From Micro to Macroscopic Brain Connectivity Using Multiple Modalities
View this Special IssueReview Article  Open Access
Roberto C. Sotero, "Modeling the Generation of PhaseAmplitude Coupling in Cortical Circuits: From Detailed Networks to Neural Mass Models", BioMed Research International, vol. 2015, Article ID 915606, 12 pages, 2015. https://doi.org/10.1155/2015/915606
Modeling the Generation of PhaseAmplitude Coupling in Cortical Circuits: From Detailed Networks to Neural Mass Models
Abstract
Phaseamplitude coupling (PAC), the phenomenon where the amplitude of a high frequency oscillation is modulated by the phase of a lower frequency oscillation, is attracting an increasing interest in the neuroscience community due to its potential relevance for understanding healthy and pathological information processing in the brain. PAC is a diverse phenomenon, having been experimentally detected in at least ten combinations of rhythms: deltatheta, deltaalpha, deltabeta, deltagamma, thetaalpha, thetabeta, thetagamma, alphabeta, alphagamma, and betagamma. However, a complete understanding of the biophysical mechanisms generating this diversity is lacking. Here we review computational models of PAC generation that range from detailed models of neuronal networks, where each cell is described by HodgkinHuxleytype equations, to neural mass models (NMMs) where only the average activities of neuronal populations are considered. We argue that NMMs are an appropriate mathematical framework (due to the small number of parameters and variables involved and the richness of the dynamics they can generate) to study the PAC phenomenon.
1. Introduction
From the theory of signal processing we know that if an inputstateoutput system is linear its output will have the same frequency content as its inputs. Conversely, in nonlinear systems, the energy at one frequency in the inputs appears at different frequencies in the outputs. This induces crossfrequency coupling (CFC) between any two sources, when the output of one serves as the input to the other [1]. It has been shown that pyramidal cells produce a varied set of intrinsic dynamics based only on the type and compartmental localization of intrinsic conductances [2]. A combination of sodium, potassium, and calcium conductances produces coexistent gamma (~40 Hz) and theta (~6 Hz) rhythms on tonic depolarization. In contrast, combinations of persistent sodium and potassium channels in the soma produce a usedependent transition between regular spiking at ~10 Hz and a repetitive, brief burst generation at ~20 Hz [2]. Since cortical columns and brain areas generating different brain rhythms are interconnected, the presence of CFC should not be surprising, even if the exact mechanisms responsible for its generation remain imprecise. The question is then whether CFC is only a mechanistic result of the way the brain is constructed or if it also has a role in brain computations. At least six types of CFC have been documented [3, 4]: amplitudeamplitude coupling (AAC), phasephase coupling (PPC), frequencyfrequency coupling (FFC), phaseamplitude coupling (PAC), phasefrequency coupling (PFC), and frequencyamplitude coupling (FAC). PAC, the type of CFC that occurs when the phase of a low frequency oscillation modulates the amplitude of a higher frequency oscillation, has received a lot of attention in the last decade due to its potential relevance for understanding healthy and pathological brain function [5–11]. PAC has been hypothesized to be the carrier mechanism for the interaction of local and global processes and therefore being directly linked to the integration of distributed information in the brain [12]. For instance, it has been suggested that thetagamma PAC is used as a coding scheme for multiitem shortterm memory in the hippocampus, where different spatial information is represented in different gamma subcycles of a theta cycle [13, 14]. Recent experimental evidence also suggests that PAC links local bloodoxygenleveldependent (BOLD) signals to BOLD correlation across distributed networks [15].
In parallel to the experimental study of the PAC phenomenon, computational models have been proposed in order to clarify the neurobiological mechanism underlying its generation [16–23]. Here we review these models, going from the detailed description of each cell (via the HodgkinHuxley formalism) in neuronal networks to neural mass models (NMMs), which are a type of mean field description that focuses on the dynamics of the average activity in a neuronal population while neglecting the secondorder statistics (variance and covariances) and from models only focusing on generation of the thetagamma PAC in the hippocampus to the most recent models capable of simultaneously generating several PAC combinations.
This review is structured as follows. First, in Section 2, we show that there is evidence for at least ten different PAC combinations (of a low and a higher frequency oscillation). Computational models of the PAC phenomenon can be divided into two types: detailed and NMMs. The main characteristics of these two types are briefly discussed in Section 3, followed by two sections describing specific models of both types.
2. Experimental Evidence of the Diversity of the PAC Phenomenon
The classic example of PAC was demonstrated in the CA1 region of the hippocampus [24] where the phase of the theta rhythm was found to modulate the power of gamma oscillations. Later studies found that PAC is neither restricted to thetagamma coupling nor to the hippocampus. For instance, PAC has also been reported in the frontal, posterior, and parietal human cortices during auditory, visual, linguistic, and memory tasks [25–27], in monkey auditory and visual cortices [15, 28, 29] and rodent olfactory bulb [30]. In addition to Bragin et al.’s study [24], other studies have confirmed the existence of thetagamma coupling in the hippocampus [31–34] and other brain areas [35–44]. Other PAC combinations of low and high frequency rhythms have also been detected: deltatheta [37, 45], deltaalpha [46, 47], deltabeta [44, 46], deltagamma [34, 35, 38, 41, 44], thetaalpha [46], thetabeta [44, 46], alphabeta [45], alphagamma [15, 26, 27, 35, 46, 48, 49], and betagamma [7, 15].
