# Talk:Coherent activity in excitatory pulse-coupled networks

## Reviewer A (Dr. Michael Rosenblum)

The paper is well-written. I am convinced that it will be useful and timely contribution to Scolarpedia. I have, however, a number of minor comments.

1. I encourage the authors to pay more attention to two paragraphs before the Introduction, to ensure that they can be understood by readers who are not familiar with the topic. In particular, please make clear the difference between coherent oscillatory activity, collective oscillations, and fully synchronized regimes. Both coherent activity and synchronized states are oscillations, right? How to distinguish them? Please also make clear that first you speak about real brain networks and then concentrate on identical LIF neurons.

2. Parkinson disease shall be Parkinson's disease

3. Pionieristic - shall it not be pioneering?

4. ...this is not the case (Luccioli et al., 2012). Since in this latter case the fluctuations in the synaptic current do not vanish, even for extremely large networks.

Shall it not be ...this is not the case (Luccioli et al., 2012), since in this latter case the fluctuations in the synaptic current do not vanish, even for extremely large networks.

More important, at this point it is not clear what is "the fluctuations in the synaptic current" (it has not been explained yet).

5. Sparse and massively connected networks reveal even more striking differences at the microscopic level associated to the membrane potentials' evolution, chaotic evolution has been observed in both cases for finite networks.

Maybe: Sparse and massively connected networks reveal even more striking differences at the microscopic level associated to the membrane potentials' evolution. So, chaotic evolution has been observed in both cases for finite networks.

...sparse networks remains chaotic shall be sparse networks remain chaotic

6. Averaged fields: whot is the meaning of taking P=\alpha E+\dot E? Why not simply \dot E?

7. Splay state, the first equation. Why x=A\cos()? There seem to be no harmonic process here.

8. Notice that R is sometiomes calle the order parameter and sometimes the modulus of the order parameter; please, do it in a unique way.

9. Fig 3a: in caption shall be "Averaged modulus of the order parameter R". Next, 3b: in text it is \bar P, \bar E, but in the figure simply P and E (\bar E=E, \bar P = P here, but it is confusing).

10. Since the splay state is characterized by a constant field E, while PS is associated to a periodically oscillating field.

I assume, this sentence is not quite correct grammatically.

11. Caption Fig 4:

The black curve correspond

shall be

The black curve corresponds

12. Sparse networks. irrispectively shall be irrespectively

What is intensive variable? Maybe you can add one or two sentences explaining a bit more the extensivity property?

## Response to Reviewer A

We thank the Referee for his positive remarks and useful comments.

Here are our answers to all the points he raised:

1. I encourage the authors to pay more attention to two paragraphs before the Introduction, to ensure that they can be understood by readers who are not familiar with the topic. In particular, please make clear the difference between coherent oscillatory activity, collective oscillations, and fully synchronized regimes. Both coherent activity and synchronized states are oscillations, right? How to distinguish them? Please also make clear that first you speak about real brain networks and then concentrate on identical LIF neurons.

Answer: We have tried to clarify these issues by rewriting the first sentence as follows:

Neural collective oscillations have been observed in very many contexts in brain circuits, ranging from ubiquitous $\gamma$-oscillations to $\theta$-rhythm in the hippocampus. The origin of these oscillations is commonly associated to the balance between excitation and inhibition in the network, while purely excitatory circuits are believed to lead to “unstructured population bursts” (Buzsàki, 2006). However, coherent activity patterns have been observed also in “in vivo” measurements of the developing rodent neocortex and hippocampus for a short period after birth, despite the fact that at this early stage the nature of the involved synapses is essentially excitatory, while inhibitory synapses will develop only later (Allene et al., 2008). Of particular interest are the so-called giant depolarizing potentials (GDPs), recurrent oscillations which repeatedly synchronizes a relatively small assembly of neurons and whose degree of synchrony is orchestrate by hub neurons (Bonifazi et al., 2009). These experimental results suggest that the macroscopic dynamics of excitatory networks can reveal unexpected behaviors.

Furthermore, since our article will appear on the computational neuroscience section, we think it is sufficiently clear we are not dealing with the "real brain circuits". However, we are ready to write this explicitly in the article if needed.

2. Parkinson disease shall be Parkinson's disease

3. Pionieristic - shall it not be pioneering?

Answer : We have corrected the text as suggested by the Referee.

4. ...this is not the case (Luccioli et al., 2012). Since in this latter case the fluctuations in the synaptic current do not vanish, even for extremely large networks.

