Hecke Schrobsdorff | Max-Planck Institute for Dynamics and Self-Organisation (original) (raw)

Papers by Hecke Schrobsdorff

Research paper thumbnail of Turbulent-laminar patterns in plane Poiseuille flow

Physics of Fluids, 2014

Turbulent-laminar banded patterns in plane Poiseuille flow are studied via direct numerical simul... more Turbulent-laminar banded patterns in plane Poiseuille flow are studied via direct numerical simulations in a tilted and translating computational domain using a parallel version of the pseudospectral code Channelflow. 3D visualizations via the streamwise vorticity of an instantaneous and a time-averaged pattern are presented, as well as 2D visualizations of the average velocity field and the turbulent kinetic energy. Simulations for 2300 ≥ Re m ≥ 700 show the gradual development from uniform turbulence to a pattern with wavelength 20 half-gaps at Re m ≈ 1900, to a pattern with wavelength 40 at Re m ≈ 1300 and finally to laminar flow at Re m ≈ 800. These transitions are tracked quantitatively via diagnostics using the amplitude and phase of the Fourier transform and its probability distribution. The propagation velocity of the pattern is approximately that of the mean flux and is a decreasing function of Reynolds number. Examination of the time-averaged flow shows that a turbulent band is associated with two counter-rotating cells stacked in the cross-channel direction and that the turbulence is highly concentrated near the walls. Near the wall, the Reynolds stress force accelerates the fluid through a turbulent band while viscosity decelerates it; advection by the laminar profile acts in both directions. In the center, the Reynolds stress force decelerates the fluid through a turbulent band while advection by the laminar profile accelerates it. These characteristics are compared with those of turbulent-laminar banded patterns in plane Couette flow.

Research paper thumbnail of Learning as a Cause for Aging Impairments

Research paper thumbnail of Modeling Negative Priming Using EEG-Data

A continuous stream of trials is processed. ⋆ Two subsequent displays form a prime-probe pair.

Research paper thumbnail of Cognitive Aging as a Loss of Criticality

Research paper thumbnail of Localized Solutions in a Simple Neural Field Model

We investigate analytically properties like stability and existence of solutions of the two dimen... more We investigate analytically properties like stability and existence of solutions of the two dimensional neural field equation as proposed by in [1] as a model of macroscopic activation dynamics in neural tissue. While the one dimensional case has been treated comprehensively, for the two dimensional case only the existence of circular solutions was shown, and stability was as well only considered for radially symmetric perturbations. We introduce a simplified neural field model, defined by a piecewise constant interaction kernel. First we show that in this setting the existence of noncircular solutions can be negated except for small perturbations of a circle. Secondly in a special case, the dynamics of the boundary of an activated area is considered, which leads to a proof, that in this special case, solutions are necessarily circular.

Research paper thumbnail of Localization of neural activity

In this work, the phenomenon of localization of neural activity is approached from two different ... more In this work, the phenomenon of localization of neural activity is approached from two different sides, a mean field like approach and a network ansatz. Thus, the chapters are organized in two separated parts after a general introduction. In a main discussion and outlook the results are brought together. An index of the most important symbols used in this thesis follows in the appendix, as well as the bibliography, a publication list and acknowledgments.

Research paper thumbnail of Modeling selective attention using EEG data

Research paper thumbnail of The General Model for Negative Priming

Negative priming is characterized by longer reaction times when responding to stimuli which have ... more Negative priming is characterized by longer reaction times when responding to stimuli which have been actively ignored recently. A central problem of the interpretation of the NP effect is the lack of agreement about the underlying mechanisms. Over the past 20 years, various theoretical accounts have been developed to explain NP. However, empirical evidence does not clearly favour one theory over the others. One of the reasons why it is so hard to decide between the different theoretical accounts, is the lack of a concrete computational formulation. We therefore developed a general model for stimulus-based action selection which attempts to incorporate all mechanisms relevant to selective attention. The concrete implementation incorporates building blocks for feature detection, feature binding, semantic representation, action planning and episodic memory as well as a control unit that keeps track of higher goals. Reaction time differences in various priming conditions emerge by an interplay of all model components. We present different paradigms of behavioral negative priming experiments and review how the major theories explain decision making and negative priming. After the introduction of the model structure together with its implementation, we point to the behavior of the general model in the different priming conditions and the integration of different theories.