It should be noted that the studies mentioned above do not always use the same frequency values for the boundaries of the different brain rhythms [50] and that sometimes the gamma band is divided into different subbands such as lowgamma, middlegamma, and fastgamma, with boundaries that can differ between different studies. Thus, subdivisions of classical bands can potentially increase the number of PAC combinations to be studied. Additionally, a high number of mathematical methods for detecting PAC have been proposed [3, 12, 51–57], each with advantages and caveats, and no gold standard has emerged. Furthermore, those methods are not exempted of spurious results, that is, identifying PAC that is not related to true modulations between neuronal subsystems. These issues (reviewed recently in [58]) are out of the scope of this review, but we mention them here to highlight the fact that the experimental study of the PAC phenomenon is far from being complete and new methods and results in the upcoming years will be necessary to complement, inform, and refine past and future computational models of the phenomenon.
3. Detailed Mathematical Models versus Neural Mass Models
There are two main approaches to modeling the dynamics of neuronal populations. One approach is to realistically model each cell in the network, using multiple compartments for the soma, axon, and dendrites. The most prominent example of this approach is the Blue Brain Project [59], which aims to achieve in the next decade a full simulation of human brain dynamics (a network of approximately 86 billion neurons) in a supercomputer. A practical disadvantage of such realistic modeling is that it requires high computational power. For this reason, simplified versions of such models in which only one compartment is taken into account have been used [16, 60]. However, even in this case, the use of such detailed models makes it difficult to determine the influence of each model parameter on the generated average network characteristics. The second approach is based on the use of NMMs, which constitute a class of biophysical models that captures the average activity of neuronal ensembles, rather than modeling each neuron in the network individually [61, 62]. NMMs are described by nonlinear differential equations and can be rigorously obtained from mean field approaches [63–65] after neglecting the secondorder moments. For instance, the WilsonCowan neural mass model [61] can be obtained from a mean field approximation of two coupled networks of FitzHughNagumo neurons [63]. An alternative way of constructing the NMM formalism is to consider that each neuronal population performs two mathematical operations [62]. The first is the conversion of postsynaptic potentials (PSP) into an average density of action potentials (AP) which is modeled using a sigmoid function. The second operation is the conversion of AP into PSP, which is done by means of a linear convolution with an impulse response function. The WilsonCowan model is obtained when the impulse response function is , which produces a system of firstorder differential equations describing the activity in each population. A more recent neural mass model, the JansenRit model [62], is obtained when the impulse response function has the form . This results in a system of secondorder differential equations describing the dynamics of PSPs in each population. Computational models based on WilsonCowan and JansenRit models have provided the mathematical framework for simulating the generation of electrical activity in the brain during resting state [62, 66–71], stimulation [62, 72–74], and disease [67, 75–77].
4. Detailed Mathematical Models
Detailed mathematical models of PAC generation [16, 18] have focused on the thetagamma interaction observed in the hippocampus [24]. These models consist of either purely inhibitory networks [16] or networks with both excitatory and inhibitory cells [18–20] and are based on models previously developed to study the generation of theta and gamma rhythms separately [23].
4.1. InhibitoryInhibitory () Network
A simulated inhibitory network in the hippocampus containing fast and slow interneurons was shown to generate thetagamma coupling under restricted conditions [16]. The network comprised single compartment neurons modeled with the HodgkinHuxley formalism:where index , counts the cells in the network, is the applied current, and is a normally distributed noise. The sodium , potassium , leak , and synaptic currents are given byThe cell population was divided into half on the basis of fast and slow synaptic dynamics. Synaptic conductances had one of four possible values depending on the types of the cells connected: fast cell to fast cell, fast cell to slow cell, slow cell to slow cell, and slow cell to fast cell. Connectivity was all to all. Equations for the gating variables , , and , as well as parameter values can be found in [16]. The numerical simulations performed in [16] showed that the model can generate PAC under restricted conditions that included strong connections within the same populations, weaker connections between populations (especially from fast to slow populations), and carefully tuned inputs.
4.2. ExcitatoryInhibitory () Networks
Hippocampal networks producing thetagamma PAC also have pyramidal cells. To consider this situation, a model comprising three neuronal populations was proposed in [18] and was shown to produce thetagamma PAC [23]. The three populations are pyramidal cells, fastspiking basket cells, and oriens lacunosummoleculare (OLM) interneurons. The outputs of the OLM cells are projected as slow inhibitory postsynaptic potentials (IPSP) onto the distal apical dendrites of pyramidal cells [18].
