Center Publications

Neural circuits display complex activity patterns both spontaneously and when responding to a stimulus or generating a motor output. How are these two forms of activity related? We develop a procedure called FORCE learning for modifying synaptic strengths either external to or within a model neural network to change chaotic spontaneous activity into a wide variety of desired activity patterns. FORCE learning works even though the networks we train are spontaneously chaotic and we leave feedback loops intact and unclamped during learning. Using this approach, we construct networks that produce a wide variety of complex output patterns, input-output transformations that require memory, multiple outputs that can be switched by control inputs, and motor patterns matching human motion capture data. Our results reproduce data on premovement activity in motor and premotor cortex, and suggest that synaptic plasticity may be a more rapid and powerful modulator of network activity than generally appreciated.

A key question in the analysis of hippocampal memory relates to how attention modulates the encoding and long-term retrieval of spatial and nonspatial representations in this region. To address this question, we recorded from single cells over a period of 5 days in the CA1 region of the dorsal hippocampus while mice acquired one of two goal-oriented tasks. These tasks required the animals to find a hidden food reward by attending to either the visuospatial environment or a particular odor presented in shifting spatial locations. Attention to the visuospatial environment increased the stability of visuospatial representations and phase locking to gamma oscillations - a form of neuronal synchronization thought to underlie the attentional mechanism necessary for processing task-relevant information. Attention to a spatially shifting olfactory cue compromised the stability of place fields and increased the stability of reward-associated odor representations, which were most consistently retrieved during periods of sniffing and digging when animals were restricted to the cup locations. Together, these results suggest that attention selectively modulates the encoding and retrieval of hippocampal representations by enhancing physiological responses to task-relevant information.

Early in development, the cat primary visual cortex (V1) is dominated by inputs driven by the contralateral eye. The pattern then reorganizes into ocular dominance columns that are roughly equally distributed between inputs serving the two eyes. This reorganization does not occur if the eyes are kept closed. The mechanism of this equalization is unknown. It has been argued that it is unlikely to involve Hebbian activity-dependent learning rules, on the assumption that these would favor an initially dominant eye. The reorganization occurs at the onset of the critical period (CP) for monocular deprivation (MD), the period when MD can cause a shift of cortical innervation in favor of the nondeprived eye. In mice, the CP is opened by the maturation of cortical inhibition, which does not occur if the eyes are kept closed. Here we show how these observations can be united: under Hebbian rules of activity-dependent synaptic modification, strengthening of intracortical inhibition can lead to equalization of the two eyes' inputs. Furthermore, when the effects of homeostatic synaptic plasticity or certain other mechanisms are incorporated, activity-dependent learning can also explain how MD causes a shift toward the open eye during the CP despite the drive by inhibition toward equalization of the two eyes' inputs. Thus, assuming similar mechanisms underlie the onset of the CP in cats as in mice, this and activity-dependent learning rules can explain the interocular equalization observed in cat V1 and its failure to occur without visual experience.

In what regime does the cortical circuit operate? Our intracellular studies of surround suppression in cat primary visual cortex (V1) provide strong evidence on this question. Although suppression has been thought to arise from an increase in lateral inhibition, we find that the inhibition that cells receive is reduced, not increased, by a surround stimulus. Instead, suppression is mediated by a withdrawal of excitation. Thalamic recordings and previous work show that these effects cannot be explained by a withdrawal of thalamic input. We find in theoretical work that this behavior can only arise if V1 operates as an inhibition-stabilized network (ISN), in which excitatory recurrence alone is strong enough to destabilize visual responses but feedback inhibition maintains stability. We confirm two strong tests of this scenario experimentally and show through simulation that observed cell-to-cell variability in surround effects, from facilitation to suppression, can arise naturally from variability in the ISN.

