Cortical and subcortical neural activity has been modeled for decades by means of recurrent networks of integrate and fire neurons (IFN). Such networks have proved to be able to capture a variety of neural functions ranging from sensory processing to short term memory and decision making. Some network scale phenomena, however, were considered too complex to be simulated with simple basic components as IFN. Namely, Local Field Potentials (LFP) were simulated with multi-compartmental models, as IFN lacked the ability to cope with the spatial features of the signal, and thalamic oscillations were modeled with Hodgkin and Huxley and similar neuron models, as IFN were thought not be able to mimic the rebound properties of the neurons of the area. Here we will show instead how it is possible to capture both phenomena with IFN networks. First, in a series of works spanning almost a decade, we were able to build flexible LFP proxies simulating the extracellular signal from IFN network dynamics. We identified several key properties of the neuronal structure, based on the dipolar approximation of LFP generation, and we implemented them in the IFN model. This led to a dimensionality reduction of the system parameters of two order of magnitudes, while capturing almost entirely the LFP dynamic. Actually, the resulting model of LFP is so efficient to be able to reconstruct spiking dynamics from LFP recorded in the primary visual cortex. Second, we were able to reproduce with IFN two dynamic regimes of the thalamus displaying different characteristic oscillations bands and different information processing properties. The key advancement was to highlight the connection between single neuron dynamics and network regimes. Spindle oscillations preventing information transmission to the cortex occur when the neurons fire exclusively because of rebound from inhibition. When the external excitatory inputs are sufficiently strong, instead, thalamocortical relay neurons start firing because of depolarization, weakening spindle oscillations and leading to information transmission. Starting from these results we will discuss the possibility of capturing the whole process of encoding of sensory stimuli by means of a chain of spiking neuron model covering the whole path from peripheral sensors to primary sensory cortex. Building the whole structure with simple and elegant IFN will make the analysis and the comparison with experimental results sharper.

In this survey, we review recent results concerning the canonical dispersive flow e itH led by a Schrodinger Hamiltonian H. We study, in particular, how the time decay of space L p -norms depends on the frequency localization of the initial datum with respect to the some suitable spherical expansion. A quite complete description of the phenomenon is given in terms of the eigenvalues and eigenfunctions of the restriction of H to the unit sphere, and a comparison with some uncertainty inequality is presented.