Balanced networks near bifurcation?

[Alireza]

Reading the following paper [1] I recalled few ideas I have been thinking in recent years. That is a balanced network works near a bifurcation point which is presumably a Hopf bifurcation. There are several cues for this arguments, first with a suitable set of parameters the system is in irregular-asynchronous state -- a fixed point in a mean-field perspective. Second in this regime system shows damped oscillations in response to external stimuli, say a synchronous pulse packet as you see in Fig.2 of the paper below. Third is with changing the parameters, the system begins to oscillate and produce rhythm, that is a limit cycle in mean-field view. 

The third observation can be a result of another type of bifurcation, say SNIC but the second indicates it should be a Hopf one. For me the first question is why always Hopf bifurcation when we have no prior constraint on the constituent neurons? and when most of the simulations use LIF neurons which are sort of type-I. 

The answer may be the feedback system of excitatory-inhibitory which is usually believed to be origin of the rhythms (although it is not always). It is interesting to inspect how presence of feedback changes the type of an excitable creature. Specific problems:

1- to study the bifurcation properties of an excitable system, say a model neuron of either type with a feedback (an autaptic connection). It can be interesting to show if both neuronal types result in Hopf bifurcation in presence of feedback! (see also [2])

2- If we accept the common sense that the inh-exc feedback loop is responsible for rhythmogenesis and it works simply like a two components loop, the feedback delay time is a decisive parameter. I haven't seen a modeling study with distributed delay times, have you? with constant delays it seems reasonable for feedback system to work but what happens in case of distributed delays?

[1] Hahn, Bujan, Fregnac, Aertsen, and Kumar, PLoS Comp Biol 10(8), e1003811 (2014)
[2] Hashemi, Valizadeh, and Azizi, PRE 85(2), 021917 (2012)

[Zahra]

The third observation can be a result of another type of bifurcation, say SNIC but the second indicates it should be a Hopf one.

I am afraid, It would not be a SNIC bifurcation. This actually a Hopf one, at first with a damped oscillation to the stable fixed point and then the stability of the point and limit cycle switch and we see oscillation.

LIF neurons which are sort of type-I

LIF neurons have a positive non-periodic PRC and this leads to categorize them in type-I neurons, but it is obvious that they are not near a SNIC bifurcation, even QIFs with manual resetting! 

I haven't seen a modeling study with distributed delay times, have you?

Brunel, 2000
In brief, a uniform distribution of delays, just convert the synchronized state to desynchrony (Fig. 4), no matter it is regular or not. 

[Alireza]

About the bifurcation, I agree that it is a Hopf. But from PRC perspective a population of type one oscillators is itself type-I with a positive PRC. But for the balanced networks presence of feedback by inhibition may change the story. That's why I suggested to study a single oscillator with a feedback to see if the feedback can change the type of excitability of an oscillator.

see also these two:
Ledoux and Brunel, Front Comp Neuro 2011
Veltz and Sejnowski, Neural comp 2015


LIF is not a limit cycle oscillator, agreed. But PRC can be defined and the only problem with its PRC is that it is not continuous.

About the delay, that's why I suggest to re-study the origin of rhythms to see how much it is robust against discrepancy of delay times.