2nd Dolomites Winter School

I will introduce a continuous time probabilistic model for systems of interacting and spiking neurons. In this process, neurons spike at a rate depending on their membrane potential value. When spiking, they have a direct influence on their post-synaptic partners, namely, a fixed value, called "synaptic weight", is added to the potential of the postsynaptic neurons. In between successive spikes, due to some leakage effects, the membrane potential process follows a deterministic flow. Firstly, I will discuss the construction, well-posedness and the longtime behavior of the process, for a finite number of neurons and for infinite systems of neurons, both in the case with and without reset of the spiking neuron. I will then discuss mean field limits for the Hawkes description (without reset) of the model. In particular we will see how in the limit an ODE describing the evolution of the mean firing rate appears and how this approach allows to describe for example oscillatory behavior. If time permits, I will also discuss the influence of delay in the synaptic transmission and quickly speak about short term memory. The final part of the course will be devoted to the more difficult case with reset (the membrane potential of the spiking neuron goes back to a resting value, inducing discontinuities in the model). We will see how the limit process and its longtime behavior help us to explain important phenomena in neuroscience such as "metastability".