Simulating the effect of PC synchrony on downstream neurons

Overview

The code in this repository seeks to understand the effect of non-zero covariance (synchronous firing) in a population of Purkinje cells on a downstream neuron in the deep cerebellar nucleus. Current recording methodolgies allow us to record for pairs (or sometimes triplets) of Purkinje cells simultaneously, but estimates suggest that approximately 40-50 Purkinje cells synapse on a single cerebellar nucleus neuron (c.f., Person and Raman, 2012). As we cannot record from the entire presynaptic Purkinje cell population simultaneously, we use simulation to estimate the combined effect of synchronous spiking across this population of 40-50 Purkinje cells.

To answer this question, the code in the repository constructs two populations of simulated Purkinje cells. These Purkinje cells are modeled as Poisson distributed point-process neurons. In the first "independent" population, we assume that all simulated Purkinje cells have zero covariance. In the second population, we bootstrap a covariance matrix by choosing random covariance values between pairs of Purkinje cells from the distribution that we actually measured. By keeping the mean firing rates of the two populations the same, we can ask what the effect of changes in the temporal spiking patterns might have on the downstream nuclei.

Data requirements

Raw data necessary for these simulations are stored in the Open Science Framework repository located here. In particular, you will need an HDF5 file called pairwise_covariance.h5. This file contains the measured pair-wise covariance values from our Purkinje cell population. This HDF5 file should be placed in the top-level directory of this package after checkout.

Software requirements

The simulations were performed and tested in Julia v1.8, but should be compatible with any version of Julia greater than v1.0. Following installation of Julia, several additional Julia packages will need to be installed to read from the HDF5 file and perform plotting.

All commands necessary to run the simulation, including the installation of additional packages, can be found in the Jupyter notebook (pc_model.ipynb). Note that you do not need to install Jupyter to run these commands. Rather, these commands can be copy-and-pasted directly into a Julia terminal in the order seen in the notebook.

Estimating the fraction of fully synchronous Purkinje cells

To aid in comparison with past results, namely those from Person and Raman (2012), our goal was to describe how many Purkinje cells would need to have identical spiketrains to produce the average distribution of spikes in the independent and non-zero covariance populations. If all Purkinje cellss are independent, then one spike in one Purkinje cell would be accompanied, on average, by $(N-1)\bar{M}\Delta{}t$ additional synchronous spikes from the point of view of a downstream neuron. Here, $N$ is the number of neurons in the Purkinje cell population (e.g., N=40), $\bar{M}$ is the mean firing rate of the Purkinje cells in spikes/s, and $\Delta{}t$ is the temporal resolution (1 ms). If all PCs are fully synchronous, then the firing of one PC completely predicts the firing of the rest of the population, resulting in $(N-1)$ spikes arriving at the downstream neuron simultaneously. We solved this model for all values of synchrony. Let $x$ be the fraction of the population that is fully synchronous, ranging from 0 (fully independent) to 1 (completely synchronous). Then, the firing of a single PC in the population predicts the arrival of additional spikes at the downstream neuron according to the following equation:

$$x \left[ (Nx-1)+(N-Nx) \bar{M}\Delta{}t\right] + (1-x)(N-1)\bar{M}\Delta{}t$$