The processing of synaptic potentials by neuronal dendrites depends on both their passive cable properties and active voltage-gated channels, which can generate complex effects as a result of their nonlinear properties. We characterized the actions of HCN (hyperpolarization-activated cyclic nucleotide-gated cation) channels on dendritic processing of subthreshold excitatory postsynaptic potentials (EPSPs) in mouse CA1 hippocampal neurons. The HCN channels generated an excitatory inward current (Ih) that exerted a direct depolarizing effect on the peak voltage of weak EPSPs, but produced a paradoxical hyperpolarizing effect on the peak voltage of stronger, but still subthreshold, EPSPs. Using a combined modeling and experimental approach, we found that the inhibitory action of Ih was caused by its interaction with the delayed-rectifier M-type K1 current. In this manner, Ih can enhance spike firing in response to an EPSP when spike threshold is low and can inhibit firing when spike threshold is high.

Recent theoretical work has provided a basic understanding of signal propagation in networks of spiking neurons, but mechanisms for gating and controlling these signals have not been investigated previously. Here we introduce an idea for the gating of multiple signals in cortical networks that combines principles of signal propagation with aspects of balanced networks. Specifically, we studied networks in which incoming excitatory signals are normally cancelled by locally evoked inhibition, leaving the targeted layer unresponsive. Transmission can be gated 'on' by modulating excitatory and inhibitory gains to upset this detailed balance. We illustrate gating through detailed balance in large networks of integrate-and-fire neurons. We show successful gating of multiple signals and study failure modes that produce effects reminiscent of clinically observed pathologies. Provided that the individual signals are detectable, detailed balance has a large capacity for gating multiple signals.

A number of important data analysis problems in neuroscience can be solved using state-space models. In this article, we describe fast methods for computing the exact maximum a posteriori (MAP) path of the hidden state variable in these models, given spike train observations. If the state transition density is log-concave and the observation model satisfies certain standard assumptions, then the optimization problem is strictly concave and can be solved rapidly with Newton-Raphson methods, because the Hessian of the loglikelihood is block tridiagonal. We can further exploit this block-tridiagonal structure to develop efficient parameter estimation methods for these models. We describe applications of this approach to neural decoding problems, with a focus on the classic integrate-and-fire model as a key example.

Signal to noise ratios in physical systems can be signi cantly degraded if the output of a system is highly variable. Biological processes for which highly stereotyped signal generation is a necessary feature appear to have reduced their signal variabilities by employing multiple processing steps. To better understand why this multi-step cascade structure might be desirable, we prove that the reliability of a signal generated by a multi-state system with no memory (i.e. a Markov chain) is maximal if and only if the system topology is such that the process steps irreversibly through each state, with transition rates chosen such that an equal fraction of the total signal is generated in each state. Furthermore, our result indicates that by increasing the number of states, it is possible to arbitrarily increase the reliability of the system. In a physical system, however, there is an energy cost associated with maintaining irreversible transitions, and this cost increases with the number of such transitions (i.e. the number of states). Thus an infinite length chain, which would be perfectly reliable, is infeasible. To model the e ects of energy demands on the maximally reliable solution, we numerically optimize the topology under two distinct energy functions that penalize either irreversible transitions or incommunicability between states respectively. In both cases, the solutions are essentially irreversible linear chains, but with upper bounds on the number of states set by the amount of available energy. We therefore conclude that a physical system for which signal reliability is important should employ a linear architecture with the number of states (and thus the reliability) determined by the intrinsic energy constraints of the system.

In cerebral cortex, ongoing activity absent a stimulus can resemble stimulus-driven activity in size and structure. In particular, spontaneous activity in cat primary visual cortex (V1) has structure significantly correlated with evoked responses to oriented stimuli. This suggests that, from unstructured input, cortical circuits selectively amplify specific activity patterns. Current understanding of selective amplification involves elongation of a neural assembly's lifetime by mutual excitation among its neurons. We introduce a new mechanism for selective amplification without elongation of lifretime:'balanced amplification'. Strong balanced amplification arises when feedback inhibition stabilizes strong recurrent excitation, a pattern likely to be typical of cortex. Thus, balanced amplification should ubiquitously contribute to cortical activity. Balanced amplification depends on the fact that individual neurons project only excitatory or only inhibitory synapses. This leads to a hidden feedforward connectivity between activity patterns. We show in a detailed biophysical model that this can explain the cat V1 observations.