Basket cells were modeled with a single compartment, using the fastspiking interneuron model proposed in [78], similar to (1) and (2). OLM cells were also modeled with a single compartment. In addition to sodium, potassium, leak, and synaptic current, two other currents were considered: the hcurrent and the A current [17, 18]. Pyramidal cells were modeled by 5 compartments: 1 for basal dendrites, 1 for soma, and 3 for apical dendrites. The equation for each compartment () iswhere is the current due to electrical coupling between compartments. The expressions for the ionic and synaptic currents as well as the parameter values to simulate the model can be found in the supplementary information section in [18]. Different simulations were performed in [18], but the one with the highest number of cells comprised a total of 180 cells. Their results showed that OLM cells alone can coordinate cell assemblies and that the same theta rhythm can coordinate different cell assemblies with different frequencies in the gamma range [18, 23].
5. Neural Mass Models
In this section we review three NMMs that are able to generate PAC. The first two studies [21, 22] are based on the works of WilsonCowan and JansenRit and only focus on the generation of one PAC combination. The last study [79] is also based on the JansenRit model but is able to simultaneously generate different PAC combinations.
5.1. PAC Generation Using the WilsonCowan Model
Onslow et al. [21] used the WilsonCowan model to study the generation of thetagamma PAC in a brain region not necessarily restricted to the hippocampus. The model comprises two coupled populations (Figure 1(a)), one excitatory and one inhibitory. The system of firstorder differential equations describing the model iswhere and are the average activity levels of excitatory and inhibitory populations, respectively [61] and and are the external inputs to the two populations. The weight of the connection from the excitatory population to the inhibitory population is and from the inhibitory to the excitatory population is , and the selfconnections are and . and are the time constants for each population. The nonlinearity in the model is introduced by means of a sigmoid function:where parameter determines the steepness of the sigmoid curve, determines the position of the sigmoid function, and determines the amplitude of the response.
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System (4) is capable of producing oscillations due to the reciprocal connections between the two populations. Numerical simulations showed [21] that this model generates gamma oscillations that are locked to a certain phase of theta oscillations when considering oscillatory inputs.
Figure 2(a) shows a realization of the model where the phase of a 4 Hz oscillation modulates the amplitude of a 55 Hz oscillation. The parameter values used in this simulation were , , , , , , , , , , and . Additional simulations showed [21] that the amplitude, frequency, and phaselocking characteristics of the PAC activity generated were dependent on the strength of the connectivity parameters and on the amplitude and mean value of the low frequency input signal. It was possible to tune the parameters of the model to produce different frequencies of activity phaselocked to different phases of the theta rhythm [21].
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5.2. The JansenRit Model of a Cortical Column
The JansenRit model of a cortical column [62] comprises three neuronal populations (Figure 1(b)): pyramidal cells, excitatory interneurons, and inhibitory interneurons. The model is mathematically described by a system of secondorder differential equations:where is the excitatory postsynaptic potential (EPSP) that feeds into the two populations of interneurons and and are EPSP and inhibitory postsynaptic potentials (IPSP) that enter into the pyramidal cell population, respectively. , , , and are the connection strengths between the populations. In this model, the electroencephalography (EEG) signal is considered to be proportional to .
Figure 2(b) shows a realization of model (6) where delta (3 Hz)alpha (11 Hz) PAC is produced when considering an oscillatory input . The parameter values used in this simulation were , , , , , , , , , , , and .
Alternatively, EEG signals presenting PAC can be obtained by coupling multiple JansenRit models (see Figures 5, 8, and 9 in [67]). In a more recent work [22], several JansenRit models were also coupled and the crossfrequency transfer was studied in a setting where oscillators (generating the different rhythms) were coupled unidirectionally and thus the driving between them was passive. This study showed that this passive driving can also account for CFC in the brain as a result of the complex nonlinear dynamics of the underlying neuronal activity.
5.3. Cortical Column Model Comprising 4 Layers and 14 Neuronal Populations
A more complex neural mass model of the cortical column was recently proposed [79] in which 4 cortical layers and 14 neuronal populations are considered. Figure 1(c) shows the model obtained by distributing four cell classes in four cortical layers (L2/3, L4, L5, and L6). This produced 14 different neuronal populations, since not all cell classes are present in every layer [80]. Excitatory neurons were either regular spiking (RS) or intrinsically bursting (IB) ones, and inhibitory neurons were either fastspiking (FS) or lowthreshold spiking (LTS) neurons.
The model is based on the JansenRit model and the dynamics of the average PSP in each neuronal population is obtained by solving a system of 14 secondorder differential equations:where , and . The populations are numbered from 1 to 14 following the order: [L2RS, L2IB, L2LTS, L2FS, L4RS, L4LTS, L4FS, L5RS, L5IB, L5LTS, L5FS, L6RS, L6LTS, L6FS]. Layer 2/3 was labelled as 2. As can be seen in (7), neuronal populations interact via the connectivity matrix (Figure 1(d)). External inputs are accounted for via which can be any arbitrary function including white noise [62]. The “damping” parameter critically determines the behavior of the system. For (which corresponds to the JansenRit model) an individual population is not capable of oscillating, and it is the presence of interpopulation connections (nonzero , ) that produces oscillatory behavior that mimics observed EEG signals. To account for the possibility of an oscillatory population [78, 81] a nonzero value for was used.
