**by Amin Safaeesirat, Saman Moghimi-Araghi** (Physics Department, Sharif University of Technology)

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*The presence of both critical behavior and oscillating patterns in brain dynamics is a very interesting issue. In this work, we have considered a model for a neuron population, where each neuron is modeled by an over-damped rotator. We have found that in the space of external parameters, there exist some regions that system shows synchronization. Interestingly, just at the transition point, the avalanche statistics show a power-law behavior. Also, in the case of small systems, the (partially) synchronization and power-law behavior can happen at the same time.*

Criticality and oscillatory patterns of activity are two well-known behavior in neuronal systems. The presence of these two behaviour at the same time could be controversial. On the one hand, synchronization introduces a scale to the system; on the other hand, the system should be scale-free due to the criticality. Our main concern in this work is to show how they can happen together simultaneously. We have used the over-damped rotator as our single neuron model and put them on a random network consists of both inhibitory and excitatory neurons. Tuning inhibitory strength, axonal delay, and external stimulation, we observed synchronized and asynchronized activity in the system.

We have defined a suitable quantitative order parameter to identify synchrony in the network. Using the order parameter, we have found some synchronous areas in the system’s phase diagram. We have recognized that the synchrony results from the axonal delay and areas’ locations change as the value of axonal delay varies. We have plotted probability avalanche size as well as period distribution functions for the system. Interestingly, the distributions become power-law somewhere between synchronous and asynchronous regions closer to the asynchronous area.

We have changed the system’s size and found that the scale of power-law behavior increases with the system’s size. In addition, the critical point gets closer to asynchronous states as the system becomes bigger. We expect that the order parameter for an infinitely large system is zero (the hallmark of second-order transition). In other words, for large systems, the critical point occurs at the onset of synchronization. For further investigation of the system in terms of criticality, we have performed finite-size scaling analysis for our system and checked the self-consistency of calculated exponents. The exponents are close to experimental results.

Thank you for your nice comments on our poster.

Our dynamic doesn’t have noise explicitly, but it does implicitly due to the random structure of the network and external stimulation, which is a Poisson process. We have changed external stimulation frequency as one of our variables (somewhat the noise intensity). The transition occurs in the vicinity of some synchronous areas (brighter areas in the phase-space) but not necessarily at low external stimulation (low noise), and they are continuous (second-order transition). Moreover, for having synchronous collective oscillation, the delay plays a crucial role.

About the criticality, we have investigated a constant inhibition line in phase-space (g=8) in terms of criticality. Surprisingly, in this specific line, the (pseudo) critical points are located in low external stimulation (low noise), as you mentioned in your comments.

Thank you for your nice engaging poster (#28). It is very attractive for me, especially the discrete phase- transition and the critical behavior that occurs between synchronous and asynchronous states (very similar to our results!). I will appreciate it If you provide more information about this work.

For more information, I refer to our recent work, which is my master thesis (I’ve recently graduated in Physics). I’ve attached the pdf file, and here is the arxiv link:

http://arxiv.org/abs/2010.01493

Best,

Amin

Dear Amin Safaeesirat, Saman Moghimi-Araghi

thanks for your very interesting nice poster,

have you tryed to change the noise frequency and noise intensity?

At lower values of noise, are there values of inhibition, delays etc where a transition between syncrony and asyncronous state occurs ? Is the transition, at lower values of noise frequency and intensity, different from the

continuos one that you observe at critical value of noise? Is there a regime where transition is discontinuos?

I ask you this question since, as you see in my poster 28

https://braincriticality.org/?s=scarpetta , a similar model with critical scale-free beheviour near the edge of syncronous collective oscillations and asyncronous poisson activity shows that al lower values of noise the transition changes! and a discontinuos behavior appear.

That’s why I ask how your model works in the phase-space at different values of noise inhibition etc ….

please send my any preprint of your recent works,

Thanks for your reply,

Best regards

Silvia

sscarpetta@unisa.it

Thank you for your nice comments on our poster.

Our dynamic doesn’t have noise explicitly, but it does implicitly due to the random structure of the network and external stimulation, which is a Poisson process. We have changed external stimulation frequency as one of our variables (somewhat the noise intensity). The transition occurs in the vicinity of some synchronous areas (brighter areas in the phase-space) but not necessarily at low external stimulation (low noise), and they are continuous (second-order transition). Moreover, for having synchronous collective oscillation, the delay plays a crucial role.

About the criticality, we have investigated a constant inhibition line in phase-space (g=8) in terms of criticality. Surprisingly, in this specific line, the (pseudo) critical points are located in low external stimulation (low noise), as you mentioned in your comments.

Thank you for your nice engaging poster (#28). It is very attractive for me, especially the discrete phase- transition and the critical behavior that occurs between synchronous and asynchronous states (very similar to our results!). I will appreciate it If you provide more information about this work.

Best,

Amin

Aminsafaeesirat@gmail.com

Dear Silvia Scarpetta,

Thank you for your nice comments on our poster.

Our dynamic doesn’t have noise explicitly, but it does implicitly due to the random structure of the network and external stimulation, which is a Poisson process. We have changed external stimulation frequency as one of our variables (somewhat the noise intensity). The transition occurs in the vicinity of some synchronous areas (brighter areas in the phase-space) but not necessarily at low external stimulation (low noise), and they are continuous (second-order transition). Moreover, for having synchronous collective oscillation, the delay plays a crucial role.

About the criticality, we have investigated a constant inhibition line in phase-space (g=8) in terms of criticality. Surprisingly, in this specific line, the (pseudo) critical points are located in low external stimulation (low noise), as you mentioned in your comments.

Thank you for your nice engaging poster (#28). It is very attractive for me, especially the discrete phase- transition and the critical behavior that occurs between synchronous and asynchronous states (very similar to our results!). I will appreciate it If you provide more information about this work.

For more information, I refer to our recent work, which is my master thesis (I’ve recently graduated in Physics). Here is the arxiv link:

http://arxiv.org/abs/2010.01493

Best regard,

Amin