Adaptively optimizing experiments has the potential to significantly reduce the number of trials needed to build parametric statistical models of neural systems. However, application of adaptive methods to neurophysiology has been limited by severe computational challenges. Since most neurons are high-dimensional systems, optimizing neurophysiology experiments requires computing high-dimensional integrations and optimizations in real time. Here we present a fast algorithm for choosing the most informative stimulus by maximizing the mutual information between the data and the unknown parameters of a generalized linear model (GLM) that we want to fit to the neuron's activity. We rely on important log concavity and asymptotic normality properties of the posterior to facilitate the required computations. Our algorithm requires only low-rank matrix manipulations and a two-dimensional search to choose the optimal stimulus. The average running time of these operations scales quadratically with the dimensionality of the GLM, making real-time adaptive experimental design feasible even for high-dimensional stimulus and parameter spaces. For example, we require roughly 10 milliseconds on a desktop computer to optimize a 100-dimensional stimulus. Despite using some approximations to make the algorithm efficient, our algorithm asymptotically decreases the uncertainty about the model parameters at a rate equal to the maximum rate predicted by an asymptotic analysis. Simulation results show that picking stimuli by maximizing the mutual information can speed up convergence to the optimal values of the parameters by an order of magnitude compared to using random (nonadaptive) stimuli. Finally, applying our design procedure to real neurophysiology experiments requires addressing the nonstationarities that we would expect to see in neural responses; our algorithm can efficiently handle both fast adaptation due to spike history effects and slow, nonsystematic drifts in a neuron's activity.

Experimental research seeking to quantify neuronal structure constantly contends with restrictions on image resolution and variability. In particular, experimentalists often need to analyze images with very low signal-to-noise ratio (SNR). In many experiments dye toxicity scales with the light intensity; this leads experimentalists to reduce image SNR in order to preserve the viability of the specimen. In this work we present a Bayesian approach for estimating the neuronal shape given low-SNR observations. This Bayesian framework has two major advantages. First, the method effectively incorporates known facts about 1) the image formation process, including blur and the Poisson nature of image noise at low intensities, and 2) dendritic shape, including the fact that dendrites are simply-connected geometric structures with smooth boundaries. Second, we may employ standard Markov chain Monte Carlo (MCMC) techniques for quantifying the posterior uncertainty in our estimate of the dendritic shape. We describe an efficient computational implementation of these methods and demonstrate the algorithm's performance on simulated noisy two-photon laser-scanning microscopy images.

Background: Biophysically detailed models of single cells are difficult to fit to real data. Recent advances in imaging techniques allow simultaneous access to various intracellular variables, and these data can be used to significantly facilitate the modelling task. These data, however, are noisy, and current approaches to building biophysically detailed models are not designed to deal with this.
Methodology: We extend previous techniques to take the noisy nature of the measurements into account. Sequential Monte Carlo methods in combination with a detailed biophysical description of a cell are used for principled, model-based smoothing of noisy recording data. We also provide an alternative formulation of smoothing where the neural nonlinearities are estimated in a nonparametric manner.
Conclusions / Significance: Biophysically important parameters of detailed models (such as channel densities, intercompartmental conductances, input resistances and observation noise) are inferred automatically from noisy data via expectation-maximisation. Overall, we find that model-based smoothing is a powerful, robust technique for smoothing of noisy biophysical data and for inference of biophysical parameters in the face of recording noise.

Theoretical neuroscience has experienced explosive growth over the past 20 years. In addition to bringing new researchers into the field with backgrounds in physics, mathematics, computer science, and engineering, theoretical approaches have helped to introduce new ideas and shape directions of neuroscience research. This review presents some of the developments that have occurred and the lessons they have taught us.