Research paper thumbnail of The Time Course of Negative Priming

Research paper thumbnail of Computational modelling as an approach to neural mechanisms of Negative Priming

Research paper thumbnail of Event-related brain potential correlates of identity negative priming from overlapping pictures

Psychophysiology, 2010

Event-related potentials (ERPs) were obtained from an identity priming task, where a green target... more Event-related potentials (ERPs) were obtained from an identity priming task, where a green target had to be selected against a superimposed red distractor. Several priming conditions were realized in a mix of control (CO), negative priming (NP), and positive priming (PP) trials. PP and NP effects in reaction times (RTs) were significant. ERP results conceptually replicate earlier findings of left-posterior P300 reduction in PP and NP trials compared to CO. This ERP effect may reflect the detection of prime-probe similarity corresponding to the concept of a retrieval cue. A novel finding concerned amplitude increase of the frontal late positive complex (LPC) in the order NP, CO, and PP. NP therefore seemed to induce brain activity related to cognitive control and/or memory processes, with reduced LPC amplitude indicating effortful processing. Overall, retrieval-based explanations of identity NP are supported.

Research paper thumbnail of Identity Negative Priming: A Phenomenon of Perception, Recognition or Selection?

PLoS ONE, 2012

The present study addresses the problem whether negative priming (NP) is due to information proce... more The present study addresses the problem whether negative priming (NP) is due to information processing in perception, recognition or selection. We argue that most NP studies confound priming and perceptual similarity of prime-probe episodes and implement a color-switch paradigm in order to resolve the issue. In a series of three identity negative priming experiments with verbal naming response, we determined when NP and positive priming (PP) occur during a trial. The first experiment assessed the impact of target color on priming effects. It consisted of two blocks, each with a different fixed target color. With respect to target color no differential priming effects were found. In Experiment 2 the target color was indicated by a cue for each trial. Here we resolved the confounding of perceptual similarity and priming condition. In trials with coinciding colors for prime and probe, we found priming effects similar to Experiment 1. However, trials with a target color switch showed such effects only in trials with role-reversal (distractor-to-target or target-to-distractor), whereas the positive priming (PP) effect in the target-repetition trials disappeared. Finally, Experiment 3 split trial processing into two phases by presenting the trial-wise color cue only after the stimulus objects had been recognized. We found recognition in every priming condition to be faster than in control trials. We were hence led to the conclusion that PP is strongly affected by perception, in contrast to NP which emerges during selection, i.e., the two effects cannot be explained by a single mechanism.

Research paper thumbnail of Localized activations in a simple neural field model

Neurocomputing, 2005

A quarter of a century ago Amari (Biol. Cybernet. 27 (1977) 77–87) has presented a comprehensive ... more A quarter of a century ago Amari (Biol. Cybernet. 27 (1977) 77–87) has presented a comprehensive and very elegant solution of the one-dimensional neural field equation. In the two-dimensional case analytical results on localized solutions are available under the assumption of rotational invariance, as numerical evidence indicates that no other stable solutions exist. We present analytic results for a special case of a “tophat” interaction function, which partially justifies the implicit assumption of circular solutions and allows us ...

Research paper thumbnail of A feature-binding model with localized excitations

Neurocomputing, 2007

We study a model of feature binding in prefrontal cortex which defers specific perceptual informa... more We study a model of feature binding in prefrontal cortex which defers specific perceptual information to lower areas and merely maintains the identity of the combination. The model consists of three layers of pulse-coupled leaky integrate-and-fire neurons. Features are encoded by the location of sustained activity in the subordinate layers. The feature layers are excitatorily coupled to a superordinate layer that represents combinations of features by means of an oscillatory dynamics. The model accounts for effects such as the memorization of an object that was perceived only for a short period, illusory binding of simultaneous stimuli, and the limit of attentional capacity. The present paper discusses conditions for localized excitations in networks of integrate-and-fire neurons and considers the application to a dynamic link architecture.

