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Serial Editor Vincent Walsh Institute of Cognitive Neuroscience University College London 17 Queen Square London WC1N 3AR UK Editorial Board Mark Bear, Cambridge, USA Medicine & Translational Neuroscience Hamed Ekhtiari, Tehran, Iran Addiction Hajime Hirase, Wako, Japan Neuronal Microcircuitry Freda Miller, Toronto, Canada Developmental Neurobiology Shane O’Mara, Dublin, Ireland Systems Neuroscience Susan Rossell, Swinburne, Australia Clinical Psychology & Neuropsychiatry Nathalie Rouach, Paris, France Neuroglia Barbara Sahakian, Cambridge, UK Cognition & Neuroethics Bettina Studer, Dusseldorf, Germany Neurorehabilitation Xiao-Jing Wang, New York, USA Computational Neuroscience Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, USA First edition 2016 Copyright # 2016 Elsevier B.V All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein) Notices Knowledge and best practice in this field are constantly changing As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein ISBN: 978-0-444-63698-0 ISSN: 0079-6123 For information on all Elsevier publications visit our website at https://www.elsevier.com/ Publisher: Zoe Kruze Acquisition Editor: Kirsten Shankland Editorial Project Manager: Hannah Colford Production Project Manager: Magesh Kumar Mahalingam Cover Designer: Greg Harris Typeset by SPi Global, India Contributors D Ansari Numerical Cognition Laboratory, University of Western Ontario, London, ON, Canada I Berteletti University of Illinois at Urbana–Champaign, Champaign, IL, United States B Butterworth Institute of Cognitive Neuroscience, University College London, London, United Kingdom; Melbourne School of Psychological Sciences, University of Melbourne, Melbourne, Vic, Australia; Research Center for Mind, Brain, and Learning, National Chengchi University, Taipei, Taiwan R Cohen Kadosh University of Oxford, Oxford, United Kingdom M.D de Hevia Universit e Paris Descartes, Sorbonne Paris Cite; Laboratoire Psychologie de la Perception, CNRS UMR 8242, Paris, France B De Smedt Parenting and Special Education Research Unit, Faculty of Psychology and Educational Sciences, University of Leuven, Leuven, Belgium A De Visscher Psychological Sciences Research Institute, Universite catholique de Louvain (UCL), Louvain-la-Neuve, Belgium M D’Onofrio Universit a degli Studi di Roma ‘Sapienza’; Fondazione Santa Lucia IRCCS, Rome, Italy F Doricchi Universit a degli Studi di Roma ‘Sapienza’; Fondazione Santa Lucia IRCCS, Rome, Italy E Eger INSERM Cognitive Neuroimaging Unit, NeuroSpin Center, CEA DSV/I2BM, Universit e Paris-Sud, Universit e Paris-Saclay, Gif/Yvette, France E Fattorini Universit a degli Studi di Roma ‘Sapienza’; Fondazione Santa Lucia IRCCS, Rome, Italy D.C Geary University of Missouri, Columbia, MO, United States T Hinault Aix-Marseille University & CNRS, Marseille, France v vi Contributors D.C Hyde University of Illinois at Urbana–Champaign, Champaign, IL, United States T Iuculano Stanford Cognitive and Systems Neuroscience Laboratory, Stanford University School of Medicine, Palo Alto, CA, United States V Karolis Institute of Psychiatry, Psychology and Neuroscience, King’s College London, London, United Kingdom P Lemaire Aix-Marseille University & CNRS, Marseille, France C.Y Looi University of Oxford, Oxford, United Kingdom I.M Lyons Numerical Cognition Laboratory, University of Western Ontario, London, ON, Canada V Menon Stanford Cognitive and Systems Neuroscience Laboratory, Palo Alto, CA S Merola Universit a degli Studi di Roma ‘Sapienza’; Fondazione Santa Lucia IRCCS, Rome, Italy A.M Moore University of Missouri, Columbia, MO, United States Y Mou University of Illinois at Urbana–Champaign, Champaign, IL, United States M.-P Noeăl Psychological Sciences Research Institute, Universite catholique de Louvain (UCL), Louvain-la-Neuve, Belgium M Pinto Universit a degli Studi di Roma ‘Sapienza’; Fondazione Santa Lucia IRCCS, Rome, Italy D Szu˝cs University of Cambridge, Cambridge, United Kingdom K Vanbinst Parenting and Special Education Research Unit, Faculty of Psychology and Educational Sciences, University of Leuven, Leuven, Belgium S.E Vogel University of Graz, Graz, Austria Preface Mathematical proficiency is essential for social life (eg, sharing a bill at a restaurant), health (eg, examining whether your blood pressure is too high), and work (eg, calculating your salary) among other things This is why mathematical abilities have been widely studied in the last three decades, from babies to monkey, from congenital and acquired pathology to intervention, from child to elderly This volume aims to provide a comprehensive and critical overview of the mathematical brain across the life span, with an emphasis on learning and on the impact of intervention Two main questions are put to scrutiny: first, what are the numerical and nonnumerical abilities that support the development and the maintenance of mathematical abilities in the lifetime Two views will be presented, one promoting the idea that mathematical abilities are “core,” innate skills, based on the approximate number system (ANS) and suggesting that they are predominantly independent from other cognitive abilities; the other view highlighting the intrinsic and critical role of language, working memory, and cognitive control functions in the development, pathology, as well as normal functioning of mathematical abilities The second question addressed in this volume is to what extent mathematical abilities are trainable, and if so what exactly can be trained, what are the neuronal correlates of learning, whether training can be a valuable option for developmental disorders of maths, and how it relates to education Evidence suggesting the promising possibility to improve some numerical abilities will be presented, albeit leaving open the question of the generalizability of the training effects A clear and comprehensive introduction to numerical abilities in terms of the ANS is offered by Eger’s chapter on the neuronal foundations of human numerical representations, which emphasizes how basic numerical principles are