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Harnessing fast periodic visual stimulation to study face cognition: Sub‐processes, brain‐behavior relationships, and objectivity

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Jeff Nador at University of Fribourg
Jeff Nador
Meike Ramon at University of Lausanne
Meike Ramon
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Abstract

Rossion et al. (2020) review over a decade of work investigating the neural basis of unfamiliar face individuation (FI) ‐ the brain’s ability to distinguish unfamiliar face identity ‐ using fast periodic visual stimulation (FPVS). Though FPVS measures rapid, automatic processing, its value for studying vision and face cognition could be increased by addressing three important aspects.
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DR. MEIKE RAMON (Orcid ID : 0000-0001-5753-5493)
Article type : Featured Paper Commentary
Corresponding author mail id: meike.ramon@gmail.com
Harnessing fast periodic visual stimulation to study face cognition:
sub-processes, brain-behavior relationships, and objectivity
Jeffrey D. Nador & Meike Ramon
Applied Face Cognition Lab, Department of Psychology, University of Fribourg, Fribourg, Switzerland
Key words: EEG; FPVS; Face cognition; Face individuation; Methodology
Corresponding author:
Meike Ramon
University of Fribourg
Applied Face Cognition Lab
Department of Psychology
Faucigny 2
1700 Fribourg
Switzerland
Accepted Article
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1Rossion et al. (2020) review over a decade of work investigating the neural basis of unfamiliar
2face individuation (FI) - the brain’s ability to distinguish unfamiliar face identity - using fast
3periodic visual stimulation (FPVS). Though FPVS measures rapid, automatic processing, its value
4for studying vision and face cognition could be increased by addressing three important aspects.
5Processing levels. Most FPVS studies of face cognition to date have focussed on face vs.
6object categorization or FI, and collectively support the ability of FPVS to assess face cognition
7sub-processes of primarily visual nature (i.e., detection, discrimination; Ramon & Gobbini, 2018).
8However, FI critically contributes to further sub-processes (Rossion et al., 2020), i.e. recognition
9and identification, which involve a mnemonic component investigated in only few published
10 studies (Campbell et al 2020; Verosky et al., 2020; Yan et al., 2020; Zimmerman et al., 2019).
11 Thus, systematic characterization of the relationships between FPVS signatures of different
12 processing levels (cf. Quek et al., 2020), and their modulations via task demands and stimulus
13 predictability (cf. Ramon, 2018; Ramon et al., 2019) require further scrutiny to better inform our
14 conceptualization of brain-behavior relationships in this domain. Such research will require
15 increased paradigmatic flexibility, in order to adapt the FPVS methodology to this end.
16 Brain-behavior relationships. Studies measuring the FI response have contrasted
17 neurotypical and atypical/impaired populations to characterize the mechanisms of face cognition
18 and development of FI. Among neurotypical observers, findings are limited to modest correlations
19 between FI response amplitude and behavioral test performance (Xu et al., 2017; Dzhelyova et al.,
20 2020; Rossion et al., 2020). However, it is questionable “whether these behavioral tests possess
21 the virtues [...] for adequately measuring FI” (Rossion et al., 2020). The tests considered include
22 the Cambridge Face Memory Test (CFMT; Duchaine & Nakayama, 2006) and Benton Face
23 Recognition Test (BFRT; Benton & Van Allen, 1968). Since both were designed to distinguish
24 typical from impaired populations, their usefulness in investigating brain-behavior relationships
25 amongst neurotypical individuals is questionable. This aligns with the reasoning that previously
26 observed significant, weak correlations may be driven by low performing individuals (Rossion et
27 al., 2020). Moreover, by relying on simultaneous matching only (BFRT), or perception and
28 recognition (CFMT), performance on these tests relies, to a nontrivial extent, on
29 cognitive/perceptual abilities other than specifically FI.