Research paper thumbnail of Inhibition in the Dynamics of Selective Attention: An Integrative Model for Negative Priming

Frontiers in Psychology, 2012

We introduce a computational model of the negative priming (NP) effect that includes perception, ... more We introduce a computational model of the negative priming (NP) effect that includes perception, memory, attention, decision making, and action. The model is designed to provide a coherent picture across competing theories of NP. The model is formulated in terms of abstract dynamics for the activations of features, their binding into object entities, their semantic categorization as well as related memories and appropriate reactions. The dynamic variables interact in a connectionist network which is shown to be adaptable to a variety of experimental paradigms. We find that selective attention can be modeled by means of inhibitory processes and by a threshold dynamics. From the necessity of quantifying the experimental paradigms, we conclude that the specificity of the experimental paradigm must be taken into account when predicting the nature of the NP effect.

Research paper thumbnail of Aging Deficits as a Side-Effect of Optimization

Frontiers in Computational Neuroscience, 2008

Aging Deficits as a Side-Effect of Optimization ... Aging Deficits as a Side-Effect of Optimizati... more Aging Deficits as a Side-Effect of Optimization ... Aging Deficits as a Side-Effect of Optimization ... Hecke Schrobsdorff1,2, Matthias Ihrke1,3, J��rg Behrendt1,3, Marcus Hasselhorn1,3, J. Michael Herrmann1,4 ... 1 Bernstein Center for Computational Neuroscience G��ttingen 2 University of G��ttingen, Institute for Nonlinear Dynamics 3 University of G��ttingen, Georg-Elias-M��ller Institute for Educational Psychology 4 University of Edinburgh, School of Informatics ... Aging Deficits as a Side-Effect of Optimization ... Advanced Averaging ERPs Single Trial of Aging Biology

Research paper thumbnail of Negative Priming Persists in the Absence of Response-Retrieval

Experimental Psychology (formerly Zeitschrift für Experimentelle Psychologie), 2013

The hypothesis that retrieval of the prime response is responsible for the negative priming (NP) ... more The hypothesis that retrieval of the prime response is responsible for the negative priming (NP) effect has gained popularity in recent studies of visual identity NP. In the current study we report an experiment in which we aimed to remove the response from the prime memory trace by means of spatio-temporal separation. Compared to an identical experiment without this separation (Ihrke et al., 2011), we find that the response-retrieval-specific interaction is absent indicating that the separation was successful in preventing response-retrieval. Still, both negative and positive priming are present as main effects which show that processes other than response-retrieval can produce NP. In addition, based on recordings of the eye-movements during task processing, we localize the NP effect in a target-selection process while positive priming manifests in facilitated response-selection. Our results are in line with a multiple-route view of NP.

Research paper thumbnail of A computational approach to negative priming

Connection Science, 2007

Priming is characterized by a sensitivity of reaction times to the sequence of stimuli in psychop... more Priming is characterized by a sensitivity of reaction times to the sequence of stimuli in psychophysical experiments. The reduction of the reaction time observed in positive priming is well-known and experimentally understood . Negative primingthe opposite effect -is experimentally less tangible . The dependence on subtle parameter changes (such as response-stimulus interval) usually varies. The sensitivity of the negative priming effect bears great potential for applications in research in fields such as memory, selective attention, and aging effects.

Research paper thumbnail of Are age-related cognitive effects caused by optimization?

BMC Neuroscience, 2009

Cognitive aging seems to be a process of global degradation. Performance in psychological tests o... more Cognitive aging seems to be a process of global degradation. Performance in psychological tests of fluid intelligence, such as Raven's Advanced Progressive Matrices, tends to decrease with age [1]. These results are strongly contrasted by performance improvements in everyday situations . We therefore hypothesize that the observed aging deficits are partly caused by the optimization of cognitive functions due to learning.

Research paper thumbnail of 1 Experimental Psychology: A Place for Innovative Research and Methodological Developments

Experimental Psychology, 2011

Experimental Psychology 2011; Vol. 58(6):509–516 DOI: 10.1027/1618-3169/a000140 ... 171 Can Emoti... more Experimental Psychology 2011; Vol. 58(6):509–516 DOI: 10.1027/1618-3169/a000140 ... 171 Can Emotion Modulate Attention? Evidence for Reciprocal Links in the Attentional Network Test Noga Cohen, Avishai Henik, and Nilly Mor 180 Parallel Constraint Satisfaction in Memory-Based Decisions Andreas Glöckner and Sara D. Hodges 196 Multiple-Object Tracking: The Binding of Spatial Location and Featural Identity K. Botterill, R. Allen, and P. McGeorge 201 Holistic Representation of Unit Fractions Dana Ganor-Stern, Irina Karasik-Rivkin, and Joseph Tzelgov 207 ...

