## Section: New Results

### Modeling and simulating single neurons

#### Spiking dynamics of bidimensional integrate-and-fire neurons

Participants : Romain Brette, Jonathan Touboul.

Spiking neuron models are hybrid dynamical systems combining differential equations and discrete resets, which generate complex dynamics. Several two-dimensional spiking models have been recently introduced, modelling the membrane potential and an additional variable, and where spikes are defined by the divergence to infinity of the membrane potential variable. These simple models reproduce a large number of electrophysiological features displayed by real neurons, such as spike frequency adaptation and bursting. The patterns of spikes, which are the discontinuity points of the hybrid dynamical system, have been mainly studied numerically. Here we show that the spike patterns are related to orbits under a discrete map, the adaptation map, and we study its dynamics and bifurcations. Regular spiking corresponds to fixed points of the adaptation map while bursting corresponds to periodic orbits. We find that the models undergo a transition to chaos via a cascade of period adding bifurcations. Finally, we discuss the physiological relevance of our results with regard to electrophysiological classes.

This work has appeared in SIAM Dynamical Systems [24]

This work was partially supported by the EC IP project FP6-015879, FACETS and the Fondation d'Entreprise EADS.

#### Sensitivity to the cutoff value in the quadratic adaptive integrate-and-fire model

Participant : Jonathan Touboul.

The quadratic adaptive integrate-and-fire model is recognized as very interesting for its computational efficiency and its ability to reproduce many behaviors observed in cortical neurons. For this reason it is currently widely used, in particular for large scale simulations of neural networks. This model emulates the dynamics of the membrane potential of a neuron together with an adaptation variable. The subthreshold dynamics is governed by a two-parameter differential equation, and a spike is emitted when the membrane potential variable reaches a given cutoff value. Subsequently the membrane potential is reset, and the adaptation variable is added a fixed value called the spike-triggered adaptation parameter. We show in this note that when the system does not converge to an equilibrium point, both variables of the subthreshold dynamical system blow up in finite time whatever the parameters of the dynamics. The cutoff is therefore essential for the model to be well defined and simulated. The divergence of the adaptation variable makes the system very sensitive to the cutoff. Changing this parameter dramatically changes the spike patterns produced. Furthermore from a computational viewpoint, the fact that the adaptation variable blows up and the very sharp slope it has when the spike is emitted implies that the time step of the numerical simulation needs to be very small (or adaptive) in order to catch an accurate value of the adaptation at the time of the spike. It is not the case for the similar quartic and exponential models whose adaptation variable does not blow up in finite time, and which are therefore very robust to changes in the cutoff value.

This work has appeared in Neural Computation [25]

This work was partially supported by the EC IP project FP6-015879, FACETS, , and the Fondation d'Entreprise EADS.