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doi: 10.1016/j.neuron.2015.12.018
Phase-relation diversity
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Diverse phase relations among neuronal rhythms
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and their potential function
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Eric Maris*, Pascal Fries#*, Freek van Ede*^
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*Radboud
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EZ, Nijmegen, The Netherlands
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#Ernst
8
Planck Society, 60528, Frankfurt, Germany
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^Oxford Centre for Human Brain Activity, Department of Psychiatry, University of
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University, Donders Institute for Brain, Cognition and Behaviour, 6525
Strüngmann Institute (ESI) for Neuroscience in Cooperation with Max
Oxford, OX3 7JX, Oxford, United Kingdom
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12
Corresponding Author:
13
Maris E. ([email protected])
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15
Keywords: neuronal oscillations, correlated neuronal activity, phase relations,
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travelling waves, selective neuronal communication.
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The total number of words of the manuscript and the original three boxes,
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excluding abstract, glossary, new box (nr 1) and figure legends: 4773
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The number of words of the abstract: 106
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The number of boxes (excluding Glossary): 4
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The number of figures: 3
1
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doi: 10.1016/j.neuron.2015.12.018
Phase-relation diversity
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Page nr. 2 of 38
Key figure: Number 3
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Acknowledgements:

and very useful input.
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28

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The authors would like to thank the two expert reviewers for their sharp
FvE was supported by The British Academy and The Royal Society
(Newton International Fellowship).

PF was supported by the Human Connectome Project (WU-Minn
31
Consortium, NIH grant 1U54MH091657), a European Young Investigator
32
Award, the European Union (HEALTH F2 2008 200728) and the LOEWE
33
program (“Neuronale Koordination Forschungsschwerpunkt Frankfurt”)
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2
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Abstract
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Neuronal oscillations at nearby sites in the brain often show phase relations that
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are consistent across time, yet diverse across space. We discuss recent
38
demonstrations of this phase-relation diversity, and show that, contrary to
39
earlier beliefs, this diversity is a general property of oscillations that is neither
40
restricted to low-frequency oscillations, nor to periods outside of stimulus
41
processing. Arguing for the computational relevance of phase-relation diversity,
42
we discuss that it can be modulated by sensory and motor events, and put
43
forward the idea that phase-relation diversity may support effective neuronal
44
communication by (1) enhancing selectivity and (2) allowing for the concurrent
45
segregation of multiple information streams.
46
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Existence and potential relevance of phase-relation
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diversity
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A central question in neuroscience is how the billions of neurons in the human
50
brain are coordinated such that they perform useful computations. Looking for
51
an answer, it is obvious to consider the fact that neuronal activity is usually
52
coordinated across time and space. This holds both for sub-threshold membrane
53
potentials (reflecting the inputs to a neuron) and action potentials (spikes, the
54
neuronal output). Correlations among neuronal output, such as spike synchrony
55
and relative spike timing, have a substantial impact on neuronal function [1-7].
56
Correlations across time often occur within a limited frequency band, as typically
57
identified by a rhythmic pattern in the auto-correlogram (see Glossary). This
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rhythmic neuronal activity is typically denoted as a neuronal oscillation. Neuronal
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oscillations are involved in a whole range of sensory [8], motor [9, 10], and
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cognitive processes [11-13]. They have been described in terms of their
61
frequency, amplitude, synchronization and between-site phase relations. Here,
62
we focus on the surprising diversity in their between-site phase relations, even
63
between nearby sites (see Box 1, for a description of how these phase relations
64
can be assessed empirically). Historically, this diversity has mainly been studied
65
in the context of traveling waves, and was considered typical for ongoing
66
oscillations in the absence of stimulus processing. In this paper, we discuss
67
recent demonstrations of phase-relation diversity, showing that it is a general
68
property of oscillations that is neither restricted to low-frequency oscillations,
69
nor to periods outside of stimulus processing.
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To argue for the functional relevance of phase-relation diversity, it is important
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to demonstrate (1) that it can be modulated by sensory and motor events, and
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(2) that it plays a role in the flow of neuronal information. With respect to the
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role of neuronal oscillations in the flow of neuronal information, much attention
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has gone to interactions between communicating sites. Specifically, it has been
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proposed that effective communication between two oscillating neuronal
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populations depends, at least in part, on their phase relation [14-17].
78
Communication is most effective when the neuronal output of the sending
79
population arrives at the receiving population at its most excitable phase.
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Because oscillations in both the sending and the receiving populations may be
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characterized by local diversity in their phase relations (as we will review), it
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makes sense to incorporate this type of diversity into models of how neuronal
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oscillations contribute to selective routing of information. As a first attempt to
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this, toward the end of this review, we will propose two ways by which such
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diversity may support effective neuronal communication: (1) by enhancing
86
selectivity and (2) by allowing for the concurrent segregation of multiple
87
information streams.
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In the following, we will first discuss recent observations demonstrating that
90
phase-relation diversity is a general property of oscillations. Next, we highlight
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several recent demonstrations of phase-relation diversity being modulated by
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sensory and motor events. Finally, we review recent studies that investigated the
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potential role of phase relations in selective neuronal communication.
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Phase-relation diversity is a general property of neuronal
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oscillations
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Neuronal activity is often correlated across space. Strong evidence for correlated
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neuronal input comes from optical recordings using voltage-sensitive dyes [VSDs;
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18] and from dual patch-clamp recordings [19], two methods which are only
100
weakly or not affected by volume conduction and therefore ideal to assess
101
correlation. VSD images show orderly spatio-temporal patterns in subthreshold
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membrane potentials, of which travelling waves are the most intensively studied
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[20].
