Working Paper Article Version 1 This version is not peer-reviewed

Towards Random Walks Underlying Neuronal Spikes

Version 1 : Received: 3 December 2019 / Approved: 4 December 2019 / Online: 4 December 2019 (04:43:17 CET)

How to cite: Tozzi, A.; Yurkin, A.; Peters, J.F. Towards Random Walks Underlying Neuronal Spikes. Preprints 2019, 2019120041 Tozzi, A.; Yurkin, A.; Peters, J.F. Towards Random Walks Underlying Neuronal Spikes. Preprints 2019, 2019120041


The brain, rather than being homogeneous, displays an almost infinite topological genus, because is punctured with a very high number of topological vortexes, i.e., .e., nesting, non-concentric brain signal cycles resulting from inhibitory neurons devoid of excitatory oscillations. Starting from this observation, we show that the occurrence of topological vortexes is constrained by random walks taking place during self-organized brain activity. We introduce a visual model, based on the Pascal’s triangle and linear and nonlinear arithmetic octahedrons, that describes three-dimensional random walks of excitatory spike activity propagating throughout the brain tissue. In case of nonlinear 3D paths, the trajectories in brains crossed by spiking oscillations can be depicted as the operation of filling the numbers of the octahedrons in the form of “islands of numbers”: this leads to excitatory neuronal assemblies, spaced out by empty area of inhibitory neuronal assemblies. These procedures allow us to describe the topology of a brain of infinite genus, to assess inhibitory neurons in terms of Betti numbers, and to highlight how non-linear random walks cause spike diffusion in neural tissues when tiny groups of excitatory neurons start to fire.


arithmetic figures; Pascal’s triangle; geometrization of brain; central nervous system; nonlinear


Biology and Life Sciences, Neuroscience and Neurology

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