shared across species and ages The more specific forms a “core” number system can take are presented in terms of a computational approach in Butterworth and Karolis’ chapter This is the starting ground against which the discussion of numerical abilities across the life span unfolds De Hevia provides a detailed account of the core number abilities in infants, while Geary and Moore complement this view discussing the importance of domain-general abilities and how they may interact with the core ones at the early stages of development From infancy to childhood, De Smedt discusses individual variability in children’s mathematical abilities, and focuses specifically on how symbols are progressively linked to magnitudes, and on the role of domain-general functions like working memory, executive control, and language The importance of memory skills, especially in learning arithmetic problems, is then discussed by De Visscher and Noeăl at the cognitive level, and by Menon at the neuronal level A different aspect of magnitude processing is discussed in two chapters focusing on adulthood Lyons and colleagues introduce ordinality as an important type of xv xvi Preface numerical function, besides magnitude Doricchi and colleagues add to this the discussion of how number abilities are associated with space Hinault and Lemaire then discuss the role of executive control in arithmetical abilities, with a focus on aging The role of domain-general abilities is subsequently discussed in two chapters focusing on mathematical disabilities and dyscalculia by Szucs and Iuculano, respectively The importance of intervention programs in dyscalculia leads to the two final chapters by Hyde and colleagues, and by Looi and Kadosh, respectively, discussing training programs in terms of the ANS in early development and comparing mathematical training of core and noncore skills Overall, this volume addresses open questions and controversial issues in mathematical cognition across the life span, and it offers an overview of the promising new avenue of learning to both improve and better characterize mathematical cognition itself Marinella Cappelletti Wim Fias CHAPTER Neuronal foundations of human numerical representations E Eger1 INSERM Cognitive Neuroimaging Unit, NeuroSpin Center, CEA DSV/I2BM, Universit e Paris-Sud, Universit e Paris-Saclay, Gif/Yvette, France Corresponding author: Tel.: +33-1-69 08 19 06; Fax: +33-1-69 08 79 73, e-mail address: evelyn.eger@gmail.com Abstract The human species has developed complex mathematical skills which likely emerge from a combination of multiple foundational abilities One of them seems to be a preverbal capacity to extract and manipulate the numerosity of sets of objects which is shared with other species and in humans is thought to be integrated with symbolic knowledge to result in a more abstract representation of numerical concepts For what concerns the functional neuroanatomy of this capacity, neuropsychology and functional imaging have localized key substrates of numerical processing in parietal and frontal cortex However, traditional fMRI mapping relying on a simple subtraction approach to compare numerical and nonnumerical conditions is limited to tackle with sufficient precision and detail the issue of the underlying code for number, a question which more easily lends itself to investigation by methods with higher spatial resolution, such as neurophysiology In recent years, progress has been made through the introduction of approaches sensitive to within-category discrimination in combination with fMRI (adaptation and multivariate pattern recognition), and the present review summarizes what these have revealed so far about the neural coding of individual numbers in the human brain, the format of these representations and parallels between human and monkey neurophysiology findings Keywords Number representation, fMRI, Parietal cortex, Adaptation, Multivariate decoding INTRODUCTION High-level numerical abilities appear at the heart of many inventions of technologically advanced human societies It is, therefore, not surprising that a substantial amount of neuroscientific effort is dedicated to understanding what a “number” is for the human brain Answering this question is made complex in the first place Progress in Brain Research, Volume 227, ISSN 0079-6123, http://dx.doi.org/10.1016/bs.pbr.2016.04.015 © 2016 Elsevier B.V All rights reserved CHAPTER Neuronal foundations of human numerical representations by the multiple meanings in which we use the term number: in its most basic sense, “number” refers to a property characterizing any set of concrete objects, such as its cardinality (numerosity) Humans, nonhuman primates, and many other animals share the ability to rapidly extract and compare the numerosity of sets of objects in an approximate fashion, and the behavior of both human and nonhuman primates in such tasks is characterized by Weber’s law: the accuracy with which the numerosity of two sets of items can be discriminated depends linearly on their ratio, at least over an intermediate range of (not too small and not too large) numerosities (eg, Cantlon and Brannon, 2006; Piazza et al., 2004) It has been suggested that numerosity is not a mere abstract concept but a perceptual property, since it is subject to adaptation after-effects in a similar way as other visual features, for example, orientation, color, motion (Burr and Ross, 2008).a Numerosity, however, is a more complicated property in the sense that it is not bound to any single input modality or presentation mode, and the way it is extracted by sensory systems is far less understood than it is for the other features mentioned Interestingly, perceptual adaptation to numerosity can occur across changes in sensory modality (visual, auditory) and presentation mode (simultaneous vs sequential) (Arrighi et al., 2014), suggesting that the neuronal populations coding for it within each modality may be at least intricately connected, if not feeding into a common representation The second meaning of the term “number” is an abstract mathematical object referred