Neurons in vivo receive a large amount of internally generated "background" activity in addition to synaptic input directly driven by an external stimulus. Stimulus-driven and background synaptic inputs interact, through the nonlinearities of neuronal integration, in interesting ways. The dynamic clamp can be used in vitro to duplicate background input, allowing the experimenter to take advantage of the accessibility of neurons in vitro while still studying them under in vivo conditions. In this chapter we discuss some results from experiments in which a neuron is driven by current injection that simulates a stimulusdriven input as well as dynamic-clamp-generated background activity. One of the effects uncovered in this way is multiplicative gain modulation, achieved by varying the level of background synaptic input. We discuss how the dynamic clamp was used to discover this effect and also how to choose parameters to simulate in vivo background synaptic input in slice neurons.

Sequential optimal design methods hold great promise for improving the efficiency of neurophysiology experiments. However, previous methods for optimal experimental design have incorporated only weak prior information about the underlying neural system (e.g., the sparseness or smoothness of the receptive field). Here we describe how to use stronger prior information, in the form of parametric models of the receptive field, in order to construct optimal stimuli and further improve the efficiency of our experiments. For example, if we believe that the receptive field is well-approximated by a Gabor function, then our method constructs stimuli that optimally constrain the Gabor parameters (orientation, spatial frequency, etc.) using as few experimental trials as possible. More generally, we may believe a priori that the receptive field lies near a known sub-manifold of the full parameter space; in this case, our method chooses stimuli in order to reduce the uncertainty along the tangent space of this sub-manifold as rapidly as possible. Applications to simulated and real data indicate that these methods may in many cases improve the experimental efficiency by an order of magnitude.

To perform nontrivial, real-time computations on a sensory input stream, biological systems must retain a short-term memory trace of their recent inputs. It has been proposed that generic highdimensional dynamical systems could retain a memory trace for past inputs in their current state. This raises important questions about the fundamental limits of such memory traces and the properties required of dynamical systems to achieve these limits. We address these issues by applying Fisher information theory to dynamical systems driven by time-dependent signals corrupted by noise. We introduce the Fisher Memory Curve (FMC) as a measure of the signal-to-noise ratio (SNR) embedded in the dynamical state relative to the input SNR. The integrated FMC indicates the total memory capacity. We apply this theory to linear neuronal networks and show that the capacity of networks with normal connectivity matrices is exactly 1 and that of any network of N neurons is, at most, N. A nonnormal network achieving this bound is subject to stringent design constraints: It must have a hidden feedforward architecture that superlinearly amplifies its input for a time of order N, and the input connectivity must optimally match this architecture. The memory capacity of networks subject to saturating nonlinearities is further limited, and cannot exceed N. This limit can be realized by feedforward structures with divergent fan out that distributes the signal across neurons, thereby avoiding saturation. We illustrate the generality of the theory by showing that memory in fluid systems can be sustained by transient nonnormal amplification due to convective instability or the onset of turbulence.

An important task of the brain is to represent the outside world. It is unclear how the brain may do this, however, as it can only rely on neural responses and has no independent access to external stimuli in order to ?decode? what those responses mean. We investigate what can be learned about a space of stimuli using only the action potentials (spikes) of cells with stereotyped?but unknown?receptive fields. Using hippocampal place cells as a model system, we show that one can (1) extract global features of the environment and (2) construct an accurate representation of space, up to an overall scale factor, that can be used to track the animal's position. Unlike previous approaches to reconstructing position from place cell activity, this information is derived without knowing place fields or any other functions relating neural responses to position. We find that simply knowing which groups of cells fire together reveals a surprising amount of structure in the underlying stimulus space; this may enable the brain to construct its own internal representations.

A long-standing conjecture in neuroscience is that aspects of cognition depend on the brain's ability to self-generate sequential neuronal activity. We found that reliably and continually changing cell assemblies in the rat hippocampus appeared not only during spatial navigation but also in the absence of changing environmental or body-derived inputs. During the delay period of a memory task, each moment in time was characterized by the activity of a particular assembly of neurons. Identical initial conditions triggered a similar assembly sequence, whereas different conditions gave rise to different sequences, thereby predicting behavioral choices, including errors. Such sequences were not formed in control (nonmemory) tasks. We hypothesize that neuronal representations, evolved for encoding distance in spatial navigation, also support episodic recall and the planning of action sequences.

