NEURAL SYSTEMS by. STEPHEN JAMES GOTTS ... One couldn't ask for a more vibrant and engaging research community. Among ...... decreases in reaction time with each repetition (e.g. Brown et al., 1996; Logan, 1990;. Ostergaard, 1998 ...
MECHANISMS UNDERLYING ENHANCED PROCESSING EFFICIENCY IN NEURAL SYSTEMS
by STEPHEN JAMES GOTTS
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A Dissertation Presented to the FACULTY OF THE DEPARTMENT OF PSYCHOLOGY CARNEGIE MELLON UNIVERSITY In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Psychology)
May 2003
Dedication This thesis is dedicated to Dr. Frederick J. Bremner, who first ignited my curiosity about the relationship between brain and mind.
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Acknowledgements On leaving my home for the past six years, I feel very fortunate to have been surrounded by so many exceptional, gifted, and giving teachers. First and foremost among these, I would like to thank the members of my dissertation committee: David Plaut was the model advisor, giving me just the right amount of guidance, feedback, and space to let me develop my own ideas; aside from teaching me basic scientific and critical thinking skills, he imparted to me an invaluable meta-perspective on how one goes about developing an explanation of something. Carson Chow, in teaching me the basics of biophysics and computational neuroscience on which much of the current work is based, really acted as a second advisor - exposing me to the valuable perspectives of physics and mathematics. Jay McClelland, in addition to his always on-point gut reactions, provided an ideal example of the excellence and humility that all scientists should strive to acheive. Bobby Klatzky kept me honest by pushing me to think hard about the empirical implications of my ideas and encouraging me to pursue them, something that I'm just starting to do now. I also owe a deep debt of gratitude to Lisa Cipolotti and Tim Shallice, who graciously allowed me to spend a summer at the National Hospital in London so that I could stand over their shoulders and learn firsthand how Neuropsychology is done. Much of my graduate education was the result of more informal discussions and interactions with friends and collegues in the PDP research group, the CMU Psychology department, and the CNBC. One couldn't ask for a more vibrant and engaging research community. Among these collegues, I’d particularly like to thank Marlene Behrmann, Brad Best, Bard Ermentrout, Lisa Gershkoff-Stowe, Mike Harm, David Noelle, Alex Petrov, Lynne Reder, Tim Rogers, Doug Rohde, and David Wood for pushing my mind in many different healthy directions. No one can get through graduate school without close friends. There could be none closer than Craig Haimson and Thomas Kang. I'm honored to have shared the long ride with yinz guys at 5934 Phillips Ave. Many thanks are also due to DJ Bolger, John Connelly, John Klein, and Suzy Scherf for pulling me out of the house/lab periodically for a breath of fresh air and beer. Finally, special thanks to Carol, Louie, and Madison for their generous patience, loving support, and understanding during the months of long hours I spent working on this - you kept me sane, happy, and balanced.
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CONTENTS
Dedication
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Acknowledgements
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Abstract
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Introduction 1.1 Neural Adaptation and Synaptic Depression . . . . . . . . . . . . . . . . . . . 1.2 Neural Activity Decreases in Behaving Humans and Non-human Primates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Repetition Suppression and Behavioral Priming ............. 1.4 Previous Accounts of Repetition Priming . . . . . . . . . . . . . . . . . . . 1.4.1 Spreading Activation Accounts ................... 1.4.2 Episodic Retrieval Accounts . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Distributed Connectionist Accounts . . . . . . . . . . . . . . . . . . . Neural Synchrony, Behavioral Priming and Processing Efficiency 2.1 The Importance of Metabolic Efficiency in Neural Processing 2.2 Mechanisms Underlying Spike Synchrony in Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Synchrony in Networks with Excitatory or Inhibitory Synapses ..................................... 2.2.2 Synchrony in Networks with Both Excitation and Inhibition 2.2.3 The Impact of Short-term Plasticity on Synchronization 2.3 Empirical Evidence for Neural Synchrony . . . . . . . . . . . . . . . . . . . 2.4 Basic Hypothesis and Overview of Simulations: Short-term Plasticity Enhances Neural Synchrony and Processing Efficiency ....... 2.4.1 Short-term Plasticity, Synchrony, Priming, and Processing Efficiency (Simulations 1-5) ............. 2.4.2 Interaction of Short-term Plasticity and Neuromodulation (Simulations 6-8) ...............................
1 2 5 8 9 10 13 15 27 31 32 36 39 42 45 51 53 54
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Spiking Neuron Simulations of the Relationships Among Repetition Suppression, Neural Synchronization, and Behavioral Priming 3.1 General Simulation Methods: Simulations 1-5 ............. 3.1.1 Synaptic Currents ............................... 3.1.2 Firing-rate Adaptation and Synaptic Depression ....... 3.1.3 Stimuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Measuring Firing Rate and Synchrony ............. 3.1.4.1 Firing Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4.2 Synchrony ............................... 3.2 Simulation 1: The Effect of Stimulus Repetition on Firing Rate 3.2.1 Short-term Repetition Suppression in Anesthetized Monkeys (Miller, Gochin, & Gross, 1991) ................... 3.2.2 Short-term Repetition Suppression in Awake Monkeys Performing a Delay Match-to-Sample (DMS) task (Miller, Li, & Desimone, 1993) ................... 3.2.3 Short-term Repetition Suppression in Human Adults Performing Delay Match-to-Sample (Jiang, Haxby, Martin, Ungerleider, & Parasuraman, 2000) 3.2.4 Short-term Repetition Suppression in Human Adults During an fMR-Adaptation Paradigm (Grill-Spector & Malach, 2001) .................. 3.2.5 Comment ..................................... 3.3 Simulation 2: The Effect of Stimulus Repetition on Synchronization 3.3.1 The Effect of Stimulus Repetition on Spike Synchronization and its Tolerance to Input Heterogeneity ............. 3.3.2 The Impact of Synaptic Delays and Interactions with Other Cortical Regions on Repetition-Related Changes in Spike Synchronization ............................... 3.3.3 Impact of Inter-Stimulus Interval on Repetition-Related Changes in Spike Synchronization . . . . . . . . . . . . . . . . . . . 3.3.4 Comment ..................................... 3.4 Simulation 3: Contribution of Inhibition, Adaptation, and Synaptic Depression to Synchronization ................... 3.4.1 Effect of Blocking Inhibition, Adaptation, and Synaptic Depression on Firing Rate and Synchrony . . . . . . . . . . . . . 3.4.2 Contributions of Inhibition, Adaptation, and Synaptic Depression to Synchrony that are Unrelated to Changes in Firing Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Comment ..................................... 3.5 Simulation 4: Effect of Firing Rate and Synchronization on Reaction Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Effect of Input Firing Rate, Input Synchrony, and Balanced Excitation/Inhibition on Output Firing Rate . . . . . . . . . . . . . 3.5.2 Effect of Input Firing Rate and Synchrony on Reaction Time 3.5.3 Estimating Reaction Times for Simulation 2 (Section 3.3)
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3.5.4 Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Simulation 5: Quantification of Efficiency . . . . . . . . . . . . . . . . . . . 150 3.6.1 Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Firing-Rate and Spiking Neuron Simulations of the Interaction Between Short-Term Plasticity and Neuromodulation 4.1 Semantic Knowledge and the Distinction Between Access/Refractory and Degraded-Store Semantic Impairments 4.1.1 Access/Refractory and Degraded-Store Semantic Impairments: Patient Data ......................... 4.2 General Simulation Methods: Simulations 6-7 ............. 4.2.1 Network Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Synaptic Depression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Neuromodulation ............................... 4.2.4 Training Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4.1 Input Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4.2 Target Patterns ......................... 4.2.4.3 Time Course of a Single Training Pattern . . . . . . . 4.2.5 Lesioning Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6 Testing Procedure ............................... 4.3 Simulation 6: Accounting for the Basic Contrast Between Access/Refractory and Degraded-Store Patient Performance 4.3.1 Rate Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Semantic Relatedness Effects . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Frequency Effects ............................... 4.3.4 Serial Position Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Consistency Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Simulation 7: Accounting for Exceptions to the Access/Refractory and Degraded-Store Performance Patterns . . . . . . . . . . . . . . . . . . . 4.4.1 Frequency Effects without Consistency ............. 4.4.2 Consistency without Frequency Effects ............. 4.4.3 Access/Refractory Pattern without Serial Position Effect 4.4.4 Comment: Simulations 6 and 7 ................... 4.5 Simulation 8: Dependence of Priming vs. Habituation Effects on Neuromodulatory Level in Spiking Neural Networks ....... 4.5.1 Effect of Neuromodulation on Firing Rate and Synchrony 4.5.2 Effect of Neuromodulation on Reaction Time and Efficiency 4.5.3 Comment .....................................