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105
When neuronal activity is correlated across space, this correlation can include
106
systematic time- and phase relations. Importantly, neuronal activity can be cross-
107
correlated (“cross” denotes “across locations”) with zero or non-zero time
108
relations; in the latter case, the cross-correlations are said to be time-lagged.
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Depending on their frequency content, these patterns can be better described in
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either the time or the frequency domain. A single-waveform (non-rhythmic)
111
travelling wave can be better described by its trajectory in the time domain,
112
whereas rhythmic standing and travelling waves can be better described in the
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frequency domain. We will use the terms time and phase relations
114
interchangeably; when using the latter, their dependence on frequency will be
115
implicit (see Box 2). In the current review, we will consider traveling waves as a
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special case of the more general phenomenon of phase-relation diversity, in
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which phase relations between sites are diverse, regardless of whether they vary
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systematically with space (see Box 3). Systematic phase relations can be ongoing
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or can be evoked by a sensory or motor event. In the latter case, the associated
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neuronal activity is often denoted as an evoked response, which is to be
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distinguished from ongoing neuronal activity of which the phases are not locked
122
to some event. We will not distinguish between systematic phase relations in
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evoked or ongoing neuronal activity, although their functions may be different.
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Reliable phase-relation diversity (schematically depicted in Fig. 1A) has been
126
shown on multiple spatial scales, ranging from a few millimeters (using VSD
127
imaging and extracellular wire recordings) to over ten centimeters (using electro-
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corticography, electro-encephalography, and magneto-encephalography). In
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investigations of travelling waves, the focus has been on the low-frequency
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oscillations that are prominent in the absence of task-relevant sensory
131
processing. For example, studies based on extracranial recordings typically
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focused on travelling alpha waves [21-26], which are prominent in the absence of
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visual stimulation. These waves appear to be widespread, travelling from
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occipital to frontal sites over more than ten centimeters, although they are likely
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to also contain volume conduction effects from much more local travelling waves
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of primary currents in occipito-parietal cortex [21]. A prominent review article on
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the subject expresses the view that this type of phase-relation diversity is
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“typically present during periods outside of stimulation, while synchronous
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activity dominates in the presence of a strong stimulus” [20]. In contrast to this
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view, we now review recent studies that demonstrate robust phase-relation
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diversity during sensory stimulation and in periods of cognitive engagement.
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First, phase-relation diversity was demonstrated for band-limited oscillations in
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the gamma-frequency range during sensory processing of task-relevant visual
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stimuli [27, 28]. Specifically, under similar visual stimulation conditions, gamma-
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band phase-relation diversity was observed both on a very small spatial scale
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(inter-electrode distance <900 𝜇m) in macaque V4 [27], and on a large spatial
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scale (using magneto-encephalography) in humans [28; see Fig. 1A and 1B].
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Importantly, this gamma-band phase-relation diversity was not only observed
150
between the site-specific local field potentials (LFPs), but also between the multi-
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unit spiking activity observed at these same sites [27], implying that the phase-
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relation diversity is also visible in the synaptic input to the receiving areas.
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Second, gamma-band phase-relation diversity was also observed across the
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layers of a V1 cortical column [29], and this diversity was moreover shown to
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have a spatial pattern that is consistent with feed-forward processing: current
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sinks occurred first in input layer 4 and propagated to deep and superficial layers
158
of cortex [30]. In addition, the latter study showed that the temporal progression
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across the layers was the opposite for alpha-band oscillations: current sinks
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occurred first in the feedback-recipient layers 1, 2 and 5 and propagated to the
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input layer 4. Thus, the spatial pattern in the phase-relation diversity across
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layers provided evidence for the gamma-band rhythm indexing feed-forward and
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the alpha-band rhythm indexing feedback processing.
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Third, phase-relation diversity was demonstrated in one of the most prominent
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mammalian rhythms: the hippocampal theta rhythm. More precisely, it was
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demonstrated that the hippocampal theta rhythm in the CA1 layer is a travelling
168
wave with dominant movement direction along the septo-temporal axis [31],
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with the maximum spatial phase difference (between the septal and the
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temporal pole) ranging from 0 to 𝜋 [32]. Crucially, it was also shown that, due to
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the locking of the CA1 neurons’ spikes to their local theta phase, exactly the
172
same travelling wave was also observed in the spiking activity [31]. Because the
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different hippocampal subregions along the septo-temporal axis project to
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different targets (retrosplenial and perirhinal cortex for the septal subregion and
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hypothalamus, lateral septum, amygdala and medial prefrontal cortex for the
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temporal one), travelling waves ensure that these distinct hippocampal targets
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receive peak CA1 input in a particular order. This can be relevant for how these
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target areas integrate hippocampal input (see also Phase-relation Diversity and
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the Flow of Neuronal Information).
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Fourth and last, there is not only diversity in the between-site phase relations at
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a particular frequency, but also in the between-site phase relations involved in
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phase-amplitude coupling (see Fig. 1C). Using ECoG, recorded in humans during
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working-memory encoding and maintenance, strong diversity was observed in
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the between-site phase relations between low-frequency phases and high-
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frequency amplitude envelopes [33]. The diversity in these cross-frequency
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coupling phases was mainly due to diversity in the low-frequency phase
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relations; bursts of high-frequency amplitudes tended to be synchronized [33].