to by symbols and used to count, measure, or rank virtually everything Although this might appear quite removed from the perceptual property of numerosity, a lot of evidence has accumulated to show that across the whole lifespan, in humans there exists a profound link between the capacity to enumerate/compare concrete sets and more abstract numerical/mathematical abilities: behavioral performance for distinguishing two symbolic numerals, although usually more precise overall than the one to distinguish two nonsymbolic numerical stimuli, is less precise and more slow for numerical quantities separated by a smaller ratio, suggesting that the system for comparing the numerical magnitude of symbols is inheriting parts of its metric from the processing of nonsymbolic numerical input (Buckley and Gillman, 1974; Dehaene et al., 1990) Interindividual differences in the precision with which numerosity is discriminated can be correlated with, and even longitudinally predictive of children’s success in symbolic skills such as numerical comparison and calculation (eg, Gilmore et al., 2007; Halberda et al., 2008), even though sensitivity to numerosity is not necessarily the only significant predictor and also other visuospatial abilities (eg, sensitivity to orientation) have been found to correlate with mathematical performance (Tibber et al., 2013) In some children suffering from dyscalculia, the capacity to discriminate visual numerosity can be strongly impaired with respect to age and intelligence matched controls (eg, Mazzocco et al., 2011; Mussolin et al., 2010; Piazza et al., 2010), and interestingly, the a After prolonged exposure (adaptation) to a given numerosity, a set of items of smaller numerosity than the one adapted to is perceived as smaller than its actual value and the opposite for a larger one A core numerical representation in parietal cortex impairment seems to be mainly related to situations where other properties of the stimuli such as, for example, size or area covered provide incongruent magnitude information and have to be discarded to extract a rather abstract representation of cardinality (Bugden and Ansari, 2015; Szucs et al., 2013) Training on approximate additions and subtractions of dot numerosities appeared to have positive transfer effects onto performance in symbolic numerical tasks (Park and Brannon, 2013), while reciprocally, learning symbols for number and/or learning to count has been suggested to enhance the precision of visual numerosity discrimination (Piazza et al., 2013) However, other studies did not find a relation between nonsymbolic and symbolic numerical skills (see, eg, De Smedt et al., 2013, for a review), it has been observed that the relation between numerosity discrimination capacities and mathematical skills is weaker than other relations, for example, the one between symbolic comparison and calculation (Schneider et al., 2016), and some developmental studies did not find a relation between nonsymbolic processing capacities and acquisition of numerical symbols (Sasanguie et al., 2014) Taken together, even though no definitive consensus has been achieved, there is some evidence to suggest that the cognitive systems for processing nonsymbolic numerical input and more abstract (symbolic) numerical concepts may share some common resources This raises the questions of whether and how in the human brain the representations of nonsymbolic and symbolic numerical information may be linked, and what is the nature of the neuronal code of numerical magnitude The present review will give an overview of neuroscientific findings related to the underpinnings of numerical representations in humans, with a particular focus on functional imaging methods Starting by outlining the regions that have emerged as important substrates of numerical processing and placing them into the context of the more general functional neuroanatomy, the review will then focus on what techniques providing enhanced sensitivity to finer-scale brain representations in combination with fMRI have so far revealed about some crucial stages of the representation of individual numerical magnitudes within these key regions A CORE NUMERICAL REPRESENTATION IN PARIETAL CORTEX 2.1 NUMERICAL PROCESSING AND THE COARSE SCALE FUNCTIONAL NEUROANATOMY OF PARIETAL CORTEX Long before the introduction of functional brain imaging methods, neuropsychology had already demonstrated that damage to preferentially left-sided parts of the parietal lobe can result in profound deficits in calculation and other tasks requiring to represent and manipulate numerical information (eg, Cipolotti et al., 1991; Dehaene et al., 1998) Since then, the implication of parts of the parietal (and frontal) lobes in different numerical tasks has been studied extensively with fMRI Synthesizing findings from neuropsychology and early fMRI studies, it has been hypothesized that CHAPTER Neuronal foundations of human numerical representations central parts of the human intraparietal sulcus (IPS) constitute a key node for the abstract representation of numerical magnitude (Dehaene et al., 2003) Intraparietal cortex is recruited during a wide range of symbolic and nonsymbolic numerical tasks and is one of the most consistently activated regions in a recent metaanalysis of fMRI studies of numerical processing, both for nonarithmetic and arithmetic tasks (Arsalidou and Taylor, 2011), see Fig 1A As part of high-level association cortex, the IPS is endowed with a rather complex functionality beyond the domain of numerical cognition This includes, for example, spatial and action-related aspects of perception (Culham and Valyear, 2006), multisensory, and sensory-motor integration Sensory-motor integration is achieved within a series of spatial field maps which are characterized by coding for space by a progression of reference frames (see, eg, Hubbard et al., 2005; Sereno and Huang, 2014) Superior parts of the intraparietal cortex further play a crucial role B A AS PS CS IPS 30 20 Addition Subtraction Multiplication 10 % Number responsive neurons FIG Cortical regions important for numerical