In rodent hippocampus, neuronal activity is organized by a 6-10 Hz theta oscillation. The spike timing of hippocampal pyramidal cells with respect to the theta rhythm correlates with an animal's position in space. This correlation has been suggested to indicate an explicit temporal code for position. Alternatively, it may be interpreted as a byproduct of theta-dependent dynamics of spatial information flow in hippocampus. Here we show that place cell activity on different phases of theta reflects positions shifted into the future or past along the animal's trajectory in a two-dimensional environment. The phases encoding future and past positions are consistent across recorded CA1 place cells, indicating a coherent representation at the network level. Consistent theta-dependent time offsets are not simply a consequence of phase-position correlation (phase precession), because they are no longer seen after data randomization that preserves the phase-position relationship. The scale of these time offsets, 100?300 ms, is similar to the latencies of hippocampal activity after sensory input and before motor output, suggesting that offset activity may maintain coherent brain activity in the face of information processing delays.

Gating deficits and hallucinatory sensations are prominent symptoms of schizophrenia. Comparing these abnormalities with the failure modes of network models is an interesting way to explore how they arise. We present a network model that can both propagate and gate signals. The model exhibits effects reminiscent of clinically observed pathologies when the balance between excitation and inhibition that it requires is not properly maintained.

We evaluate the capacity and performance of a perceptron discriminator operating in a highly sparse regime where classic perceptron results do not apply. The perceptron is constructed to respond to a specified set of q stimuli, with only statistical information provided about other stimuli to which it is not supposed to respond. We compute the probability of both false-positive and false-negative errors and determine the capacity of the system for not responding to nonselected stimuli and for responding to selected stimuli in the presence of noise. If q is a sublinear function of N, the number of inputs to the perceptron, these capacities are exponential in N=q.

There has recently been a great deal of interest in inferring network connectivity from the spike trains in populations of neurons. One class of useful models which can be fit easily to spiking data is based on generalized linear point process models from statistics. Once the parameters for these models are fit, the analyst is left with a nonlinear spiking network model with delays, which in general may be very difficult to understand analytically. Here we develop mean-field methods for approximating the stimulus-driven firing rates (both in the time-varying and steady-state case), auto- and cross-correlations, and stimulus-dependent filtering properties of these networks. These approximations are valid when the contributions of individual network coupling terms are small and, hence, the total input to a neuron is approximately Gaussian. These approximations lead to deterministic ordinary differential equations that are much easier to solve and analyze than direct Monte Carlo simulation of the network activity. These approximations also provide analytical way to evaluate the linear input-output filter of neurons and how the filters are modulated by network interactions and some stimulus feature. Finally, in the case of strong refractory effects, the mean-field approximations in the generalized linear model become inaccurate; therefore we introduce a model that captures strong refractoriness, retains all of the easy fitting properties of the standard generalized linear model, and leads to much more accurate approximations of mean firing rates and cross-correlations that retain fine temporal behaviors.