155 159 162 172 174 175 179 183 183 183 184 185 185 186 194 194 195 196 197 201 202 202 205 208 216 219 222 226
Conclusions and Future Directions 228 5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 5.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
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References
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Appendix A
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Appendix B
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Appendix C
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Appendix D
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LIST OF TABLES
4.1
Summary of "degraded-store" (above line) and "access/refractory" (below line) patient data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.2 Effect of serial position in the model and in patient data from Warrington & Cipolotti (1996). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4.3 Effect of response consistency in the model and in patient data from Warrington & Cipolotti (1996). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
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LIST OF FIGURES 1.1
Evidence that neural firing rates (spikes/sec) return to baseline just after the offset of a stimulus: (A) Extracellular recordings of a monkey inferotemporal cortex neuron responding to 9 repeated presentations of a visual object (separated by delays of 20 seconds), as reported by Miller, Gochin, and Gross (1991). Firing rates (in both rasters and PSTH) return to baseline levels after the offset of the stimulus (duration of 1 sec). From "Habituation-like decrease in the responses of neurons in inferior temporal cortex of the macaque," by E. Miller, P. Gochin, and C. Gross, 1991, Visual Neuroscience, 7, p. 359. Copyright 1991 by Cambridge University Press. Permission pending. (B) Firing rate of a complex cell in monkey V1 to repeated presentations of a stationary sinusoidal grating (averaged over 40 runs), as reported by Muller, Metha, Krauskopf, and Lennie (1999). Firing rates return to baseline rapidly after the offset of each of the 3 stimuli shown. From "Rapid adaptation in visual cortex to the structure of images," by J. Muller, A. Metha, J. Krauskopf, and P. Lennie, 1999, Science, 285, p. 1405. Copyright 1999 by the American Association for the Advancement of Science. Permission pending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2 (A) A combination of unit activity increases and decreases induced by gradual learning lead the dot product of target and actual unit activities to increase with practice (compare familiar with unfamiliar patterns), as reported by McClelland and Rumelhart (1985, p. 175). Processing is shown as a function of time during an individual stimulus presentation. (B) Throughout learning, the number of processing cycles required to reach a fixed criterion of accuracy decreases for both familiar and unfamiliar stimuli (McClelland and Rumelhart, 1985, p. 179). From "Distributed memory and the representation of general and specific information," by J. McClelland and D. Rumelhart, 1985, Journal of Experimental Psychology: General, 114. Copyright 1985 by the American Psychological Association, Inc. Permission pending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
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(A) Firing rates of neurons in prefrontal cortex of monkeys viewing familiar or novel visual objects, as reported by Rainer and Miller (2000). Firing rates were lower throughout stimulus processing for familiar relative to novel stimuli. From "Effects of visual experience on the representation of objects in the prefrontal cortex," by G. Rainer and E. Miller, 2000, Neuron, 27, p. 181. Copyright 2000 by Cell Press. Permission pending. (B) Firing rates of inferotemporal neurons in monkeys performing a delay match-to-sample task show similar decreases as a function of repetition at short delays (several seconds: Sample vs. Match) and longer delays (several minutes: 1st vs. 2nd presentation), as reported by Li, Miller, and Desimone (1993). From "The representation of stimulus familiarity in anterior inferior temporal cortex," by L. Li, E. Miller, and R. Desimone, 1993, Journal of Neurophysiology, 69, p.1925. Copyright 1993 by the American Physiological Society. Permission pending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Firing rates of neurons in monkey motor cortex (FEF) during the performance of a saccade initiation task, as reported by Schall (2001). Reaction times were reliably associated with the amount of time required for movement-related cells in FEF to reach a fixed firing-rate threshold. From "Neural basis of deciding, choosing, and acting," by J. Schall, 2001, Nature Reviews Neuroscience, 2, p. 38. Copyright 2001 by Macmillan Magazines Ltd. Permission pending. ...................................... 1.5 Short-term repetition priming effects in human subjects performing a lexical decision task, as reported by McKone (1998). Priming effects show large reductions in magnitude over several seconds, regardless of whether intervening trials are present during the prime-probe delay. From "The decay of short-term implicit memory: unpacking lag," by E. McKone, 1998, Memory & Cognition, 26, p.1181. Copyright 1998 by Psychonomic Society, Inc. Data re-graphed, permission pending. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Spike synchrony across a population of randomly spiking cells leads to fluctuations in the population-averaged firing rate. In contrast, asynchronous or independent random firing across the same population leads to a much more stable estimate of firing rate. The results shown are for 1000 spiking cells, each firing at 50 Hz with correlated random (partially synchronized: every cell shares identical spike times with 5% of the other cells) or independent random spike times generated from a homogenous Poisson distribution. Firing rate across the population was calculated in 10-ms bins at increments of 10 ms. The asynchronous case better meets the stationary requirement of input spiking for the derivation of firing-rate from spiking networks (e.g. Amit & Tsodyks, 1991; Gerstner, 1995; Wilson & Cowan, 1972). Firing rate networks will therefore be better approximations of spiking networks in this case. ................................
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(A) Synchrony is unstable in Type I model cells with excitatory coupling, and (B) synchrony is stable, as is anti-synchrony in this case, for Type I cells with inhibitory coupling. The stability of particular phase-locked solutions is determined by the slope of Hodd where it crosses ω1-ω2: Positive slope indicates stability and negative slope indicates instability (see text for explanation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heterogeneity of firing rate (indicated by non-zero ω1-ω2) degrades the stability of the synchronous solution in two Type I neurons with inhibitory coupling. Since Hodd is always 0 at the beginning and end of the period of firing, it cannot be equal to ω1-ω2. It should be noted that this depiction is only for illustrative purposes and is not precise because the calculation of Hodd here actually assumes equal firing rates. .................... Network architecture assumed in analyzing the interaction of coupled excitatory (e) and inhibitory (i) neuron pairs (e.g. Ermentrout et al., 2001). . . Short-term repetition of odor puffs to the antennae of a locust leads to decreased spike rates in antennal lobe neurons while at the same time leading to enhanced synchronization, as reported by Stopfer and Laurent (1999): (A) local field potentials (LFPs) and voltage traces of projection neurons (PNs) initially exhibit relatively random timing relationships, yet after 12-15 odor puffs, spikes in the PNs are locked to the peaks of the LFP; (B) Firing rates in the PNs decrease across repeated odor puffs, while simultaneously showing greater coherence with the LFP - indicating better spike synchrony across the population. From "Short-term memory in olfactory network dynamics," by M. Stopfer and G. Laurent, 1999, Nature, 402, p. 664. Copyright 1999 by Macmillan Magazines Ltd. Permission pending. .................... Architecture of the basic model used for Simulations 1-5. Poisson-spiking excitatory input neurons (N=1000) provide input to 250 excitatory and 50 inhibitory spiking cells that are sparsely interconnected. .............. The x and s gating variables that mediate AMPA synaptic currents are shown in (A) and (B) for a random Poisson input spike train. The resultant EPSPs induced in a post-synaptic cell are shown in (C). (D)-(F) show the same information for a GABAergic input. .......................... (A) The impact of firing-rate adaptation on a cell's spiking responses when driven with a constant input current. Adaptation leads later spikes to be spread further apart relative to initial spikes, and the removal of current reveals a slowly recovering after-hyperpolarization. (B) shows the same cell and applied current as in (A), except with adaptation blocked. Spikes have been added to the voltage traces in order to aid visualization. . . . . . . . . Dynamics of synaptic depression at excitatory versus inhibitory synapses. (A) shows that depression is stronger at excitatory (De) than at inhibitory synapses (Di) for the same fixed firing rate. (B) shows the reductions in the heights of EPSPs due to synaptic depression at AMPA synapses, and (C) shows the same information for inhibitory synapses. Consistent with (A), reductions are more severe at excitatory synapses. . . . . . . . . . . . . . . . . . . . .
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A sample stimulus presented to the model. (A) shows the sine-modulated input spike trains across all 1000 excitatory input neurons (each dot corresponds to the occurrence of an individual spike), and (B) shows more detail for neurons 1-20. ...................................... Simple method for calculating coherence between two spike trains used by White, Chow, Ritt, Soto-Trevino, and Kopell (1998). Two spike trains shown in (A) and (B) are replaced by square pulses with a width of 20% of the period of the fastest firing cell (T1 here). The shaded area in D shows the overlap between the two square-pulse trains that is summed over time and used to calculate coherence. See text for more details. From "Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons," by J. White, C. Chow, J. Ritt, C. Soto-Trevino, and N. Kopell, 1998, Journal of Computational Neuroscience, 5, p. 7. Copyright 1998 by Kluwer Academic Publishers. Permission pending. . . . . . . . . . . . . . . A comparision between the simple measure of coherence used by White et al. (1998) and the modification used in the current simulations that involved a time-varying estimate of firing rate. Two sample spike trains are shown in (A) over a short time window. The pulse trains and overlap generated using time-varying firing rate are shown in (B) and the trains using fixed firing rate (as in White et al., 1998) are shown in (C). Short bursts of firing rate that are asynchronous give less overlap and lower coherence values using the method in (B) than in (C). . . . . . . . . Effect stimulus repetition and inter-stimulus interval (ISI) on firing rates in monkey inferotemporal cortex (A) reported by Miller, Gochin, and Gross (1991) and in the model (B). The changes in firing rate that build up across repetitions in both cases are smaller with longer ISIs. From "Habituationlike decrease in the responses of neurons in inferior temporal cortex of the macaque," by E. Miller, P. Gochin, and C. Gross, 1991, Visual Neuroscience, 7, p. 360. Copyright 1991 by Cambridge University Press. Permission pending. .................................................. Schematic diagram of the delay match-to-sample (DMS) paradigm used with macaque monkeys by Miller, Li, and Desimone (1993). A sample stimulus is presented for 500 ms, followed by between 0-4 distractor or "nonmatching" stimuli (each also presented for 500 ms and separated by 700-ms delays), and finally by the matching stimulus that required the monkey to respond by releasing a bar. From "Activity of neurons in anterior inferior temporal cortex during a short-term memory task," by E. Miller, L. Li, and R. Desimone, 1993, Journal of Neuroscience, 13, p. 1461. Copyright 1993 by the Society for Neuroscience. Permission pending. . . . . . . . . . . . . . .
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3.10 (A) Effect of repeating a stimulus (Sample to Match) on the firing rates of monkey inferotemporal cortex cells as a function of the number of intervening nonmatching stimuli in the DMS task, as reported by Miller, Li, and Desimone (1993). Responses to nonmatching stimuli are also shown. (B) Analogous results for the model. The difference between match and nonmatching stimuli decreased with more intervening stimuli. From "Activity of neurons in anterior inferior temporal cortex during a short-term memory task," by E. Miller, L. Li, and R. Desimone, 1993, Journal of Neuroscience, 13, p. 1465. Copyright 1993 by the Society for Neuroscience. Permission pending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 This figure re-graphs the differences between firing rates to match and nonmatch stimuli shown in Figure 3.10 for (A) the empirical data (Miller, Li, and Desimone, 1993) and (B) the model. These differences decrease similarly for both as a function of the number of intervening nonmatch stimuli. (A) also plots a dotted line corresponding to the monkeys' level of correct performance in the DMS task, showing that performance is correlated with the difference between match and nonmatch firing rates. From "Activity of neurons in anterior inferior temporal cortex during a short-term memory task," by E. Miller, L. Li, and R. Desimone, 1993, Journal of Neuroscience, 13, p. 1467. Copyright 1993 by the Society for Neuroscience. Permission pending. .................................................. 3.12 Distributions of repetition suppression indices calculated for the empirical data (A-D), as reported by Miller, Li, and Desimone (1993), and for the model (E). Negative values of the suppression index indicate greater firing rates to the nonmatch and sample stimuli than to the matching stimuli. (A) match-nonmatch effects for stimuli that evoked match-nonmatch differences, (B) match-sample effects for stimuli that evoked match-nonmatch differences, (C) match-nonmatch effects for stimuli that evoked abovebaseline responses, (D) match-sample effects for stimuli that evoked abovebaseline responses, (E) match-sample effects in the model. From "Activity of neurons in anterior inferior temporal cortex during a short-term memory task," by E. Miller, L. Li, and R. Desimone, 1993, Journal of Neuroscience, 13, p. 1464. Copyright 1993 by the Society for Neuroscience. Permission pending. .................................................. 3.13 The degree of repetition suppression in the model from Stimulus 1 to Stimulus 2 was larger for larger Stimulus 1 firing rates. The negative suppression indices observed in Figure 3.12 were for smaller firing rates to Stimulus 1. This indicates that for the model, firing rate decreases will be largest for cells that respond most vigorously to a stimulus, leading to decreased stimulus selectivity. ................................