30 Given the importance of describing the relationship between FI response and behavior, we
31 advance two methodological alternatives to abandoning this pursuit. Firstly, correlations should be
32 performed between the FI FPVS response and performance for behavioral tests that are
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1conceptually closer to FI, avoiding the pitfalls introduced by differing sources of variability
2between tests. Secondly, the approach to investigating FI-behavior correspondence should be
3reconsidered by improving the flexibility of the FPVS paradigm. Given the FI response’s
4reliability (Dzhelyova et al., 2019, Stacchi et al., 2019) and fundamental role for recognition and
5identification, it could be used as a reference to probe the value of a given behavioral test (or
6performance measure), by control or systematic variation of task/stimulus factors during FPVS.
7Characterizing the relationships between the objective FI response and different tests of face
8cognition could provide an objective means to ascertain differential test validity, and behavioral
9specificity.
10 Objectivity. Notwithstanding the FI response’s advantages, there are two aspects related to
11 the FPVS applications advocated by Rossion et al. (2020) that limit its adaptability to research on
12 individual differences, and its objectivity in capturing these. First, the inflexibility of eliminating
13 the contributions of other cognitive processes to the FI response hampers the advancement of
14 theoretically based questions regarding the relationship between sub-processes involved in face
15 cognition, their neural basis, and the degree to which these vary inter-individually and are
16 differentially modulated. The authors’ suggested procedure for measuring individual differences
17 involves taking a unitary FI measure, and correlating it with a range of behaviors or participant
18 attributes. We suggest that it is time for an agnostic approach to investigating FI response
19 modulation, employing systematic variations of task- and stimulus-related features to provide
20 theoretically important insights into how the FI process unfolds across individuals.
21 The second aspect concerns agnostic electrode selection. Objectivity in psychological
22 research generally implies maximizing available degrees of freedom, while limiting controlled
23 task- and stimulus-related attributes to those necessary for eliminating potential confounds.
24 Rossion et al. (2020) suggest reducing the degrees of freedom in FI response analysis by
25 downsampling to occipital, temporal and parietal electrodes. Yet, this practice can inflate false-
26 positive statistical inferences, by homogenizing them based on potentially circular selection
27 criteria (Kriegeskorte et al., 2009; Simons, 2011). Conducting analyses as agnostically as possible,
28 with respect to the spatial distribution of the FI response, can maximize the available degrees of
29 freedom, for instance by retaining all channels in analyses or conducting whole-brain analyses
30 (Kriegeskorte et al., 2009; Simmons, et al., 2011; Veldkamp et al., 2017; Stacchi et al., 2019),
31 rather than focusing on a selection of occipito-temporal ROIs.
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1We appreciate the reasoning that “fluctuations of amplitude outside of the ROIs associated
2with the maximal response may be essentially due to signal leakage, which is problematic for
3MVPA”. However, we contest the view that limited reliability observed for “outside-ROIs” should
4be interpreted as “no evidence that they could account for significant modulations of the FI
5response” (Rossion et al., 2020). In our opinion, combining systematically varied task- and
6stimulus-related attributes with electrode-agnostic analyses could unveil important individual
7differences in the FI response’s spatial distribution. This is particularly important if cognitive
8processes involved in or contributing to the FI response occur at different cortical loci.
9Expanding on the volume of previous work, we anticipate that with increasing,
10 independent usage of FPVS, individual differences in FI responses could provide crucial insights
11 into brain-behavior relationships. Systematically characterizing the relationship between behavior
12 and (task/stimulus related modulation) of FI with a more agnostic view of its spatial distribution,
13 we aim to examine the sub-processes of face cognition holistically - in the same way that FI is
14 conceptualized to proceed.
15 ACKNOWLEDGEMENT
16 MR is supported by a Swiss National Science Foundation PRIMA (Promoting Women in
17 Academia) grant (PR00P1_179872).
18 CONFLICT OF INTEREST
19 The authors declare no conflicts of interest.
20 ORCID
21 Jeffrey Daniel Nador: https://orcid.org/0000-0002-6502-7226
22 Meike Ramon: https://orcid.org/0000-0001-5753-5493
23 REFERENCES
24 Benton, A. L., & Van Allen, M. W. (1968). Impairment in facial recognition in patients with
25 cerebral disease. Transactions of the American Neurological Association, 93, 38–42.