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Some of the diversity in these low-frequency phase relations has a travelling
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wave pattern [34].
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Interactions with sensory and motor events
193
Evidence for the functional relevance of phase-relation diversity comes from
194
studies that show how phase-relation diversity is affected by sensory input and
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motor output. Related to this, we consider the question how well a stimulus
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configuration can be decoded from the pattern of between-site phase relations.
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A relevant variable in this respect is stimulus strength, and we therefore also
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discuss the results of recent modeling work on the relation between stimulus
199
strength, neuronal oscillation frequency, and phase-relation diversity. Although
200
all reviewed studies demonstrate that phase-relation diversity exists, not all of
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them also quantified its strength [specifically, this was not the case for 35, 36,
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37].
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In one study, we investigated the effect of a visual grating stimulus (versus a
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blank screen) on phase-relation diversity in macaque area V4 [27]. It is well
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known that, in visual cortical areas, the presentation of a visual stimulus reduces
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the alpha band and increases the gamma band power [38]. Using LFPs recorded
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from four electrodes at the corners of a 650 by 650 µm square, we observed a
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decrease in the alpha-band phase-relation diversity and an increase in the
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gamma-band phase-relation diversity. That is, phase-relation diversity increased
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with power, opposite to what would be expected if, at this spatial scale, power
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increases would only result from increases in lag-zero synchronization.
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In a related study, gamma-band phase-relation diversity was demonstrated at
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the level of isolated single units recorded in V1 [39]. The results of this study
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pertain to spike-field coherence, the phenomenon that neuronal spiking is
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concentrated at a particular phase of the LFP. For the gamma-band LFP, spikes
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preferably occur at the trough of the gamma phase [40-43]. This study
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investigated the preferred phases of isolated single units relative to an aggregate
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gamma band LFP obtained from the other electrodes (i.e., excluding the
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electrode with which the unit was recorded). Subsequently, it demonstrated that
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the diversity in these spike-LFP phase relations could be predicted by variables
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that index neuronal activation relative to the local population: stimulus
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orientation (relative to the unit’s preferred orientation), time after stimulus
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onset, and spike density (calculated over a 250 ms window centered at the unit
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spike) [39]. Specifically, for all these indices, with increasing neuronal activation,
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the preferred phase shifted forward in the gamma cycle.
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Stimulus-induced shifts in spike-LFP phase are not always observed. For instance,
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they were not observed in a study that evaluated the effects of stimulus contrast
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on different properties of gamma band synchronization [44]. They were also not
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observed in a study that investigated the spiking of putatitve inhibitory neurons
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in relation to a pooled gamma phase (pooled across multiple recording sites in
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V4) [45]. On the positive side, besides the two studies that were already
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mentioned [27, 39], two more studies reported shifts in spike-LFP phase as a
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function of visual stimulation [46, 47].
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Phase-relation diversity is also triggered by motor events. This was demonstrated
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in a recent study that investigated neuronal activity in macaque V4 as a function
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of its time relative to saccadic eye movements [35]. This study showed that
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saccadic eye movements evoke a wave with a single wavefront, travelling across
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the V4 retinotopic map, originating from the foveal representation and travelling
243
towards the periphery (Fig. 2A). Importantly, these waves co-occur with a
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reorganization of the post-saccadic firing rates, which follow the same
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retinotopic pattern as the LFPs, and therefore may contribute to a temporal
246
pattern in the prioritizing of visual stimuli.
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248
Highly relevant for a possible functional role of phase-relation diversity would be
249
a quantification of the degree to which a stimulus configuration can be decoded
250
from the pattern of between-site phase relations. Such a quantification of the
251
decoding performance was recently performed using recordings from the CA1
252
region in rat hippocampus [36]. The hippocampus receives input from all sensory
253
modalities and uses this information, amongst others, to determine the animal’s
254
position relative to landmarks in the environment. The firing rate of the
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hippocampal neurons varies as a function of the animal’s position, forming so-
256
called place fields [48], allowing a decoder to determine the animal’s position
257
from the joint activity of multiple hippocampal neurons [49]. Crucially, it has now
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been demonstrated that the animal’s position could be equally well determined
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from the diversity in the theta-band phase relations [36; see Fig. 2B].
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Finally, there is recent modeling work on the relation between stimulus strength,
262
neuronal oscillation frequency, and phase-relation diversity. Here, the starting
263
point is the observation that the frequency of gamma-band oscillations in visual
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cortex increases with stimulus strength, as implemented experimentally by
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stimulus contrast [44, 50, 51]. Modeling studies have shown that differences in
266
excitatory drive not only determine gamma peak frequency but also between-
267
site phase relations [52-54]. Specifically, varying a whole range of model details
268
(involving conductance-based, Izhikevitch, and phase oscillator network models),
269
it was demonstrated that, within a population of model neurons oscillating at the
270
same frequency, neurons that received the stronger excitatory drive were phase-
271
advanced relative to those that received a weaker drive [53]. This demonstrates
272
a possible computational mechanism via which differences in excitatory drive are
273
translated into phase-relation diversity.
