Towards a low-dimensional dynamics of spiking neuron networks
Modelling brain functions as an emergent collective phenomenon of neuronal networks needs to take into account its intrinsic multiscale organization. Local microcircuits distributed across different layers are hardwired as a mesoscopic web of connections making up the single cortical areas, which eventually interplay to give rise to the macroscopic orchestration underlying brain computation. Bridging the gap between these multiple levels of description requires the development of a theoretical framework in which the huge-dimensionality of the system may be progressively reduced allowing to climb up such hierarchy of scales. Here I will talk about a viable approach in which microscopic details of neuronal networks can be effectively captured following the stochastic evolution of the membrane potentials and other single-neuron degrees of freedom, like spike-frequency adaptation and non-instantaneous synaptic transmission. I will briefly review this population density approach, focusing on a particular implementation in which a spectral expansion method is exploited. I will conclude providing evidence of the feasibility and the effectiveness of a dimensional reduction of the system dynamics in which the mesoscopic network states are characterized by the instantaneous rate of spikes emitted by the whole population and other few activity-dependent variables.