The brain uses its own previous activity to adapt to an ever-changing environment. This history
dependent adaptation takes place at all scales of organization of the nervous system. The objective
of this project is to develop a common theoretical formalism to be applied to multiple history
dependent phenomena, from the biochemical reactions that underlie synaptic plasticity, to the
emergent patterns in complex neural networks. At the core of this formalism is the recognition that
most models of neuronal activity are based on the classical reaction-diffusion equation. Aim 1 will
prove that a generalization of this equation, the fractional order reaction-diffusion equation, provides
all the natural mathematical tools to incorporate history dependence into neuronal function analysis.
The fractional order exponent of the fractional derivative is a parameter that captures the emergent
interactions of multiple elements that cause the history dependent process. Systems of equations will
be developed for specific cases of activation of membrane conductances, the membrane voltage, and
firing rate activity. The objective of Aim 2 is to demonstrate the significance of using this formalism in
four applications across neurobiological scales, from synaptic plasticity to sensory processing.
Application 1 will focus in understanding history dependence in the biochemical reactions that
underlie synaptic long-term depression in cerebellar Purkinje cells as a function of the intracellular
structure of dendritic spines. Application 2 will establish a collaboration with a group at the Allen Brain
Institute to build fractional order models that can replicate the variability observed in their Cell Types
Database of mouse cortical neurons. Application 3 will be a collaboration with a group at the Max
Plank Institute for Dynamics and Self-Organization to test the hypothesis that history-dependent
neuronal elements give rise to robust reverberating networks. Finally, Application 4 is a collaboration
with a group at McGill University. In this case, experiments in awake weakly electric fish will be
conducted to determine the cellular and network biophysical substrates that implement optimal coding
of sensory input due to fractional differentiation. Overall, this project will provide a unified theoretical
framework and develop applications to study, analyze, and design experiments of history dependent
neuronal activity across scales.