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Local anaxonic neurons with graded potential release are importantingredients of nervous systems, present in the olfactory bulb system ofmammalians, in the human visual system, as well as in arthropods and nematodes.We develop a neuronal network model including both axonic and anaxonic neuronsand monitor the activity tuned by the following parameters: The decay length ofthe graded potential in local neurons, the fraction of local neurons, thelargest eigenvalue of the adjacency matrix and the range of connections of thelocal neurons. Tuning the fraction of local neurons, we derive the phasediagram including two transition lines: A critical line separating subcriticaland supercritical regions, characterized by power law distributions ofavalanche sizes and durations, and a bifurcation line. We find that the overallbehavior of the system is controlled by a parameter tuning the relevance oflocal neuron transmission with respect to the axonal one. The statisticalproperties of spontaneous activity are affected by local neurons at largefractions and in the condition that the graded potential transmission dominatesthe axonal one. In this case the scaling properties of spontaneous activityexhibit continuously varying exponents, rather than the mean field branchingmodel universality class.

On the role of anaxonic local neurons in the crossover to continuously varying exponents for avalanche activity