The readings listed below are the foundation of this course. Where available, journal article abstracts from
PubMed (an online database providing access to citations from biomedical literature) are included.
Textbooks
Recommended: Hertz, J., A. Krogh, and R. G. Palmer. Introduction to the Theory of Neural Computation. Addison-Wesley, 1991.
Definitions of Computational Neuroscience and Neural Networks. Classical Neural Network Equations. Integrate-and-Fire Model Neurons and Reduction by the Method of Averaging.
Ermentrout, Bard. "Reduction of Conductance-Based Models with Slow Synapses to Neural Nets." In Neural Computation. Vol. 6, 1995, pp. 679-695.
Koch, Christof. Biophysics of Computation. New York, Oxford: Oxford University Press, 1999, Section 14.2, pp. 335-341.
Perceptron as Feature Detector. Visual Receptive Fields.
Hubel, D. H. Eye, Brain, and Vision. New York: Scientific American Library, 1988-1995, Chap. 3, pp. 39-46.
Marr, D. Vision. New York: W. H. Freeman and Company, 1982. Section 2.2, pp. 54-79.
The Problem of Credit Assignment. Perceptron Learning Rule. Convergence Theorem. Learning by Gradient Following. Online learning.
Hertz, Krogh, and Palmer. Chap. 5.
Multilayer Perceptrons and Backpropagation.
Hertz, Krogh, and Palmer. Chap. 6.
LeCun, Y., L. Bottou, G. B. Orr, and K. R. Muller. "Efficient backprop." In Neural Networks: Tricks of the Trade, by G. Orr and K. Muller. Springer, 1998.
Backpropagation Applications. LeNet and the Visual System.
LeCun, Y., Y. Bengio, Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. "Gradient-Based Learning Applied to Document Recognition." Proc. IEEE, Nov. 1998.
Robinson, David A. "Implications of Neural Networks for How We Think About Brain Function." Behav. Brain. Sci. 15 (1992): 644-55.
The Capacity of the Perceptron, Statistical Learning Theory.
Cover, T. M. "Geometrical and Statistical Properties of Systems of Linear Inequalities with Applications in Pattern Recognition." IEEE Trans. Electronic. Comput. 14 (1965): 326-34.
Hertz, Krogh, and Palmer. Section 5.7.
Vapnik, V. The Nature of Statistical Learning Theory. Springer, 1995.
Feedback in Linear Networks. Eigenmode Analysis, Amplification and Attenuation, Gain-bandwidth Theorem.
Strang, G. Introduction to Applied Mathematics. Wellesley, Massachusetts: Wellesley-Cambridge Press, 1986, Section 1.5, pp. 47-68.
Neural Network Models of the Retina.
Adelson, E. H. "Lightness Perception and Lightness Illusions."
Boahen, K. A. "Computation and Neural Systems Program." In Spatio-temporal Sensitivity of the Retina: A Physical Model. Pasadena, CA: California Institute of Technology, 1991, pp. 1-19.
Hartline, H. K., and F. Ratliff. "Inhibitory Interaction in the Retina of Limulus." In Physiology of Photoreceptor Organs. Edited by M. G. F. Fuortes. Berlin, Heidelberg, New York: Springer-Verlag, 1972, pp. 382-447.
Kandel, E. R., J. H. Schwartz, and T. M. Jessell. Principles of Neural Science. 4/e, McGraw-Hill, 2000, Part V, Chap. 26, pp. 507-522.
Mead, C. Analog VLSI and Neural Systems. Addison-Wesley Publishing Company, 1989, Appendix C, pp. 339-351.
Press, W. H., et al. Numerical Recipes in C. 2nd ed. Chap. 12 and 13.
Strang, G. Introduction to Applied Mathematics. Wellesley, Massachusetts: Wellesley-Cambridge Press, 1986, Section 4.2, pp. 290-309.
Hybrid Analog-digital Computation. Permitted and Forbidden Sets.
Hahnloser, R., R. Sarpeshkar, M. Mahowald, R. J. Douglas, and H. S. Seung. "Digital Selection and Analog Amplification Coexist in a Cortex-Inspired Silicon Circuit." Nature 405 (2000): 947-51.