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3.14 Match suppression effects, as reported by Miller, Li, and Desimone (1993), were observed for stimuli that evoked larger firing rates (A), whereas match enhancement effects were observed for stimuli that evoked smaller firing rates (B). This is potentially analogous to the results shown for the model in Figure 3.13. From "Activity of neurons in anterior inferior temporal cortex during a short-term memory task," by E. Miller, L. Li, and R. Desimone, 1993, Journal of Neuroscience, 13, p. 1466. Copyright 1993 by the Society for Neuroscience. Permission pending. . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.15 Stimulus-selectivity in individual IT cells as a function of short-term stimulus repetition (Match vs. Nonmatch), as reported by Miller, Li, and Desimone (1993). Repetition suppression effects were not largest for poor stimuli (as defined by the firing rate elicited to each stimulus as Sample). Cells either showed comparable decreases for all stimuli or showed disproportionately large decreases for the best stimuli. This indicates that stimulus repetition led to overall decreases in stimulus selectivity. From "Activity of neurons in anterior inferior temporal cortex during a short-term memory task," by E. Miller, L. Li, and R. Desimone, 1993, Journal of Neuroscience, 13, p. 1463. Copyright 1993 by the Society for Neuroscience. Permission pending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.16 Repetition suppression effects observed in an fMRI study of a delay matchto-sample (DMS) task in humans, as reported by Jiang, Haxby, Martin, Ungerleider, and Parasuraman (2000) (A, B), and similar effects in the model (C). Short-term repetition of targets and distractors led to decreased hemodynamic responses in ventral temporal cortices (A) but not in frontal/insular cortices (B). Decreased activity in the model was comparable in magnitude and time course to that observed for distractor stimuli in (A). From "Complementary neural mechanisms for tracking items in human working memory," by Y. Jiang, J. Haxby, A. Martin, L. Ungerleider, and R. Parasuraman, 2000, Science, 287, p. 644. Copyright 2000 by the American Association for the Advancement of Science. Permission pending. . . . . . . . . 94 3.17 Schematic diagram of the fMR-Adaptation paradigm used by Grill-Spector and Malach (2001). Scans were for a total of 32 seconds during which 32 picture stimuli were presented at 1 per second. Different numbers of picture stimuli were presented repeatedly during the scan (1, 2, 4, 8, and 32) to yield different amounts of fMR-Adaptation. One stimulus presented repeatedly over the 32 seconds corresponded to 32 identical repetitions, whereas 32 stimuli presented over the 32 seconds implied a single presentation for each stimulus. From "fMR-adaptation: a tool for studying the functional properties of human cortical neurons," by K. Grill-Spector and R. Malach, 2001, Acta Psychologica, 107, p. 298. Copyright 2001 by Elsevier Science B.V. Permission pending. . . . . . . . . . . . . . . . . . . . . . . . . . . 98
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3.18 fMR-Adaptation effects as a function of time for the different repetition conditions used in Grill-Spector and Malach (2001). (A) The greatest hemodynamic response decreases in humans were observed in the 1-stimulus condition, and the weakest decreases were observed in the 32-stimulus condition. (B) The model produced similar decreases in average firing rate. See text for explanation. From "fMR-adaptation: a tool for studying the functional properties of human cortical neurons," by K. Grill-Spector and R. Malach, 2001, Acta Psychologica, 107, p. 303. Copyright 2001 by Elsevier Science B.V. Permission pending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.19 Average activity decreases in the fMR-Adaptation paradigm for the different repetition conditions. The adaptation ratio corresponds to the hemodynamic response (or firing rate) when normalized by the responses in the 32-stimulus condition. Results are plotted in as a function of the number of repetitions used in each condition rather than as a function of number of stimuli; (A) shows results for the human data in Grill-Spector and Malach (2001), and (B) shows the same results calculated for the model. From "fMRadaptation: a tool for studying the functional properties of human cortical neurons," by K. Grill-Spector and R. Malach, 2001, Acta Psychologica, 107, p. 305. Copyright 2001 by Elsevier Science B.V. Permission pending. . . 102 3.20 Both the means (A) and the standard deviations (B) of firing rates in the model decrease as a function of stimulus repetition. (B) also shows that increases in input heterogeneity from 0 to 60% of the mean number of excitatory inputs leads to increases in the standard deviation of firing rates. 108 3.21 Coherence values among the 250 excitatory cells in the model as a function of stimulus repetition and level of input heterogeneity (0-60%). Repetition leads to increases in coherence for all of the heterogeneity conditions, but the increases are smaller for larger values of heterogeneity. . . . . . . . . . . . . . . 110 3.22 Raster plots on a sample run of the model for Stimulus 1 (A), Stimulus 2 (B), Stimulus 5 (C), and Stimulus 10 (D). The spike times of the 300 cells (1-250 excitatory and 251-300 inhibitory) over the course of each stimulus (1000-1500 ms) are indicated in the plots by individual black dots. It is apparent that large changes in spike synchronization occur from Stimulus 1 to Stimulus 2 (A vs. B). High levels of synchronization are maintained across stimulus repetitions (B-D). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.23 The impact of synaptic delays (0-2 ms) on repetition-related changes in coherence. (A) shows changes in coherence among the excitatory cells, whereas (B) shows the changes among the inhibitory cells. (A) shows that coherence increases are robust for each synaptic delay condition, whereas (B) shows a more complicated relationship for the inhibitory cells. See text for explanation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
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3.24 Changes in firing rate and coherence for two inter-connected cortical regions separated by synaptic delays of 5 ms (within-region delays = 1 ms). Decreases in firing rate across stimulus repetitions are observed in both regions. Increases in coherence are also observed in both regions as long as the excitatory feedback from Region 2 to Region 1 is relatively weak (0-10%). Coherence increases in Region 1 are observed regardless of the amount of feedback. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.25 The effect of inter-stimulus interval (ISI) on the magnitude of repetition priming effects in humans (A), as reported by McKone (1998), and the effect of ISI on changes in coherence observed in the model for simulations in Section 3.2.2 (B) and Section 3.2.4 (C). Repetition priming effects and coherence changes are reduced by longer delays between Stimulus 1 and 2, although this trend in the model is more rapid than for human priming data. From "The decay of short-term implicit memory: unpacking lag," by E. McKone, 1998, Memory & Cognition, 26, p.1181. Copyright 1998 by Psychonomic Society, Inc. Data re-graphed, permission pending. . . . . . . . . 3.26 The effect of blocking inhibition (IGABA), firing-rate adaptation (IK(Ca)), and synaptic depression on firing rates (A) and coherence values (B). Blocking all three mechanisms led to increases in the means and standard deviations of firing rate and decreases in coherence. .......................... 3.27 The effect of blocking inhibition (IGABA), firing-rate adaptation (IK(Ca)), and synaptic depression on coherence values when asymptotic firing rates (mean and standard deviation) were matched across conditions. Blocking adaptation or synaptic depression reduced coherence values, whereas blocking inhibition actually increased them. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.28 The impact of changes in input firing rate, input coherence (for the posterior cortical inputs only), and balanced excitatory/inhibitory synaptic currents on the firing rate of a single output cell that represents motor cortical responses. (A) shows the results for different levels of excitatory/inhibitory balance with adaptation and synaptic depression completely blocked (evenly balanced: gEI=0.3, stronger excitation: gEI=0.15), whereas (B) shows the same results for 20% adaptation and synaptic depression. See text for explanation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.29 The impact of changes in input firing rate, input coherence (for all inputs), and balanced excitatory/inhibitory synaptic currents on the firing rate of a single output cell that represents motor cortical responses. (A) shows the results for different levels of excitatory/inhibitory balance with adaptation and synaptic depression completely blocked (evenly balanced: gEI=0.3, stronger excitation: gEI=0.15), whereas (B) shows the same results for 20% adaptation and synaptic depression. See text for explanation. ........
