# Talk:Phase model

## Round 2 of review

This is a great article. I only have a few small suggestions for now: - Add units for phase in the third panel of Fig. 1. It seems that phase is in terms of the period T of oscillation. Since there are three definitions of phase of oscillation mentioned in the main text, maybe the figure caption should state which one is used. - The phase of oscillation is actually the geometric phase. - Although largely arbitrary, the zero-phase point can be chosen to correspond to an event of interest.

Authors: Done. thank you.

I think this is a very nice article. It is concise, and offers the most important facts about phase models. A couple of suggestions: the user is pointed to Fig. 1 for the definition of the phase of oscillation, but the symbol for the phase does not appear in the figure, so this may be confusing.

Authors: The caption was changed to reflect that; thank you.

The models reviewed are very nice - however, I really missed pulse coupled oscillators, or perhaps something more like Haken's lighthouse model. As these are very useful, it may be good to mention them explicitly.

Possible extension: a phase reduction sometimes also makes sense in chaotic systems, and it may be worth mentioning that the assumption of periodicity may be relaxed. The same is true for stochastic system (Stratonovich has some nice chapters on this). However, if the goal is to be concise, these can probably be omitted.

Authors: We mention that the phase of oscillation could be defined for chaotic oscillators and make a link to Pikovsky et al. book.). Mentioning stochastic systems is way outside the scope of this article.

Carmen's contribution "They are stable if the slope of the graph is negative at the intersection." The previous statement is given without explanation. Please explain why a negative slope implies stability..

Authors: Carmen, the stability of $$x'=H(x)$$ follows from the condition $$H'<0$$. I am not sure what is confusing in this statement. I changed it a bit, hoping that it would be clear what slope of what graph I mean here.

H(chi) = -omega Eugene the above is only true for small delta phi.

Authors: The wording is changed.

When you measure the phase resetting it is in units of time, or normalized time. This is not equal to frequency or normalized frequency.

Authors: This is true. We removed the comment that the H function corresponds to phase resetting (inserted by a reviewer). The H function has the units of phase/time, i.e., the frequency.

Say you have a normalized intrinsic period of 1 and an intrinsic frequency of 1. Then say you have a phase reset (delta phi) of 0.1 so that your new period is 0.9. Then your change in frequency is not delta phi but delta w: (delta phi)/(1 - delta phi) = 0.1111 not 0.1 However, for weak coupling , delta phi and delta w are both small so it works, but why not write it in terms of delta phi or at least state that delta phi and delta w are the same for small delta phi/delta w

Authors: These points are valid for pulse-coupled oscillators (and for the Windree's method of finding Q), but, as you say, they are not important when the coupling strength is infinitezimal. The article Phase response curve is a better place to discuss these issues.

Reviewer A response: If you convolve the iPRC with the coupling waveform (synaptic current), what you get is the phase resetting curve obtained using the current waveform as the perturbation, provided that you are in the linear weak coupling regime. Netoff et al Beyond two-cell networks: experimental measurements of neuronal responses to multiple synaptic inputs, J Computational Neuroscience, clearly show this ( see the equation top left on page 290) and should optimally be cited. This is an important point that aids intuitive understanding and should not be omitted, in my opinion, and other than that the article looks great. It is also clear from this article that the appropriate units for the H function are phase, and based on argument 2 above, the units for the H function is not frequency but phase, however as you say in the article if w is small it doesn't matter. This is a required assumption for the analysis to work. Some confusion results because the units of the iPRC are different from those of the PRC itself. Preyer and Butera Physical Review letters 95, 138103 (2005) make the same point in their Equation 2 except that their iPRC is computed with respect to conductance. This is the only thing I feel very strongly about, that the relationship of the H function to the PRC should be made clear. Netoff et al did an excellent job on that point.

Author (EMI): I added a reference to Netoff's article. However, the equation on page 290 does not have normalization by time (the 1/T factor), so the units are phases, not phase/time=frequency. In any case, I added a paragraph (starting "Computational neuroscience...") to mention the meaning of H in computational neuroscience.

My major suggestion is that I think that it would be much clearer to use an illustrative example with Q(phi_i), g(phi_i,phi_j) and H(chi) all illustrated as in Bard and John's chapter in the Koch book on methods in computational neuroscience. The units need to be clear and labeled on every graph.

Authors: Different components of the vector Q(theta) have different units of measument. This could be seen from the normalization condition Q f = 1. Since each component of f has its own units/time, the vector Q has units (unit1/time, unit2/time, ... unitn/time). It probably be too confusing to plot all of them in the figures. The function H has units of frequency, so the caption to fig.3 is modified accordingly.

This is just a comment, but another thing I have never understood is why you allow g(phi_i,phi_j) to depend on both the forced and forcing oscillator. To be consistent with linear systems theory, it seems like it should not depend upon the system that produces the impulse response. Wouldn't it be better to look at an infinitessimal perturbation in conductance rather than current, because the conductance depends only on the presynaptic oscillator and not the postsynaptic oscillator? I realize that you then could not utilize the adjoint, but you could still use Winfree's criterion.

Authors: The linear coupling is a particular case of the general case, and the theory works for the general case. In the example that you sugges, i.e., perturbation to the conductance, the effect depends on the timing of the perturbation and on the voltage of the postsynaptic neuron, as the conductance is multipled by the (V-E) term, where E is the Nerst potential for the ionic current whose conductancei s perturbed. Thus, even this simple case requires both variables, pre- and post-synaptic. By the way, the adjoint approach works whether or not the coupling involves the post-synaptic (receiving) oscilaltor. The coupling function does not participate in the adjoint equation, but only in the intergral for H.

Reviewer A response: The effect of a perturbation in conductance does depend upon both the pre and postsynaptic voltage, but the effect of the postsynaptic voltage is built in when you use a perturbation in conductance, if you assume that since the perturbation is given at a fixed phase, the membrane potential at that phase for the postsynaptic oscillator will be the same as it was when the phase resetting curve was generated. This is the assumption in Preyer and Butera and in Maran and Canavier, J. computational Neuroscience 2008. Therefore you do not have to consider the voltage in the postsynaptic oscillator in the simplest case, and you do not need the adjoint because you use the iPRC to conductance rather than current.

Author (EMI): This is an important point. We do not consider the simplest case, but the most general case. However, even in the simplest case, you still need the adjoint, or at least its voltage component, because it is the iPRC in response to the infinitezimal perturbation of the voltage equation. The iPRC is the same whether you perturb conductance, current, or anything else in the voltage equation. The nature of perturbation will only affect what you are convolving with the iPRC. Let me illustrate this using the equation on page 290 of Netoff et al. It is

${\rm PRC}(t)=\int_0^T {\rm iPRC}(\tau)g_{\rm syn}(\tau-t)[V_m(\tau)-V_s]d\tau$

Using the notations of this article, $Q={\rm iPRC}$ and $g_{ij}=g_{\rm syn}(\tau-t)[V_m(\tau)-V_s]$ Apparently, the function $$g_{ij}$$ must depend on the pre- and post-synaptic (forcing and forced) oscillator. If you require that $$g_{ij}$$ be independent of the post-synaptic oscillator, then you will only consider additive influences (current perturbation).