Introducing a Comprehensive Framework to Measure Spike-LFP Coupling

Abstract

Measuring the coupling of single neuron's spiking activities to the local field potentials (LFPs) is a method to investigate neuronal synchronization. The most important synchronization measures are phase locking value (PLV), spike field coherence (SFC), pairwise phase consistency (PPC), and spike-triggered correlation matrix synchronization (SCMS). Synchronization is generally quantified using the PLV and SFC. PLV and SFC methods are either biased on the spike rates or the number of trials. To resolve these problems the PPC measure has been introduced. However, there are some shortcomings associated with the PPC measure which is unbiased only for very high spike rates. However evaluating spike-LFP phase coupling (SPC) for short trials or low number of spikes is a challenge in many studies. Lastly, SCMS measures the correlation in terms of phase in regions around the spikes inclusive of the non-spiking events which is the major difference between SCMS and SPC. This study proposes a new framework for predicting a more reliable SPC by modeling and introducing appropriate machine learning algorithms namely least squares, Lasso, and neural networks algorithms where through an initial trend of the spike rates, the ideal SPC is predicted for neurons with low spike rates. Furthermore, comparing the performance of these three algorithms shows that the least squares approach provided the best performance with a correlation of 0.99214 and R ² of 0.9563 in the training phase, and correlation of 0.95969 and R ² of 0.8842 in the test phase. Hence, the results show that the proposed framework significantly enhances the accuracy and provides a bias-free basis for small number of spikes for SPC as compared to the conventional methods such as PLV method. As such, it has the general ability to correct for the bias on the number of spike rates.

Keywords: local field potentials; pairwise phase consistency; phase locking value; spike field coherence; spike-LFP phase coupling.

Figure 1

Dependence of the spike-LFP phase coupling (SPC) on the number of spikes for 100 neurons. SPC curves exhibit an exponential descending behavior and are biased on the spike rates.

Figure 2

Estimation of the ideal SPC value using exponential modeling. (A) For a sample of 4 neurons, PLV based SPC exhibits an exponential descending behavior and is biased on the spike rates (the blue curves). The red curves show an estimated SPC through modeling with two exponential functions and an asymptote. (B) Depicts the average and error bar of RMSE across 100 trials among 100 neurons. The values are very small which illustrates high accuracy for estimation of SPC (0.0003±0.01). (C) Shows the distribution function of RMSE across 100 trials.

Figure 3

Proposed scheme to predict ideal spike-LFP phase coupling (SPC) based on the trend of the first 20 points (SR = 20) of trials and the ideal SPC of each trial. In the training phase, the first 20 points of each 150 trials as input 1, along with the asymptote of each of the trials or ideal SPC as input 2, are designated as the inputs to the machine learning algorithms of least squares, Lasso, and ELM. The output of the learning system or the corrected SPC is shown to be quite close to the ideal SPC. This model is trained based on the trend of the first 20 points of trials and the ideal SPC of each trial.

Figure 4

Predicting a more reliable SPC for trials with small spike rates. (A) Each of the bars shows the correlation between the ideal SPC and the corrected SPC across 150 trials. The correlation is performed using three separate algorithms of least squares, Lasso, and ELM (the red, blue, and green bars, respectively). (B) Shows that the corrected SPC can accurately predict the ideal SPC based on the first 20 points of each test trials with a high correlation. (C) Shows the correlation between the ideal SPC and the corrected SPC for 25 trials which are selected from 25 independent neurons. Least squares approach has the best performance, with a correlation of 0.99214, 0.95969, and 0.95817, for (A–C), respectively. (D) Shows the correlation between the ideal and the corrected SPC curves which is modeled based on different spike rates across 150 trials. Statistically, the least squares algorithm is significantly different from the ELM algorithm (p < 0.05^*; sign test) while it is not significantly (“ns”) different from the Lasso algorithm.

Figure 5

Comparison of the error between the SPC based on “N-spike” and proposed corrected SPC with the ideal SPC. (A) For a sample trial of neuron, shows that the corrected SPC is very close to the ideal SPC. The difference between the SPC based on the N-spike (green line) and the ideal SPC (turquoise line) is much higher than the difference between the corrected SPC (dash red line) and the ideal SPC. (B) Depicts the histogram distribution of the difference between the N-spike SPC and the ideal SPC, as well as the difference between corrected SPC and ideal SPC, across 50 test trials (p < < 0.001; sign test).

Figure 6

A bias-free framework for SPC and comparing the proposed framework with the conventional method of PLV. (A) For a sample of 4 neurons, these curves are provided based on the PLV method with no correction. There is a systematic relation of strictly descending between the spike rates and the PLV (blue curves). The red lines show the linear estimations. (B) For a sample of 4 neurons, for each of the test trials and for each of their spike rates, the value of the corrected SPC (blue curves) is computed. The red lines show the linear estimations. (C) The blue histogram shows the distribution for all linear estimations across 100 trials among 100 neurons which suggests no systematic relation between the spike rates and the corrected SPC and is symmetrically distributed about zero. The red histogram shows the distribution for all linear estimations across 100 trials among 100 neurons which suggests a systematic relation between the spike rates and the PLV, and is strictly descending (p < < 0.001; sign test).