Shall it not be ...this is not the case (Luccioli et al., 2012), since in this latter case the fluctuations in the synaptic current do not vanish, even for extremely large networks.

More important, at this point it is not clear what is "the fluctuations in the synaptic current" (it has not been explained yet).

Answer: We have rewritten the sentence as follows:

... this is not the case (Luccioli et al., 2012). This is due to the fact that, for sufficiently large networks, the synaptic currents, driving the dynamics of the single neurons, become essentially identical for massively connected networks, while the differences among them do not vanish for sparse networks.

5. Sparse and massively connected networks reveal even more striking differences at the microscopic level associated to the membrane potentials' evolution, chaotic evolution has been observed in both cases for finite networks.

Maybe: Sparse and massively connected networks reveal even more striking differences at the microscopic level associated to the membrane potentials' evolution. So, chaotic evolution has been observed in both cases for finite networks.

...sparse networks remains chaotic shall be sparse networks remain chaotic

Answer: We have rewritten the phrase as follows:

Sparse and massively connected networks reveal even more striking differences at the microscopic level associated to the membrane potentials' dynamics. As a matter of fact, for finite networks chaotic evolution has been observed in both cases. However, this chaos is weak in the massively connected networks, vanishing for sufficiently large system sizes, while sparse networks remain chaotic for any large number of neurons and the chaotic dynamics is extensive.

6. Averaged fields: what is the meaning of taking P=\alpha E+\dot E? Why not simply \dot E?

Answer: This choice is simply dictated by the form of the equation ruling the dynamics of the field, with this choice the second order differential equation for the field can be rewritten as two simple first order ODEs. Furthermore, we have performed this choice for historical reasons by following the seminal paper by Van Vreeswijk.

7. Splay state, the first equation. Why x=A\cos()? There seem to be no harmonic process here.

Answer: The Referee is right, this is somehow misleading. We were thinking to split phase and amplitude dynamics. We have removed the harmonic part for clarity.

8. Notice that R is sometimes called the order parameter and sometimes the modulus of the order parameter; please, do it in a unique way.

9. Fig 3a: in caption shall be "Averaged modulus of the order parameter R". Next, 3b: in text it is \bar P, \bar E, but in the figure simply P and E (\bar E=E, \bar P = P here, but it is confusing).

Answer: We have performed the requested modifications.

10. Since the splay state is characterized by a constant field E, while PS is associated to a periodically oscillating field.

I assume, this sentence is not quite correct grammatically.

Answer: We have rewritten the phrase as follows:

Since the field $E$ is constant for splay states and periodically oscillating in the PS regime.

11. Caption Fig 4: The black curve correspond shall be

The black curve corresponds

12. Sparse networks. irrispectively shall be irrespectively

13. What is intensive variable? Maybe you can add one or two sentences explaining a bit more the extensivity property?

Answer: We have hopefully clarified this issue by adding the following sentences:

Furthermore, the dynamics is characterized by extensive high-dimensional chaos (Ruelle, 1982; Grassberger, 1989), i.e. the number of active degrees of freedom, measured by the fractal dimension, increases proportionally to the system size. Extensive chaos has been usually observed in diffusive coupled systems (Livi et al., 1986; Grassberger, 1989; Paul et al., 2007), where the system can be easily decomposed in weakly interacting sub-systems. Whenever the system is chaotically extensive the associated spectra of the Lyapunov exponents $$\{\lambda_i\}$$ collapse onto one another, when they are plotted versus the rescaled index $$i/N$$, as shown in Fig. 7b (Livi et al., 1986).

## Reviewer B (Dr. Oleksandr V. Popovych)

The article explains the emergence of splay states and partial synchronization in ensembles of pulse-coupled neurons under variety of conditions including variation of parameters and coupling topology. It is clearly written and very inviting introduction to the topic.

I have only a few minor remarks which might facilitate the reading of the article by non-specialists and contribute to the educational purposes of it as an encyclopedic and scholarly article.

1. It would probably be reasonable to include at the very beginning (as the first sentence) a definition of the subject (formulated in the title), as many other articles in Scholarpedia do. Then it becomes clear at the very beginning what this article is about.

2. Several terms used in the article like "purely excitatory circuits", "coherent activity patterns", "dilution in networks", directed/undirected "Erdös-Renyi graph", "in-degree connectivity", "massively connected networks", "sparse networks", "fully coupled system", etc might be explained at the places where they appear for the first time.

3. Please define "the average standard deviation $\bar \sigma$" mentioned in section "Massively Connected Networks". Is it the time average of the standard deviation defined in section "Model and Indicators" or something else?