Figure 2(c) presents the temporal evolution of the average PSP in each neuronal population. Time series colored in red correspond to excitatory PSP (EPSP) whereas inhibitory PSP (IPSP) are presented in blue. As seen in the figure, both EPSP and IPSP present the characteristic “waxing and waning” (i.e., amplitude modulation) observed in real EEG signals. Parameters values are presented in Tables 1 and 2. To quantify the PAC phenomenon, a causality measure between time series, the information flow [82], was computed using phases and amplitudes of the signals shown in Figure 2(c). Figure 3 shows the information flow from the phase to the amplitude for nine different combinations of phases and amplitudes: deltatheta, deltaalpha, deltabeta, deltagamma, thetaalpha, thetabeta, thetagamma, alphabeta, and alphagamma. A negative value of the information flow means that the phase tends to stabilize the amplitude whereas a positive value means that the phase tends to make the amplitude more uncertain. An exploratory analysis of the influence of the parameters on PAC showed that changes in external input and time constants produced thetagamma PAC values higher than alphagamma PAC, whereas changes in connectivity produced higher alphagamma PAC values. Additional information can be found in [79].
6. Conclusions
In conclusion, we have shown that PAC is a diverse phenomenon, not restricted to the thetagamma coupling in the hippocampus. In order to model the complexity of the PAC phenomenon, which is hypothesized to bridge local and global scales in the brain [12, 15], reduced models of neuronal activity such as NMMs are needed, since detailed models are computationally expensive and their results are difficult to interpret due to the high number of variables and parameters involved. An open problem to be explored with NMMs is how the different PAC combinations are related.
While both types of models reviewed here, detailed models and NMMs, are capable of generating signals reflecting PAC, only in a few studies a quantitative measure of the phenomenon has been provided. This is probably related to the lack of a gold standard for PAC detection, which has resulted in the development of numerous methods.
The computational models summarized here focused on the mechanistic generation of the PAC phenomenon. NMMs are simple (in the sense of the few variables and parameters involved) but complex (in the sense of the richness of the dynamics they can generate) enough to approach important questions related to the functional role of the PAC phenomenon.
Conflict of Interests
The author declares that no conflict of interests exists.
References
 C. C. Chen, S. J. Kiebel, and K. J. Friston, “Dynamic causal modelling of induced responses,” NeuroImage, vol. 41, no. 4, pp. 1293–1312, 2008. View at: Publisher Site  Google Scholar
 R. D. Traub, E. H. Buhl, T. Gloveli, and M. A. Whittington, “Fast rhythmic bursting can be induced in layer 2/3 cortical neurons by enhancing persistent Na^{+} conductance or by blocking BK channels,” Journal of Neurophysiology, vol. 89, no. 2, pp. 909–921, 2003. View at: Publisher Site  Google Scholar
 V. Jirsa and V. Müller, “Crossfrequency coupling in real and virtual brain networks,” Frontiers in Computational Neuroscience, 2013. View at: Publisher Site  Google Scholar
 O. Jensen and L. L. Colgin, “Crossfrequency coupling between neuronal oscillations,” Trends in Cognitive Sciences, vol. 11, no. 7, pp. 267–269, 2007. View at: Publisher Site  Google Scholar
 J. LópezAzcárate, M. Tainta, M. C. RodríguezOroz et al., “Coupling between beta and highfrequency activity in the human subthalamic nucleus may be a pathophysiological mechanism in Parkinson's disease,” Journal of Neuroscience, vol. 30, no. 19, pp. 6667–6677, 2010. View at: Publisher Site  Google Scholar
 S. A. Shimamoto, E. S. RyapolovaWebb, J. L. Ostrem, N. B. Galifianakis, K. J. Miller, and P. A. Starr, “Subthalamic nucleus neurons are synchronized to primary motor cortex local field potentials in Parkinson's disease,” Journal of Neuroscience, vol. 33, no. 17, pp. 7220–7233, 2013. View at: Publisher Site  Google Scholar
 C. de Hemptinne, E. S. RyapolovaWebb, E. L. Air et al., “Exaggerated phaseamplitude coupling in the primary motor cortex in Parkinson disease,” Proceedings of the National Academy of Sciences of the United States of America, vol. 110, no. 12, pp. 4780–4785, 2013. View at: Publisher Site  Google Scholar