PubMed abstract: Digital circuits such as the flip-flop use feedback to achieve multistability and nonlinearity to restore signals to logical levels, for example 0 and 1. Analogue feedback circuits are generally designed to operate linearly, so that signals are over a range, and the response is unique. By contrast, the response of cortical circuits to sensory stimulation can be both multistable and graded. We propose that the neocortex combines digital selection of an active set of neurons with analogue response by dynamically varying the positive feedback inherent in its recurrent connections. Strong positive feedback causes differential instabilities that drive the selection of a set of active neurons under the constraints embedded in the synaptic weights. Once selected, the active neurons generate weaker, stable feedback that provides analogue amplification of the input. Here we present our model of cortical processing as an electronic circuit that emulates this hybrid operation, and so is able to perform computations that are similar to stimulus selection, gain modulation and spatiotemporal pattern generation in the neocortex.
Intra-group Excitation and Global Inhibition. Marr-Poggio Model of Stereopsis. Complex Cell Model.
Marr, D. Vision New York: W.H. Freeman and Company, Section 3.3 (1982): 111-159.
Lateral Excitation and Global Inhibition. Gain Fields and Stimulus Selection.
Ben-Yishai, R., R. L. Bar-Or, and H. Sompolinsky. "Theory of Orientation Tuning in Visual Cortex." PNAS 92 (1995): 3844.
PubMed abstract: The role of intrinsic cortical connections in processing sensory input and in generating behavioral output is poorly understood. We have examined this issue in the context of the tuning of neuronal responses in cortex to the orientation of a visual stimulus. We analytically study a simple network model that incorporates both orientation-selective input from the lateral geniculate nucleus and orientation-specific cortical interactions. Depending on the model parameters, the network exhibits orientation selectivity that originates from within the cortex, by a symmetry-breaking mechanism. In this case, the width of the orientation tuning can be sharp even if the lateral geniculate nucleus inputs are only weakly anisotropic. By using our model, several experimental consequences of this cortical mechanism of orientation tuning are derived. The tuning width is relatively independent of the contrast and angular anisotropy of the visual stimulus. The transient population response to changing of the stimulus orientation exhibits a slow "virtual rotation." Neuronal cross-correlations exhibit long time tails, the sign of which depends on the preferred orientations of the cells and the stimulus orientation.
Salinas, E., and L. F. Abbott. "A Model of Multiplicative Responses in Parietal Cortex." PNAS 93 (1996): 11956-61.
PubMed abstract: Visual responses of neurons in parietal area 7a are modulated by a combined eye and head position signal in a multiplicative manner. Neurons with multiplicative responses can act as powerful computational elements in neural networks. In the case of parietal cortex, multiplicative gain modulation appears to play a crucial role in the transformation of object locations from retinal to body-centered coordinates. It has proven difficult to uncover single-neuron mechanisms that account for neuronal multiplication. Here we show that multiplicative responses can arise in a network model through population effects. Specifically, neurons in a recurrently connected network with excitatory connections between similarly tuned neurons and inhibitory connections between differently tuned neurons can perform a product operation on additive synaptic inputs. The results suggest that parietal responses may be based on this architecture.
Models of Associative Memory.
Hopfield, J. J. "Neural Networks and Physical Systems with Emergent Collective Computational Abilities." Proc. Natl. Acad. Sci. USA 79 (1982): 2554-58.
Summary of Lyapunov analysis: Chapter 3 from Slotine,1991.
Tsodyks, M. V., and M. Feigelman.
Delay Activity. Griniasty-Tsodyks-Amit Model.
Amit, D. J. "The Hebbian Paradigm Reintegrated: Local Reverberations as Internal Representations." Behav. Brain Sci. 18 (1995): 617-26.
Amit, D. J., N. Brunel, and M. V. Tsodyks. "Correlations Of Cortical Hebbian Reverberations: Theory Versus Experiment." J. Neurosci. 14 (1994): 6435-45.
PubMed abstract: Interpreting recent single-unit recordings of delay activities in delayed match-to-sample experiments in anterior ventral temporal (AVT) cortex of monkeys in terms of reverberation dynamics, we present a model neural network of quasi-realistic elements that reproduces the empirical results in great detail. Information about the contiguity of successive stimuli in the training sequence, representing the fact that training is done on a set of uncorrelated stimuli presented in a fixed temporal sequence, is embedded in the synaptic structure. The model reproduces quite accurately the correlations between delay activity distributions corresponding to stimulation with the uncorrelated stimuli used for training. It reproduces also the activity distributions of spike rates on sample cells as a function of the stimulating pattern. It is, in our view, the first time that a computational phenomenon, represented on the neurophysiological level, is reproduced in all its quantitative aspects. The model is then used to make predictions about further features of the physiology of such experiments. Those include further properties of the correlations, features of selective cells as discriminators of stimuli provoking different delay activity distributions, and activity distributions among the neurons in a delay activity produced by a given pattern. The model has predictive implications also for the dependence of the delay activities on different training protocols. Finally, we discuss the perspectives of the interplay between such models and neurophysiology as well as its limitations and possible extensions.
