Branching fBM as a Model of Serotonergic Axons*

Fractional Brownian Motion (fBM) has recently been used as a mathematical model for the structure of the random neural network that propagates serotonin throughout the brains of vertebrates. Density maps of superdiffusive reflecting fBM agree well with the densities of serotonergic axons in embryonic transgenic mouse mid-brains. In this talk, we introduce branching events in an fBM trajectory to improve the model. We examine the geometry of the branching event and its resemblance to branching events in our biological data. Then we study trajectories with a non-zero branching rate to determine whether the mean square displacement and boundary density accumulations are retained from the non-branching reflected fBM. As fractional Brownian Motion is the result of long-term correlations between the increments, we investigate the effects of erasing or preserving the ‘memory’ of a trajectory after a branching event. Three general models are discussed here: a) both resulting trajectories preserving long-term correlations, b) one trajectory preserving and one trajectory erasing their long-term correlations, and c) both trajectories having their correlations erased. In cases b) and c), a finite branching rate effectively introduces a tempering of the fBM correlations. As a result, density accumulations close to boundaries are restricted to a finite boundary region.