processing in the human and macaque monkey brain (A) Overview of regions revealed by a recent metaanalysis of human fMRI studies of numerical processing, separately for nonarithmetic tasks (top) and arithmetic tasks (bottom), in that case color coding separately different types of arithmetic operations (B) Overview of regions of the macaque monkey brain where different percentages of numerically selective neurons have been found during delayed match-to-sample tasks with visual numerosities While the similar regions found across the two species suggests a close homology, it is important to bear in mind that rather different kinds of comparisons provided the basis for the different figures: discrimination within dimension (between individual numerosities) in the case of the neurophysiological findings, and in most cases subtractions between numerical and nonnumerical control conditions in the fMRI findings, where controls differed not only in the type of stimulus but also different instrumental processes recruited Panel (A) Adapted 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J Learn Disabil 34 (3), 237–248 http://dx.doi org/10.1177/002221940103400304 Witt, M., 2011 School based working memory training: preliminary finding of improvement in children’s mathematical performance Adv Cogn Psychol 7, 7–15 Wood, G., Nuerk, H.-C., Willmes, K., 2006 Neural representations of two-digit numbers: a parametric fMRI study Neuroimage 29 (2), 358–367 Xu, F., Arriaga, R.I., 2007 Number discrimination in 10-month-old infants Br J Dev Psychol 25, 103–108 Xu, F., Spelke, E.S., 2000 Large number discrimination in 6-month-old infants Cognition 74 (1), B1–B11 Zacks, J.M., 2008 Neuroimaging studies of mental rotation: a meta-analysis and review J Cogn Neurosci 20 (1), 1–19 Zamarian, L., Ischebeck, A., Delazer, M., 2009 Neuroscience of learning arithmetic— evidence from brain imaging studies Neurosci Biobehav Rev 33 (6), 909–925 Zorzi, M., Butterworth, B., 1999 A Computational Model of Number Comparison Erlbaum, Mahwah, NJ Index Note: Page numbers followed by “f ” indicate figures, “t” indicate tables, “b” indicate boxes, and “np” indicate footnotes A Accumulator, 30–31, 30f model, 77 Age in arithmetic, strategic variations, 259–270 Age-related changes in strategic variations, 259–260 in strategy distribution, 259–260 in strategy execution, 260 Age-related differences, during arithmetic problem solving, 258–260 Aging, 261–270 Aging effects on arithmetic performance, 258–259 on sequential modulations, 268–269 on strategy selection, 260 Analogue magnitude system (AMS) computational models, 43 construction, 42 integrator, 41–42 “leaky” accumulator, 42, 43f log vs linear scale, 46–48 method, 44 numerical consequences, 40 stochastic cascades, 46, 47f utility, 41 Angular gyrus (ANG), 263–264 ANS See Approximate number system (ANS) Anterior cingulate cortex (ACC), 263–264 Approximate number system (ANS), 77, 79, 336–337, 343, 355–356 acuity, 80–81, 83 vs and mathematical achievement, 84–86, 84f brain system, 79–80 development, 80–83, 82f vs mathematics, 339–342, 340–341f in symbolic mathematics, 55 vs symbolic number system, 344–347, 345f Arithmetical computation, on numerosities, 57–59 Arithmetic fact retrieval, 111, 114, 118–122 Arithmetic facts defined, 132 learning/retrieving deficit, 143–145 Arithmetic facts network atypical development, 141–150 models, 132–136 similarity interference in arithmetic facts, brain-imaging evidence, 140–141 through development, 136–140 typical development, 132 Arithmetic in children, neurocognitive development, 108–110 Arithmetic performance, aging effects on, 258–259 Arithmetic problem solving age-related differences during, 258–260 memory and cognitive control circuits, 164–166, 168f, 176–179, 177f Arithmetic strategy selection, 262–264 use, 261–262 Attention, 354, 358, 360–361 Attentional SNARC (Att-SNARC) CTIs, 227–229 eight-digit cues, 236–238, 237f ERPs, 228–229 four-digit cues, 233–236, 235f MC-SNARC effects, 230–231 psychophysical studies, 230 task demands, 232–233 Attention-deficit hyperactivity disorder, 360–361 B Backward Corsi span test, 281–284 Baddeley memory model, 113–114, 116, 279–280 Binomial accumulator, 36–37, 36f Blood oxygenation level-dependent (BOLD) response, 365 Brain numerical order in, 196–198 ordinal and cardinal processing in, 190–191, 191f stimulation, 363–366 and mathematical training, 366–373 Brain plasticity, 305–306, 311, 317–320 Brown–Peterson task, 145–146 C Campbell’s theory, 151 Canonical distance effects, 192–193, 192np, 198–199 389 390 Index Cardinality, 188–189, 189f vs ordinality, 190–195 Cardinal processing in brain, 190–191, 191f distance effects, 192–194 Children with dyscalculia, 119–120 formal mathematical development, 86 neurocognitive development of arithmetic in, 108–110 numerical capacity development, 77 Cognitive control systems dynamic hippocampal–frontal control signals, 176–179 dynamic parietal–frontal control signals, 176 flexible hubs, 175 schematic diagram, 161f Cognitive determinants, 106 Cognitive factors, 343–344 Cognitive flexibility, 261 Cognitive foundation, for mathematical abilities, 336–344, 336f Cognitive function, complex, 53–54 Cognitive load theory, 92 Cognitive study, DD, 316–317 Cognitive system, nonverbal, 77 approximate number system, 77, 79–83 numerical representation, 54–57 object tracking system, 77–79 Cognitive training, 339–340, 354–355, 362 Complex cognitive functions, 53–54 Complex ordinal process, in nonhuman animals, 200–203 Congruency effects, 267 Core and noncore parietal systems, 162 Core skills, 354–358, 361–362 See also Noncore skills Cross-modal numerosity, 13 Cue target intervals (CTIs), 227–228 Cytoarchitectonic maps, 164 D DD See Developmental dyscalculia (DD) Declarative memory systems, 160 See also Hippocampal–frontal declarative memory system Developmental dyscalculia (DD), 142, 278 brain plasticity, 305–306, 311, 317–320 cognitive impairments, 305–306 domain-general processing deficits, 310–311 dorsal and ventral streams deficits, 312–313 etiology, 321–322 frontoparietal deficits, 313–314 math anxiety, 311 memory deficits, 