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The relation between neurons’ intrinsic frequencies and their between-neuron
276
phase relations is well understood in the theory of weakly coupled oscillators [for
277
a review, see 55]. The basic ingredients of the theory are (1) a drive to each of
278
the network nodes (model neurons) which manifests itself as a node-specific
279
intrinsic frequency, and (2) a coupling strength between all model neurons that
280
co-determines whether a given neuron pair will synchronize (possibly at a non-
281
zero phase, as determined by their relative intrinsic frequencies). Thus, via
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network-level synchronization, different input strengths may be translated into
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non-zero phase relations. The quantitative relation between input strength,
284
synchronization frequency, and between-neuron phase relations, is
285
characterized by so-called Arnold tongues [55; see Fig. 2C]. An important
286
question is whether Arnold tongues can also be identified in vivo and, if so,
287
whether synchronization frequency and between-neuron phase relations are
288
modulated by input strength as specified by the theory of weakly coupled
289
oscillators. And further, because selective attention increases both firing rates
290
[56] and gamma peak frequency [57], the question is whether selective attention
291
modulates between-neuron phase relations as prescribed by the Arnold tongues.
292
Both questions require further empirical research.
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Phase-relation diversity and the flow of neuronal
295
information
296
The influence of sensory and motor events on phase-relation diversity calls for an
297
investigation of the role such diversity may have in the routing of information in
298
the brain. A recent study investigated how the activity of a local neuronal group
299
depends on its preceding phase relation to another group [58]. The authors
300
investigated gamma band (50-80Hz) phase relations in macaque area V1 (with
301
electrodes sampling within a 16 mm2 cortical patch) while the animal watched
302
short video clips. They used a direction-specific information-theoretic measure
303
[transfer entropy; 59] with which they quantified how much a between-
304
population gamma band phase relation (the gamma phase of a sending
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population relative to the one of a receiving population) influences the spiking
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activity of the receiving population above and beyond what can be predicted by
307
the past firing rate of the receiving population itself. The study demonstrates
308
positive transfer entropy, indicating a causal effect (in the Wiener-Granger
309
sense) of the gamma band phase relation (sending-versus-receiving) on the firing
310
rate in the receiving location. Moreover, the study showed that Granger-
311
causation predominantly flows from the neuronal group that leads the phase
312
relation to the one that lags. These effects are likely functionally relevant,
313
because visually induced changes in gamma band phase relations resulted in an
314
increase in transfer entropy when they were in same direction as the average
315
phase relation. This study thus suggests that transient shifts in gamma band
316
phase relations mediate a dynamic reconfiguration of the pattern of causal
317
interactions (but see Box 4).
318
319
This study [58] also raises the question of how a neuronal network with fixed
320
anatomical connections (and thus also fixed axonal conduction delays) is able to
321
produce phase relation shifts on such a short time scale (i.e., the time scale at
322
which different movie segments trigger different phase relations). A recent
323
computational study showed that such stimulus-induced shifts in phase relations
324
can be produced by a network of connected neuronal populations if the intra-
325
population inhibition is larger than the inter-population inhibition [60]. This
326
study showed further, that the sign of the phase relation indicates the dominant
327
direction of information flow: information predominantly flows from the location
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with the earlier phase to the location with the later phase, as confirmed
329
experimentally in [58].
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331
Phase relations cannot only be affected by bottom-up stimulation, but also by
332
top-down cognitive control. We will briefly consider work that has revealed an
333
important role for the alignment (and misalignment) of oscillatory phase in the
334
filtering of relevant and irrelevant information [37, 61]. In this work, the authors
335
investigated neuronal entrainment to stimuli repeating quasi-rhythmically at 1.5-
336
2 Hz, a form of temporal attention that optimizes stimulus processing for the
337
time window in which a relevant stimulus will appear [37]. They demonstrated
338
phase-relation diversity across different frequency-tuned sites in primary
339
auditory cortex (A1) and – crucially – its modulation by top-down attention. This
340
work built on previous work that demonstrated a frequency-specific phase reset
341
in A1: pure tones that correspond to a site’s preferred or non-preferred
342
frequency reset local oscillatory activity to its high, respectively, low excitability
343
phase [61]. Now, using dual rhythmic auditory streams, of which only one was
344
attended, it was demonstrated that the temporal structure of the LFP phase
345
reflected the attended frequency: differently tuned A1 regions were entrained
346
with opposite phases, with the phase of the attended frequency corresponding
347
to high neural excitability (as reflected by a firing rate increase) [37]. Thus, this
348
pattern of phase-relation diversity results in both stronger and more selective
349
neural responses at the attended time points, effectively functioning as a
350
spectro-temporal filter.
351
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The study described in the previous paragraph has revealed that phase-relation
353
diversity of neural oscillations can separate relevant from irrelevant external
354
stimuli by virtue of the temporal predictability of the input streams. Although
355
this remains largely untested, the same concept may also apply in the context of
356
a neuronal population that receives input from both a relevant and an irrelevant
357
lower-area population. This requires two ingredients: (1) the higher-area
358
population must be able to predict the timing of inputs from the lower-area
359
population, and (2) temporally segregated outputs of the lower-area
360
populations. Predictability in the higher-area population follows from the lower-
361
area populations sending their output rhythmically. And temporal segregation in
362
the lower-area populations’ output follows from phase-relation diversity across
363
these populations. This configuration may allow a higher-area population to filter
364
the incoming information from multiple phase-diverse lower-area populations by
365
means of a phase alignment to the input from the relevant lower-area
366
population.
