Note: Joint work with Carl van Vreeswijk (Paris Descartes Univeristy and CNRS, Paris, France).
We study a novel approach for the analysis of the asynchronous state in the ensemble of neurons receive i.i.d. post-synaptic external input and with an arbitrary strength of the feedback coupling. We consider a master equation of the limiting probability density for the age distribution using the so called hazard function . We solve a detailed linearized perturbation expansion for the population density of the leaky integrate-fire network by applying the Fokker-Plank approach. We derive the analytical description of the hazard function and formally study the effect of perturbations on the instantaneous firing rate of ensemble. Here, we elaborate on the analytical expression of input dependent hazard function and study its deformation under the effect of small perturbation in the external input. Thereafter, we determine the equilibrium distribution of ages given the hazard expression and investigate the stability of asynchronous state beyond the renewal master equation, following the method develop in , in a case augmented with a slow adaptation conductance and synaptic depression. We derive eigenvalue equation of the system, and it is shown that the synchronous state for the network with adaptation and feedback coupling can be unstable as this state can be destabilised in directions orthogonal to the limit cycle, which may change the units behaviour qualitatively. This confirms the result previously shown in a case with constant current input by . The approach introduced here ties the conventional Fokker-Plank perturbation analysis of the network activity to a point process master equation, which provides a simplified constructional method to study the synchronisation of a coupled neuronal ensemble in the presence of external noisy input. This approach introduces a new class of the population density models which does not required the diffusion term in the master equation. Moreover, it provides a fairly simple way for the introduction of heterogeneity into the network.  Cox, D. R. (1962). Renewal theory. London: Methuen.  Muller E and Buesing L. and Schemmel J. and Meier K. (2007). Spike-Frequency Adapting Neural Ensembles: Beyond Mean Adaptation and Renewal Theories. Neural Comp. 19 2958-3010.  van Vreeswijk C. (2000). Analysis of the asynchronous state in networks of strongly coupled oscillators. Phys Rev Lett 84 5110-5113.