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Neural Activity Propagation in an Unfolded Hippocampal Preparation with a Penetrating Micro-electrode Array

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Non-Gaussian propagator for elephant random walks.

M A A da Silva¹, J C Cressoni, Gunter M Schütz

¹Departamento de Física e Química, FCFRP, Universidade de São Paulo, 14040-903 Ribeirão Preto, SP, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|September 17, 2013

Summary

The elephant random walk (ERW) model

Area of Science:

Statistical physics
Complex systems modeling

Background:

The elephant random walk (ERW) model is widely used in statistical physics.
A long-standing consensus suggested its propagator is Gaussian.

Purpose of the Study:

To investigate the nature of the ERW propagator.
To challenge the prevailing Gaussian assumption.

Main Methods:

Numerical simulations to gather evidence.
Mathematical analysis of the associated Fokker-Planck equation.
Calculation of skewness to determine propagator properties.

Main Results:

Strong numerical evidence indicates the ERW propagator is generally non-Gaussian.
The propagator was found to be non-Lévy.

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A second, non-Gaussian solution to the Fokker-Planck equation was identified.

Mathematical proof confirms a non-Gaussian propagator in the superdiffusive regime.

Conclusions:

The established Gaussian propagator assumption for the ERW model is incorrect.
The ERW model exhibits complex behavior beyond Gaussian or Lévy distributions.
Further investigation into higher-order terms of the Fokker-Planck equation is warranted.