Spiking neurons make inviting targets for analytical methods based on stochastic processes: spike trains carry information in their temporal patterning, yet they are often highly irregular across time and across experimental replications. The bulk of this volume is devoted to mathematical and biophysical models useful in understanding neurophysiological processes. In this chapter we consider statistical models for analyzing spike train data. Strictly speaking, what we would call a statistical model for spike trains is simply a probabilistic description of the sequence of spikes. But it is somewhat misleading to ignore the data-analytical context of these models. In particular, we want to make use of these probabilistic tools for the purpose of scientific inference. The leap from simple descriptive uses of probability to inferential applications is worth emphasizing for two reasons. First, this leap was one of the great conceptual advances in science, taking roughly two hundred years. It was not until the late 1700s that there emerged any clear notion of inductive (or what we would now call statistical) reasoning; it was not until the first half of the twentieth century that modern methods began to be developed systematically; and it was only in the second half of the twentieth century that these methods became well understood in terms of both theory and practice. Second, the focus on inference changes the way one goes about the modeling process. It is this change in perspective we want to highlight here, and we will do so by discussing one of the most important models in neuroscience, the stochastic integrate-and-fire (IF) model for spike trains. The stochastic IF model has a long history (Gerstein and Mandelbrot, 1964; Stein, 1965; Knight, 1972; Burkitt, 2006): it is the simplest dynamical model that captures the basic properties of neurons, including the temporal integration of noisy subthreshold inputs, all- or-none spiking, and refractoriness. Of course, the IF model is a caricature of true neural dynamics (see, e.g., (Ermentrout and Kopell, 1986; Brunel and Latham, 2003; Izhikevich, 2007) for more elaborate models) but, as demonstrated in this book and others (Ricciardi, 1977; Tuckwell, 1989; Gerstner and Kistler, 2002), it has provided much insight into the behavior of single neurons and neural populations. In this chapter we explore some of the key statistical questions that arise when we use this model to perform inference with real neuronal spike train data. How can we efficiently fit the model to spike train data? Once we have estimated the model parameters, what can the model tell us about the encoding properties of the observed neuron? We also briefly consider some more general approaches to statistical modeling of spike train data. We begin, in section 1, by discussing three distinct useful ways of approaching the IF model, via the language of stochastic (diffusion) processes, hidden Markov models, and point processes, respectively. Each of these viewpoints comes equipped with its own specialized analytical tools, and the power of the IF model is most evident when all of these tools may be brought to bear simultaneously. We discuss three applications of these methods in section 2, and then close in 3 by indicating the scope of the general point process framework of which the IF model is a part, and the possibilities for solving some key outstanding data-analytic problems in systems neuroscience.

How many independent samples N do we need from a distribution p to decide that p is epsilon-distant from uniform in an L1 sense, Sigma(i=1 to m) |p(i) - 1/m| > epsilon? (Here m is the number of bins on which the distribution is supported, and is assumed known a priori.) Somewhat surprisingly, we only need N*epsilon^2 >> m^1/2 to make this decision reliably (this condition is both sufficient and necessary). The test for uniformity introduced here is based on the number of observed "coincidences" (samples that fall into the same bin), the mean and variance of which may be computed explicitly for the uniform distribution and bounded nonparametrically for any distribution that is known to be epsilon-distant from uniform. Some connections to the classical birthday problem are noted.

Statistical dependencies in the responses of sensory neurons govern both the amount of stimulus information conveyed and the means by which downstream neurons can extract it. Although a variety of measurements indicate the existence of such dependencies, their origin and importance for neural coding are poorly understood. Here we analyse the functional significance of correlated firing in a complete population of macaque parasol retinal ganglion cells using a model of multi-neuron spike responses. The model, with parameters fit directly to physiological data, simultaneously captures both the stimulus dependence and detailed spatio-temporal correlations in population responses, and provides two insights into the structure of the neural code. First, neural encoding at the population level is less noisy than one would expect from the variability of individual neurons: spike times are more precise, and can be predicted more accurately when the spiking of neighbouring neurons is taken into account. Second, correlations provide additional sensory information: optimal, model-based decoding that exploits the response correlation structure extracts 20% more information about the visual scene than decoding under the assumption of independence, and preserves 40% more visual information than optimal linear decoding. This model-based approach reveals the role of correlated activity in the retinal coding of visual stimuli, and provides a general framework for understanding the importance of correlated activity in populations of neurons.

We develop a "plug-in" kernel estimator for the differential entropy that is consistent even if the kernel width tends to zero as quickly as 1/N, where N is the number of independent and identically distributed (i.i.d.) samples. Thus, accurate density estimates are not required for accurate kernel entropy estimates; in fact, it is a good idea when estimating entropy to sacrifice some accuracy in the quality of the corresponding density estimate.