4. Please clarify the definition of the phases of neurons (section "Model and Indicators"). Why should the phase of neuron j depend on the spike times of some other neuron q? The definition should be valid also in the case of uncoupled (independent) neurons.

5. Please check the following sentences for plural/singular verbs:

-- "Similarly to what observed ..." (section "Massively Connected Networks")

-- "The chaotic motion ... " (section "Massively Connected Networks")

-- "Similarly to what observed ..." (section "Sparse Networks")

-- "The most striking difference ..." (section "Sparse Networks")

The last sentence is far too long and difficult to read. Could it be split in parts?

## Response to Reviewer B

We thank the Referee for his kind comments and the fast review process.

1. It would probably be reasonable to include at the very beginning (as the first sentence) a definition of the subject (formulated in the title), as many other articles in Scholarpedia do. Then it becomes clear at the very beginning what this article is about.

Answer: We have added the following phrase at the very beginning of the article

An excitatory pulse-coupled neural network is a network composed of neurons coupled via excitatory synapses, where the coupling among the neurons is mediated by the transmission of excitatory post-synaptic potentials (EPSPs). The coherent activity of a neuronal population usually indicates that some form of correlation is present in the firing of the considered neurons. The article focuses on the influence of dilution on the collective dynamics of these networks: a diluted network is a network where connections have been randomly pruned. Two kind of dilution are examined: massively connected versus sparse networks. A massively (sparse) connected network is characterized by an average connectivity which grows proportionally to (does not depend on) the system size.

2. Several terms used in the article like "purely excitatory circuits", "coherent activity patterns", "dilution in networks", directed/undirected "Erdös-Renyi graph", "in-degree connectivity", "massively connected networks", "sparse networks", "fully coupled system", etc might be explained at the places where they appear for the first time.

i) we have added a first sentence where the meaning of excitatory networks, coherent activity, dilution,massively connected and sparse networks has been clarified;

ii) We have introduced the concept of fully coupled systems as follows:

Van Vreeswijk in 1996 has extended these analysis to globally (or fully) coupled excitatory networks of leaky integrate-and-fire (LIF) neurons, where each neuron is connected to all the others. This analysis has confirmed that for slow synapses the collective dynamics is asynchronous ( Splay States ) while for sufficiently fast synaptic responses a quite peculiar coherent regime emerges, characterized by partial synchronization at the population level, while single neurons perform quasi-periodic motions (van Vreeswijk, 1996).

iii) The concept of directed and undirected Erdös-Renyi network have been introduced as follows

In general, the connectivity matrix $$C$$ is non-symmetric. The random network associated to such connectivity matrix is termed directed Erdös-Renyi network and it is characterized by an average (in-degree) connectivity $<k> = p \times N$. An undirected network has a symmetric connectivity matrix.

iv) The in-degree connectivity is defined as the number of afferent synapses

3. Please define "the average standard deviation $\bar σ$" mentioned in section "Massively Connected Networks". Is it the time average of the standard deviation defined in section "Model and Indicators" or something else?

Answer: Yes it is the time average of the standard deviation defined in section "Model and Indicators", we have clarified this point.

4. Please clarify the definition of the phases of neurons (section "Model and Indicators"). Why should the phase of neuron j depend on the spike times of some other neuron q? The definition should be valid also in the case of uncoupled (independent) neurons.

Answer: This definition can be used in a network where all neurons fires with firing rates not too dissimilar. We have used this definition in some papers, because it is easier to implement and it does not require to store all the spiking times of the neurons. We have also verified that this definition does not give results substantially different from the classical one in the studied examples. However, for simplicity, we have now modified the definition of the phase in the article reporting the standard one.

5. Please check the following sentences for plural/singular verbs:

-- "Similarly to what observed ..." (section "Massively Connected Networks")

-- "The chaotic motion ... " (section "Massively Connected Networks")

-- "Similarly to what observed ..." (section "Sparse Networks")

-- "The most striking difference ..." (section "Sparse Networks")

6. The last sentence is far too long and difficult to read. Could it be split in parts?

Answer: We have rewritten the sentence as follows, hopefully now it is more clear

The extensivity property is highly nontrivial in sparse networks, since in this case the dynamics is non additive. Contrary to what happens in spatially extended systems with diffusive coupling, where the dynamical evolution of the whole system can be approximated by the juxtaposition of almost independent sub-structures (Grassberger, 1989; Paul et al., 2007). Extensive chaos has not been observed in globally coupled networks, which exhibit a non-extensive component in the Lyapunov spectrum (Takeuchi et al., 2011).