308–309 MTL deficits, 314–315 network-level deficits, 315–316 number sense deficits, 307–308 numerical mapping deficits, 309–310 ordinality deficits, 309–310 remediation embodied intervention, 320–321 intervention outcomes, 319 neuroimaging studies, 317–319 pedagogical and cognitive studies, 316–317 systems neuroscience, 319, 321–322 vs typically developing children, 308–309 Distance effects canonical, 192–193, 192np, 198–199 classic, 192np ordinal and cardinal process, 192–194 reverse, 192, 198–199 sensitivity of, 193–194 DLPFC See Dorsolateral prefrontal cortex (DLPFC) Domain-general abilities executive functions and working memory, 88–90 mathematics learning, 86–87 potential evolutionary mechanisms, 90–93, 91f Domain-general cognitive factors, 117 Domain-general processing deficits, 310–311 Domain-specific models, 116 Dominant theory, 110–111 Dorsal and ventral streams deficits, 312–313 Dorsolateral prefrontal cortex (DLPFC), 263–264 Doubly stochastic process, 38–39, 39f Dyscalculia, 2–3, 106, 119–120, 142, 149–150 See also Developmental dyscalculia (DD) Dyslexia, 118–120 E Embodied intervention, DD, 320–321 ENS See Exact number systems (ENS) Event related potentials (ERPs), 193 modulations of, 269 Exact number systems (ENS), 355–358 Executive control process brain imaging data, 263–264 correlational data, 262 experimental data, 262–263 sequential modulations of poorer strategy effects, 267–270, 268f in strategic variations, 261 strategy sequential difficulty effects, 265–267, 266f Index strategy switch costs, 264, 265f Executive control, role, 261–270 Executive functions (EFs), 354, 358–359 and intelligence, 86–88 and mathematics learning, 87–88 task, 90 and working memory, 88–90 F Fact retrieval arithmetic, 111, 114, 118–122 description, 108–110, 114–115, 118–119 phonological processes during, 109 Feature-matching process, 134 Feature overlap theory, 137–138 Fine-scale neuronal representation, with fMRI, 9b Fine-scale representation of numerical information fMRI in humans, 8–15 macaque neurophysiology, 6–8 Fluid intelligence, 89, 91–92 fMRI See Functional magnetic resonance imaging (fMRI) Formal mathematical development, children, 86 Frequency effect, 133 Frontal–parietal network, 92–93 Frontoparietal deficits, 313–314 Functional magnetic resonance imaging (fMRI) adaptation technique, 9, 10f, 14–15 fine-scale neuronal representations with, 9b fine-scale representation of numerical information, 8–15 pattern recognition methods, 12f, 14–15 G The Graphogame-Math, 317 H Hippocampal–frontal declarative memory system, 171 children’s mathematical skill development, 173–175 coactivation, 171–173, 172f longitudinal developmental changes, 171, 172f medial temporal lobe, 169–171, 174f role, 179 schematic diagram, 161f ventrolateral and dorsolateral PFC, 176–179, 178f Human development, ordinal processing in, 204–206 Human intraparietal cortex, numerical stimuli in, 10f Hyperactive parietal–frontal working memory circuits, 167–169 Hyperactivity disorder, attention-deficit, 360–361 Hypersensitivity-to-interference, in memory hypothesis, 139–140 dyscalculia, 149–150 group study, 148–149 single-case study, 145–147 I Imagery attentional SNARC eight-digit cues, 238–241, 240f four-digit cues, 238–241, 239f Infants arithmetical computations on numerosities, 57–59 mappings across quantitative dimensions, 60–65, 62–63f numerical abilities, 59 spatially oriented representation of number in, 65–67, 66f, 68f Influential neurocognitive model, 118 Inhibition, 261 Inhibitory control, 359 Intelligence, 88, 90 executive functions and, 86–88 fluid, 89, 91–92 verbal, 93 Intel Pentium PC running E-Prime software, 233 Interference effect, 140 modulation of brain by, 140–141, 141f Interference parameter, 137–140, 139f, 147–149, 151 Interference theory, similarity-based, 137 Intraparietal cortex multifaceted functionality of, numerical stimuli in, 10f Intraparietal sulcus (IPS), 4–5, 79–80, 82–83, 109, 160, 179, 336, 336f, 356–357 activity, 106–107, 109 preferential activation of, Introspective number forms, 224–225 Intuitions of number, 76–83 IPS See Intraparietal sulcus (IPS) IQ testing, MLD, 287–295, 288f Item working memory (IWM), 208 L Lateral intraparietal area (LIP), 6–7, 16–19, 21 Lateralized response potentials (LRP), 226–227 Learning arithmetic facts, difficulties in, 107 Learning order effect, 135–136 Linear accumulator models, 39–40 LIP See Lateral intraparietal area (LIP) Long-term memory, 132, 136–137, 259–260 391 392 Index M Macaque neurophysiology, 6–8 Magnitude attentional SNARC eight-digit cues, 246–249 four-digit cues, 244–246, 247f Magnitude-based mechanism, ordinal processing, 207–208 Math anxiety, 311 Mathematical abilities, cognitive foundations for, 336–344, 336f Mathematical achievement, 354 Mathematical cognition, 354–355, 359, 362 neural bases of, 362 typical and atypical, 373–374 Mathematical disability (MD) aberrant parietal–frontal response, 167, 170f IPS, 166 parietal hyperconnectivity, 167–169, 170f Mathematical learning disability (MLD), 84, 86, 355–356, 360–361, 367–368 ANCOVA, 281–284, 298 vs control groups, 291 developmental pathways, 292–293, 292f potential memory impairments, 295–297, 296f processing networks, 295 study data analysis ability-matched young controls, 294–295 characteristics, 281, 282t matching reading and IQ, 287–295, 288f mean effect sizes, 287, 289t power, 284–286, 285f standardized effect sizes, 287, 288f, 299 subtypes, 295–299 task difficulty, 295 vs typically developing children, 279 verbal and visual memory deficits, 281–284 WM models, 279–280 Mathematical training, 354np brain stimulation and, 366–373 Mathematics achievement, 84–86, 88, 93, 106, 338–339, 343 Mathematics learning, 88–90 in evolutionary context, 86–87 Mathematics vs approximate number system, 339–342, 340–341f MD See Mathematical disability (MD) Medial temporal lobe (MTL) deficits, 314–315 Memory-based problem-solving strategies, 171, 172f, 173 Memory deficits, 308–309 Memory hypothesis hypersensitivity-to-interference in, 139–140 dyscalculia, 149–150 group study, 148–149 single-case study, 145–147 Mental number line (MNL), 54, 