin-Huxley models, the *adaptive integrate-and-fire model* (1-4).

With Sarah Goethals, we have updated the theory to take into account the spatial extent of the AIS (19). We find a formula for the spike threshold, as a function of geometrical parameters of the AIS and sodium conductance density. Quantitatively, the theory implies that the effect of AIS displacements on excitability are quite small (for physiological displacements). However, resistive coupling theory also predicts that the current backpropagated to the soma at spike initiation scales inversely with the distance of the AIS, and this effect can be quite substantial. We have shown with Maarten Kole and Sarah Goethals that this property seems to be used to normalize somatic spikes in the face of variations in somatodendritic geometry (14). In other words, the geometry of the AIS is involved not only in tuning the neuron’s excitability, but also its ability to transmit the spike to the soma.

We are currently testing some of our theoretical predictions with patch clamp experiments and immunochemical labeling of the AIS, as well as developing the theory further, in particular to understand how the neuron might “learn” to tune the geometry of the AIS.

Related presentations: Neural geometry and excitability (Collège de France, 2017); What is the most realistic single-compartment neuron model? (Barcelona 2015).

Threshold dynamics

An often neglected aspect is that spike threshold is not a fixed quantity, but varies on short and long timescales. On short timescales, we found with Jonathan Platkiewicz, using biophysical models, that most data could be explained by the inactivation of Na channels, possibly in conjunction with the subthreshold opening of K channels, and we proposed a threshold equation that predicts the value of spike threshold as a function of biophysical properties (6). Functionally, it implies that the threshold adapts to the membrane potential, which enhances the neuron's coincidence detection properties. We introduced the concept of the « effective postsynaptic potential » (difference between PSP and threshold) to understand the integrative properties with an adaptive threshold, such as enhanced coincidence detection (8). The threshold equation, initially derived in single-compartment models, was amended to take into account axonal spike initiation (11). Recently, our collaborator Dominique Debanne and colleagues found that the inactivation of Na channels also modulates post-synaptic potentials (18); thus it affects not only spike initiation but also spike transmission.

We then showed that our adaptive threshold model could predict spike threshold dynamics in vivo (12). I also showed how a model with adaptive threshold can produce responses that do not depend on input amplitude (9). We used this model to predict the responses of auditory neurons to sounds with various levels (10).

Related presentation: Dynamics of neural excitability (U. Antwerp 2014).

*Relevant publications* (chronological order):

- Brette, R. and W. Gerstner (2005).
*Adaptive exponential integrate-and-fire model as an effective description of neuronal activity*. - Touboul, J. and R. Brette (2008). Dynamics and bifurcations of the adaptive exponential integrate-and-fire model. (code)
- Gerstner, W. and R. Brette (2009) Adaptive exponential integrate-and-fire model.
- Touboul, J. and R. Brette (2009). Spiking dynamics of bidimensional integrate-and-fire neurons.
- Rossant C, Goodman DF, Platkiewicz J and Brette R (2010). Automatic fitting of spiking neuron models to electrophysiological recordings.
- Platkiewicz J, Brette R (2010) A Threshold Equation for Action Potential Initiation.
- Rossant C, Goodman DF, Fontaine B, Platkiewicz J, Magnusson AK and Brette R (2011). Fitting neuron models to spike trains.
- Platkiewicz J, Brette R (2011). Impact of Fast Sodium Channel Inactivation on Spike Threshold Dynamics and Synaptic Integration.
- Brette R (2012). Spiking models for level-invariant encoding.
- Fontaine B, Benichoux V, Joris PX and Brette R (2013). Predicting spike timing in highly synchronous auditory neurons at different sound levels.
- Brette R (2013). Sharpness of spike initiation in neurons explained by compartmentalization.
- Fontaine B, Peña JL, Brette R (2014). Spike-threshold adaptation predicted by membrane potential dynamics in vivo.
- Brette R (2015). What Is the Most Realistic Single-Compartment Model of Spike Initiation?
- Hamada M, Goethals S, de Vries S, Brette R, Kole M (2016). Covariation of axon initial segment location and dendritic tree normalizes the somatic action potential.
- Teleńczuk M, Fontaine B, Brette R (2017). The basis of sharp spike onset in standard biophysical models. (Code and binder)
- Teleńczuk M, Brette R, Destexhe A, Teleńczuk B (2018). Contribution of the axon initial segment to action potentials recorded extracellularly.
- Kole MHP and Brette R (2018). The electrical significance of axon location diversity.
- Zbili M, Rama S, Yger P, Inglebert Y, Boumedine-Guignon N, Fronzaroli-Moliniere L, Brette R, Russier M, and Debanne D (2019). Axonal Na+ Channels Detect and Transmit Levels of Input Synchrony in Local Brain Circuits.
- Goethals S, Brette R (2019). Theoretical relation between axon initial segment geometry and excitability. (code)

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