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3.30 The impact of changes in input firing rate and input coherence (in the posterior cortical inputs only) on reaction time for evenly balanced excitatory/inhibitory synaptic currents and no adaptation or synaptic depression. (A) shows that increases in input firing rate generally lead to faster reaction times, as do increases in input coherence, shown in (B). (C) shows that the coherence-related changes in reaction times happen mainly in the higher range of input firing rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.31 The impact of stimulus repetition and input heterogeneity on reaction time for the simulations presented in Section 3.3.1. Reaction times are decreased strongly for all heterogeneity conditions from Stimulus 1 to Stimulus 2, showing significant repetition priming effects. Reaction times increase slightly in all conditions for later stimulus repetitions, although they are generally still faster than the response to the first stimulus. . . . . . . . . . . . . . . 3.32 Changes in a simple measure of metabolic inefficiency, the average number of spikes in each cell prior to a response, are shown for the four levels of input heterogeneity explored in Simulations 2-4 (+/- 0, 20, 40, and 60%). Inefficiency was reduced similarly for each heterogeneity condition across stimulus repetitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.33 Changes in the scale-free measure of metabolic efficiency, ∆Efficiency, across stimulus repetitions are shown for the four levels of input heterogeneity explored in Simulations 2-4 (+/- 0, 20, 40, and 60%). Efficiency was increased similarly for each heterogeneity condition; changes in all cases are relative to the firing rates and reaction times to Stimulus 1. . . 4.1 Network architecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Effect of time-varying pre-synaptic firing rate on synaptic depression in the spiking Varela et al. (1999) model and the firing-rate connectionist approximation. ............................................ 4.3 Pre-synaptic effect of neuromodulation on transmitter release and the buildup of synaptic depression. Depicted on the y-axis is the post-synaptic net input (η) for a single synapse (equal to pre-synaptic activity, sinusoidally modulated, multiplied by a weight of 1.0 and scaled by the release factor, ρ(M), and the depression factor, D). Two different levels of neuromodulation are shown for comparison, very low (M=-4.0) and moderately high (M=2.0). ...................................... 4.4 Effect of neuromodulation on post-synaptic activity. Shown on the y-axis is unit activity, a(η) (function of net input, η, with the gain, g( ), dependent on neuromodulation, M). The increase in gain/sensitivity is apparent for moderately high levels of neuromodulation (M=2.0) compared to very low levels (M=-4.0). ............................................ 4.5 Effect of rate in the model under different damage combinations and the same effect in the patient data. The values of M and % lesioned connections are listed for each damage combination. .......................... 4.6 Effect of semantic relatedness in the model under different damage combinations and the same effect in the patient data. .............. 4.7 Effect of frequency in the model under different damage combinations and the same effect in the patient data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.8 4.9
4.10
4.11
4.12
4.13
Effect of serial position in the model under different damage combinations (fast presentation rate conditions, RSI = 1 sec). .................... Surface plots showing the directions and magnitudes of the rate, semantic relatedness, frequency, and serial position effects for the entire space of damage combinations. For each effect, the x- and y-axes represent the two damage types (neuromodulation, M, and damage to connections, % Lesion), and the z-axis represents the magnitude of the effect in terms of proportion correct (e.g. the slow minus the fast condition for the rate effect, distant - close for the semantic relatedness effect, etc.). The magnitudes of the effects are also represented in grayscale, where white corresponds to the maximum effect value and black corresponds to the minimum value; colorbars to the right of each plot indicate scale. .................... The impact of stimulus repetition and neuromodulatory level, M, on firing rate means (A) and firing rate standard deviations (B). Stimulus repetition reduced both the means and standard deviations of firing rates, although rates were larger overall for higher levels of M. .................... Impact of stimulus repetition and neuromodulatory level, M, on coherence values among the excitatory cells. Coherence increased strongly across repetitions for moderate values of M but not for low values (little or no increases for M d[k] Dj[k]
(0 < d[k] < 1)
dDj[k]/dt = (1 - Dj[k]) / τ[k]
where d[k] determines the amount of depression for each pre-synaptic spike (0.78 for the fast component, 0.97 for the slow component), and τ[k] determines the rate of recovery back to a value of 1 (.634 sec for the fast component, 9.3 sec for the slow component). These two equations were approximated with a single equation, weighting the depression and recovery terms by an estimate of the likelihood of a spike versus no spike for a given value of aj:
dDj[k]/dt = amax . aj . ρ(M) (d[k] - 1) Dj[k] + (1 - amax . aj . ρ(M)) (1 - Dj[k]) / τ[k]
where amax is the maximum firing rate (spikes/millisecond = 0.03 or 30 Hz) to which a unit activity value of 1 is supposed to correspond. Values for both d[k] and τ[k] were taken directly from the Varela et al. model, and as such, are not free parameters. For simplicity, different depression factors were not implemented at positive versus negative connections (to represent weaker depression at inhibitory synapses), and the firing-rate approximation for depression at excitatory synapses was used at all connections. Note that Dj actually follows the product aj . ρ(M) rather than aj by itself, roughly instantiating the notion that synaptic depression depends on transmitter release21.
21
Given that the dynamics of each depression factor at excitatory synapses depend only on itself and not on . . the product of the two factors, ρ(M) D[f] D[s] is not precisely instantaneous release probability but rather an approximation of it.
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Figure 4.2: Effect of time-varying pre-synaptic firing rate on synaptic depression in the spiking Varela et al. (1999) model and the firing-rate connectionist approximation.
Figure 4.2 shows that this implementation of synaptic depression produces depression dynamics that are virtually identical to those of the original Varela et al. (1999) implementation. Firing rate of a single pre-synaptic cell was sine-modulated with a period of 500 ms and amplitudes ranging between 0 and 30 Hz; spikes were Poissondistributed for runs of the Varela et al. (1999) model, and depression values were averaged over a total of 100 runs.
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4.2.3
Neuromodulation
As discussed above, two of the main actions of acetylcholine and norepinephrine in the neocortex are to suppress transmitter release pre-synaptically and to increase the sensitivity of cells post-synaptically to excitatory inputs. Both of these actions were implemented, as they were expected to influence the extent to which network dynamics were dominated by synaptic depression. When transmitter release is suppressed, the amount of synaptic depression is reduced. Because the sensitivity of post-synaptic cells to excitatory inputs is enhanced at the same time, neural firing should be sustained with less attenuation. If the concentration of neuromodulators is reduced due to damage, the network should be more dominated by synaptic depression and should exhibit refractorylike effects. The pre-synaptic effect of neuromodulation is to scale down the value of presynaptic activity, simulating pre-synaptic suppression of transmitter release and reducing the buildup of synaptic depression. The release scaling factor, ρ, follows a decreasing sigmoid function of the level of neuromodulator, M, from a maximum value of 1 for large negative values of M down to a minimum value, ρmin, of 0.2 for large positive values of M:
ρ(M) = ρmin + (1- ρmin) e-M / (1 - e-M)
ρmin here was chosen to be consistent with the experimental results of Hasselmo and Bower (1992), which showed that transmitter release under high concentrations of acetylcholine saturated at 20-30% of the levels measured in the absence of acetylcholine. At large positive values of M, ρ scales down the pre-synaptic unit activity by 0.2, leading
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Figure 4.3: Pre-synaptic effect of neuromodulation on transmitter release and the buildup of synaptic depression. Depicted on the y-axis is the post-synaptic net input (η) for a single synapse (equal to pre-synaptic activity, sinusoidally modulated, multiplied by a weight of 1.0 and scaled by the release factor, ρ(M), and the depression factor, D). Two different levels of neuromodulation are shown for comparison, very low (M=-4.0) and moderately high (M=2.0).
to a weaker buildup of synaptic depression. However, this also has the effect of scaling down the impact of the pre-synaptic activity on other units in the network. Therefore, at high levels of neuromodulation, synaptic depression is less extreme, although input activity to subsequent units is reduced overall. This is depicted graphically in Figure 4.3 for a single synapse with time-varying sinusoidal pre-synaptic activity. In contrast, for large negative values of M (low levels of neuromodulation), synaptic depression is more extreme and the resulting input activity to other units is initially much higher because transmitter release is not suppressed. However, after repeated stimulation, the input to 180
other units is comparable to that for high levels of neuromodulation and can even cross over to lower input values. The post-synaptic effect of neuromodulation is to block firing-rate adaptation effects that otherwise reduce sensitivity to excitatory inputs and attenuate spiking. This means that neuromodulation leads to higher and more sustained spiking than when given identical input in the absence of neuromodulation. These changes in sensitivity were instantiated by including a multiplicative scaling factor, g (for "gain"), on the net input,
ηi, to each unit. The value of g was an increasing sigmoid function of M, the level of neuromodulation; g ranged between a minimum of gmin for large negative values of M and gmax for large positive values of M:
g(ηi, M) = ηi (gmin + (gmax - gmin)/(1 + e-M)) + βi
where βi is a bias term for the ith unit (< -3.0 in our simulations to set low baseline unit activity). To remain consistent with observations that acetylcholine and norepinephrine enhance spiking responses mainly to depolarizing/excitatory input, g( ) was only applied if ηi > 0; g( ) simply returned ηi + βi for ηi < 0. Choices of gmin and gmax were relatively unconstrained, although a ratio of gmax / gmin was chosen that was plausible given experimental results (15/7.5 = 2.0 for our simulations). For comparison, Barkai and Hasselmo (1994) observed 2-3 times the control spiking response following the introduction of acetylcholine22.
22
Barkai & Hasselmo (1994) emphasized the importance of dynamic changes in post-synaptic excitability, such as those due to firing-rate adaptation used in Simulations 1-4. For simplicity, we have ignored these dynamics in the current context because the larger changes in excitability that they consider build up/recover nearly completely within 100 ms (somewhat slower and considerably weaker changes recover within 500-600 ms). These changes are nearly instantaneous relative to the time scale of updates to unit
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Figure 4.4: Effect of neuromodulation on post-synaptic activity. Shown on the y-axis is unit activity, a(η) (function of net input, η, with the gain, g( ), dependent on neuromodulation, M). The increase in gain/sensitivity is apparent for moderately high levels of neuromodulation (M=2.0) compared to very low levels (M=-4.0).
The enhancing effect of neuromodulation on post-synaptic sensitivity is shown in Figure 4.4 for different values of M. While this implementation of neuromodulation is highly simplified, it is broadly consistent with a range of empirical findings and previous approaches to simulating neuromodulatory mechanisms (e.g. Cohen & Servan-Schreiber, 1992; Gil et al., 1997; Hasselmo & Bower, 1992; Tsodyks & Markram, 1997; although see Barkai & Hasselmo, 1994).
activities in our simulations (every 50 ms), and the refractory phenomenon that we consider occurs on the longer time scale of seconds. As discussed in Section 1.1, firing-rate adaptation effects can be much slower (Sanchez-Vives, Nowak, & McCormick, 2000), although adding such effects would not substantially alter the behavior of the model because they are expected to be stronger under low levels of neuromodulation and weaker under normal levels.
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4.2.4
Training Procedure
4.2.4.1 Input Patterns One hundred twenty-eight artificial "words", each composed of a pattern of on and off unit activities, were constructed to be presented to the network. For each word, 4 of the 30 phonological units were randomly selected to be active. The only constraint on the randomized pattern generation was that each pattern be unique.
4.2.4.2 Target Patterns One hundred twenty-eight target artificial semantic patterns, each corresponding to a single input word, were designed by picking 10 of the 150 semantic units to be on. These units were chosen such that patterns clustered into 8 different semantic categories, 16 patterns per category. Within a category, 8 of the 16 patterns were randomly generated to be "closely" related to one another (average pair-wise normalized dot product of 0.493), and 8 were generated to be "distantly" related (average pair-wise normalized dot product of 0.266). The relatedness between different categories was much lower (average pairwise normalized dot product of 0.044). Each target semantic pattern was then paired at random with an input word to instantiate the assumption that the phonology of a word is more or less arbitrarily related to its meaning (see Plaut & Shallice, 1993b, for discussion). The current model was trained using an iterative version of the back-propagation learning procedure known as back-propagation through time (Rumelhart, Hinton, & Williams, 1986; Williams & Peng, 1990). Half of the 128 training patterns were assigned
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to be high frequency (presented 20 times as often during training) and half low frequency, crossing training frequency with semantic relatedness (close vs. distant). The level of neuromodulation (M) during training was set to a moderate positive value (+2.0 in our simulations) so that there was a moderate degree of synaptic depression being applied throughout learning.
Training was considered complete when the activity for
each semantic unit at the end of stimulus presentation was on the correct side of 0.5 (either greater than 0.5 for an "on" unit or less than 0.5 for an "off" unit). The network reached this level of accuracy after approximately 500 passes through the entire training set.