 E. A. Allen, J. Liu, K. A. Kiehl et al., “Components of crossfrequency modulation in health and disease,” Frontiers in Systems Neuroscience, vol. 5, article 59, 2011. View at: Publisher Site  Google Scholar
 L. V. Moran and L. E. Hong, “High vs low frequency neural oscillations in schizophrenia,” Schizophrenia Bulletin, vol. 37, no. 4, pp. 659–663, 2011. View at: Publisher Site  Google Scholar
 K. Kirihara, A. J. Rissling, N. R. Swerdlow, D. L. Braff, and G. A. Light, “Hierarchical organization of gamma and theta oscillatory dynamics in schizophrenia,” Biological Psychiatry, vol. 71, no. 10, pp. 873–880, 2012. View at: Publisher Site  Google Scholar
 V. Miskovic, D. A. Moscovitch, D. L. Santesso, R. E. McCabe, M. M. Antony, and L. A. Schmidt, “Changes in EEG crossfrequency coupling during cognitive behavioral therapy for social anxiety disorder,” Psychological Science, vol. 22, no. 4, pp. 507–516, 2011. View at: Publisher Site  Google Scholar
 R. T. Canolty and R. T. Knight, “The functional role of crossfrequency coupling,” Trends in Cognitive Sciences, vol. 14, no. 11, pp. 506–515, 2010. View at: Publisher Site  Google Scholar
 J. E. Lisman and M. A. P. Idiart, “Storage of 7+/2 shortterm memories in oscillatory subcycles,” Science, vol. 267, no. 5203, pp. 1512–1515, 1995. View at: Google Scholar
 J. E. Lisman and O. Jensen, “The thetagamma neural code,” Neuron, vol. 77, no. 6, pp. 1002–1016, 2013. View at: Publisher Site  Google Scholar
 L. Wang, Y. B. Saalmann, M. A. Pinsk, M. J. Arcaro, and S. Kastner, “Electrophysiological lowfrequency coherence and crossfrequency coupling contribute to BOLD connectivity,” Neuron, vol. 76, no. 5, pp. 1010–1020, 2012. View at: Publisher Site  Google Scholar
 J. A. White, M. I. Banks, R. A. Pearce, and N. J. Kopell, “Networks of interneurons with fast and slow γaminobutyric acid type A (${\text{GABA}}_{\text{A}}$) kinetics provide substrate for mixed gammatheta rhythm,” Proceedings of the National Academy of Sciences of the United States of America, vol. 97, no. 14, pp. 8128–8133, 2000. View at: Publisher Site  Google Scholar
 H. G. Rotstein, D. D. Pervouchine, C. D. Acker et al., “Slow and fast inhibition and an Hcurrent interact to create a theta rhythm in a model of CA1 interneuron network,” Journal of Neurophysiology, vol. 94, no. 2, pp. 1509–1518, 2005. View at: Publisher Site  Google Scholar
 A. B. L. Tort, H. G. Rotstein, T. Dugladze, T. Gloveli, and N. J. Kopell, “On the formation of gammacoherent cell assemblies by oriens lacunosummoleculare interneurons in the hippocampus,” Proceedings of the National Academy of Sciences of the United States of America, vol. 104, no. 33, pp. 13490–13495, 2007. View at: Publisher Site  Google Scholar
 X. Zhang, K. M. Kendrick, H. Zhou, Y. Zhan, and J. Feng, “A computational study on altered thetagamma coupling during learning and phase coding,” PLoS ONE, vol. 7, no. 6, Article ID e36472, 2012. View at: Publisher Site  Google Scholar
 E. Spaak, M. Zeitler, and S. Gielen, “Hippocampal theta modulation of neocortical spike times and gamma rhythm: a biophysical model study,” PLoS ONE, vol. 7, no. 10, Article ID e45688, 2012. View at: Publisher Site  Google Scholar
 A. C. Onslow, M. W. Jones, R. Bogacz, and A. B. Tort, “A canonical circuit for generating phaseamplitude coupling,” PLoS ONE, vol. 9, no. 8, Article ID e102591, 2014. View at: Publisher Site  Google Scholar
 M. Jedynak, A. J. Pons, and J. GarciaOjalvo, “Crossfrequency transfer in a stochastically driven mesoscopic neuronal model,” Frontiers in Computational Neuroscience, vol. 9, article 14, 2015. View at: Publisher Site  Google Scholar
 N. Kopell, C. Börgers, D. Pervouchine, P. Malerba, and A. Tort, “Gamma and theta rhythms in biophysical models of hippocampal circuits,” in Hippocampal Microcircuits: A Computational Modeler's Resource Book, V. Cutsuridis, B. P. Graham, S. Cobb, and I. Vida, Eds., Springer, New York, NY, USA, 2010. View at: Google Scholar
 A. Bragin, G. Jandó, Z. Nádasdy, J. Hetke, K. Wise, and G. Buzsáki, “Gamma (40–100 Hz) oscillation in the hippocampus of the behaving rat,” The Journal of Neuroscience, vol. 15, no. 1, pp. 47–60, 1995. View at: Google Scholar
 R. T. Canolty, E. Edwards, S. S. Dalal et al., “High gamma power is phaselocked to theta oscillations in human neocortex,” Science, vol. 313, no. 5793, pp. 1626–1628, 2006. View at: Publisher Site  Google Scholar
 D. Osipova, D. Hermes, and O. Jensen, “Gamma power is phaselocked to posterior alpha activity,” PLoS ONE, vol. 3, no. 12, Article ID e3990, 2008. View at: Publisher Site  Google Scholar