367
368
Phase-relation diversity may thus provide a mechanism for achieving selectivity
369
in inter-areal communication. This may be particularly relevant for oscillations
370
that are coherent over a wide area. For such widely coherent oscillations, if there
371
would be no diversity in the phase relations across space, it would be difficult to
372
conceive how selectivity could be achieved. This is also depicted schematically in
373
Figure 3A. However, in the presence of phase-relation diversity – either in the
374
lower- (as depicted in Fig. 3B) or the higher-area population – the information
375
could potentially be filtered for a smaller population within the larger population
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that oscillates coherently. As such, phase-relation diversity may introduce
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selectivity in inter-areal communication. Finally, if we consider that phase-
378
relation diversity may be prominent in both the lower- and the higher-area
379
populations, then this leads to the possibility that such diversity may support the
380
concurrent segregation of multiple communication channels, as schematically
381
depicted in Figure 3C.
382
383
This hypothesis is untested and therefore speculative. However, two recent
384
studies have provided support for the notion that the alignment – as well as the
385
misalignment – of phases in distant populations may indeed shape inter-areal
386
communication and vary dynamically as a function of task-demands.
387
Interestingly, both of these studies have demonstrated this for oscillations in the
388
beta frequency band that are coherent across wide parts of the cortex and in
389
particular between frontal and parietal brain areas. One study employed a visual
390
working memory task requiring an occulo-motor response [62]. This task had
391
previously been shown to engage coherent beta oscillations between prefrontal
392
and posterior parietal areas [63]. Interestingly, this study showed that this beta-
393
band inter-areal coherence exhibits phase-relation diversity, with the phase
394
relations forming a bimodal distribution, with modes at 0 and ±𝜋 [62].
395
Moreover, it was demonstrated that these inter-areal phase relations vary with
396
task-demands. Another study also recorded from parietal and frontal areas (i.e.,
397
the parietal reach region and the dorsal premotor area), but now in the context
398
of a planned movement task that had to be performed with either a hand or the
399
eyes [64]. In this task, monkeys maintained a movement plan for one second
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400
until a go signal instructed movement execution. These authors also observed
401
pronounced beta-band inter-areal coherence, but in their dataset this occurred
402
predominantly at phase relations close to 0. Critically, however, when
403
statistically controlling for the contribution of the dominant parietal LFP
404
component (using a partial coherence analysis), they found that there are
405
periods of the task in which the spikes of parietal neurons actually show a phase-
406
opposition with the premotor cortex LFP. This pattern was particularly
407
pronounced during the movement planning phase and was much stronger when
408
a saccade was planned as compared to when a hand movement was planned.
409
This phase-opposition may contribute to the down-regulation of the
410
communication between these parietal and frontal areas at the stage of the
411
motor planning (when the movement plan should not yet be communicated to
412
output areas). This study therefore suggests that the dynamic regulation of inter-
413
areal phase-relation diversity may not only serve to enhance selectivity and
414
segregate information streams, but also to temporarily and selectively inhibit
415
effective inter-areal interactions.
416
417
An outstanding question is to what extent these observations generalize to
418
oscillations in other frequency bands and/or brain areas. A crucial requirement
419
for the proposed mechanisms (selection and segregation, depicted in Fig. 3B and
420
3C) is coherence between neuronal populations, both the interacting and the
421
non-interacting ones. Because of this requirement, it is uncertain whether these
422
mechanisms can hold for some oscillations, such as gamma-band oscillations
423
over cortical areas that span more than a few millimeters (e.g., V1, V2 and V4
19
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424
combined). For the latter oscillations, part of the available data suggests that
425
selectivity is not implemented through phase-relation diversity, but through the
426
selective presence of coherence (see Box 4).
427
428
Conclusion
429
We have reviewed a series of recent studies that demonstrated that phase-
430
relation diversity is a general property of oscillations that is neither restricted to
431
low-frequency oscillations, nor to periods outside of stimulus processing, as
432
suggested by the earlier studies. Further, arguing for the functional role of
433
phase-relation diversity, we have discussed that (1) this phase-relation diversity
434
is modulated by sensory and motor events, to the extent that a stimulus
435
configuration can be decoded from it, and (2) that it is a potentially relevant
436
parameter in determining the flow of neuronal information, allowing for flexible
437
information selection and segregation mechanisms. One core target for future
438
research will be the identification of the precise conditions (in terms of
439
anatomical connectivity motif and network input) under which phase-relation
440
diversity is relevant for effective inter-areal communication (see Outstanding
441
Questions Box).
442
443
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Boxes
630
Glossary Box
Page nr. 29 of 38
631
632
Auto-correlogram: correlation across time between a signal and a time-shifted
633
copy of that same signal.
634
Coherence: degree to which the phase relations between two signals are
635
consistent across non-overlapping time windows.
636
Cross-correlogram: correlation across time between a signal and a time-shifted
637
copy of another signal.
638
Excitable phase: phase at which a neuron with an oscillating membrane
639
potential has the highest probability of generating an action potential in
640
response to excitatory synaptic input.
641
Decoding: data-analytic technique that estimates a variable (e.g., an animal’s
642
current position) on the basis of a set of other variables (e.g., neuronal signals).
643
Membrane potential: the difference in electric potential between the interior
644
and the exterior of a cell. The membrane potential is especially relevant for cells
645
with voltage-sensitive ion channels, such as neurons: the more positive the
646
membrane potential, the higher the probability that an excitatory synaptic input
647
results in an action potential (produced by a massive opening of voltage sensitive
648
sodium channels).
649
Neuronal oscillation: neuronal signal with a repeating waveform. Such a signal is
650
also said to exhibit periodicity or rhythmicity, and it is reflected in the fact that
651
the auto-correlogram has multiple peaks.
