Self-similar network model for fractional-order neuronal spiking: implications of dendritic spine functions

Fractional-order derivatives have been widely used to describe the spiking patterns of neurons, without considering their self-similar dendritic structures. In this study, a self-similar resistor-capacitor network is proposed to relate the spiny dendritic structure with fractional spiking properties. In order to achieve this goal, two types of networks comprising recursively staggered resistors and capacitors were developed to model the functional properties of smooth and spiny dendrites, respectively. Their overall electrotonic properties can be described by fractional order temporal operators derived by Heaviside operational calculus. According to this operator method, spiking patterns of spiny dendrites were controlled by the standard 0.5-order derivative, whereas an exponential modulation term was added in the governing fractional operator of the smooth dendrites. The application of these fractional operators in a leaky integrate-and-fire model reveals that the dendritic spine plays an important role in alternations of the spiking properties, including first-spike latency, firing rate adaptation, and afterhyperpolarization conductance. Further, the multilevel assembly of this network indicates that the fractional spiking behaviors of spiny neurons might originate from their hierarchical substructures, thereby highlighting possible functional consequences of alterations to dendritic self-similarity.