Research paper thumbnail of Turbulent-laminar patterns in plane Poiseuille flow

Physics of Fluids, 2014

Turbulent-laminar banded patterns in plane Poiseuille flow are studied via direct numerical simul... more Turbulent-laminar banded patterns in plane Poiseuille flow are studied via direct numerical simulations in a tilted and translating computational domain using a parallel version of the pseudospectral code Channelflow. 3D visualizations via the streamwise vorticity of an instantaneous and a time-averaged pattern are presented, as well as 2D visualizations of the average velocity field and the turbulent kinetic energy. Simulations for 2300 ≥ Re m ≥ 700 show the gradual development from uniform turbulence to a pattern with wavelength 20 half-gaps at Re m ≈ 1900, to a pattern with wavelength 40 at Re m ≈ 1300 and finally to laminar flow at Re m ≈ 800. These transitions are tracked quantitatively via diagnostics using the amplitude and phase of the Fourier transform and its probability distribution. The propagation velocity of the pattern is approximately that of the mean flux and is a decreasing function of Reynolds number. Examination of the time-averaged flow shows that a turbulent band is associated with two counter-rotating cells stacked in the cross-channel direction and that the turbulence is highly concentrated near the walls. Near the wall, the Reynolds stress force accelerates the fluid through a turbulent band while viscosity decelerates it; advection by the laminar profile acts in both directions. In the center, the Reynolds stress force decelerates the fluid through a turbulent band while advection by the laminar profile accelerates it. These characteristics are compared with those of turbulent-laminar banded patterns in plane Couette flow.

Research paper thumbnail of Learning as a Cause for Aging Impairments

Research paper thumbnail of Modeling Negative Priming Using EEG-Data

A continuous stream of trials is processed. ⋆ Two subsequent displays form a prime-probe pair.

Research paper thumbnail of Cognitive Aging as a Loss of Criticality

Research paper thumbnail of Localized Solutions in a Simple Neural Field Model

We investigate analytically properties like stability and existence of solutions of the two dimen... more We investigate analytically properties like stability and existence of solutions of the two dimensional neural field equation as proposed by in [1] as a model of macroscopic activation dynamics in neural tissue. While the one dimensional case has been treated comprehensively, for the two dimensional case only the existence of circular solutions was shown, and stability was as well only considered for radially symmetric perturbations. We introduce a simplified neural field model, defined by a piecewise constant interaction kernel. First we show that in this setting the existence of noncircular solutions can be negated except for small perturbations of a circle. Secondly in a special case, the dynamics of the boundary of an activated area is considered, which leads to a proof, that in this special case, solutions are necessarily circular.

Research paper thumbnail of Localization of neural activity

In this work, the phenomenon of localization of neural activity is approached from two different ... more In this work, the phenomenon of localization of neural activity is approached from two different sides, a mean field like approach and a network ansatz. Thus, the chapters are organized in two separated parts after a general introduction. In a main discussion and outlook the results are brought together. An index of the most important symbols used in this thesis follows in the appendix, as well as the bibliography, a publication list and acknowledgments.

Research paper thumbnail of Modeling selective attention using EEG data

Research paper thumbnail of The General Model for Negative Priming

Negative priming is characterized by longer reaction times when responding to stimuli which have ... more Negative priming is characterized by longer reaction times when responding to stimuli which have been actively ignored recently. A central problem of the interpretation of the NP effect is the lack of agreement about the underlying mechanisms. Over the past 20 years, various theoretical accounts have been developed to explain NP. However, empirical evidence does not clearly favour one theory over the others. One of the reasons why it is so hard to decide between the different theoretical accounts, is the lack of a concrete computational formulation. We therefore developed a general model for stimulus-based action selection which attempts to incorporate all mechanisms relevant to selective attention. The concrete implementation incorporates building blocks for feature detection, feature binding, semantic representation, action planning and episodic memory as well as a control unit that keeps track of higher goals. Reaction time differences in various priming conditions emerge by an interplay of all model components. We present different paradigms of behavioral negative priming experiments and review how the major theories explain decision making and negative priming. After the introduction of the model structure together with its implementation, we point to the behavior of the general model in the different priming conditions and the integration of different theories.