29
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652
Phase-amplitude coupling: particular relation between a low- and a high-
653
frequency signal component, involving that the time-varying amplitude of the
654
high-frequency component predominantly occurs a particular phase of the low-
655
frequency component.
656
Phase relation: time relation between two periodic signal components,
657
expressed as the fraction of the repeating waveform by which the signals are
658
shifted.
659
Phase-relation diversity: variability across signal pairs with respect to their phase
660
relations.
661
Traveling wave: waveform over (as a function of) space that changes its position
662
over time.
663
Voltage-sensitive dye: dye that changes its spectral properties in response to
664
voltage changes, allowing it to be used for optical imaging of potential
665
distributions.
666
667
Box 1. Calculating phase relations, their diversity, and their coherence
668
In a first step, the phases of single-sites are obtained from a Fourier (or wavelet)
669
transform of some epoch of the recorded signal (see Box Fig. IA). The Fourier
670
coefficients are typically handled as real and imaginary components of
671
frequency-indexed complex numbers whose phase angle and magnitude are the
672
ingredients for several derived quantities. First, from the single-site phases
673
obtained from multiple sites, one can calculate the between-site phase relations
674
(see Box Fig. IB). In Box Figure IC, we consider an example situation involving
675
three sites (color-coded blue, green and red), resulting in two site pairs (blue30
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Phase-relation diversity
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676
green and green-red), for which we obtained recordings in a series of
677
trials/epochs. The phase relations are shown in the grey middle columns. They
678
are obtained as the product of the immediate left and the conjugate of the
679
immediate right column. This operation produces complex numbers whose
680
phase angles equal the phase differences (phase relations) between the two
681
sites, and magnitudes that equal the product between the magnitudes of the
682
single-site coefficients. Importantly, one must distinguish the epoch-specific
683
phase relations from their average across all epochs, which is also called the
684
average phase relation. Phase-relation diversity pertains to the diversity in the
685
average phase relations across the site pairs (here illustrated for 2 pairs – the
686
minimum number). Crucially, this across-site-pair diversity only makes sense if
687
the average phase relations are also reliable, implying that the corresponding
688
epoch-wise phase relations are similar to the average phase relation. This
689
reliability can be quantified using a split-half procedure (see Fig. 1A), as well as
690
by measures of coherence, which is obtained by normalizing the cross-spectrum
691
(the average of the complex numbers in the middle columns in Fig. IB) by the
692
square-roots of the sites’ power (the average of the squared magnitudes of the
693
numbers in the outer columns). Thus, phase-relation diversity is only relevant for
694
sites that are coherent, since otherwise the diverse average phase relations are
695
not representative of the epoch-wise phase relations.
696
697
Box 2. Phase-relation diversity across frequencies
698
The main focus of this review is on phase-relation diversity across space.
699
However, phase-relation diversity across frequencies also turns out to be an
31
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Phase-relation diversity
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700
important concept, and it is important to distinguish between the two. Diversity
701
across frequencies in the phase relations between two populations corresponds
702
to the conduction time delay between these populations, which may be positive
703
or negative. This time delay can be used to infer the directionality of the coupling
704
between the populations. Consider two populations that are coherently
705
oscillating in a range of frequencies, say between 15 and 30 Hz. Provided there is
706
a fixed time delay between these populations, the phase relations between
707
these populations will vary linearly with frequency (at least, within the frequency
708
band in which they are coherent). Critically, with increasing frequency, the phase
709
lags will increase when the flow of information is from population A to B (a
710
positive time delay), whereas they will decrease when the flow is in the opposite
711
direction (a negative time delay). As such, directionality of the coupling between
712
two oscillating populations can be inferred from the pattern of the across-
713
frequency phase-relation diversity. This rationale form the basis of the phase
714
slope index [65] that has proven to be highly informative in studying the roles of
715
neuronal oscillations in the flow of neuronal communication [43, 66, 67].
716
717
Box 3. Phase-relation diversity and traveling waves
718
The defining feature of a traveling wave is that its phase varies monotonically as
719
a function of anatomical space – such as along the septo-temporal axis of the
720
hippocampus [31, 32]. As such, traveling waves are always characterized by
721
phase-relation diversity. However, the reverse is not necessarily true: phase-
722
relation diversity need not always reflect a traveling wave. For example, phase-
723
relation diversity may also emerge when (1) populations with different (non32
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724
spatial) tuning profiles (e.g., V1 populations that differ with respect to their
725
orientation tuning), (2) nearby cortical areas (e.g., V1, V2, and V3) or (3) different
726
layers, oscillate at different phases. In all these cases, phase relations will be
727
diverse across space, but will not necessarily vary monotonically as a function of
728
anatomical space. Importantly, in such cases, oscillatory phase may still vary
729
systematically as a function of a latent coding space (e.g., orientation tuning) and
730
may thus serve important functional roles in neuronal information processing
731
(see Phase-relation diversity and the flow of neuronal information). Thus, if the
732
underlying connectivity is mostly patchy rather than continuous, then wave-like
733
propagation across a latent coding space will produce non-wave-like phase
734
diversity across anatomical space.