224–229, 251–252 Metaphorical thinking, 60–61 MLDs See Mathematical learning disabilities (MLDs) MNL See Mental number line (MNL) Model-based clustering approach, 111–112 Modulation of brain by interference effect, 140–141, 141f of ERPs, 269 Multiple cognitive functions, 306 Multiple distributed neural processes, 160 Multiple parietal–frontal working memory circuits, 162–164 IPS, 179 schematic diagram, 161f, 163f Multiplication verification task, 140 Multivariate decoding, 11–15 Multivariate pattern recognition, 9, 11–13 Multivariate searchlight analysis, 11–13 Multivoxel pattern analysis approach, 196–197 Multivoxel response pattern, 12f N NDE See Numerical distance effect (NDE) Network-level deficits, 315–316 Network retrieval models, 133 Neural distance effect, 313–314 Neurocognitive development of arithmetic, 108–110 Neurocognitive model, influential, 118 Neurocognitive systems and DD brain plasticity deficits, 311 dorsal and ventral streams deficits, 312–313 frontoparietal deficits, 313–314 MTL deficits, 314–315 network-level deficits, 315–316 Neuroimaging studies, DD, 317–319 NIBS See Noninvasive brain stimulation (NIBS) N-methyl-D-aspartate (NMDA) receptors, 364 Noise characteristics binomial accumulator, 36–37, 36f doubly stochastic accumulator, 38–39, 39f Poisson accumulator, 37, 38f Noncore skills, 354, 358–362 Nonhuman animals, complex ordinal processing in, 200–203 Noninvasive brain stimulation (NIBS), 354–355, 363, 366–367 Nonnumerical ordinality, 199–200 Nonsymbolic numerical skills, 2–3 Nonsymbolic numerical stimuli, Nonsymbolic ordinal process, 194–195 Index Nonsymbolic stimuli, 11, 197 Nonverbal cognitive systems, 77 approximate number system, 77, 79–83 object tracking system, 77–79 Nonverbal numerical representation, cognitive system for, 54–57 Number from symbols, extraction of, 19–21 Number learning and representation model, 31, 32f The Number Race software, 317 Number sense, 54–55 deficits, 307–308 Number synestheses, 224–225 Numerical abilities, 338–339 high-level, 1–2 infants, 59 Numerical capacity development, children, 77 Numerical distance effect (NDE), 87 Numerical magnitude process role of, 113–114 symbolic, 110–113, 120 Numerical magnitudes, 337–338 Numerical mapping deficits, 309–310 Numerical order, 195 in brain, 196–198 Numerical ordinality, 188–189, 189f, 199 Numerical ordinal processing mechanisms, 206–207 magnitude-based mechanisms, 207–208 serial-order WM, 208–209 spatial mechanisms, 209–211 Numerical problem solving, 160, 162, 165–169, 176, 179 Numerical representation, triple-code model, 19–21 Numerical skills nonsymbolic, 2–3 symbolic, 2–3 Numerical stimuli in human intraparietal cortex, 10f nonsymbolic, symbolic, 11 Numerons, 30–31 Numerosity from concrete sets of objects, 15–19, 18f cross-modal, 13 detection system model, 31–32, 33f extraction of, 16, 18f infants’ arithmetical computations on, 57–59 visual, O Object-tracking system (OTS), 55–58, 77–79 Operational Overlap Hypothesis, 344–345 Order working memory (OWM), 208 Ordinality, 188–189, 189f vs cardinality, 190–195 and implications, 212–215 Ordinal number deficits, 309–310 Ordinal processing acquisition and access of ordinal associations, 211–212 in brain, 190–191, 191f distance effects, 192–194 in human development and learning, 204–206 magnitude-based mechanisms, 207–208 in nonhuman animals, 200–203 serial-order WM, 208–209 spatial mechanisms, 209–211 symbolic vs nonsymbolic, 194–195 OTS See Object-tracking system (OTS) OWM See Order working memory (OWM) P Pana-Math program, 84 Parietal cortex, 109 numerical processing and coarse scale functional neuroanatomy of, 3–6, 4f Parietal–frontal working memory systems core and noncore parietal systems, 162 mathematical cognition, 164–166 multiple circuits (see Multiple parietal–frontal working memory circuits) PFC control signals, 176, 177f Parieto-frontal integration theory (P-FIT), 91–92 Pedagogical studies, DD, 316–317 Persistent DD, 319–320 PFC See Prefrontal cortex (PFC) P-FIT See Parieto-frontal integration theory (P-FIT) Phonological loop, 89, 114–115 Phonological process, 118–120 during fact retrieval, 109 skills, 118–119 PMd, 190, 197np PMv, 197np, 198 Poisson accumulator, 37, 38f Poorer strategy effects, sequential modulations of, 267–270, 268f Posterior parietal cortex (PPC), 6–7, 356–357, 371–373 deficits, 312–313, 319–320 PPC See Posterior parietal cortex (PPC) Prefrontal cortex (PFC), 6–7, 109, 356–357 Premotor cortex, 197–198 Presupplementary motor area (pre-SMA), 197–198 Preverbal system accumulator, 30–31, 30f binomial accumulator, 36–37, 36f doubly stochastic accumulator, 38–39, 39f 393 394 Index Preverbal system (Continued) log scaling, 34 monkey intraparietal sulcus, 34 number learning and representation model, 31, 32f numerons, 30–31 numerosity detection system model, 31–32, 33f Poisson accumulator, 37, 38f Proactive interference, 135–138, 145–146, 150 Problem size effect, 132–135, 140–141 Q Quantitative dimensions, 59–60 infants’ mappings across, 60–65, 62–63f R Repetition-related memory phenomenon, Representational Overlap Hypothesis, 346 Reverse distance effects, 192, 198–199 Robust effect, 132, 151–152 S Sequential modulations aging effects on, 268–269 of poorer strategy effects, 267–270, 268f of strategy execution, 265–266 Serial-order WM mechanism, ordinal processing, 208–209 Short-term memory, 136–137 Siegler’s Distribution of Association model, 133–134 Similarity-based interference theory, 137 Similarity interference, arithmetic facts network development, 136–140 Skills core, 354–358, 361–362 noncore, 354, 358–362 nonsymbolic numerical, 2–3 phonological processing, 118–119 spatial, 354, 360 spatial cognitive, 358 symbolic numerical, 2–3 SNARC effect “response-related” interpretation, 226–227 “small/large” codes, 226 spatial codes, 225–226 t-tests, 249–251, 250t working memory, 227 SNS See Symbolic number system (SNS) Spatial attentional SNARC eight-digit cues, 242–244, 245f, 248f four-digit cues, 241–244, 243f Spatial cognitive skills, 358 Spatial mechanism, ordinal processing mechanisms, 209–211 Spatial skills, 354, 360 Spearman–Brown correction, 242, 244, 246, 