Fractals in the Nervous System: Conceptual Implications for Theoretical Neuroscience Gerhard Werner gwer1@mail.utexas.edu Department of Biomedical Engineering University of Texas at Austin, TX. Abstract This essay aims to document the prevalence of fractals across all levels of the nervous system, supporting their functional relevance, and to highlight unresolved issues regarding the relationships among power law scaling, self-similarity, and self-organized criticality. It emphasizes the significance of allometric control processes and the adaptability of dynamic fractals across scales. The final section reflects on the implications of these findings for future research in theoretical neuroscience. Contents 1. Introduction Fractals, introduced by Mandelbrot in 1977, are self-similar geometric objects with features on infinite scales. They describe highly intermittent temporal behavior without a characteristic time scale. Their statistical analysis provides insights into complex systems. Fractals can be characterized by power functions with non-integer exponents, known as fractal dimensions. This essay focuses on random fractals with stochastic elements. Fractals are signals with scale-invariant, self-similar behavior. They can be analyzed by decomposing signals into temporal and spatial scales. Monofractal processes are characterized by a single scaling exponent, while multifractal signals have multiple exponents. Power law scaling and fractal patterns are observed at all levels of neural organization. Recent advances in measurement methods have led to new data, prompting integration of fractality with other insights into brain organization and complexity, particularly in the context of criticality. 2. Power-law scaling in neuronal structures and processes This section summarizes essential aspects of fractal properties at each level of neuronal organization. Observations remain diverse, with varying conditions and methods. Recent studies have established stringent criteria for identifying fractal properties. Some apparent power law scaling may not be supported by rigorous statistical criteria. The majority of experimental data on fractals in neural structures suggest widespread functional significance. 2.1 Neuronal morphology Mandelbrot (1977) noted that neurons, such as Purkinje cells, may be fractal. Studies show fractal dimensions in dendritic trees, glial cells, and neuron types in different laminae. Fractal analysis reveals differences in dendrite branching structures and functional capacities. Fractal properties differentiate between primary and secondary visual areas in macaque cortex. Fractal dimensions of dendrite arbors are universal, suggesting similar processes in their construction. 2.2 The peripheral nervous system: ion channels, point process analysis of activity in peripheral nerves and individual neurons Ion channel gating and neuronal discharge patterns exhibit fractal features. The kinetics of ion transport across neuronal membranes involve ion channels. The switching probabilities of ion channels at different time scales display fractal scaling. The fractal activity at the neuromuscular junction is supported by data showing fractal scaling in MEPPs. 2.2.Fractals in the Nervous System: Conceptual Implications for Theoretical Neuroscience Gerhard Werner gwer1@mail.utexas.edu Department of Biomedical Engineering University of Texas at Austin, TX. Abstract This essay aims to document the prevalence of fractals across all levels of the nervous system, supporting their functional relevance, and to highlight unresolved issues regarding the relationships among power law scaling, self-similarity, and self-organized criticality. It emphasizes the significance of allometric control processes and the adaptability of dynamic fractals across scales. The final section reflects on the implications of these findings for future research in theoretical neuroscience. Contents 1. Introduction Fractals, introduced by Mandelbrot in 1977, are self-similar geometric objects with features on infinite scales. They describe highly intermittent temporal behavior without a characteristic time scale. Their statistical analysis provides insights into complex systems. Fractals can be characterized by power functions with non-integer exponents, known as fractal dimensions. This essay focuses on random fractals with stochastic elements. Fractals are signals with scale-invariant, self-similar behavior. They can be analyzed by decomposing signals into temporal and spatial scales. Monofractal processes are characterized by a single scaling exponent, while multifractal signals have multiple exponents. Power law scaling and fractal patterns are observed at all levels of neural organization. Recent advances in measurement methods have led to new data, prompting integration of fractality with other insights into brain organization and complexity, particularly in the context of criticality. 2. Power-law scaling in neuronal structures and processes This section summarizes essential aspects of fractal properties at each level of neuronal organization. Observations remain diverse, with varying conditions and methods. Recent studies have established stringent criteria for identifying fractal properties. Some apparent power law scaling may not be supported by rigorous statistical criteria. The majority of experimental data on fractals in neural structures suggest widespread functional significance. 2.1 Neuronal morphology Mandelbrot (1977) noted that neurons, such as Purkinje cells, may be fractal. Studies show fractal dimensions in dendritic trees, glial cells, and neuron types in different laminae. Fractal analysis reveals differences in dendrite branching structures and functional capacities. Fractal properties differentiate between primary and secondary visual areas in macaque cortex. Fractal dimensions of dendrite arbors are universal, suggesting similar processes in their construction. 2.2 The peripheral nervous system: ion channels, point process analysis of activity in peripheral nerves and individual neurons Ion channel gating and neuronal discharge patterns exhibit fractal features. The kinetics of ion transport across neuronal membranes involve ion channels. The switching probabilities of ion channels at different time scales display fractal scaling. The fractal activity at the neuromuscular junction is supported by data showing fractal scaling in MEPPs. 2.2.