Research paper thumbnail of The Time Course of Negative Priming

Research paper thumbnail of Computational modelling as an approach to neural mechanisms of Negative Priming

Research paper thumbnail of Event-related brain potential correlates of identity negative priming from overlapping pictures

Psychophysiology, 2010

Event-related potentials (ERPs) were obtained from an identity priming task, where a green target... more Event-related potentials (ERPs) were obtained from an identity priming task, where a green target had to be selected against a superimposed red distractor. Several priming conditions were realized in a mix of control (CO), negative priming (NP), and positive priming (PP) trials. PP and NP effects in reaction times (RTs) were significant. ERP results conceptually replicate earlier findings of left-posterior P300 reduction in PP and NP trials compared to CO. This ERP effect may reflect the detection of prime-probe similarity corresponding to the concept of a retrieval cue. A novel finding concerned amplitude increase of the frontal late positive complex (LPC) in the order NP, CO, and PP. NP therefore seemed to induce brain activity related to cognitive control and/or memory processes, with reduced LPC amplitude indicating effortful processing. Overall, retrieval-based explanations of identity NP are supported.

Research paper thumbnail of Identity Negative Priming: A Phenomenon of Perception, Recognition or Selection?

PLoS ONE, 2012

The present study addresses the problem whether negative priming (NP) is due to information proce... more The present study addresses the problem whether negative priming (NP) is due to information processing in perception, recognition or selection. We argue that most NP studies confound priming and perceptual similarity of prime-probe episodes and implement a color-switch paradigm in order to resolve the issue. In a series of three identity negative priming experiments with verbal naming response, we determined when NP and positive priming (PP) occur during a trial. The first experiment assessed the impact of target color on priming effects. It consisted of two blocks, each with a different fixed target color. With respect to target color no differential priming effects were found. In Experiment 2 the target color was indicated by a cue for each trial. Here we resolved the confounding of perceptual similarity and priming condition. In trials with coinciding colors for prime and probe, we found priming effects similar to Experiment 1. However, trials with a target color switch showed such effects only in trials with role-reversal (distractor-to-target or target-to-distractor), whereas the positive priming (PP) effect in the target-repetition trials disappeared. Finally, Experiment 3 split trial processing into two phases by presenting the trial-wise color cue only after the stimulus objects had been recognized. We found recognition in every priming condition to be faster than in control trials. We were hence led to the conclusion that PP is strongly affected by perception, in contrast to NP which emerges during selection, i.e., the two effects cannot be explained by a single mechanism.

Research paper thumbnail of Localized activations in a simple neural field model

Neurocomputing, 2005

A quarter of a century ago Amari (Biol. Cybernet. 27 (1977) 77–87) has presented a comprehensive ... more A quarter of a century ago Amari (Biol. Cybernet. 27 (1977) 77–87) has presented a comprehensive and very elegant solution of the one-dimensional neural field equation. In the two-dimensional case analytical results on localized solutions are available under the assumption of rotational invariance, as numerical evidence indicates that no other stable solutions exist. We present analytic results for a special case of a “tophat” interaction function, which partially justifies the implicit assumption of circular solutions and allows us ...

Research paper thumbnail of A feature-binding model with localized excitations

Neurocomputing, 2007

We study a model of feature binding in prefrontal cortex which defers specific perceptual informa... more We study a model of feature binding in prefrontal cortex which defers specific perceptual information to lower areas and merely maintains the identity of the combination. The model consists of three layers of pulse-coupled leaky integrate-and-fire neurons. Features are encoded by the location of sustained activity in the subordinate layers. The feature layers are excitatorily coupled to a superordinate layer that represents combinations of features by means of an oscillatory dynamics. The model accounts for effects such as the memorization of an object that was perceived only for a short period, illusory binding of simultaneous stimuli, and the limit of attentional capacity. The present paper discusses conditions for localized excitations in networks of integrate-and-fire neurons and considers the application to a dynamic link architecture.

Research paper thumbnail of Inhibition in the Dynamics of Selective Attention: An Integrative Model for Negative Priming

Frontiers in Psychology, 2012

We introduce a computational model of the negative priming (NP) effect that includes perception, ... more We introduce a computational model of the negative priming (NP) effect that includes perception, memory, attention, decision making, and action. The model is designed to provide a coherent picture across competing theories of NP. The model is formulated in terms of abstract dynamics for the activations of features, their binding into object entities, their semantic categorization as well as related memories and appropriate reactions. The dynamic variables interact in a connectionist network which is shown to be adaptable to a variety of experimental paradigms. We find that selective attention can be modeled by means of inhibitory processes and by a threshold dynamics. From the necessity of quantifying the experimental paradigms, we conclude that the specificity of the experimental paradigm must be taken into account when predicting the nature of the NP effect.