Griniasty, M., M. V. Tsodyks, and D. J. Amit. "Conversion Of Temporal Correlations Between Stimuli To Spatial Correlations Between Attractors." Neural Comput. 5 (1993): 1-17.
Miyashita, Y. "Neuronal Correlate Of Visual Associative Long-Term Memory In The Primate Temporal Cortex." Nature 335 (1998): 817-20.
PubMed abstract: In human long-term memory, ideas and concepts become associated in the learning process. No neuronal correlate for this cognitive function has so far been described, except that memory traces are thought to be localized in the cerebral cortex; the temporal lobe has been assigned as the site for visual experience because electric stimulation of this area results in imagery recall and lesions produce deficits in visual recognition of objects. We previously reported that in the anterior ventral temporal cortex of monkeys, individual neurons have a sustained activity that is highly selective for a few of the 100 coloured fractal patterns used in a visual working-memory task. Here I report the development of this selectivity through repeated trials involving the working memory. The few patterns for which a neuron was conjointly selective were frequently related to each other through stimulus-stimulus association imposed during training. The results indicate that the selectivity acquired by these cells represents a neuronal correlate of the associative long-term memory of pictures.
Zipser, D., B. Kehoe, G. Littlewort, and J. Fuster. "A Spiking Network Model of Short-Term Active Memory." J. Neurosci. 13 (1993): 3406-3420.
PubMed abstract: Studies of cortical neurons in monkeys performing short-term memory tasks have shown that information about a stimulus can be maintained by persistent neuron firing for periods of many seconds after removal of the stimulus. The mechanism by which this sustained activity is initiated and maintained is unknown. In this article we present a spiking neural network model of short-term memory and use it to investigate the hypothesis that recurrent, or "re-entrant," networks with constant connection strengths are sufficient to store graded information temporarily. The synaptic weights that enable the network to mimic the input-output characteristics of an active memory module are computed using an optimization procedure for recurrent networks with non-spiking neurons. This network is then transformed into one with spiking neurons by interpreting the continuous output values of the nonspiking model neurons as spiking probabilities. The behavior of the model neurons in this spiking network is compared with that of 179 single units previously recorded in monkey inferotemporal (IT) cortex during the performance of a short-term memory task. The spiking patterns of almost every model neuron are found to resemble closely those of IT neurons. About 40% of the IT neuron firing patterns are also found to be of the same types as those of model neurons. A property of the spiking model is that the neurons cannot maintain precise graded activity levels indefinitely, but eventually relax to one of a few constant activities called fixed-point attractors. The noise introduced into the model by the randomness of spiking causes the network to jump between these attractors. This switching between attractor states generates spike trains with a characteristic statistical temporal structure. We found evidence for the same kind of structure in the spike trains from about half of the IT neurons in our test set. These results show that the behavior of many real cortical memory neurons is consistent with an active storage mechanism based on recurrent activity in networks with fixed synaptic strengths.
Neural Integrators.
Camperi, M., and X.-J. Wang. "A Model of Visuospatial Working Memory in Prefrontal Cortex: Recurrent Network and Cellular Bistability." J. Comput. Neurosci. 5 (1998): 383-405.
PubMed abstract: We report a computer simulation of the visuospatial delayed-response experiments of Funahashi et al. (1989), using a firing-rate model that combines intrinsic cellular bistability with the recurrent local network architecture of the neocortex. In our model, the visuospatial working memory is stored in the form of a continuum of network activity profiles that coexist with a spontaneous activity state. These neuronal firing patterns provide a population code for the cue position in a graded manner. We show that neuronal persistent activity and tuning curves of delay-period activity (memory fields) can be generated by an excitatory feedback circuit and recurrent synaptic inhibition. However, if the memory fields are constructed solely by network mechanisms, noise may induce a random drift over time in the encoded cue position, so that the working memory storage becomes unreliable. Furthermore, a "distraction" stimulus presented during the delay period produces a systematic shift in the encoded cue position. We found that the working memory performance can be rendered robust against noise and distraction stimuli if single neurons are endowed with cellular bistability (presumably due to intrinsic ion channel mechanisms) that is conditional and realized only with sustained synaptic inputs from the recurrent network. We discuss how cellular bistability at the single cell level may be detected by analysis of spike trains recorded during delay-period activity and how local modulation of intrinsic cell properties and/or synaptic transmission can alter the memory fields of individual neurons in the prefrontal cortex.