26 Campbell A, Louw R, Michniak E, Tanaka JW (2020). Identity-specific neural responses to three
27 categories of face familiarity (own, friend, stranger) using fast periodic visual stimulation.
28 Neuropsychologia, 141, 107415.
29 Duchaine, B.C., & Nakayama, K. (2006). The Cambridge face memory test: Results for
30 neurologically intact individuals and an investigation of its validity using inverted face stimuli
31 and prosopagnosic participants. Neuropsychologia, 44, 576-585.
Accepted Article
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1Dzhelyova, M. Schiltz, C., & Rossion, B. (2020). The relationship between the benton face
2recognition test and electrophysiological unfamiliar face individuation response as revealed
3by fast periodic visual Stimulation. I-Perception.
4Kriegeskorte, N., Simmons, W. K., Bellgowan, P. S., & Baker, C. I. (2009). Circular analysis in
5systems neuroscience: the dangers of double dipping. Nature neuroscience, 12(5), 535.
6Quek GL, Rossion B & Liu-Shuang J (2020). Critical information thresholds underlying
7concurrent face recognition functions. bioRxiv 2020.06.22.163584; doi:
8https://doi.org/10.1101/2020.06.22.163584
9Ramon M (2018). The power of how – lessons learned from neuropsychology and face processing.
10 Cognitive Neuropsychology, 35, 83-6.
11 Ramon M, Sokhn N & Caldara R (2019). Decisional space modulates visual categorization –
12 evidence from saccadic reaction times. Cognition, 186, 42-9.
13 Ramon, M., & Gobbini, M. I. (2018). Familiarity matters: A review on prioritized processing of
14 personally familiar faces. Visual Cognition, 26(3), 179–195.
15 Rossion B, Retter TL, Liu-Shuang J (2020). Understanding human individuation of unfamiliar
16 faces with oddball fast periodic visual stimulation and electroencephalography. Eur J
17 Neurosci, 10.1111/ejn.14865.
18 Simmons, J. P., Nelson, L. D., & Simonsohn, U. (2011). False-positive psychology: Undisclosed
19 flexibility in data collection and analysis allows presenting anything as significant.
20 Psychological science, 22(11), 1359-1366.
21 Stacchi, L., Liu-Shuang, J., Ramon, M., & Caldara, R. (2019). Reliability of individual differences
22 in neural face identity discrimination. NeuroImage, 189, 468–475.
23 Veldkamp, C. L. S., Bakker, M., van Assen, M. A. L. M., Crompvoets, E. A. V., Ong, H. H.,
24 Soderberg, C. K., & Wicherts, J. M. (2017). Restriction of opportunistic use of researcher
25 degrees of freedom in pre-registrations on the Open Science Framework. Preprint retrieved
26 from https://psyarxiv. com/g8cjq.
27 Verosky, S.C., Zoner, K.A., Marble, C.W., Sammon, M.M., & Babarinsa, C.O. (2020).
28 Familiarization increases face individuation measured with fast periodic visual stimulation.
29 Biol Psychol. 153:107883.
30 Yan X, Zimmermann FG, Rossion B (2020). An implicit neural familiar face identity recognition
31 response across widely variable natural views in the human brain. Cogn Neurosci, 11, 143-
32 156.
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1Zimmermann FGS, Yan X, Rossion B (2019). An objective, sensitive and ecologically valid
2neural measure of rapid human individual face recognition. R Soc Open Sci, 6, 181904.
Accepted Article
... Fig. 3). Crucially, individuals' performances across independent tests should never be estimated by simple raw score summation (see Nador and Ramon, 2021). ...
... Going one step further, if the goal is to characterize differences between typical and high-performing Super-Recognizers (Ramon, 2021), the analytical focus has yet again to change. For instance, in some scenarios, inter-observer differences in ability may be reflected more accurately via measured differences in performance consistency (e.g., Nador et al., 2021). It is clear that such questions may require the development of novel analytical procedures (e.g., Nador et al., 2021b). ...
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Understanding human individuation of unfamiliar faces with oddball fast periodic visual stimulation and electroencephalography
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