# Synchrony-Division Neural Multiplexing: An Encoding Model

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. Different Temporal Filters Map Distinct Features of a Mixed Stimulus

#### 2.2. Low-Dimensional Feature Space of the Neural Response Can Be Characterized by the STAs of Synchronous and Asynchronous Spikes

#### 2.3. Different Nonlinear Functions Are Associated with Synchronous and Asynchronous Spikes

#### 2.4. An Augmented LNL Cascade Model for Synchrony-Division Multiplexing

## 3. Discussion

#### 3.1. Subspace Feature Extractors: iSTAC vs. STC

#### 3.2. Choice of Static Nonlinearity in the LNL Model

#### 3.3. Generalized Linear Model (GLM) for Augmented LNL

## 4. Materials and Methods

#### 4.1. Stimulated Mixed Input

#### 4.2. Simulated Neural Ensemble and Its Response to the Mixed Input

#### 4.3. Generalized Linear Model (GLM) Details

#### 4.4. STA and STC Estimators

#### 4.5. iSTAC Estimator

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Slow and fast features of a mixed signal can be inferred from responses of a homogeneous ensemble of neurons using the iSTAC method. (

**A**) Slow and fast signals comprising a mixed signal. (Bottom) Sample raster plot of 30 model neurons receiving the common mixed signal (and independent noise). Spikes evoked by the fast and slow signals cannot be distinguished visually. (

**B**) The iSTAC method was applied to spike-triggered mixed signal and eigenvalues and eigenvectors were obtained (see Methods). (

**Left**) The eigenvalues of the iSTAC matrix reveal two significant components of the population code. (

**Right**) The projection of spike-triggered mixed signal onto the main eigenvectors of the iSTAC matrix. Two clusters can be visually distinguished. (

**C**) The 1st and 2nd eigenvectors of the iSTAC matrix, V1 and V2, respectively, are shown against the spike-triggered average (STA). V1 resembles the STA filter reflecting slowly-varying changes in the signal. Unlike V1, V2 resembles a high-pass filter (differentiator) that reflects fast features of the mixed signal.

**Figure 2.**Synchronous and asynchronous spikes represent slow and fast features of the mixed signal, respectively. (

**A**) Synchronous (red) and asynchronous (blue) spikes are distinguished by setting a threshold on the instantaneous firing rate calculated by a narrow kernel (see Methods). Synchronous spikes evoked by the fast signals can be distinguished visually. (

**B**) The projection of spike-triggered mixed signal onto the STA

_{Sync}and STA

_{Async}. Two (visually) distinguishable clusters belong to asynchronous spikes representing the slow feature of the signal (blue dots) and synchronous spikes representing the fast features (red circles). (

**C**) The spike-triggered average of synchronous (red) and asynchronous (blue) spikes, namely, STA

_{Sync}and STA

_{Async}, respectively, is shown against the STA of all spikes (similar to Figure 1C).

**Figure 3.**Block diagram of decoding the mixed signal from spikes using (

**A**) the STA filter (light brown), (

**B**) a weighted sum of the 1st and 2nd eigenvectors of the iSTAC method (green), and (

**C**) a weighted sum of filtered asynchronous spikes (by STA

_{Async}) and filtered synchronous spikes (by STA

_{Sync}) (purple). Original mixed signal (black) is overlaid with reconstructed signal (color) in the plots. As can be seen in these figures, the reconstructed signal based on STA

_{Sync}and STA

_{Async}—similar to that obtained by eigenvectors of iSTAC method—can capture both slow and fast components of the signal accurately. “*” indicates the spiking signals.

