In primates, GPi and SNr neurons fire in a tonic manner, keeping the motor thalamus inhibited, and momentary reductions of their discharge rate facilitate movement. The fundament of the classic model of pathophysiology of Parkinson’s disease lies on an excessive discharge rate of the output nuclei of the basal ganglia: the globus pallidus interna (GPi), and substantia nigra reticulata (SNr) ( Albin et al., 1989 DeLong, 1990). The lack of a description of the complex properties of basal ganglia neuronal activity might be one cause of this failure. However, no current model allows to predict and control deep brain stimulation (DBS), one of the major therapeutic approaches to Parkinson’s disease ( Montgomery, 2016).
Current models of the basal ganglia are partly successful in the prediction of neurophysiologic alterations occurring in movement disorders, including Parkinson’s disease. The basal ganglia are a circuit of densely interconnected subcortical nuclei, whose disease is related to human movement disorders ( Obeso et al., 2008). The results obtained from the model predict that passive electric properties of neuronal activity, including ephaptic coupling, are far more relevant to the human brain than what is usually considered: passive electric properties determine the temporal and spatial organization of neuronal activity in the neural tissue. The model reproduces the behavior of human basal ganglia neurons and shows that it is like that of turbulent fluids. Multifractality is present in the model for a range of diffusion coefficients, and multifractal temporal properties are mirrored into space. I introduce a neural field model that includes a non-linear gradient field, representing neuronal excitability, and a diffusive term to consider the physical properties of the electric field. I propose a new approach to the study of neuronal signals: to study spiking activity in terms of the velocity of spikes, defining it as the inverse function of the instantaneous frequency. To achieve an accurate model of such multifractality is critical for understanding the basal ganglia, since other non-linear properties, such as entropy, depend on the fractal properties of complex systems. Here, I apply high order structure functions to the analysis of neuronal signals recorded from parkinsonian patients during functional neurosurgery, recovering multifractal properties. Complex temporal properties, which are characteristic of neuronal systems, can only be described with the appropriate, complex mathematical tools. Neuronal signals are usually characterized in terms of their discharge rate, a description inadequate to account for the complex temporal organization of spike trains.