 B. Voytek, R. T. Canolty, A. Shestyuk, N. E. Crone, J. Parvizi, and R. T. Knight, “Shifts in gamma phaseamplitude coupling frequency from theta to alpha over posterior cortex during visual tasks,” Frontiers in Human Neuroscience, vol. 4, article 191, 2010. View at: Publisher Site  Google Scholar
 P. Lakatos, C.M. Chen, M. N. O'Connell, A. Mills, and C. E. Schroeder, “Neuronal oscillations and multisensory interaction in primary auditory cortex,” Neuron, vol. 53, no. 2, pp. 279–292, 2007. View at: Publisher Site  Google Scholar
 P. Lakatos, G. Karmos, A. D. Mehta, I. Ulbert, and C. E. Schroeder, “Entrainment of neuronal oscillations as a mechanism of attentional selection,” Science, vol. 320, no. 5872, pp. 110–113, 2008. View at: Publisher Site  Google Scholar
 D. RojasLíbano and L. M. Kay, “Olfactory system gamma oscillations: the physiological dissection of a cognitive neural system,” Cognitive Neurodynamics, vol. 2, no. 3, pp. 179–194, 2008. View at: Publisher Site  Google Scholar
 R. SchefferTeixeira, H. Belchior, F. V. Caixeta, B. C. Souza, S. Ribeiro, and A. B. L. Tort, “Theta phase modulates multiple layerspecific oscillations in the CA1 region,” Cerebral Cortex, vol. 22, no. 10, pp. 2404–2414, 2012. View at: Publisher Site  Google Scholar
 M. J. Gillies, R. D. Traub, F. E. N. LeBeau et al., “A model of atropineresistant theta oscillations in rat hippocampal area CA1,” Journal of Physiology, vol. 543, no. 3, pp. 779–793, 2002. View at: Publisher Site  Google Scholar
 B. Lega, J. Burke, J. Jacobs, and M. J. Kahana, “Slowthetatogamma phaseamplitude coupling in human hippocampus supports the formation of new episodic memories,” Cerebral Cortex, 2014. View at: Publisher Site  Google Scholar
 J. Gross, N. Hoogenboom, G. Thut et al., “Speech rhythms and multiplexed oscillatory sensory coding in the human brain,” PLoS Biology, vol. 11, no. 12, Article ID e1001752, 2013. View at: Publisher Site  Google Scholar
 E. Florin and S. Baillet, “The brain's restingstate activity is shaped by synchronized crossfrequency coupling of neural oscillations,” NeuroImage, vol. 111, pp. 26–35, 2015. View at: Publisher Site  Google Scholar
 S. Dürschmid, T. Zaehle, K. Kopitzki et al., “Phaseamplitude crossfrequency coupling in the human nucleus accumbens tracks action monitoring during cognitive control,” Frontiers in Human Neuroscience, vol. 7, article 635, 2013. View at: Publisher Site  Google Scholar
 P. Lakatos, A. S. Shah, K. H. Knuth, I. Ulbert, G. Karmos, and C. E. Schroeder, “An oscillatory hierarchy controlling neuronal excitability and stimulus processing in the auditory cortex,” Journal of Neurophysiology, vol. 94, no. 3, pp. 1904–1911, 2005. View at: Publisher Site  Google Scholar
 J. Lee and J. Jeong, “Correlation of risktaking propensity with crossfrequency phaseamplitude coupling in the resting EEG,” Clinical Neurophysiology, vol. 124, no. 11, pp. 2172–2180, 2013. View at: Publisher Site  Google Scholar
 R. J. McGinn and T. A. Valiante, “Phaseamplitude coupling and interlaminar synchrony are correlated in human neocortex,” Journal of Neuroscience, vol. 34, no. 48, pp. 15923–15930, 2014. View at: Publisher Site  Google Scholar
 T. Demiralp, Z. Bayraktaroglu, D. Lenz et al., “Gamma amplitudes are coupled to theta phase in human EEG during visual perception,” International Journal of Psychophysiology, vol. 64, no. 1, pp. 24–30, 2007. View at: Publisher Site  Google Scholar
 S. M. Szczepanski, N. E. Crone, R. A. Kuperman et al., “Dynamic changes in phaseamplitude coupling facilitate spatial attention control in frontoparietal cortex,” PLoS Biology, vol. 12, no. 8, Article ID e1001936, 2014. View at: Publisher Site  Google Scholar
 M. van Wingerden, R. van der Meij, T. Kalenscher, E. Maris, and C. M. A. Pennartz, “Phaseamplitude coupling in rat orbitofrontal cortex discriminates between correct and incorrect decisions during associative learning,” The Journal of Neuroscience, vol. 34, no. 2, pp. 493–505, 2014. View at: Publisher Site  Google Scholar
 J. Wang, D. Li, X. Li et al., “Phaseamplitude coupling between theta and gamma oscillations during nociception in rat electroencephalography,” Neuroscience Letters, vol. 499, no. 2, pp. 84–87, 2011. View at: Publisher Site  Google Scholar
 C. Nakatani, A. Raffone, and C. van Leeuwen, “Efficiency of conscious access improves with coupling of slow and fast neural oscillations,” Journal of Cognitive Neuroscience, vol. 26, no. 5, pp. 1168–1179, 2014. View at: Publisher Site  Google Scholar
