Update paper.md

paper.md

@@ -59,20 +59,17 @@ These two accounts describe the origin of alpha waves as a phenomenon relying on
 
 Mathematical models of human brain activity have provided significant insights into neural processes at multiple scales [@deco2008dynamic]. The present study focuses on the alpha rhythm using a 'top-down' approach, whereby a mathematical expression is used that represents the collective activity of neuron groups rather than individual cells [@cook2021neural;@cooray2023global]. A general term for this set of techniques is neural ensemble or _neural population models_ (NPMs). NPMs consider the aggregate activity of neuron populations with common synaptic connectivity (excitatory or inhibitory), assuming uncorrelated states across the ensemble, and capturing emergent properties of neural tissue patches [@breakspear2017dynamic]. This method is effective for modelling oscillatory activity like the alpha rhythm, aligning with the spatial scales of EEG channels and approximating local field potentials [@coombes2014neural;@evertz2022alpha]. 
 
-NPMs include neural mass models (NMMs), mean-field models (MFMs), and neural field models (NFMs; {cite:alp}'deco2008dynamic;Nonebojak2014neural') - however it should be noted that the terminology for these NPM sub-types is not used consistently across the published literature. In one prominent strand of work [@deco2008dynamic;@moran2013neural] the term `MFM' is reserved specifically for NPMs that simplify neural population activity using a diffusion approximation, defining it as a standard normal probability distribution characterized by the mean and variance of the firing rate, with stochastic dynamics governed by Fokker-Planck equations. In this schema, NMMs are defined as a specific type of NPM where the variance of the state variables (e.g. average firing rate) across the population is fixed, allowing a simpler representation of the dynamics than MFMs with fewer equations [@breakspear2017dynamic]. NMMs are 'point process' models, i.e. they describe neural population activity without any explicit spatial information. In contrast, NFMs include local spatial information by considering the cortex as a smooth and continuously connected sheet, supporting phenomena such as propagating activity waves, often described by damped wave equations [@pinotsis2014neural;@breakspear2017dynamic]. Both NMMs and NFMs can be used to simulate whole-brain activity, by coupling local neural populations according to a discrete weighted connectivity matrix ('anatomical connectome'), or a continuous cortical surface manifold, respectively [@breakspear2017dynamic;@schirner2018inferring;@glomb2021computational;@robinson2016eigenmodes;@nunez2006electric;@visser2017standing]. For a detailed review of NPM development and whole-brain modelling, see {cite:t}`griffiths2022whole` and {cite:t}`chow2020before`, and additional remarks given in S.10. 
+NPMs include neural mass models (NMMs), mean-field models (MFMs), and neural field models (NFMs; {cite:alp}'deco2008dynamic;Nonebojak2014neural') - however it should be noted that the terminology for these NPM sub-types is not used consistently across the published literature. In one prominent strand of work [@deco2008dynamic;@moran2013neural] the term 'MFM' is reserved specifically for NPMs that simplify neural population activity using a diffusion approximation, defining it as a standard normal probability distribution characterized by the mean and variance of the firing rate, with stochastic dynamics governed by Fokker-Planck equations. In this schema, NMMs are defined as a specific type of NPM where the variance of the state variables (e.g. average firing rate) across the population is fixed, allowing a simpler representation of the dynamics than MFMs with fewer equations [@breakspear2017dynamic]. NMMs are 'point process' models, i.e. they describe neural population activity without any explicit spatial information. In contrast, NFMs include local spatial information by considering the cortex as a smooth and continuously connected sheet, supporting phenomena such as propagating activity waves, often described by damped wave equations [@pinotsis2014neural;@breakspear2017dynamic]. Both NMMs and NFMs can be used to simulate whole-brain activity, by coupling local neural populations according to a discrete weighted connectivity matrix ('anatomical connectome'), or a continuous cortical surface manifold, respectively [@breakspear2017dynamic;@schirner2018inferring;@glomb2021computational;@robinson2016eigenmodes;@nunez2006electric;@visser2017standing]. For a detailed review of NPM development and whole-brain modelling, see {cite:t}`griffiths2022whole` and {cite:t}`chow2020before`, and additional remarks given in S.10. 
 
-## Classification of NPMs and mathematical characteristics of \\ convolution-based models
 
-The three NPM variants discussed above represent three different approaches to the treatment of heterogeneity in mesoscale activity patterns - with NFMs describing variation over space, MFMs describing variation within a point process as a statistical distribution, and NMMs simplifying both of these by assuming no spatial or statistical variation. NPMs can also be categorized according to their general approach to the mathematical description of neural population-level activity, with the main distinctions being convolution vs. conductance-based and voltage vs. activity-based models. Activity-based models, such as the Wilson-Cowan equations [@wilson1972excitatory;@cowan2016wilson] have state variables representing the proportion of active cells in the population, while voltage-based models represent the population-average membrane potential within various neuron classes. Conductance-based NPMs assume very high coherence between neurons, to the extent that the dynamics of neuron population resembles the dynamics of each single neuron, allowing the use of equations that follow the same structure as single neuron conductance-based models [@marreiros2010dynamic;@breakspear2017dynamic], with distinct ionic current types and corresponding channel kinetics. We refer the reader to {cite:t}`marreiros2010dynamic;Nonepinotsis2013conductance;Nonemoran2011vivo;Nonemoran2013neural` for further discussion on the nuances of conductance-based and activity-based modelling approaches, and restrict the remainder of our discussion here to convolutional voltage-based NPM type, of which the four models focused on in this paper are all variants.
+## Classification of NPMs and mathematical characteristics of convolution-based models
 
+The three NPM variants discussed above represent three different approaches to the treatment of heterogeneity in mesoscale activity patterns - with NFMs describing variation over space, MFMs describing variation within a point process as a statistical distribution, and NMMs simplifying both of these by assuming no spatial or statistical variation. NPMs can also be categorized according to their general approach to the mathematical description of neural population-level activity, with the main distinctions being convolution vs. conductance-based and voltage vs. activity-based models. Activity-based models, such as the Wilson-Cowan equations [@wilson1972excitatory;@cowan2016wilson] have state variables representing the proportion of active cells in the population, while voltage-based models represent the population-average membrane potential within various neuron classes. Conductance-based NPMs assume very high coherence between neurons, to the extent that the dynamics of neuron population resembles the dynamics of each single neuron, allowing the use of equations that follow the same structure as single neuron conductance-based models [@marreiros2010dynamic;@breakspear2017dynamic], with distinct ionic current types and corresponding channel kinetics. We refer the reader to {cite:t}`marreiros2010dynamic;Nonepinotsis2013conductance;Nonemoran2011vivo;Nonemoran2013neural` for further discussion on the nuances of conductance-based and activity-based modelling approaches, and restrict the remainder of our discussion here to convolutional voltage-based NPM type, of which the four models focused on in this paper are all variants.
 
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 Even though convolution-based NMM share a common framework with two core operators, three key factors sets the models apart: 1) the number of neural population modelled, 2) the degree of physiological complexity associated with each neural population, and 3) the connectivity between them. By exploring these differences, we aim to uncover how each model uniquely captures the dynamics of alpha oscillations, providing insights into the description of the underlying mechanisms.