A model of multifractality as observed in the neuronal activity of the human basal ganglia
Neuronal signals are usually characterized in terms of their discharge rate. However, this level of description is inadequate to account for the complex temporal organization of spike trains. In particular, fractal and multifractal patterns can be described using methods based on the so-called structure functions. Here, increasing order temporal structure functions are applied to neuronal signals obtained from the globus pallidus interna (GPi) of patients with Parkinsons disease. It is shown that these spike trains exhibit a characteristic multifractal spectrum, which is interpreted as a hallmark of the neuronal activity of the human, parkinsonian basal ganglia. Further, a neural field model is proposed to study the observed multifractality. The model consists of a partial differential equation that relates temporal to spatial activity considering a gradient field and a diffusive term. As the model is perturbed with stochastic initial conditions, the following is observed: 1. multifractality depends on the diffusive term; 2. multifractality is present for a range of diffusion coefficients; and 3. multifractal temporal properties are mirrored in space, which can be explained from the physical properties of the electric field governing neuronal activity. These results predict that passive electric properties of neuronal activity are far more relevant to the human brain than what has been usually considered.