**Figure 4.**Static nonlinearities underlying asynchronous spikes. (

**A**) Block diagram of LNL model for asynchronous spikes. (

**B**) Static nonlinearity calculated for asynchronous spikes is obtained by mapping the output of filtered stimulus to the instantaneous rate of asynchronous spikes (calculated with a wide kernel, σ = 25 msec). Static nonlinearity calculated based on 1st eigenvectors of the iSTAC method, ${v}_{1}$, (

**Left**) and STA

_{Async}(

**Right**). The solid black shows fitted rectifiers. (

**C**) The PSTHs constructed using the fitted nonlinearities based on ${v}_{1}$ were drawn against the PSTH of asynchronous spikes. (

**D**) The PSTHs constructed using the fitted nonlinearities based on STA

_{Async}were drawn against the PSTH of asynchronous spikes.

**Figure 5.**Static nonlinearities underlying synchronous spikes. (

**A**) Block diagram of LNL model for synchronous spikes. (

**B**) Static nonlinearity calculated for the synchronous spikes is obtained by mapping the output of filtered stimulus to the instantaneous synchronous events (calculated with a narrow kernel, σ = 1 msec). Static nonlinearity calculated based on 2nd eigenvectors of the iSTAC method, ${v}_{2}$, (

**Left**) and STA

_{Sync}(

**Right**). The solid black shows fitted sigmoid functions. (

**C**) The PSTHs constructed using the fitted nonlinearities based on ${v}_{2}$ were drawn against the PSTH of synchronous spikes. (

**D**) The PSTHs constructed using the fitted nonlinearities based on STA

_{Sync}were drawn against the PSTH of synchronous spikes.

**Figure 6.**Two-stream LNL model, referred to as augmented LNL model, enables co-existence of temporal and rate codes. (

**A**) Block diagram of the augmented LNL model for combining rate of asynchronous and synchronous spike events. (

**B**) The PSTHs estimated using a conventional Poisson GLM (red) are shown against the original PSTH (calculated with a 1 msec Gaussian kernel). (

**C**) The PSTHs estimated using the segmented LNL using temporal filters of iSTAC method. (

**D**) The PSTHs estimated using LNL using the segmented LNL using STA

_{Async}and STA

_{Sync}.

**Table 1.**Mean absolute error (MAE) and root mean squared error (RMSE) performance measure comparison between Poisson GLM and augmented LNL model on test data.

(MAE) | $\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$ | |||||
---|---|---|---|---|---|---|

Sync | Async | Mixed | Sync | Async | Mixed | |

Poisson GLM | 0.245 | 0.223 | 0.228 | 0.338 | 0.313 | 0.326 |

Augmented LNL ($\mathit{S}\mathit{T}{\mathit{A}}_{{V}_{1},}\mathit{S}\mathit{T}{\mathit{A}}_{{V}_{2}}$) | 0.101 | 0.104 | 0.102 | 0.137 | 0.134 | 0.135 |

Augmented LNL ($\mathit{S}\mathit{T}{\mathit{A}}_{\mathit{A}\mathit{S}\mathit{y}\mathit{n}\mathit{c}},\mathit{S}\mathit{T}{\mathit{A}}_{\mathit{S}\mathit{y}\mathit{n}\mathit{c}}$) | 0.103 | 0.107 | 0.106 | 0.140 | 0.151 | 0.149 |

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**MDPI and ACS Style**

Rezaei, M.R.; Saadati Fard, R.; Popovic, M.R.; Prescott, S.A.; Lankarany, M.
Synchrony-Division Neural Multiplexing: An Encoding Model. *Entropy* **2023**, *25*, 589.
https://doi.org/10.3390/e25040589

**AMA Style**

Rezaei MR, Saadati Fard R, Popovic MR, Prescott SA, Lankarany M.
Synchrony-Division Neural Multiplexing: An Encoding Model. *Entropy*. 2023; 25(4):589.
https://doi.org/10.3390/e25040589

**Chicago/Turabian Style**

Rezaei, Mohammad R., Reza Saadati Fard, Milos R. Popovic, Steven A. Prescott, and Milad Lankarany.
2023. "Synchrony-Division Neural Multiplexing: An Encoding Model" *Entropy* 25, no. 4: 589.
https://doi.org/10.3390/e25040589