Research paper thumbnail of Aging Deficits as a Side-Effect of Optimization

Frontiers in Computational Neuroscience, 2008

Aging Deficits as a Side-Effect of Optimization ... Aging Deficits as a Side-Effect of Optimizati... more Aging Deficits as a Side-Effect of Optimization ... Aging Deficits as a Side-Effect of Optimization ... Hecke Schrobsdorff1,2, Matthias Ihrke1,3, J��rg Behrendt1,3, Marcus Hasselhorn1,3, J. Michael Herrmann1,4 ... 1 Bernstein Center for Computational Neuroscience G��ttingen 2 University of G��ttingen, Institute for Nonlinear Dynamics 3 University of G��ttingen, Georg-Elias-M��ller Institute for Educational Psychology 4 University of Edinburgh, School of Informatics ... Aging Deficits as a Side-Effect of Optimization ... Advanced Averaging ERPs Single Trial of Aging Biology

Research paper thumbnail of Negative Priming Persists in the Absence of Response-Retrieval

Experimental Psychology (formerly Zeitschrift für Experimentelle Psychologie), 2013

The hypothesis that retrieval of the prime response is responsible for the negative priming (NP) ... more The hypothesis that retrieval of the prime response is responsible for the negative priming (NP) effect has gained popularity in recent studies of visual identity NP. In the current study we report an experiment in which we aimed to remove the response from the prime memory trace by means of spatio-temporal separation. Compared to an identical experiment without this separation (Ihrke et al., 2011), we find that the response-retrieval-specific interaction is absent indicating that the separation was successful in preventing response-retrieval. Still, both negative and positive priming are present as main effects which show that processes other than response-retrieval can produce NP. In addition, based on recordings of the eye-movements during task processing, we localize the NP effect in a target-selection process while positive priming manifests in facilitated response-selection. Our results are in line with a multiple-route view of NP.

Research paper thumbnail of A computational approach to negative priming

Connection Science, 2007

Priming is characterized by a sensitivity of reaction times to the sequence of stimuli in psychop... more Priming is characterized by a sensitivity of reaction times to the sequence of stimuli in psychophysical experiments. The reduction of the reaction time observed in positive priming is well-known and experimentally understood . Negative primingthe opposite effect -is experimentally less tangible . The dependence on subtle parameter changes (such as response-stimulus interval) usually varies. The sensitivity of the negative priming effect bears great potential for applications in research in fields such as memory, selective attention, and aging effects.

Research paper thumbnail of Are age-related cognitive effects caused by optimization?

BMC Neuroscience, 2009

Cognitive aging seems to be a process of global degradation. Performance in psychological tests o... more Cognitive aging seems to be a process of global degradation. Performance in psychological tests of fluid intelligence, such as Raven's Advanced Progressive Matrices, tends to decrease with age [1]. These results are strongly contrasted by performance improvements in everyday situations . We therefore hypothesize that the observed aging deficits are partly caused by the optimization of cognitive functions due to learning.

Research paper thumbnail of 1 Experimental Psychology: A Place for Innovative Research and Methodological Developments

Experimental Psychology, 2011

Experimental Psychology 2011; Vol. 58(6):509–516 DOI: 10.1027/1618-3169/a000140 ... 171 Can Emoti... more Experimental Psychology 2011; Vol. 58(6):509–516 DOI: 10.1027/1618-3169/a000140 ... 171 Can Emotion Modulate Attention? Evidence for Reciprocal Links in the Attentional Network Test Noga Cohen, Avishai Henik, and Nilly Mor 180 Parallel Constraint Satisfaction in Memory-Based Decisions Andreas Glöckner and Sara D. Hodges 196 Multiple-Object Tracking: The Binding of Spatial Location and Featural Identity K. Botterill, R. Allen, and P. McGeorge 201 Holistic Representation of Unit Fractions Dana Ganor-Stern, Irina Karasik-Rivkin, and Joseph Tzelgov 207 ...