 R. C. Sotero, A. Bortel, S. Naaman et al., “Laminar distribution of crossfrequency coupling during spontaneous activity in rat area S1Fl,” in Proceedings of the 43rd Annual Meeting of the SocietyforNeuroscience, San Diego, Calif, USA, 2013. View at: Google Scholar
 M. X. Cohen, C. E. Elger, and J. Fell, “Oscillatory activity and phaseamplitude coupling in the human medial frontal cortex during decision making,” Journal of Cognitive Neuroscience, vol. 21, no. 2, pp. 390–402, 2009. View at: Publisher Site  Google Scholar
 J. Ito, P. Maldonado, and S. Grün, “Crossfrequency interaction of the eyemovement related LFP signals in V1 of freely viewing monkeys,” Frontiers in Systems Neuroscience, vol. 7, article 1, 2013. View at: Google Scholar
 E. Spaak, M. Bonnefond, A. Maier, D. A. Leopold, and O. Jensen, “Layerspecific entrainment of gammaband neural activity by the alpha rhythm in monkey visual cortex,” Current Biology, vol. 22, no. 24, pp. 2313–2318, 2012. View at: Publisher Site  Google Scholar
 T. Yanagisawa, O. Yamashita, M. Hirata et al., “Regulation of motor representation by phaseamplitude coupling in the sensorimotor cortex,” The Journal of Neuroscience, vol. 32, no. 44, pp. 15467–15475, 2012. View at: Publisher Site  Google Scholar
 C. Magri, A. Mazzoni, N. K. Logothetis, and S. Panzeri, “Optimal band separation of extracellular field potentials,” Journal of Neuroscience Methods, vol. 210, no. 1, pp. 66–78, 2012. View at: Publisher Site  Google Scholar
 W. D. Penny, E. Duzel, K. J. Miller, and J. G. Ojemann, “Testing for nested oscillation,” Journal of Neuroscience Methods, vol. 174, no. 1, pp. 50–61, 2008. View at: Publisher Site  Google Scholar
 A. B. L. Tort, R. Komorowski, H. Eichenbaum, and N. Kopell, “Measuring phaseamplitude coupling between neuronal oscillations of different frequencies,” Journal of Neurophysiology, vol. 104, no. 2, pp. 1195–1210, 2010. View at: Publisher Site  Google Scholar
 B. Voytek, M. D'Esposito, N. Crone, and R. T. Knight, “A method for eventrelated phase/amplitude coupling,” NeuroImage, vol. 64, no. 1, pp. 416–424, 2013. View at: Publisher Site  Google Scholar
 D. Dvorak and A. A. Fenton, “Toward a proper estimation of phaseamplitude coupling in neural oscillations,” Journal of Neuroscience Methods, vol. 225, pp. 42–56, 2014. View at: Publisher Site  Google Scholar
 B. PittmanPolletta, W.H. Hsieh, S. Kaur, M.T. Lo, and K. Hu, “Detecting phaseamplitude coupling with high frequency resolution using adaptive decompositions,” Journal of Neuroscience Methods, vol. 226, pp. 15–32, 2014. View at: Publisher Site  Google Scholar
 T. E. Özkurt, “Statistically reliable and fast direct estimation of phaseamplitude crossfrequency coupling,” IEEE Transactions on Biomedical Engineering, vol. 59, no. 7, pp. 1943–1950, 2012. View at: Publisher Site  Google Scholar
 M. A. Kramer and U. T. Eden, “Assessment of crossfrequency coupling with confidence using generalized linear models,” Journal of Neuroscience Methods, vol. 220, no. 1, pp. 64–74, 2013. View at: Publisher Site  Google Scholar
 J. Aru, J. Aru, V. Priesemann et al., “Untangling crossfrequency coupling in neuroscience,” Current Opinion in Neurobiology C, vol. 31, pp. 51–61, 2014. View at: Google Scholar
 R. D. Traub, R. Duncan, A. J. C. Russell et al., “Spatiotemporal patterns of electrocorticographic very fast oscillations ($>$ 80 Hz) consistent with a network model based on electrical coupling between principal neurons,” Epilepsia, vol. 51, no. 8, pp. 1587–1597, 2010. View at: Publisher Site  Google Scholar
 J. Rinzel, D. Terman, X.J. Wang, and B. Ermentrout, “Propagating activity patterns in largescale inhibitory neuronal networks,” Science, vol. 279, no. 5355, pp. 1351–1355, 1998. View at: Publisher Site  Google Scholar
 H. R. Wilson and J. D. Cowan, “Excitatory and inhibitory interactions in localized populations of model neurons,” Biophysical Journal, vol. 12, no. 1, pp. 1–24, 1972. View at: Publisher Site  Google Scholar
 B. H. Jansen and V. G. Rit, “Electroencephalogram and visualevoked potential generation in a mathematicalmodel of coupled cortical columns,” Biological Cybernetics, vol. 73, no. 4, pp. 357–366, 1995. View at: Publisher Site  Google Scholar
 H. Hasegawa, “Dynamical meanfield theory of spiking neuron ensembles: response to a single spike with independent noises,” Physical Review E, vol. 67, no. 4, Article ID 041903, 2003. View at: Publisher Site  Google Scholar