4.2.4.3 Time Course of a Single Training Pattern The time course of presentation for a single word was composed of two distinct periods. The first period served as the proxy for response-stimulus interval (RSI) in the spokenword/picture matching task, or more generally, the latency between words in the speech stream. In this period (4-60 units of network time sampled uniformly, where 4 units of network time is assumed to equal 1 actual second), zero's were presented across the phonological input units, and no target values were presented to the semantic units. During the second period of time (2 units of network time or 500 msec), the input word was presented with its corresponding target pattern presented at semantics. Presentation of the next training trial followed immediately with no unit reinitialization.
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4.2.5
Lesioning Procedure
By hypothesis, damage to connections between the hidden and semantic layers with normal levels of neuromodulation (M) should produce a degraded-store pattern of performance. Low levels of neuromodulation should instead lead to an access/refractory pattern.
"Damage" was carried out by randomly zeroing a proportion of synaptic
strengths between hidden and semantic units (ranging from 0 to 95% in steps of 5%). A neuromodulatory deficit was simulated simply by reducing the value of M (ranging from +2.0, used during training in the intact or "normal" system, down to -4.0 in steps of 1.0; the sigmoidal nature of the pre- and post-synaptic neuromodulatory functions guaranteed asymptotic
values
near
+/-
4.0).
Each
combination
of
damage
proportion/neuromodulation was repeated 80 times during testing (10 times for the testing of each of the 8 semantic categories) in order to insure a stable estimate of performance.
4.2.6
Testing Procedure
Following each instance of damage, the trained network was presented with 4 types of arrays of 4 words each, as in Warrington & Cipolotti (1996): Close/High Frequency, Close/Low Frequency, Distant/High Frequency, Distant/Low Frequency. In each testing block, all 4 words in one of the arrays were probed 3 times in a pseudorandom order and at a fixed presentation rate (RSI of either 4 or 60 time units, representing 1 sec vs. 15 secs). Each array was presented at both a fast rate and a slow rate. The pattern of semantic activity generated by an input word was compared with the target semantic patterns of all four words in the array. The best match was taken to
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be the network's response, unless the Sum Squared Error (SSE) between actual and bestmatch target was larger than an arbitrary criterion value (7.0 for our simulations). In these cases, responses were selected at random with a likelihood that was directly proportional to the amount that the criterion was exceeded (up to a likelihood of 1.0 at SSE=10.0). This appropriately penalized cases in which unit activities (i.e. firing rates) were so low or divergent from known patterns that they would not be reasonably expected to support processing at points in the cognitive system subsequent to semantics. We would suggest that patients, in the absence of reliable information available from the stimulus, may similarly "guess" in alternative-forced-choice paradigms23.
4.3 Simulation 6: Accounting for the Basic Contrast Between Access/Refractory and Degraded-Store Patient Performance This simulation demonstrated that a degraded-store pattern of impairment can be produced under severe damage to hidden=>semantics connections and normal levels of neuromodulation, whereas an access/refractory pattern can be produced under severe neuromodulatory damage and little or no damage to hidden=>semantic connections. Progressively surveying damage combinations from one extreme to the other and keeping overall correct performance in a comparable range (around 50% correct), a gradual transition was observed between the two types of performance patterns. Figures 4.5-4.8
23
A similar approach has been taken by Plaut and Shallice (1993b) in simulating omission errors in visual object naming. The basic idea is that if a pattern of activity is too far away from any known pattern (past some criterion value), it cannot support correct identification performance.
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establish these points for the stimulus factors of rate, semantic relatedness, frequency, and serial position; performance of individual patients from Warrington and Cipolotti (1996) are also shown for comparison. For example, Figure 4.5 shows the proportion correct under fast (RSI=1 sec) and slow (RSI=15 sec) presentation rates for each damage combination. The model produces rate effects under severe neuromodulatory damage (e.g. M=-4 to -2) that are comparable in magnitude to those exhibited by the access/refractory patients A1 and A2. As neuromodulatory damage becomes less severe and damage to connections becomes more severe (left to right in the figure), the rate effects diminish.
Under severe damage to connections with normal levels of
neuromodulation (M=2, % Lesion=70%), the lack of rate effect is comparable to that exhibited by degraded-store patients S1-S4.
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Figure 4.5: Effect of rate in the model under different damage combinations and the same effect in the patient data. The values of M and % lesioned connections are listed for each damage combination.
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Figure 4.6: Effect of semantic relatedness in the model under different damage combinations and the same effect in the patient data.
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The effect of semantic relatedness for different damage combinations is shown in Figure 4.6 along with patient performance.
While the effects in the model are of
somewhat smaller magnitudes than those exhibited by the patients, they show the appropriate pattern. Performance is better for semantically distant arrays relative to close arrays under severe neuromodulatory damage and little or no damage to connections. As neuromodulatory damage becomes less severe and damage to connections becomes more severe, the semantic relatedness effects gradually diminish and/or reverse. Figure 4.7 shows effects of frequency for the different damage combinations along with patient data. Effects of frequency are small for severe neuromodulatory damage, and they grow to large magnitudes that are comparable to those of patients S1S4 as neuromodulatory damage becomes less severe and damage to connections becomes more severe. Interestingly, moderate damage to both connections and neuromodulation (e.g. M=-2 to -1) produces frequency effects that are quite substantial while producing rate and semantic relatedness effects at the same time (discussed in more detail in the next section). Effects of serial position under a fast rate of presentation (RSI=1 sec) are shown in Figure 4.8 for each damage combination. It should be noted that the serial position effects for individual patients in Warrington and Cipolotti (1996) are not reported in terms of proportion correct for each within-block repetition but instead as the number of times a response changed from correct to incorrect (or vice versa) across the first two repetitions.
More direct comparisons with patient data will be made below in the
statistical analyses of serial position. For now, it is sufficient to observe that performance decreases across within-block repetitions under severe neuromodulatory damage.
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Figure 4.7: Effect of frequency in the model under different damage combinations and the same effect in the patient data.
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Figure 4.8: Effect of serial position in the model under different damage combinations (fast presentation rate conditions, RSI = 1 sec).
As neuromodulatory damage becomes less severe and damage to connections becomes more severe, serial position effects gradually diminish. Out of the seven different combinations of damage that are depicted in Figures 4.5-4.8, statistical analyses of model performance were focused on two particular combinations that typify the two patient performance patterns: M=-3, % Lesion=5% for the access/refractory pattern and M=2, % Lesion=70% for the degraded-store pattern. Repeated-measures ANOVA's were run first with both damage combinations included together to evaluate the interaction of stimulus factors with damage combination. Separate ANOVA's were then conducted for each damage combination individually. Analyses were run using both damage repetition and training items as the random factor,
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although for simplicity only the results of the analyses over item data are reported as the two methods produced comparable results and the item analyses tended to be slightly more conservative (the standard error bars shown in Figures 4.5-4.8 are based on item data). For more direct comparisons with patient results, χ2 statistics were also calculated based on the mean estimates of proportion correct over the same number of experimental trials as administered to the patients in Warrington and Cipolotti's (1996) Experiment 224. A 2 (Damage) x 2 (Rate) x 2 (Semantic Relatedness) x 2 (Frequency) repeatedmeasures ANOVA was conducted on item data with Semantic Relatedness and Frequency as between factors and Damage and Rate as within factors. This was then followed by similar 2 (Rate) x 2 (Semantic Relatedness) x 2 (Frequency) ANOVA's calculated for each damage combination separately. There was a significant main effect of Damage [F(1, 124) = 4.67, p.3], suggesting that the two damage combinations are reasonably matched for overall performance. These levels of performance are slightly lower than the average levels of Warrington and Cipolotti's (1996) patients (S1-S4: 0.66; A1-A2: 0.64), although they are within the range of the most impaired patient [A1: 0.61 correct; M=2, 70% vs. A1, p>.2; M=-3, 5% vs. A1, p>.05].
24
As discussed by Warrington and McCarthy (1983) and Warrington and Cipolotti (1996), the independence assumption of the χ2 test is unlikely to be satisfied when refractory phenomena are present at time scales longer than the duration of individual trials. These results are presented mainly for purposes of comparison with the values reported for patients.
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4.3.1
Rate Effects
The Damage x Rate interaction was highly significant [F(1, 124) = 131.39, p= IC), and τu, ga, and sa are as in (1) and (2). Figure A.1.1a below shows traces of u as a function of time for particular values of I (I=0.5, 1.0, 1.5, and 3.0 mS/cm2; A=60 and IC=0.45 mS/cm2 match data from Traub & Miles, 1991, and are taken from Ermentrout, 1998a; τu=2 ms; adaptation parameters were chosen have slow kinetics similar to sodium-activated potassium currents and yield steady-state firing-rate decreases of 40-60%, Sanchez-Vives et al., 2000: ga=0.001, αa=0.002, τa=9 s). We consider very slow adaptation (and synaptic depression) dynamics on the order of seconds, as opposed to faster dynamics, because these are the dynamics that are most relevant for the discussion of behavioral priming.
Figure A.1.1a:
For each fixed value of I, u decreases monotonically down to a steady-state value (values of u are shown out to 30 seconds). Values of u are also higher for larger I. These points are perhaps better understood in the phase-plane, shown below for u versus sa in Figure A.1.1b.
