# A concise mathematical description of signal transformations across the hippocampal apical CA3 to CA1 dendritic response

## TL;DR

This paper presents a mathematical model describing how signals are transformed from the CA3 to CA1 regions of the hippocampus in mice.

## Contribution

A new method using the Volterra expansion to derive input-output functions for hippocampal signal transmission is introduced.

## Key findings

- A 2nd order equation accurately describes the apical dendritic response with >94% accuracy.
- The model generalizes to new cases and reveals new timing rules in signal transmission.
- Basal dendrites require a higher-order model for complete characterization.

## Abstract

The synapse is the fundamental unit of communication in the nervous system. Determining how information is transferred across the synaptic interface is one of the most complex endeavors in neuroscience, owing to the large number of contributing factors and events. An approach to solving this problem involves collapsing across these complexities to derive concise mathematical formulas that fully capture the governing dynamics of synaptic transmission. We investigated the feasibility of deriving such a formula – an input-output transformation function for the CA3 to CA1 node of the hippocampus – using the Volterra expansion technique for non-linear system identification. The timecourse of the fEPSP in the apical dendrites of mouse brain slices was described with >94% accuracy by a 2nd order equation that captured the linear and non-linear influence of past inputs on current outputs. This function generalized to cases not included in its derivation and uncovered previously undetected timing rules. The basal dendrites expressed a substantially different transfer function and evidence was obtained that, unlike the apical system, a 3rd order system or higher will be needed for complete characterization. At scale, the approach will also provide information needed for the construction of biologically realistic models of brain networks.