This article reviews an earlier recursive approach for computing such reach trajectories and presents a new nonrecursive approach, with computations that may be performed analytically for the most part, leading to a significant gain in the accuracy of the inferred trajectory while imposing a very small computational burden. Extensions of the approach are discussed including the incorporation of multiple target observations at different times, and multiple possible target locations.

We describe a class of models that predict how the instantaneous firing rate of a neuron depends on a dynamic stimulus. The models utilize a learnt pointwise nonlinear transform of the stimulus, followed by a linear filter that acts on the sequence of transformed inputs. In one case, the nonlinear transform is the same at all filter lag-times. Thus, this "input nonlinearity" converts the initial numerical representation of stimulus value to a new representation that provides optimal input to the subsequent linear model. We describe algorithms that estimate both the input nonlinearity and the linear weights simultaneously; and present techniques to regularise and quantify uncertainty in the estimates. In a second approach, the model is generalized to allow a different nonlinear transform of the stimulus value at each lag-time. Although more general, this model is algorithmically more straightforward to fit. However, it has many more degrees of freedom than the first approach, thus requiring more data for accurate estimation. We test the feasibility of these methods on synthetic data, and on responses from a neuron in rodent barrel cortex. The models are shown to predict responses to novel data accurately, and to recover several important neuronal response properties.

As recent advances in calcium sensing technologies enable us to simultaneously image many neurons, complementary analytical tools must also be developed to maximize the utility of this experimental paradigm. While the observations here are fluorescence images, the signals of interest - spike trains and/or time varying intracellular calcium concentrations - are hidden. Inferring these hidden signals is often problematic due to noise, nonlinearities, slow imaging rate, and unknown biophysical parameters. We overcome these difficulties by developing a family of particle filters based on biophysical models of spiking, calcium dynamics, and fluorescence. We show that even in simple cases, the particle filters outperform the optimal linear (i.e., Wiener) filter, both by obtaining better estimates and by providing errorbars. We then relax a number of our model assumptions to incorporate nonlinear saturation of the fluorescence signal, as well external stimulus and spike history dependence of the spike trains. Using both simulations and in vitro fluorescence observations, we demonstrate superresolution by inferring when within a frame each spike occurs. Furthermore, the model parameters may be estimated using expectation-maximization with only a very limited amount of data, without the requirement of any simultaneous electrophysiology and imaging experiments.

Where we allocate our visual spatial attention depends upon a continual competition between internally gen- erated goals and external distractions. Recently it was shown that single neurons in the macaque lateral intraparietal area (LIP) can predict the amount of time a distractor can shift the locus of spatial attention away from a goal. We propose that this remarkable dy- namical correspondence between single neurons and attention can be explained by a network model in which generically high-dimensional firing-rate vectors rapidly decay to a single mode. We find direct experi- mental evidence for this model, not only in the original attentional task, but also in a very different task involv- ing perceptual decision making. These results confirm a theoretical prediction that slowly varying activity pat- terns are proportional to spontaneous activity, pose constraints on models of persistent activity, and sug- gest a network mechanism for the emergence of ro- bust behavioral timing from heterogeneous neuronal populations.

Understanding neural responses with natural stimuli has increasingly become an essential part of characterizing neural coding. Neural responses are commonly characterized by a linear-nonlinear (LN) model, in which the output of a linear filter applied to the stimulus is transformed by a static nonlinearity to determine neural response. To estimate the linear filter in the LN model, studies of responses to natural stimuli commonly use methods that are unbiased only for a linear model (in which there is no static nonlinearity): spike-triggered averages with correction for stimulus power spectrum, with or without regularization. While these methods work well for artificial stimuli, such as Gaussian white noise, we show here that they estimate neural filters of LN models from responses to natural stimuli much more poorly. We studied simple cells in cat primary visual cortex. We demonstrate that the filters computed by directly taking the nonlinearity into account have better predictive power and depend less on the stimulus than those computed under the linear model. With noise stimuli, filters computed using the linear and LN models were similar, as predicted theoretically. With natural stimuli, filters of the two models can differ profoundly. Noise and natural stimulus filters differed significantly in spatial properties, but these differences were exaggerated when filters were computed using the linear rather that the LN model. While regularization of filters computed under the linear model improved their predictive power, it also led to systematic distortions of their spatial frequency profiles, especially at low spatial and temporal frequencies.