Seung, H. S. "How The Brain Keeps The Eyes Still." Proc. Natl. Acad. Sci. USA 93 (1996): 13339-44.
PubMed abstract: The brain can hold the eyes still because it stores a memory of eye position. The brain's memory of horizontal eye position appears to be represented by persistent neural activity in a network known as the neural integrator, which is localized in the brainstem and cerebellum. Existing experimental data are reinterpreted as evidence for an "attractor hypothesis" that the persistent patterns of activity observed in this network form an attractive line of fixed points in its state space. Line attractor dynamics can be produced in linear or nonlinear neural networks by learning mechanisms that precisely tune positive feedback.
Zhang, K. "Representation Of Spatial Orientation By The Intrinsic Dynamics Of The Head-Direction Cell Ensemble: A Theory." J. Neurosci. 16 (1996): 2112-26.
PubMed abstract: The head-direction (HD) cells found in the limbic system in freely mov ing rats represent the instantaneous head direction of the animal in the horizontal plane regardless of the location of the animal. The internal direction represented by these cells uses both self-motion information for inertially based updating and familiar visual landmarks for calibration. Here, a model of the dynamics of the HD cell ensemble is presented. The stability of a localized static activity profile in the network and a dynamic shift mechanism are explained naturally by synaptic weight distribution components with even and odd symmetry, respectively. Under symmetric weights or symmetric reciprocal connections, a stable activity profile close to the known directional tuning curves will emerge. By adding a slight asymmetry to the weights, the activity profile will shift continuously without disturbances to its shape, and the shift speed can be controlled accurately by the strength of the odd-weight component. The generic formulation of the shift mechanism is determined uniquely within the current theoretical framework. The attractor dynamics of the system ensures modality-independence of the internal representation and facilitates the correction for cumulative error by the putative local-view detectors. The model offers a specific one-dimensional example of a computational mechanism in which a truly world-centered representation can be derived from observer-centered sensory inputs by integrating self-motion information.
Contrastive Hebbian Learning and Recurrent Backprop Learning.
Hertz, Krogh, and Palmer.
Hopfield, J. J. "Neurons With Graded Response Have Collective Computational Properties Like Those Of Two-State Neurons." Proc. Natl. Acad. Sci. USA 81 (1984): 3088-3092.
PubMed abstract: A model for a large network of "neurons" with a graded response (or sigmoid input-output relation) is studied. This deterministic system has collective properties in very close correspondence with the earlier stochastic model based on McCulloch - Pitts neurons. The content- addressable memory and other emergent collective properties of the original model also are present in the graded response model. The idea that such collective properties are used in biological systems is given added credence by the continued presence of such properties for more nearly biological "neurons." Collective analog electrical circuits of the kind described will certainly function. The collective states of the two models have a simple correspondence. The original model will continue to be useful for simulations, because its connection to graded response systems is established. Equations that include the effect of action potentials in the graded response system are also developed.
Slotine, J. E., W. Li. Applied Nonlinear Control. Englewood Cliffs, New Jersey: Prentice Hall, Inc., 1991, Chap. 3, Sections 3.1, 3.2, 3.4, pp. 40-47, 47-53, 57-76.
REINFORCE Algorithms. Hedonistic Neurons.
Williams 92.
Gradient Llearning of Trajectories: Backpropagation and Real-time Recurrent Learning.
Hertz, Krogh, and Palmer.
Pearlmutter, B. A. "Gradient Calculation For Dynamic Recurrent Neural Networks: A Survey." IEEE Transactions on Neural Networks 6, 5 (1995): 1212-1228.
———. "Learning State Space Trajectories in Recurrent Neural Networks." Neural Computation 1 (1989): 263-9.
Williams, R. J., and D. Zipser. "Experimental Analysis of the Real-time Recurrent Learning Algorithm." Connection Science 1 (1989): 87-111.