# A Time-Varying Information Measure for Tracking Dynamics of Neural Codes in a Neural Ensemble

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## Abstract

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## 1. Introduction

## 2. Computational Framework

#### 2.1. Responses of a Homogeneous Neural Ensemble to a Mixed Stimulus

#### 2.2. Probability Density Estimation

## 3. Results

#### 3.1. Information Underlying Synchronous and Asynchronous Spikes Are Distinctively Separable

^{5}samples of data in the simulation), K(.) is the kernel function, and h is the bandwidth, which is fixed on 0.4 based on the smoothness of the data.

#### 3.2. Different Types of Spikes in a Multiplexed Code Carry Different Amounts of Information

#### 3.3. Time-Varying Entropy (TVE) Measure

_{i}and t

_{j}.

#### 3.4. Relatinship between Mixed Stimulus and Spike Patterns

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The simulation data consist of the mixed stimulus and spiking activity of the neural ensemble. (

**A**) The mixed stimulus ${I}_{mixed}$ consists of ${I}_{fast}$ and ${I}_{slow}$ (see Section 2) (

**B**) Different patterns of spikes resulted from the neural ensemble (including 100 neurons) as response to a mixed stimulus.

**Figure 2.**The probability distributions of synchronous (top) and asynchronous (bottom) spikes. For each type of spike, the true distribution was obtained by the histogram method and is shown by blue bars (thick bars). We used a non-parametric method to approximate distributions of synchronous and asynchronous spikes, which are shown by red bars (thin bars).

**Figure 3.**Entropy of different types of spikes. (

**A**) Entropy of different patterns of spikes as a function of time-bin ($\delta t$) and word-length ($L$) (

**B**) Estimated entropy rate of spikes, for no-stimulus, slow, fast, and mixed stimuli, is plotted against the reciprocal of word length, 1/L. The dashed line and its intersection with y axis represent the value of entropy for $L\to \infty $, i.e., the minimum value of the entropy.

**Figure 4.**Time-varying entropy (TVE) measure for different type of spikes. (

**A**) TVE for different type of spikes as function of word-length and time-bins resolution. Different parameter sets for TVE enables extracting different types of information. For example, by setting $L=10,\delta t=0.05\mathrm{ms}$ TVE extracts information underlying synchronous spikes. As well, by setting $L=10,\delta t=10\mathrm{ms}$ TVE extracts information related to asynchronous spikes (

**B**) TVE measure correlation coefficient with ${I}_{fast}$, ${I}_{slow}$, and ${I}_{mixed}$. The correlation of TVE with each stimulus is aligned with the figures in panel (

**A**). For $L=10,\delta t=0.05\mathrm{ms}$ TVE is highly correlated with ${I}_{fast}$, which drives synchronous spikes. For $L=10,\delta t=10\mathrm{ms}$ TVE is highly correlated with ${I}_{slow}$, which provokes asynchronous spikes. Thus, TVE measure can extract information about the stimulus directly from the spikes (

**C**) Mean of Integration of TVE measure over time (left) and entropy of all-spikes calculated in Equation (9) (right).

**Figure 5.**Illustration of calculation of the entropy in (10) and the TVE in (11). The binary sequence of each row indicates the response of each neuron in a neural ensemble. Probability distribution of code words, p(w), over the whole length of data can be calculated based on (10). Two probability distributions underlying two time bins, t

_{i}and t

_{j}, are calculated across neurons (see (11)). The length of code words is equal to 3 and spikes are binned at a resolution ($\delta t$), equal to the sampling time of the simulation. Several code words are highlighted by red and green.

**Figure 6.**Different elements of the mixed stimulus (${I}_{fast}$, ${I}_{slow}$) and their relationship with different type of spikes. (

**A**) Reconstruction of the mixed stimulus by TVE measure. (

**B**) TVE measure spectrums for different patterns of spikes (synchronous, asynchronous, and all) given different $\delta t$ with fixed $L=10$ through time. Instantaneous firing rate of the neural ensemble is calculated with two different kernel width; green and black graphs are related to kernel width = 100 ms and kernel width = 5 ms, respectively.

**Figure 7.**Firing rate and TVE spectrum of spikes for a neural ensemble receiving weak (left, $\sigma =0.5\mathrm{pA}$), intermediate (middle, $\sigma =10\mathrm{pA}$), and strong (right, $\sigma =50\mathrm{pA}$) synaptic noises. The TVE spectrum of different types of spikes is obtained in a similar way, as explained in Figure 6.

**Figure 8.**Information decoded by synchronous and asynchronous spikes are associated with different features of the stimulus. (

**A**) Kalman-filter decoder model was developed to recunstruct the (mixed) stimulation from all spikes (top). The decoder was then applied to synchronous (middle) and asynchronous spikes (bottom). (

**B**) Similar to (

**A**) but synchronous (middle) and asynchronous (bottom) spikes were first filtered by a Gausian kernel with optimim time resolutions ($\delta t=0.05\mathrm{ms}$ for synchronous and $\delta t=10\mathrm{ms}$ for asynchronous spikes) before applying them to the Kalman-filter decoder model.

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**MDPI and ACS Style**

Rezaei, M.R.; Popovic, M.R.; Lankarany, M. A Time-Varying Information Measure for Tracking Dynamics of Neural Codes in a Neural Ensemble. *Entropy* **2020**, *22*, 880.
https://doi.org/10.3390/e22080880

**AMA Style**

Rezaei MR, Popovic MR, Lankarany M. A Time-Varying Information Measure for Tracking Dynamics of Neural Codes in a Neural Ensemble. *Entropy*. 2020; 22(8):880.
https://doi.org/10.3390/e22080880

**Chicago/Turabian Style**

Rezaei, Mohammad R., Milos R. Popovic, and Milad Lankarany. 2020. "A Time-Varying Information Measure for Tracking Dynamics of Neural Codes in a Neural Ensemble" *Entropy* 22, no. 8: 880.
https://doi.org/10.3390/e22080880