 H. Hasegawa, “Dynamical meanfield theory of noisy spiking neuron ensembles: application to the HodgkinHuxley model,” Physical Review E, vol. 68, no. 4, Article ID 041909, 2003. View at: Google Scholar
 R. Rodriguez and H. C. Tuckwell, “Noisy spiking neurons and networks: useful approximations for firing probabilities and global behavior,” BioSystems, vol. 48, no. 1–3, pp. 187–194, 1998. View at: Publisher Site  Google Scholar
 O. David and K. J. Friston, “A neural mass model for MEG/EEG: coupling and neuronal dynamics,” NeuroImage, vol. 20, no. 3, pp. 1743–1755, 2003. View at: Publisher Site  Google Scholar
 R. C. Sotero, N. J. TrujilloBarreto, Y. IturriaMedina, F. Carbonell, and J. C. Jimenez, “Realistically coupled neural mass models can generate EEG rhythms,” Neural Computation, vol. 19, no. 2, pp. 478–512, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Zavaglia, L. Astolfi, F. Babiloni, and M. Ursino, “A neural mass model for the simulation of cortical activity estimated from high resolution EEG during cognitive or motor tasks,” Journal of Neuroscience Methods, vol. 157, no. 2, pp. 317–329, 2006. View at: Publisher Site  Google Scholar
 P. A. ValdesSosa, J. M. SanchezBornot, R. C. Sotero et al., “Model driven EEG/fMRI fusion of brain oscillations,” Human Brain Mapping, vol. 30, no. 9, pp. 2701–2721, 2009. View at: Publisher Site  Google Scholar
 F. Cona, M. Zavaglia, and M. Ursino, “Binding and segmentation via a neural mass model trained with hebbian and antiHebbian mechanisms,” International Journal of Neural Systems, vol. 22, no. 2, Article ID 1250003, 2012. View at: Publisher Site  Google Scholar
 R. C. Sotero and A. Shmuel, “Energybased stochastic control of neural mass models suggests timevarying effective connectivity in the resting state,” Journal of Computational Neuroscience, vol. 32, no. 3, pp. 563–576, 2012. View at: Publisher Site  Google Scholar
 O. David, S. J. Kiebel, L. M. Harrison, J. Mattout, J. M. Kilner, and K. J. Friston, “Dynamic causal modeling of evoked responses in EEG and MEG,” NeuroImage, vol. 30, no. 4, pp. 1255–1272, 2006. View at: Publisher Site  Google Scholar
 R. C. Sotero and N. J. TrujilloBarreto, “Biophysical model for integrating neuronal activity, EEG, fMRI and metabolism,” NeuroImage, vol. 39, no. 1, pp. 290–309, 2008. View at: Publisher Site  Google Scholar
 R. C. Sotero, A. Bortel, R. MartínezCancino et al., “Anatomicallyconstrained effective connectivity among layers in a cortical column modeled and estimated from local field potentials,” Journal of Integrative Neuroscience, vol. 9, no. 4, pp. 355–379, 2010. View at: Publisher Site  Google Scholar
 F. Wendling, J. J. Bellanger, F. Bartolomei, and P. Chauvel, “Relevance of nonlinear lumpedparameter models in the analysis of depthEEG epileptic signals,” Biological Cybernetics, vol. 83, no. 4, pp. 367–378, 2000. View at: Publisher Site  Google Scholar
 F. Wendling and F. Bartolomei, “Modeling EEG signals and interpreting measures of relationship during temporallobe seizures: an approach to the study of epileptogenic networks,” Epileptic Disorders, vol. 3, no. 1, pp. SI67–SI78, 2001. View at: Google Scholar
 B. S. Bhattacharya, D. Coyle, and L. P. Maguire, “A thalamocorticothalamic neural mass model to study alpha rhythms in Alzheimer's disease,” Neural Networks, vol. 24, no. 6, pp. 631–645, 2011. View at: Publisher Site  Google Scholar
 X.J. Wang and G. Buzsáki, “Gamma oscillation by synaptic inhibition in a hippocampal interneuronal network model,” Journal of Neuroscience, vol. 16, no. 20, pp. 6402–6413, 1996. View at: Google Scholar
 R. C. Sotero, “Generation of phaseamplitude coupling of neurophysiological signals in a neural mass model of a cortical column,” BioRxiv, 2015. View at: Publisher Site  Google Scholar
 S. A. Neymotin, K. M. Jacobs, A. A. Fenton, and W. W. Lytton, “Synaptic information transfer in computer models of neocortical columns,” Journal of Computational Neuroscience, vol. 30, no. 1, pp. 69–84, 2011. View at: Publisher Site  Google Scholar
 L. Tattini, S. Olmi, and A. Torcini, “Coherent periodic activity in excitatory ErdösRenyi neural networks: the role of network connectivity,” Chaos, vol. 22, no. 2, Article ID 023133, 2012. View at: Publisher Site  Google Scholar
 X. S. Liang, “Unraveling the causeeffect relation between time series,” Physical Review E, vol. 90, no. 5, Article ID 052150, 2014. View at: Publisher Site  Google Scholar
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Copyright © 2015 Roberto C. Sotero. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.