279
Figure A.1.1b:
The nullcline for u (or sa) is a curve showing all of the values of u and sa for which du/dt=0 (dsa/dt = 0). Points of intersection between the nullclines of u and sa are referred to as “fixed points”; fixed points can be either stable (where small perturbations of either variable lead both variables back to the fixed point, and they stay there) or unstable (where small perturbations of either variable lead the variables to move away from the fixed point) (see Strogatz, 1994, for an introduction to phase-plane analyses of dynamical systems). Figure A.1.1b reveals a single stable fixed point for this system (I=1.6 mS/cm2 for this plot). Starting from initial values of u(0)=sa(0)=0, u climbs rapidly up the y-axis to its nullcline (approximately exponentially with τu = 2 ms) and then decreases along its nullcline as sa slowly builds up, stopping finally at the fixed point. sa changes so slowly in this case that it is approximately constant relative to the dynamics of u. We can see where these nullclines come from by setting du/dt = dsa/dt = 0 and solving. The nullcline for u is given by u = A I − I C − g a ⋅ sa and the nullcline for sa is given by
or
sa = α a ⋅τ a ⋅ u sa u= α a ⋅τ a
Because the nullcline of u decreases monotonically as sa builds up (determined in this case by the square-root function and the negative sign on sa), u will always decrease as a function of adaptation. This also implies that as long as the f-I curve is monotonically increasing over I, this property does not depend on the particular f-I curve. Increasing I 280
has the effect of shifting the nullcline for u to the right, whereas the nullcline for sa does not change; this means that the point of intersection between the nullclines will necessarily increase (for both u and sa), increasing the values of u for all t. Also notice that the degree of adaptation exhibited by u is not the same for each value of I (see Figure A.1.1a). Adaptation by u appears to be proportionally larger at lower values of I, whereas the absolute change increases with I over a lower range of I (I's leading to unadapted values of u < 50 Hz). Larger absolute changes due to adaptation for increases in I (and hence u) are interesting because they replicate the results of the spiking neuron simulations over the same range of firing rates shown in Figure 3.13 (Section 3.2.2), in which the degree of repetition suppression is larger for larger initial firing rates. As we will see below, this property also holds for a linear f-I curve. However, for higher values of I (u's > 50 Hz), the absolute change in u due to adaptation is more comparable for different I's (leading the proportional change becomes less at higher values of I). The amount of change in u due to adaptation is jointly determined by the position and slope of the nullcline of u and the position and slope of the nullcline of sa between sa=0 and the fixed point. When I is close to IC, the nullcline of u is shifted far over to the left; the height of the unadapted u is not very different than the height of u at the fixed point. As I is increased, u's nullcline is shifted to the right, and its steep slope at small values of u leads to larger absolute adaptation changes. Absolute adaptation changes become more comparable at larger I's because of the flatter slope of u's nullcline at large values of u. Given that u[0] serves as input to u[1] and u[0] is decreasing, this will have a decreasing effect on u[1] (u[0]*w serves as the I to u[1], and w>0; decreasing the input to u[1], as for u[0], will decrease its unadapted value, its steady-state value, and all of the values in-between). u[1] will also undergo its own adaptation due to sa[1]. Since the network architecture is feed-forward, all of the u[n] will be attenuated by adaptation. Now suppose that the stimulation paradigm is modified so that I-IC is set to a positive value for a certain amount of time (e.g. 1 second) and then to 0 for a certain amount of time (e.g. 5 seconds), allowing adaptation to recover exponentially with time constant τa. If this is taken as the proxy for one stimulus repetition and is repeated several times, adaptation will build up across repetitions as long as the delay between stimuli is short relative to τa. By defining a response metric, one can then ask whether this network will exhibit repetition priming as a consequence of adaptation. If response generation is simulated by the n-th unit (representing motor cortex neurons) reaching a fixed criterion of firing rate, it is clear that following adaptation, the n-th unit will be less rather than more likely to reach this value. If this network serves as the input to a competitive decision network (such as that studied by Usher & McClelland, 2001) where reaction time is simulated by the amount of time it takes one of the units to reach an activity threshold, it is clear that the impact of adaptation will be to weaken input to the correct unit, requiring more time to reach threshold and slowing reaction times. In other words, adaptation will cause habituation rather than priming for this network and these response mechanisms.
281
Another common choice for an f-I curve is a clipped linear function:
τu if (I-gasa >0) else
du = f (I , s a ) − u dt
(6)
f (I , s a ) = I − g a ⋅ s a f (I , s a ) = 0
where ga and sa are as in (2). Traces of u(t) for different values of I are shown in Figure A.1.1c, along with the nullclines for this system (calculated at I=1.0 mS/cm2). Figure A.1.1c:
282
In this case, the nullclines are given by
u = I − g a ⋅ sa and
or
sa = α a ⋅τ a ⋅ u sa u= α a ⋅τ a
As in Figure A.1.1b, increasing I has the effect of shifting u's nullcline to the right, leaving s's nullcline unchanged. This means that the fixed point will necessarily increase as I increases, although it will always be less than the unadapted value of u, yielding activity decreases (this is not surprising given the discussion above for the square-root f-I because the nullcline of u is a monotonically decreasing function of sa). Substituting the nullcline of sa into the first equation, we can solve for the steady-state of u: u=
I 1 + g a ⋅ α a ⋅τ a
(7)
It is clear from this equation and the plots in Figure A.1.1c that the steady-state of u following adaptation with a linear f-I is always proportional to and less than the unadapted value; the slow approach of u to its steady-state once it reaches its nullcline is very close to a simple exponential as long as τa>>τu. (where the effective τ does not depend on u0). This means that the absolute change in u due to adaptation is always larger for larger I (and larger unadapted u).
A.1.2 Synaptic depression
Is the impact of synaptic depression on firing rates similar to that of firing-rate adaptation? While they do differ in a few details, they are the same with regard to playing on overall decreasing role on activity. Figure A.1.2a shows traces of unit u[1] that receives input from unit u[0], scaled by the synaptic depression factor D[0] and a constant weight w [see eqs. (3) and (4)]; for the sake of simplicity, u[0] was set to a constant value to observe the impact of synaptic depression on u[1] (as before, A=60, τu=2 ms; parameters for synaptic depression were taken from the slow depression component at excitatory synapses in the Varela, 1997, model: d=ln(.97) and τD=9.3 s).
283
Figure A.1.2a:
Like adaptation, synaptic depression reduces monotonically the input from u[0] to u[1] over time. However, in contrast to adaptation, the rate at which it does this depends on the value of u[0] (the larger u[0], the faster the approach to the steady state). To see why this is true, we can solve the dynamics of D[0] in eq. (3) for a constant u[0] (this is possible even with a non-linear f-I curve because synaptic depression has no direct influence on pre-synaptic activity here):
D
[0]
(t ) = D
[0]
∞
D [0] ∞ =
− (D
[0]
∞
−D
[0]
0
1 d ⋅ τ D ⋅ u [0 ] + 1
)⋅ e
−
(d ⋅τ
D ⋅u
)(t −t
[ 0 ] +1
τD
0
)
(8) (9)
where D[0] approaches its steady-state value D[0] exponentially from its initial value (D[0]0=1) with time constant τ D (d ⋅ τ D ⋅ u [0 ] + 1) . Since d and τD are positive values, 0 < D[0] < 1 for u[0]>0. Increasing u[0] leads both to a faster buildup of synaptic depression and a lower value of D[0] . However, the steady-state input to u[1] (the product u[0]*w* D[0] ) is still an increasing function of u[0] (the derivative of u[0]*w* D[0] wrt u[0] is positive for u[0], w>0; revealed in Figure A.1.2a by the higher steady-state value of u[1] when u[0]=75 Hz, compared with u[0]=25 Hz). This means that as with adaptation, activities will be decreased throughout the network, and habituation would be expected to result given the assumptions about the feed-forward architecture and mechanisms for response generation. The degree of the absolute decrease with a linear f-I is larger for larger initial activity; with a nonlinear f-I, the absolute activity decreases will depend on the magnitude of the input and the particular shape of the f-I curve (as with adaptation).
284
A.2 Feed-forward network representing two competing stimuli
In this section, we consider a network architecture (shown above) in which two different stimuli are represented that compete through feed-forward inhibitory connections. Units representing the target stimulus being presented are denoted by uT[k ] and are connected in a feed-forward manner by positive synaptic strengths we. Units denoted by u D[k ] can be thought of as representing either some distractor stimuli that detract from the processing of the target or a subpopulation of neurons that on average share an opponent relation with the neurons that critically represent the target stimulus, mediated by negative synaptic strengths wi. The opponent relation between uT and uD units is mutual, and individual uD units are also connected feed-forward by positive synaptic strengths we. IT represents input currents related to the target stimulus, and ID represents input currents related either to distractor stimuli or to partial support for competing stimuli due to the target stimulus. We assume that IT>ID and IT>0. It is also necessary to assume that we>wi>0 so that in the absence of adaptation/depression uT[k ] > u D[k ] for all k in (0, 1, ..., n).
A.2.1 Firing-rate adaptation
We will first consider the impact of firing-rate adaptation on the competitive feedforward network with the choice of a linear f-I curve. For this network with IT>ID and we>wi, adaptation will always have an attenuating impact on uT[k ] for all k in (0, 1, ..., n). To see this, refer back to Figure A.1.1c. The input to uT[k ] depends on the difference between uT[k −1] and u D[k −1] (specifically, uT[k −1] ⋅ we − u D[k −1] ⋅ wi ). Since this difference decreases as the u's approach their steady states (shown in Figure A.1.1c for different values of I), the input to the subsequent target unit uT[k ] decreases (and therefore uT[k ] decreases). The input is also less selective for the target stimulus, in the sense that the inputs to uT[k ] and u D[k ] become more similar. To see this, we ignore the rapid rise of the
285
u's to their nullclines (since τa>>τu) and consider the steady-state solution for uT[k ] - u D[k ] [similar to eq. (7)]: uT[k −1] ⋅ we − u D[k −1] ⋅ wi u D[k −1] ⋅ we − uT[k −1] ⋅ wi − 1 + g a ⋅ α a ⋅τ a 1 + g a ⋅ α a ⋅τ a which rearranges to
[(u [
k −1] T
) (
)] 1 + g
⋅ we − u D[k −1] ⋅ wi − u D[k −1] ⋅ we − uT[k −1] ⋅ wi ⋅
1 a ⋅ α a ⋅τ a
In other words, the unadapted difference between uT[k ] and u D[k ] (such as when ga=0) is scaled by a value between 0 and 1 as adaptation builds up, reducing the difference (and hence stimulus selectivity). This effect combined with the reduction of uT[k ] then leads to similar activity reduction in the k+1 units, and so on. Therefore, following the discussion in Section A.1, adaptation would be expected to yield habituation effects - even in a competitive network - for a linear f-I. Do these same implications hold if one considers a non-linear f-I curve? The answer is that it depends on the nature of the curve. For example, with an f-I curve that is concave-down (such as the square-root f-I curve), a fixed amount of change in input due to adaptation can have a weaker impact at larger I's than at smaller I's if the slope at the larger I's is sufficiently flat. This could result in firing-rate activity decreases that are larger in distracting/competing units than in target units, amplifying inputs to subsequent target units. To the extent that adaptation leads to very large firing-rate decreases, this case becomes less likely because strong adaptation tends to compress all of the activities to zero. When one imposes empirical constraints on the basic model by fitting the adaptation and f-I parameters to physiological data and restricting the range of allowable I's to produce firing rates that are observed in vivo (e.g. < 100 Hz), it is unlikely that strong priming effects will ever be produced because adaptation effects are large (e.g. decreasing rates by 40-60%: Ahmed et al., 1998; Sanchez-Vives et al., 2000) and the slope of the curve is still too steep and not changing enough over the range of I. This is illustrated below in Figures A.2.1a and b (parameters for the square-root f-I curve and adaptation are as they were in Figures A.1.1a and b; IC=0 for the sake of simplicity).