249 Stimulation brain, 363–366 mathematical training and, 366–373 Strategy, defined, 257–258 Strategy effect, sequential modulations of poorer, 267–270, 268f Strategy execution, sequential modulations of, 265–266 Strategy sequential difficulty effects, 265–267, 266f Strategy switch costs, 264, 265f Stroop Color task, 261–262, 267 Symbolic mathematics, ANS in, 55 Symbolic number system (SNS), 337–338 vs approximate number system, 344–347, 345f Symbolic numerical magnitude processing, 110–113, 120 Symbolic numerical skills, 2–3 Symbolic ordinal process, 194–195 Symbolic stimuli, 11 T tDCS See Transcranial direct current stimulation (tDCS) tES See Transcranial electrical stimulation (tES) TMS See Transcranial magnetic stimulation (TMS) Trail Making Test (TMT), 261–262 Transcranial direct current stimulation (tDCS), 319–320, 363–369 Transcranial electrical stimulation (tES), 354–355, 363 Transcranial magnetic stimulation (TMS), 363 Transcranial random noise stimulation (tRNS), 363–366, 370–373 Triple-code model of numerical representation, 19–21 ventral visual cortex, 20–21 tRNS See Transcranial random noise stimulation (tRNS) V Ventral intraparietal area (VIP), 6–7 Ventral visual cortex, triple-code model, 20–21 Verbal counting, 30, 34–35 See also Preverbal system Verbal intelligence, 93 Verbal memory deficits, MLD, 281–284 Visual memory deficits, MLD, 281–284 Visual numeral area, 19–20 Index Visual numerosities, Visuospatial sketchpad, 89 Visuo-spatial sketchpad, 114–115, 117 W Weber–Fechner law, 79 Weber’s law, 1–2, 8–9, 54–55, 59–60 Working memory (WM), 113–118, 160, 206–207, 359–360 See also Multiple parietal–frontal working memory circuits Baddeley’s multicomponent model of, 113–114 DD, 308–309 executive functions and, 88–90 MLD development, 279–280 synchronization, 92 updating and monitoring of, 261 395 Other volumes in PROGRESS IN BRAIN RESEARCH Volume 167: Stress Hormones and Post Traumatic Stress Disorder: Basic Studies and Clinical Perspectives, by E.R de Kloet, M.S Oitzl and E Vermetten (Eds.) – 2008, ISBN 978-0-444-53140-7 Volume 168: Models of Brain and Mind: Physical, Computational and Psychological Approaches, by R Banerjee and B.K Chakrabarti (Eds.) – 2008, ISBN 978-0-444-53050-9 Volume 169: Essence of Memory, by W.S Sossin, J.-C Lacaille, V.F Castellucci and S Belleville (Eds.) – 2008, ISBN 978-0-444-53164-3 Volume 170: Advances in Vasopressin and Oxytocin – From Genes to Behaviour to Disease, by I.D Neumann and R Landgraf (Eds.) – 2008, ISBN 978-0-444-53201-5 Volume 171: Using Eye Movements as an Experimental Probe of Brain Function—A Symposium in Honor of Jean B€uttner-Ennever, by Christopher Kennard and R John Leigh (Eds.) – 2008, ISBN 978-0-444-53163-6 Volume 172: Serotonin–Dopamine Interaction: Experimental Evidence and Therapeutic Relevance, by Giuseppe Di Giovanni, Vincenzo Di Matteo and Ennio Esposito (Eds.) – 2008, ISBN 978-0-444-53235-0 Volume 173: Glaucoma: An Open Window to Neurodegeneration and Neuroprotection, by Carlo Nucci, Neville N Osborne, Giacinto Bagetta and Luciano Cerulli (Eds.) – 2008, ISBN 978-0-444-53256-5 Volume 174: Mind and Motion: The Bidirectional Link Between Thought and Action, by Markus Raab, Joseph G Johnson and Hauke R Heekeren (Eds.) – 2009, 978-0-444-53356-2 Volume 175: Neurotherapy: Progress in Restorative Neuroscience and Neurology — Proceedings of the 25th International Summer School of Brain Research, held at the Royal Netherlands Academy of Arts and Sciences, Amsterdam, The Netherlands, August 25–28, 2008, by J Verhaagen, E.M Hol, I Huitinga, J Wijnholds, A.A Bergen, G.J Boer and D.F Swaab (Eds.) –2009, ISBN 978-0-12-374511-8 Volume 176: Attention, by Narayanan Srinivasan (Ed.) – 2009, ISBN 978-0-444-53426-2 Volume 177: Coma Science: Clinical and Ethical Implications, by Steven Laureys, Nicholas D Schiff and Adrian M Owen (Eds.) – 2009, 978-0-444-53432-3 Volume 178: Cultural Neuroscience: Cultural Influences On Brain Function, by Joan Y Chiao (Ed.) – 2009, 978-0-444-53361-6 Volume 179: Genetic models of schizophrenia, by Akira Sawa (Ed.) – 2009, 978-0-444-53430-9 Volume 180: Nanoneuroscience and Nanoneuropharmacology, by Hari Shanker Sharma (Ed.) – 2009, 978-0-444-53431-6 Volume 181: Neuroendocrinology: The Normal Neuroendocrine System, by Luciano Martini, George P Chrousos, Fernand Labrie, Karel Pacak and Donald W Pfaff (Eds.) – 2010, 978-0-444-53617-4 Volume 182: Neuroendocrinology: Pathological Situations and Diseases, by Luciano Martini, George P Chrousos, Fernand Labrie, Karel Pacak and Donald W Pfaff (Eds.) – 2010, 978-0-444-53616-7 Volume 183: Recent Advances in Parkinson’s Disease: Basic Research, by Anders Bj€orklund and M Angela Cenci (Eds.) – 2010, 978-0-444-53614-3 Volume 184: Recent Advances in Parkinson’s Disease: Translational and Clinical Research, by Anders Bj€orklund and M Angela Cenci (Eds.) – 2010, 978-0-444-53750-8 Volume 185: Human Sleep and Cognition Part I: Basic Research, by Gerard A Kerkhof and Hans P.A Van Dongen (Eds.) – 2010, 978-0-444-53702-7 Volume 186: Sex Differences in the Human Brain, their Underpinnings and Implications, by Ivanka Savic (Ed.) – 2010, 978-0-444-53630-3 Volume 187: Breathe, Walk and Chew: The Neural Challenge: Part I, by Jean-Pierre Gossard, Rejean Dubuc and Arlette Kolta (Eds.) – 2010, 978-0-444-53613-6 Volume 188: Breathe, Walk and Chew; The Neural Challenge: Part II, by Jean-Pierre Gossard, Rejean Dubuc and Arlette Kolta (Eds.) – 2011, 978-0-444-53825-3 Volume 189: Gene Expression to Neurobiology and Behaviour: Human Brain Development and Developmental Disorders, by Oliver Braddick, Janette Atkinson and Giorgio M Innocenti (Eds.) – 2011, 978-0-444-53884-0 397 398 Other volumes in PROGRESS IN BRAIN RESEARCH Volume 190: Human Sleep and Cognition Part II: Clinical and Applied Research, by Hans P.A Van Dongen and Gerard A Kerkhof (Eds.) – 2011, 978-0-444-53817-8 Volume 191: Enhancing Performance for Action and perception: Multisensory Integration, Neuroplasticity and Neuroprosthetics: Part I, by Andrea M Green, C Elaine Chapman, John F Kalaska and Franco Lepore (Eds.) – 2011, 978-0-444-53752-2 Volume 192: Enhancing Performance