Stochastic modeling at diverse scales: from poisson to network neurons

Thisworkconcentratesonproblemsofmodelingatdiverseresolutions. Weinvestigatelimit procedureswhichleadamodelofonecategorytoamodelofanothercategory,andanalyzeobjective measures of models being close. While the methodology developedhere can be applied to any stochastic modeling problem, it is investigated in neural modeling context, which is of renewedinterest in control andinformation sciences. Weconsiderthejump-diffusionmodels,basedonacause-and-effectneurobiologicaldescription, the diffusion models obtained througha space and time averaging, and thedeterministic models, resulting from averaging out the random effects. Thethree forms of neuron models live independently in the literature, and the results obtained for one model are not moved to another form. To overcome this problem, we investigate the relations between various forms of the models. We formulate a general model driven by both marked Poisson processes and Wiener process, and identify its functions with neurobiological postulates. We then analyze the conditions under which it weakly converges to diffusion processes. We show that the convergenceis influencedby algebraic properties of theinputs. Inparticular, we provethat if the weights span the entire real space they belong to, and the jumps have deterministic amplitudes, then the model may converge only to a deterministic limit. Consequently, contrary to a common belief, the basic model cannot converge to a diffusion if its weights are modified in a learning process. We also introduce a scaling which leads the jump-diffusion models to diffusion models and show that the asymptotic models are eitherthe zero-drift diffusions or are deterministic. If separate scaling is applied to each classof inputs then the inputs can be dividedintothestochasticandthedeterministicclasses. Thelatterinfluenceonlythedriftofthe diffusion model, and the former influence only the diffusion function. These novel hypotheses call for experimental verification inreal biological systems. Thediffusionmodelsofneuronusuallyleadtoafirstpassageproblemfordiffusionprocess. We provide a uniform treatment of both analytical and numerical methods for thefirst passagetimedistributionofthediffusionprocessesthroughageneralbarrier,andapplytheresults the diffusion neuron. The results enable to further simplifythe diffusion neurons, to be able