286
Figure A.2.1a:
Figure A.2.1a shows that for a square-root f-I curve with I's leading to firing-rates < 100 Hz, adaptation is always stronger for larger I. This is made clearer in Figure A.2.1b by plotting the difference between the steady-state firing rates without versus with adaptation for different values of I. Figure A.2.1b:
287
As with the case of the linear f-I curve, the square-root f-I curve yields larger changes in firing rate for larger I's, even though its slope is decreasing with larger I's. Figures A.2.1c and d show that stimulus selectivity is analogously decreased by adaptation for the square-root f-I curve. Figure A.2.1c:
Figure A.2.1c plots the change in stimulus selectivity due to adaptation along the z-axis (steady-state of uT[0 ] - u D[0 ] without minus with adaptation), as a function of IT and ID (xand y-axes), where IT>ID. The change is always positive, meaning that stimulus selectivity is always larger without than with adaptation (i.e. adaptation decreases selectivity). Taken together with the fact that uT[0 ] always decreases with adaptation (Figures A.2.1a and b), the firing rates of subsequent target units should also be decreased. Note that the change in selectivity increases as a function of IT-ID, and the changes are largest where the stimulus selectivity without adaptation is largest (stimulus selectivity without adaptation is shown below in Figure A.2.1d).
288
Figure A.2.1d:
Therefore, it appears that in the case of a competitive feed-forward network with a concave-down f-I curve and adaptation matched to physiological data, habituation rather than priming is expected, and results are qualitatively similar to the case with a linear f-I. This is not to say that other concave-down f-I curves cannot yield priming effects; however, they would likely require a flatter curve at higher I's than what is observed in real neurons. Other non-linear f-I curves, such as those that are concave-up over the range of I being considered, would be expected to yield results similar to the linear case with even larger attenuation of firing rates (i.e. a fixed amount of change in I would lead to larger decreases at large I's compared to small I's). These issues are all modulated to a certain extent by the weight values we and wi, since they scale the changes taking place in uT[k −1] and u D[k −1] ; with we>wi, smaller changes in uT[k −1] are relatively emphasized, making priming effects even more unlikely in the case of a concave-down f-I curve.
A.2.2 Synaptic depression
With a linear f-I curve, results for synaptic depression are the like those for adaptation with the exception that u D[k ] can be transiently larger than uT[k ] (given that the kinetics of depression are faster at higher u's). As with adaptation for IT>ID and we>wi, uT[k ] and uT[k ] u D[k ] are decreased at the steady-state. To see that uT[k ] is decreased, we need to evaluate if the steady-state input to uT[k ] without synaptic depression is greater than with depression [using eq. (9)]: [k −1]
uT
[k −1]
⋅ we − u D
uT[k −1] ⋅ we u D[k −1] ⋅ wi ⋅ wi > − d ⋅ τ D ⋅ uT[k −1] + 1 d ⋅ τ D ⋅ u D[k −1] + 1
289
Rearranging, we get
1 1 uT[k −1] ⋅ we 1 − > u D[k −1] ⋅ wi 1 − [k −1] d ⋅ τ D ⋅ uT + 1 d ⋅ τ D ⋅ u D[k −1] + 1
which is true because uT[k −1] ⋅ we > u D[k −1] ⋅ wi for IT>ID and we>wi>0 and because 1 1 0< < < 1. [k −1] d ⋅ τ D ⋅ uT + 1 d ⋅ τ D ⋅ u D[k −1] + 1
An analogous calculation can be used to show that uT[k ] - u D[k ] is decreased by synaptic depression at the steady-state, leading subsequent target units to decrease their firingrates and predicting habituation rather than priming for a linear f-I. The fact that u D[k ] can be transiently larger than uT[k ] during the buildup of synaptic depression (due to faster approach to the steady-state at higher u's) might even lead errors in identification to occur. For non-linear f-I, the situation becomes more complicated, as it did for adaptation. With a concave-down f-I, priming could occur in theory because decreases in input to uT[k ] due to synaptic depression might have a weak effect if these inputs fall in a range where the f-I curve is relatively flat, whereas small decreases in input to u D[k ] might have a strong effect if the f-I curve is steep over that input range. However, the impact of slow synaptic depression when parameterized to fit physiological data still appears more likely to yield firing-rate decreases. Figure A.2.2a shows the change in firing-rate of uT[1] due to synaptic depression at different fixed values of uT[0 ] and u D[0 ] , with the square-root f-I used in earlier sections and the slow synaptic depression taken from the Varela et al. (1997; 1999) model (fit to intracellular recording data in primary visual cortex). we was chosen to yield a firing rate of ~100 Hz in uT[1] when DT[0 ] = DD[0 ] =1, uT[0 ] =100 Hz, and u D[0 ] =0 Hz (we=30). wi was arbitrarily chosen to be less than we,(so that uT[k ] > u D[k ] with no synaptic depression), yet still to yield a substantial impact of competition (wi=20=we*2/3).
290
Figure A.2.2a:
This figure shows that, as with the linear f-I, uT[1] is always larger without than with synaptic depression using a square-root f-I matched to physiological data. It is also generally true that the largest decreases are observed where uT[1] without synaptic depression is largest (i.e. where uT[0 ] - u D[0 ] is largest). Figure A.2.2b shows that stimulus selectivity is similarly decreased by synaptic depression for all values of uT[0 ] and u D[0 ] , particularly for large uT[0 ] - u D[0 ] . Figure A.2.2b:
291
The sharp ridge in the surface observed near uT[0 ] = u D[0 ] occurs where the input to the square-root function crosses over to negative, and values of zero are returned for firing rate. The input to u D[1] becomes negative when u D[0 ] we < uT[0 ] wi, which depends on the particular values of we and wi (inputs to uT[1] are always positive). When this happens, u D[1] =0 (both with and without synaptic depression), and the values shown in Figure A.2.2b become identical to those shown in Figure A.2.2a (notice the similarity of the two figures for large uT[0 ] and small u D[0 ] ). Therefore, as with adaptation discussed in Section A.2.1, synaptic depression is expected to yield habituation rather than priming, even when a concave-down f-I curve is used - provided that the parameters have been set to match physiological data. For f-I curves that are near-linear or concave-up, results are expected to match qualitatively those for a linear f-I. The assumption that we>wi and the tendency for u D[k ] to be transiently larger than uT[k ] would also be expected to reduce the possibility of priming. In summary, it is not impossible for adaptation and synaptic depression to lead to priming in competitive feed-forward networks with a non-linear, concave-down f-I curve. However, for concave-down f-I curves that have been matched to empirical data and for those that are approximately-linear or concave-up over the range of I's under consideration, adaptation and synaptic depression would always be expected to lead to habituation, given certain general assumptions about the mechanisms of response generation (discussed in Section A.1.1). To the extent that adaptation/synaptic depression are very strong or the slope of f-I is steep over all I, priming in the case of the concave-down f-I curve becomes increasingly unlikely. Simulations 6 and 7 in this thesis attest to the difficulty of demonstrating priming effects in firing-rate networks employing these short-term plasticity mechanisms (see also Simulation 8).
292
A.3 Network with locally recurrent connections representing two competing stimuli
In this section, we will briefly consider a network architecture similar to that in Section A.2 except that local recurrent interactions are allowed (shown above). Units representing a target stimulus ( uT[k ] ) excite each other in a feed-forward manner by means of positive synaptic strengths wff, and each unit excites itself through positive synaptic strengths we. Units in a particular group k that represent target or distractor stimuli mutually compete through lateral inhibitory connections wi. Note that processing within this architecture has been referred to as "biased competition" because units uT[k ] and u D[k ] compete locally and which unit wins is a function of the bias present in IT versus ID (e.g. Desimone & Duncan, 1995; see also Usher & McClelland, 2001); we assume here that IT>ID and IT>0. The multiplicative nature of synaptic depression makes it difficult to solve for a steady-state in this partially recurrent architecture, and we will therefore restrict ourselves to the effects of slow firing-rate adaptation with a linear f-I curve. We add the further constraint that we+wi u D[k ] prior to adaptation. Below we find that as in Section A.2, adaptation has an attenuating impact on firing rates of target units. Synaptic depression is expected to have a similar impact on activity, although as discussed in the previous section, the dependence of its dynamics on the value of presynaptic activity could lead u D[k ] to be transiently larger than uT[k ] - potentially causing errors of identification.
A.3.1 Firing-rate adaptation
Here we will show that for a linear f-I curve, uT[k ] and uT[k ] - u D[k ] are always decreased by firing-rate adaptation. We first solve for the steady-states of uT[0 ] and u D[0 ] with and without adaptation, holding IT and ID constant.