for Action and Perception: Multisensory Integration, Neuroplasticity and Neuroprosthetics: Part II, by Andrea M Green, C Elaine Chapman, John F Kalaska and Franco Lepore (Eds.) – 2011, 978-0-444-53355-5 Volume 193: Slow Brain Oscillations of Sleep, Resting State and Vigilance, by Eus J.W Van Someren, Ysbrand D Van Der Werf, Pieter R Roelfsema, Huibert D Mansvelder and Fernando H Lopes da Silva (Eds.) – 2011, 978-0-444-53839-0 Volume 194: Brain Machine Interfaces: Implications For Science, Clinical Practice And Society, by Jens Schouenborg, Martin Garwicz and Nils Danielsen (Eds.) – 2011, 978-0-444-53815-4 Volume 195: Evolution of the Primate Brain: From Neuron to Behavior, by Michel A Hofman and Dean Falk (Eds.) – 2012, 978-0-444-53860-4 Volume 196: Optogenetics: Tools for Controlling and Monitoring Neuronal Activity, by Thomas Kn€opfel and Edward S Boyden (Eds.) – 2012, 978-0-444-59426-6 Volume 197: Down Syndrome: From Understanding the Neurobiology to Therapy, by Mara Dierssen and Rafael De La Torre (Eds.) – 2012, 978-0-444-54299-1 Volume 198: Orexin/Hypocretin System, by Anantha Shekhar (Ed.) – 2012, 978-0-444-59489-1 Volume 199: The Neurobiology of Circadian Timing, by Andries Kalsbeek, Martha Merrow, Till Roenneberg and Russell G Foster (Eds.) – 2012, 978-0-444-59427-3 Volume 200: Functional Neural Transplantation III: Primary and stem cell therapies for brain repair, Part I, by Stephen B Dunnett and Anders Bj€orklund (Eds.) – 2012, 978-0-444-59575-1 Volume 201: Functional Neural Transplantation III: Primary and stem cell therapies for brain repair, Part II, by Stephen B Dunnett and Anders Bj€orklund (Eds.) – 2012, 978-0-444-59544-7 Volume 202: Decision Making: Neural and Behavioural Approaches, by V.S Chandrasekhar Pammi and Narayanan Srinivasan (Eds.) – 2013, 978-0-444-62604-2 Volume 203: The Fine Arts, Neurology, and Neuroscience: Neuro-Historical Dimensions, by Stanley Finger, Dahlia W Zaidel, Franc¸ois Boller and Julien Bogousslavsky (Eds.) – 2013, 978-0-444-62730-8 Volume 204: The Fine Arts, Neurology, and Neuroscience: New Discoveries and Changing Landscapes, by Stanley Finger, Dahlia W Zaidel, Franc¸ois Boller and Julien Bogousslavsky (Eds.) – 2013, 978-0-444-63287-6 Volume 205: Literature, Neurology, and Neuroscience: Historical and Literary Connections, by Anne Stiles, Stanley Finger and Franc¸ois Boller (Eds.) – 2013, 978-0-444-63273-9 Volume 206: Literature, Neurology, and Neuroscience: Neurological and Psychiatric Disorders, by Stanley Finger, Franc¸ois Boller and Anne Stiles (Eds.) – 2013, 978-0-444-63364-4 Volume 207: Changing Brains: Applying Brain Plasticity to Advance and Recover Human Ability, by Michael M Merzenich, Mor Nahum and Thomas M Van Vleet (Eds.) – 2013, 978-0-444-63327-9 Volume 208: Odor Memory and Perception, by Edi Barkai and Donald A Wilson (Eds.) – 2014, 978-0-444-63350-7 Volume 209: The Central Nervous System Control of Respiration, by Gert Holstege, Caroline M Beers and Hari H Subramanian (Eds.) – 2014, 978-0-444-63274-6 Volume 210: Cerebellar Learning, Narender Ramnani (Ed.) – 2014, 978-0-444-63356-9 Volume 211: Dopamine, by Marco Diana, Gaetano Di Chiara and Pierfranco Spano (Eds.) – 2014, 978-0-444-63425-2 Volume 212: Breathing, Emotion and Evolution, by Gert Holstege, Caroline M Beers and Hari H Subramanian (Eds.) – 2014, 978-0-444-63488-7 Volume 213: Genetics of Epilepsy, by Ortrud K Steinlein (Ed.) – 2014, 978-0-444-63326-2 Volume 214: Brain Extracellular Matrix in Health and Disease, by Asla Pitk€anen, Alexander Dityatev and Bernhard Wehrle-Haller (Eds.) – 2014, 978-0-444-63486-3 Other volumes in PROGRESS IN BRAIN RESEARCH Volume 215: The History of the Gamma Knife, by Jeremy C Ganz (Ed.) – 2014, 978-0-444-63520-4 Volume 216: Music, Neurology, and Neuroscience: Historical Connections and Perspectives, by Franc¸ois Boller, Eckart Altenm€uller, and Stanley Finger (Eds.) – 2015, 978-0-444-63399-6 Volume 217: Music, Neurology, and Neuroscience: Evolution, the Musical Brain, Medical Conditions, and Therapies, by Eckart Altenm€uller, Stanley Finger, and Franc¸ois Boller (Eds.) – 2015, 978-0-444-63551-8 Volume 218: Sensorimotor Rehabilitation: At the Crossroads of Basic and Clinical Sciences, by Numa Dancause, Sylvie Nadeau, and Serge Rossignol (Eds.) – 2015, 978-0-444-63565-5 Volume 219: The Connected Hippocampus, by Shane O’Mara and Marian Tsanov (Eds.) – 2015, 978-0-444-63549-5 Volume 220: New Trends in Basic and Clinical Research of Glaucoma: A Neurodegenerative Disease of the Visual System, by Giacinto Bagetta and Carlo Nucci (Eds.) – 2015, 978-0-444-63566-2 Volume 221: New Trends in Basic and Clinical Research of Glaucoma: A Neurodegenerative Disease of the Visual System, by Giacinto Bagetta and Carlo Nucci (Eds.) – 2015, 978-0-12-804608-1 Volume 222: Computational Neurostimulation, by Sven Bestmann (Ed.) – 2015, 978-0-444-63546-4 Volume 223: Neuroscience for Addiction Medicine: From Prevention to Rehabilitation - Constructs and Drugs, by Hamed Ekhtiari and Martin Paulus (Eds.) – 2016, 978-0-444-63545-7 Volume 224: Neuroscience for Addiction Medicine: From Prevention to Rehabilitation - Methods and Interventions, by Hamed Ekhtiari and Martin P Paulus (Eds.) – 2016, 978-0-444-63716-1 Volume 225: New Horizons in Neurovascular Coupling: A Bridge Between Brain Circulation and Neural Plasticity, by Kazuto Masamoto, Hajime Hirase, and Katsuya Yamada (Eds.) – 2016, 978-0-444-63704-8 Volume 226: Neurobiology of Epilepsy: From Genes to Networks, by Elsa Rossignol, Lionel Carmant and Jean-Claude Lacaille (Eds.) – 2016, 978-0-12-803886-4 399 ... effort is dedicated to understanding what a “number” is for the human brain Answering this question is made complex in the first place Progress in Brain Research, Volume 227, ISSN 0079-6123, http://dx.doi.org/10.1016/bs.pbr.2016.04.015... findings, where firing rates of intermingled individual neurons either increase or decrease over a rather wide range of numerosities tested (2–32 dots) In this context it is of interest that in. .. experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety
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