735
736
Box 4. Coherence in gamma-band oscillations over visual cortex
737
Gamma-band oscillations in visual cortex are strongly dependent on the
738
presence of a visual stimulus in the neuronal population’s receptive field [RF; see
739
57, supplementary material, 68]. Gamma-band oscillations are coherent
740
between neuronal populations with overlapping RFs: two studies showed this for
741
populations in V1 and V4 [57, 69], and one study for populations in V4 and FEF
742
[43]. The situation without RF overlap (but with visual input in both RFs) is
743
equally relevant for the proposed selection and segregation mechanisms (see
744
Fig. 3B and 3C), but for this configuration, gamma-band coherence is less well
745
established. Specifically, early studies in cat visual cortex [reviewed in 70]
746
demonstrated gamma-band coherence between the spiking activity of neurons
747
with non-overlapping receptive fields (RFs), but also showed that this coherence
33
Published in final edited form in:
Neurosciences 39 (2), February 2016, Pages 86-99
doi: 10.1016/j.neuron.2015.12.018
Phase-relation diversity
Page nr. 34 of 38
748
was more likely when the RFs overlapped. This pattern was replicated in
749
macaque V1 [68]. The latter study also showed that gamma-band coherence is a
750
rare phenomenon overall (8% of the single unit, and 15% of the multi-unit pairs)
751
but quite common among the single/multi-unit pairs whose firing is synchronized
752
(60% of the single unit, and 69% of the multi-unit pairs). Further, in a recent
753
study in macaques, no gamma-band coherence was found between FEF multi-
754
units and V4 LFPs when the underlying populations had non-overlapping RFs
755
[43]. However, it is unclear whether RF overlap is the essential factor that
756
determines whether gamma-band coherence is and is not observed. In fact, in
757
one of the studies we discussed, coherent gamma-band oscillation were shown
758
to exhibit non-zero phase relations that had a Granger-causal effect on the firing
759
rate [58]. Unfortunately, this study did not document RF overlap. Thus, part of
760
the results show that gamma-band coherence is limited to neuronal populations
761
that are communicating, and suggest that selectivity is implemented through the
762
selective presence of coherence. Under which conditions gamma-band
763
oscillations can also implement selectivity through phase-relation diversity
764
requires more empirical work.
765
766
767
34
Published in final edited form in:
Neurosciences 39 (2), February 2016, Pages 86-99
doi: 10.1016/j.neuron.2015.12.018
Phase-relation diversity
Page nr. 35 of 38
768
Figure Legends
769
Figure 1. Reliable phase-relation diversity occurs during sensory processing, in
770
high (>20Hz) frequency ranges, and in spatially distributed phase amplitude
771
coupling. (A) Schematic of reliable phase-relation diversity. Left panel:
772
hypothetical phase distribution across a patch of cortex, together with nine
773
hypothetical recording sites. Polar plots (see also Box 1) represent the phases in
774
each site relative to an arbitrary reference site and depict diversity in phase
775
relations across space. Right panel: a split-half procedure can be used to
776
separate systematic from unsystematic phase-relation diversity. Phase-relation
777
diversity discussed in this review refers to phase relations that are both diverse
778
(spread out across the x and y axis in the plot) as well as reliable (similar in the
779
two random partitions of the data and therefore close to the diagonal in the
780
plot). Dots represent individual site-pairs. (B) Empirical data demonstrating
781
reliable phase-relation diversity of gamma oscillations during visual stimulation
782
in monkey LFP recordings in V4 (left panel) as well as in human MEG recordings
783
from posterior sites (right panel). Scatterplots show the outcomes of the split-
784
half approach explained in A and form the basis for the depicted frequency
785
spectra of reliable diversity. (C) Phase-relation diversity is also prominent in
786
patterns of spatially-distributed phase-amplitude coupling (PAC). A
787
decomposition technique was used to extract spatial and spectral profiles of
788
distributed PAC components. For one representative component, the phase-
789
providing spatial map (depicted to the right) shows diversity in the preferred
790
coupling phase across space. In the lower panels, we zoomed in on two
791
representative site pairs. Whereas site 43 shows similar coupling strength with
35
Published in final edited form in:
Neurosciences 39 (2), February 2016, Pages 86-99
doi: 10.1016/j.neuron.2015.12.018
Phase-relation diversity
Page nr. 36 of 38
792
both site 29 and 57, the high frequency oscillations in site 43 are nested at
793
opposite phases of the low-frequency oscillations in sites 29 (left) and 57 (right).
794
Panel B adapted from [27] and [28]; panel C adapted from [33].
795
796
Figure 2. Phase-relation diversity is modulated by sensorimotor events, as well
797
as by the strength of the excitatory drive. (A) Beta oscillations following a
798
saccade are characterized by phase-relation diversity in the form of a traveling
799
wave from sites representing foveal visual input to sites representing more
800
peripheral input. (B) Upper plot: phase-relation diversity of hippocampal theta
801
oscillations is dynamic during spatial navigation. Lower plot: reconstructions of
802
the rat’s location based on spikes and LFP. Phase-relation diversity in the LFP
803
allows for a reconstruction (decoding) of the rat’s location and this
804
reconstruction is approximately as accurate as the reconstruction that is based
805
on spikes. (C) Modelling study demonstrating that the excitatory drive can
806
modulate phase-relation diversity across a neuronal population. Panel A adapted
807
from Zanos, Mineault [35]; panel B adapted from Agarwal, Stevenson [36]; panel
808
C adapted from Lowet, Roberts [53].
