We present a mathematical analysis of a networks with Integrate-and-Fire neurons and adaptive conductances. Taking into account the realistic fact that the spike time is only known within some \textit{finite} precision, we propose a model where spikes are effective at times multiple of a characteristic time scale $\delta$, where $\delta$ can be \textit{arbitrary} small (in particular, well beyond the numerical precision). We make a complete mathematical characterization of the model-dynamics and obtain the following results. The asymptotic dynamics is composed by finitely many stable periodic orbits, whose number and period can be arbitrary large and can diverge in a region of the synaptic weights space, traditionally called the ``edge of chaos'', a notion mathematically well defined in the present paper. Furthermore, except at the edge of chaos, there is a one-to-one correspondence between the membrane potential trajectories and the raster plot. This shows that the neural code is entirely ``in the spikes'' in this case. As a key tool, we introduce an order parameter, easy to compute numerically, and closely related to a natural notion of entropy, providing a relevant characterization of the computational capabilities of the network. This allows us to compare the computational capabilities of leaky and Integrate-and-Fire models and conductance based models. The present study considers networks with constant input, and without time-dependent plasticity, but the framework has been designed for both extensions.

Neural field models first appeared in the 50's, but the theory really took off in the 70's with the works of Wilson and Cowan {wilson-cowan:72,wilson-cowan:73} and Amari {amari:75,amari:77}. Neural fields are continuous networks of interacting neural masses, describing the dynamics of the cortical tissue at the population level. In this report, we study homogeneous stationary solutions (i.e independent of the spatial variable) and bump stationary solutions (i.e. localized areas of high activity) in two kinds of infinite two-dimensional neural field models composed of two neuronal layers (excitatory and inhibitory neurons). We particularly focus on bump patterns, which have been observed in the prefrontal cortex and are involved in working memory tasks {goldman-rakic:95}. We first show how to derive neural field equations from the spatialization of mesoscopic cortical column models. Then, we introduce classical techniques borrowed from Coombes {coombes:05} and Folias and Bressloff {folias-bressloff:04} to express bump solutions in a closed form and make their stability analysis. Finally we instantiate these techniques to construct stable two-dimensional bump solutions.

We propose a solution to the direct problem of VSD optical imaging based on a neural field model of a cortical area and reproduce optical signals observed in various mammals cortices. We first present a biophysical approach to neural fields and show that these easily integrate the biological knowledge on cortical structure, especially horizontal and vertical connectivity patterns. After having introduced the reader to VSD optical imaging, we propose a biophysical formula expressing the optical imaging signal in terms of the activity of the field. Then, we simulate optical signals that have been observed by experimentalists. We have chosen two experimental sets: the line-motion illusion in the visual cortex of mammals (jancke, chavane, et al. 2004} and the spread of activity in the rat barrel cortex (petersen, grinvald, et al. 2003). We begin with a structural description of both areas, with a focus on horizontal connectivity. Finally we simulate the corresponding neural field equations and extract the optical signal using the direct problem formula developed in the preceding sections. We have been able to reproduce the main experimental results with these models.