# Observable for a Large System of Globally Coupled Excitable Units

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## Abstract

**:**

## 1. Introduction

## 2. Macroscopic Evolution of a Set of Excitable Units

## 3. An Experimental System

## 4. Reconstructing the Dynamics for S from the Data

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Solutions of the proposed model for the case of a unique excitatory population. (

**a**) Bifurcation diagram of the system in the (${\eta}_{0},J$) plane for a case with $\tau =2\mathrm{ms}$ and $\mathsf{\Delta}=0.1$. The curves denote saddle node bifurcations. Three regions corresponding to the different types of solutions can be identified. (

**b**–

**d**) Simulations for different initial conditions. Regions I and III present a unique stationary attracting solution, while region II of parameter space presents bi-stability. Insets show the evolution of the order parameter of the system, with black dots representing the attractive fixed points. (

**e**) Agreement between the reduced model and a simulation of a system of oscillators (15,000 units, using a time step of $\mathsf{\Delta}t=0.001\mathrm{s})$). The rate used to drive the system was computed as the number of oscillators crossing the value $\theta =\pi $, divided by the total number of oscillators and the time step $\Delta t$. (

**f**) Evolution of the order parameter for the model and the simulation.

**Figure 2.**Raw data thresholding and multiunit activity (MUA). (

**a**) BOS sound segment (single canary song syllable). (

**b**) High-pass filtered raw data traces of the neural response to auditory presentations of BOS in anesthetized birds (see Section 3). Traces correspond to 10 trials from one protocol. Each trace consists of background electrical noise and sharp spikes corresponding to the extracellular recording of an action potential. The threshold allows the detection of spikes of multiple amplitudes. Differences in spike shape and amplitudes correspond to the electric activity generated by different neurons. Thus, the activity obtained by thresholding is multiunit in nature. After thresholding, the spikes are treated as a series of timestamps of where spiking occurred. (

**c**) Zoomed-in trace for trial 1, showing the threshold level for detection.

**Figure 3.**Single unit activity is synchronized in multiple recording sites. (

**a**) Activity profiles of spike-sorted clusters from a recording. From top to bottom: BOS sound signal, trial- and channel- averaged LFP, PSTHs for each neuron and summed single unit activity (ADD). PSTHs are histograms (15 ms bins) of the activity elicited in each isolated neuron during 20 auditory presentations of the BOS. Lastly, the bottom panel shows the multiunit activity profile (MUA), obtained by thresholding the recorded neural data (see Section 3 and Figure 2). (

**b**) Each action potential is recorded by several channels in the multielectrode array (a diagram is shown on the right). Spikes from individual, well-isolated neurons are shown with different colors. Spikes from the same neuron are simultaneously recorded as spikes of different shapes in different channels (see yellow cluster inset). The channels where each cluster was detected are shown to the right of each spike group. Color outlines indicate the maximum amplitude channel for each cluster, which corresponds with the spike shapes shown. Each cluster presents a robust response across trials (sharp peaks present in the PSTHs in (

**a**)). Additionally, these results show that registered neurons are synchronized. The sharp peaks in the ADD profile in (

**a**) result from the combination of response robustness across trials and from the synchronous firing of isolated neurons.

**Figure 4.**The reconstructed global synaptic activation captures prominent LFP features. Time traces for the global synaptic activation S (top panel, blue) and the trial- and channel- averaged LFP (bottom panel, red). The trace obtained for S approximates the measured LFP, especially where large peaks occur. The similarity between the two signals was measured using Pearson’s correlation coefficient, which yields a maximum value of ${\mathrm{c}}_{\mathrm{max}}~0.86$ by taking the regions with the peaks and ${\mathrm{c}}_{\mathrm{mean}}~0.47$ for the whole timespan. The difference in correlation strength means that the reconstructed S better approximates the LFP near signal events that correspond to the synchronous firing of multiple neurons (see Figure 3).

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**MDPI and ACS Style**

Boari, S.; Uribarri, G.; Amador, A.; Mindlin, G.B. Observable for a Large System of Globally Coupled Excitable Units. *Math. Comput. Appl.* **2019**, *24*, 37.
https://doi.org/10.3390/mca24020037

**AMA Style**

Boari S, Uribarri G, Amador A, Mindlin GB. Observable for a Large System of Globally Coupled Excitable Units. *Mathematical and Computational Applications*. 2019; 24(2):37.
https://doi.org/10.3390/mca24020037

**Chicago/Turabian Style**

Boari, Santiago, Gonzalo Uribarri, Ana Amador, and Gabriel B. Mindlin. 2019. "Observable for a Large System of Globally Coupled Excitable Units" *Mathematical and Computational Applications* 24, no. 2: 37.
https://doi.org/10.3390/mca24020037