293
without adaptation:
(1 − we ) ⋅ I T − wi ⋅ I D (1 − we )2 − wi2 (1 − we ) ⋅ I D − wi ⋅ I T u D[0 ] = (1 − we )2 − wi2 uT[0 ] =
with adaptation:
(1 − we + α a ⋅ g a ⋅τ a ) ⋅ IT − wi ⋅ I D (1 − we + α a ⋅ g a ⋅τ a )2 − wi2 (1 − we + α a ⋅ g a ⋅τ a ) ⋅ I D − wi ⋅ IT u D[0 ] = (1 − we + α a ⋅ g a ⋅τ a )2 − wi2 uT[0 ] =
(10) (11)
(12) (13)
As discussed in Section A.2.1, decreases in uT[k −1] and uT[k −1] − u D[k −1] due to adaptation will lead to decreases in uT[k ] : uT[k ] =
(1 − we ) ⋅ w ff ⋅ uT[k −1] − wi ⋅ w ff ⋅ uD[k −1] (1 − we )2 − wi2
Multiplying through by (1 − we ) − wi2 , which is a positive constant yields 2
uT[k ] ∝ (1 − we ) ⋅ w ff ⋅ uT[k −1] − wi ⋅ w ff ⋅ u D[k −1] Since we + wi < 1 , we can choose ε > 0 such that we + wi + ε = 1 . Substituting wi + ε for 1 − we : uT[k ] ∝ (wi + ε ) ⋅ w ff ⋅ uT[k −1] − wi ⋅ w ff ⋅ u D[k −1]
This simplifies to
(
)
uT[k ] ∝ wi ⋅ w ff ⋅ uT[k −1] − u D[k −1] + ε ⋅ w ff ⋅ uT[k −1]
(14)
Thus, decreases in both uT[k −1] and uT[k −1] − u D[k −1] will lead to decreases in uT[k ] . So, does adaptation decrease both of these terms? We first evaluate if adaptation decreases uT[0 ] by evaluating whether the steady-state without adaptation is greater than the steady-state with adaptation [substituting eqs. (10) and (12)]:
(1 − we ) ⋅ IT − wi ⋅ I D (1 − we )2 − wi2
>
(1 − we + α a ⋅ g a ⋅τ a ) ⋅ IT − wi ⋅ I D (1 − we + α a ⋅ g a ⋅τ a )2 − wi2
294
This will be true if the derivative of the right-hand side with respect to α a ⋅ g a ⋅τ a is negative for α a ⋅ g a ⋅τ a ≥ 0 . Letting A = α a ⋅ g a ⋅τ a ,
d
(1 − we + A) ⋅ IT − wi ⋅ I D (1 − we + A)2 − wi2
=
dA
[
]
− (1 − we + A) + wi2 ⋅ IT + 2 ⋅ (1 − we + A) ⋅ wi ⋅ I D 2
[(1 − w + A) − w ] 2
e
[
2 2 i
2 ⋅ (1 − we + A) ⋅ wi ⋅ I D
Since IT>ID, wi>0, and (1 − we + A) > 0 , this will be true as long as
(1 − we + A)2 + wi2 > 2 ⋅ (1 − we + A) ⋅ wi (1 − we + A)2 − 2 ⋅ (1 − we + A) ⋅ wi + wi2 > 0 which factors to
(1 − we + A − wi )2 > 0 This is true because 1 − we − wi > 0 and A ≥ 0 . Therefore adaptation decreases uT[0 ] for a linear f-I curve. Does adaptation also lead to decreases in uT[0] − u D[0 ] ? We evaluate whether uT[0] − u D[0 ] is greater without than with adaptation [substituting eqs. (10)-(13)]:
(1 − we ) ⋅ I T
− wi ⋅ I D − (1 − we ) ⋅ I D − wi ⋅ I T
>
(1 − we )2 − wi2 (1 − we + α a ⋅ g a ⋅τ a ) ⋅ IT − wi ⋅ I D − (1 − we + α a ⋅ g a ⋅τ a ) ⋅ I D + wi ⋅ IT (1 − we + α a ⋅ g a ⋅τ a )2 − wi2
?
Factoring the denominators and simplifying, we get IT − I D IT − I D > 1 − we − wi 1 − we − wi + α a ⋅ g a ⋅τ a
This is true for α a ⋅ g a ⋅τ a > 0 since IT>ID, IT>0, and 1 − we − wi > 0 . Thus, adaptation
decreases uT[0] − u D[0 ] . Given the decreases in uT[k −1] and uT[k −1] − u D[k −1] (where k=1), subsequent target units will be decreased and the stimulus-selectivity of units in the
295
network will also be decreased. Therefore, results for networks with partially recurrent connections, competition, and linear f-I are qualitatively similar to those obtained in Section A.2.1. Following the discussion in Section A.2, we expect that priming effects are possible for some parameterizations of the model if one considers concave-down f-I curves. However, to the extent that adaptation/synaptic depression effects are strong, or the f-I curve is approximately linear or concave-up over the range of I being considered, habituation effects should predominate.
A.3.2 Comment
Taken together, these various analyses imply - at a minimum - that it will be difficult for adaptation and synaptic depression effects to yield priming effects. This poses significant problems for firing-rate based neural network accounts of repetition suppression and repetition priming phenomena, assuming that the relatively short-term effects are due to some combination of firing-rate adaptation and synaptic depression (see Simulation 1, Section 3.2 for detailed arguments along these lines). However, it should be noted that in equation (14), increases in wff could offset decreases in uT[k −1] and uT[k −1] − u D[k −1] . In other words, these conclusions depend on the assumption that synaptic strengths are not changing radically over these time scales. As will be discussed in more detail in Simulation 4 (see Section 3.5), changes in gain due to enhanced spike synchronization would be tantamount to transiently increasing wff: They are somewhat isomorphic. It is also important to note, though, that the firing-rate characterization discussed here would give little principled guidance as to why activity decreases are related to improvements in performance. Arbitrary activity and performance changes would be possible by altering wff appropriately.
(
)
296
Appendix B
IAF network: e's:
dV = I − V − g e (t ) ⋅ (V − Ve ) − g i (t ) ⋅ (V − Vi ) dt
Corresponding Phase Model: dΦ i = ω i + H (Φ j − Φ i ) dt
dΦ j dt
= ω j + H (Φ i − Φ j )
Let ψ = Φ i − Φ j dψ = ω i − ω j + H (− ψ ) − H (ψ ) dt
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A phase-locked solution is stable if the slope of Hodd is positive where it crosses ωi - ωj. The synchronous solution corresponds to Hodd crossing ωi - ωj at t=0 and can only exist strictly if ωi - ωj = 0 (i.e. neurons i and j are firing at identical rates; see Section 2.2 for further discussion). The synchronous solution is stable in the network above, provided that the firing rates are not too large:
The synchronous solution breaks down at higher firing rates (~ 60 Hz for this system):
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It also breaks down if inhibition is removed, controlling for firing rate:
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Appendix C
Using the above architecture with Type I excitatory, e, and inhibitory, i, cells (parameters taken from excitatory cells in Gutkin et al., 2001, without IAHP), the impact of synaptic depression on synchronizing the e cells was examined using the phase-reduction method (see Section 2.2 and Appendix B). The implementation of synaptic depression was taken from Varela et al. (1999), although only the fast excitatory synaptic depression was included (τD= 634 ms; see Section 3.1.2 for details). The cells in the network were driven to fire at different rates by applying different external currents to the e cells (a fixed external current of 0.05 mS/cm2 was applied to the i cells). Rise and decay times (τ's) of the synaptic currents were 1 and 3 ms for IAMPA and 1 and 7 ms for IGABAa. The maximal synaptic conductances were set at 0.3 mS/cm2, and the maximal e to i conductance was adjusted for different levels of current applied to the e cells to ensure that a single i spike occurred for each e spike. The impact of synaptic depression on phase-locked solutions was evaluated by comparing the cases in which synaptic depression was either included or blocked. When synaptic depression was included, its impact was evaluated at the steady-state.
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The synchronous solution was stable with or without the inclusion of synaptic depression at lower rates (here ~ 10 Hz) with little difference in the shape of Hodd for the two conditions (see results below). The amplitude, however, does differ across conditions because synaptic depression scales down the synaptic currents (Hodd amplitudes are smaller with synaptic depression).
Synaptic Depression Blocked:
Synaptic Depression Present:
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The synchronous solution became unstable at higher firing rates both with and without synaptic depression, and the firing rate at which it became unstable was comparable for the two conditions. Results are shown here for ~ 50 Hz rates.
Synaptic Depression Blocked:
Synaptic Depression Present:
It seems then that the main role of synaptic depression in promoting synchronous spiking in e-i networks is to reduce the firing rates. It does not appear to contribute much of an additional dynamical effect to synchronizing the e cells by altering their phase-response curves (PRCs) as firing-rate adaptation does (see Ermentrout et al., 2001).
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Appendix D
Changes in Cellular Input Resistance Alter Neural Integrative Properties This appendix explains how changes in membrane permeability to various ions induced by neuromodulation or alterations in average synaptic currents can lead to changes in the steady-state membrane potential of a cell given a fixed input current, as well as to changes in the membrane time constant.
D.1
Dependence of Steady-State Voltage and the Membrane Time Constant on Passive Membrane Currents
Consider the following general equation for the voltage kinetics (V) of a patch of cell membrane when an external current (I0) is driven across the membrane: C⋅
dV = − g L ⋅ (V − VL ) + I 0 dt
where C is the fixed membrane capacitance, gL is the average passive leak conductance due to the open ion channels in the cell membrane and the resistances of these channels, and VL is the Nernst or "reversal" potential of the average leak current (the value of voltage at which the direction of the leak current will reverse direction). Dividing through by the average leak conductance, gL, we get C dV I ⋅ = −(V − VL ) + 0 g L dt gL
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1 = RL (often referred to as the Input Resistance of a cell) and RL ⋅ C = τ m gL (referred to as the membrane time constant), we have
Since
I dV = −(V − VL ) + 0 gL dt Rearranging and integrating both sides over corresponding intervals to solve V as a function of time t, we get:
τm ⋅
V
dV
V0
I − (V − VL ) + 0 gL
=
t
dt
t0
τm
I − (V − VL ) + 0 (t − t0 ) gL ln =− I τm − (V0 − VL ) + 0 gL
− (V − VL ) +
I0 (t − t 0 ) − gL = e τm I0 − (V0 − VL ) + gL
− I I − (V − VL ) + 0 = − (V0 − VL ) + 0 ⋅ e gL gL
(t − t 0 ) τm
I I V (t ) = VL + 0 + V0 − VL + 0 gL gL
⋅e
−
(t − t 0 ) τm
Therefore, V(t) is an exponential function of its input current I0. I V∞ = VL + 0 ("steady-state" voltage), this is simplifies to gL
V (t ) = V∞ + (V0 − V∞ ) ⋅ e
−
Substituting
(t − t 0 ) τm
I C Since V∞ and τ m are both decreasing functions of gL V∞ = VL + 0 ;τ m = , gL gL mechanisms that increase gL by activating membrane currents have the effect of reducing the impact of I0 on V∞ and leading to more rapid approach to V∞ by shortening τ m .
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D.2
The Composite Nature of the Leak Current
The average passive leak current of a cell is actually a composite of a number of different currents:
− g L ⋅ (V − VL ) = −
g i ⋅ (V − Vi ) i
where the different currents i refer to the individual passive currents of sodium, potassium, chloride, calcium, etc. When the average synaptic current due to IAMPA, IGABA, etc. is non-zero, the average leak current will also include terms corresponding to these currents. With a few simple algebraic manipulations, it is possible to express gL and VL in the following form in terms of the individual gi and Vi:
gL =
gi i
VL =
g i ⋅ Vi i
gi
=
g i ⋅ Vi i
gL
i
In other words, the total leak conductance gL is simply a sum of the individual conductances, and the total reversal potential VL of the leak current is a weighted average of the individual reversal potentials (each scaled by its corresponding leak conductance). This means that any neural mechanism, such as neuromodulation, that opens or closes various ion channels will have an impact on gL and VL, thus influencing the integrative properties of neurons (see D.1 above). Opening additional channels will increase gL and decrease the membrane time constant τ m , whereas closing channels will decrease gL and increase τ m . Since the phase-response curves (PRCs) of individual cells depend on τ m , dynamic changes in gL can alter the synchronization properties of cortical networks (see Section 2.2 and Section 4.5 for further discussion).
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