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We present a mathematical analysis of a networks with Integrate-and-Fireneurons and adaptive conductances. Taking into account the realistic fact thatthe spike time is only known within some \textit{finite} precision, we proposea model where spikes are effective at times multiple of a characteristic timescale $\delta$, where $\delta$ can be \textit{arbitrary} small (in particular,well beyond the numerical precision). We make a complete mathematicalcharacterization of the model-dynamics and obtain the following results. Theasymptotic dynamics is composed by finitely many stable periodic orbits, whosenumber and period can be arbitrary large and can diverge in a region of thesynaptic weights space, traditionally called the "edge of chaos", a notionmathematically well defined in the present paper. Furthermore, except at theedge of chaos, there is a one-to-one correspondence between the membranepotential trajectories and the raster plot. This shows that the neural code isentirely "in the spikes" in this case. As a key tool, we introduce an orderparameter, easy to compute numerically, and closely related to a natural notionof entropy, providing a relevant characterization of the computationalcapabilities of the network. This allows us to compare the computationalcapabilities of leaky and Integrate-and-Fire models and conductance basedmodels. The present study considers networks with constant input, and withouttime-dependent plasticity, but the framework has been designed for bothextensions.

On Dynamics of Integrate-and-Fire Neural Networks with Conductance Based Synapses