809
810
Figure 3. Schematic of how phase-relation diversity may contribute to the flow
811
of neuronal information. For this schematic we make two key assumptions. First,
812
oscillatory phase (here represented in polar plots; see also box 1) reflects
813
excitability fluctuations of some neuronal population. As a result, the
814
effectiveness of the communication between a lower (sending) and a higher
815
(receiving) area will be determined by their phase relation. For simplicity, we
36
Published in final edited form in:
Neurosciences 39 (2), February 2016, Pages 86-99
doi: 10.1016/j.neuron.2015.12.018
Phase-relation diversity
Page nr. 37 of 38
816
assume a conduction delay of 0 ms, such that communication is most effective
817
when the lower- and the higher-area populations have a phase relation of 0 (i.e.
818
when output that is concentrated at the most excitable phase of the sending
819
population will arrive in the most excitable phase of the receiving population). In
820
reality, the most effective phase relation will depend on the precise conduction
821
delay. Second, all depicted sites are equally coherent with each other; only the
822
average phase relations differ between the three depicted scenarios. (A)
823
Scenario without diversity in the phase relations across the lower-area
824
populations. As a consequence, output from the three depicted lower-area
825
populations will arrive in the same phase of the oscillation in the higher-area
826
population. Because coherence is assumed to be equal for all three populations,
827
their impact is equally effective. The graph on the right-hand side shows the
828
impact onto the higher-area population (Y-axis) as function of the location of the
829
lower-area population (X-axis). The flat line in this graph indicates that
830
communication is not selective. (B) Scenario in which diversity is introduced
831
across the lower-area populations. As a consequence, across the lower-area,
832
some populations will have more effective phase relations with the higher-area
833
population than others. This allows for selective communication between the
834
lower- and higher-area populations, as depicted in the right panel. This
835
selectivity can be modulated by changing the degree and the spatial extent of
836
the phase-relation diversity. (C) When diversity is also introduced across the
837
higher-area receiving populations, this may further allow for the concurrent
838
segregation of multiple information streams. This is depicted in the right-hand
839
panel, in which the colours of the lines denote the three higher-area populations.
37
Published in final edited form in:
Neurosciences 39 (2), February 2016, Pages 86-99
doi: 10.1016/j.neuron.2015.12.018
Phase-relation diversity
Page nr. 38 of 38
840
841
842
Figure I. Calculating phase relations, their diversity, and their coherence. See
843
Box 1 text.
844
38
Published in final edited form in:
Neurosciences 39 (2), February 2016, Pages 86-99
doi: 10.1016/j.neuron.2015.12.018
Outstanding Questions
What are the precise conditions (in terms of anatomical connectivity motif,
network input, …) under which phase-relation diversity is relevant for effective
inter-areal communication?
Do the modulations of phase-relation diversity by sensory and motor events
imply modulations of the flow of neuronal information?
How is phase-relation diversity generated and/or controlled at the level of the
underlying neuronal network?
What is the relation between oscillatory amplitude and phase-relation diversity
with respect to (1) how they are modulated by task- and stimulus variables, and
(2) how these parameters are determined by the underlying neuronal network?
Does phase-relation diversity observed in macroscopic EEG/MEG measurements
reflect the same neurophysiological processes as phase-relation diversity in local
LFP recordings?
How does phase-relation diversity change with cohort variables, such as ageing
and disease, and could it serve as a useful biomarker for changes in neuronal
information processing?
Published in final edited form in:
Neurosciences 39 (2), February 2016, Pages 86-99
doi: 10.1016/j.neuron.2015.12.018
Trends box

Neuronal oscillations are a prominent feature of neuronal population activity. Across
populations, oscillations exhibit stable phase-relations that can be highly diverse across
nearby sites.

Phase-relation diversity is prominent even for high frequency (>40 Hz) oscillations during
sustained visual stimulation, and is therefore neither restricted to low frequencies nor to
periods outside of sensory processing.

Arguing for its computation relevance, this diversity is modulated by both sensory and
motor events and, for hippocampal theta oscillations, it even allows for reconstructing a
rat’s location.

Phase-relation diversity may enhance selectivity of neuronal communication and allow
for the concurrent segregation of multiple information streams. This may be particularly
relevant for beta oscillations that are coherent across frontal and parietal brain areas.
A
Published in final edited form in:
Neurosciences 39
(2), February 2016, Pages 86-99magnitude
Fourier
doi: 10.1016/j.neuron.2015.12.018
Transform
phase
B
real
site 2
...
...
site 2
phase
relation
site 3
...
phase
relation
...
site 1
...
trials / epochs
C
...
phase of site x phase relation
phase of site y between site x
and site y
imaginary
average
Σ
power
crossspectrum
normalize
coherence [1,2]
power
Σ
power
phase-relation
diversity
[1,2] vs [2,3]
crossspectrum
normalize
coherence [2,3]
power
1
beta LFP
A
Published in final edited form in:
Neurosciences 39 (2), February 2016, Pages 86-99
doi: 10.1016/j.neuron.2015.12.018
foveal
foveal
peripheral
peripheral
0
-1
0
time rel. to saccade end (ms)
B
theta phase
hippocamp.
sites
coupling
strength
C
40
20
rat’s location
phase-locking
0.2
0.7
0
-0.02
1
0
∆ frequency
10 −π/2
-10
0.02 -0.02
0
∆ phase
0.02 -0.02
excitatory drive
0
π/2
0.02
impact
higher
area
lower
area
no local
diversity
impact on higher area
A
Published in final edited form in:
Neurosciences 39 (2), February 2016, Pages 86-99
doi: 10.1016/j.neuron.2015.12.018
location in
lower area
local
diversity
impact on higher area
B
C
local
diversity
local
diversity
impact on higher area
location in
lower area
location in
lower area