There are two basic problems in the statistical analysis of neural data. The ``encoding'' problem concerns how information is encoded in neural spike trains: can we predict the spike trains of a neuron (or population of neurons), given an arbitrary stimulus or observed motor response? Conversely, the ``decoding'' problem concerns how much information is in a spike train: in particular, how well can we estimate the stimulus that gave rise to the spike train? This chapter describes statistical model-based techniques that in some cases provide a unified solution to these two coding problems. These models can capture stimulus dependencies as well as spike history and interneuronal interaction effects in population spike trains, and are intimately related to biophysically-based models of integrate-and-fire type. We describe flexible, powerful likelihood-based methods for fitting these encoding models and then for using the models to perform optimal decoding. Each of these (apparently quite difficult) tasks turn out to be highly computationally tractable, due to a key concavity property of the model likelihood. Finally, we return to the encoding problem to describe how to use these models to adaptively optimize the stimuli presented to the cell on a trial-by-trial basis, in order that we may infer the optimal model parameters as efficiently as possible.

Gain modulation, in which the sensitivity of a neural response to one input is modified by a second input, is studied at single-neuron and network levels. At the single neuron level, gain modulation can arise if the two inputs are subject to a direct multiplicative interaction. Alternatively, these inputs can be summed in a linear manner by the neuron and gain modulation can arise, instead, from a nonlinear input-output relationship. We derive a mathematical constraint that can distinguish these two mechanisms even though they can look very similar, provided sufficient data of the appropriate type are available. Previously, it has been shown in coordinate transformation studies that artificial neurons with sigmoid transfer functions can acquire a nonlinear additive form of gain modulation through learning-driven adjustment of synaptic weights. We use the constraint derived for single-neuron studies to compare responses in this network with those of another network model based on a biologically inspired transfer function that can support approximately multiplicative interactions.

Receptor neurons may not encode sensory information in an efficient manner. A new paper supports the idea that the brain achieves optimal encoding downstream of sensory transduction through additional processing.

The origin of orientation selectivity in primary visual cortex (V1) is a model problem for understanding cerebral cortical circuitry. A key constraint is that orientation tuning width is invariant under changes in stimulus contrast. We have previously shown that this can arise from the combination of feedforward lateral geniculate nucleus (LGN) input and an orientation-untuned component of feedforward inhibition that dominates excitation. However, these models did not include the large background voltage noise observed in vivo. Here, we include this noise and examine a simple model of cat V1 response. Constraining our simulations to fit physiological data, our single model parameter is the strength of feedforward inhibition relative to LGN excitation. With physiological noise, the contrast invariance of orientation tuning depends little on inhibition level, although very weak or very strong inhibition leads to weak broadening or sharpening, respectively, of tuning with contrast. For any inhibition level, an alternative measure of orientation tuning -- the circular variance -- decreases with contrast as observed experimentally. These results arise primarily because the voltage noise causes large inputs to be much more strongly amplified than small ones in evoking spiking responses, relatively suppressing responses to nonpreferred stimuli. However, inhibition comparable to or stronger than excitation appears necessary to suppress spiking responses to nonpreferred orientations to the extent seen in vivo and to allow the emergence of a tuned mean voltage response. These two response properties provide the strongest constraints on model details. Antiphase inhibition from inhibitory simple cells, and not just untuned inhibition from inhibitory complex cells, appears necessary to fully explain these aspects of cortical orientation tuning.

Volitional behavior relies on the brain's ability to remap sensory flow to motor programs whenever demanded by a changed behavioral context. To investigate the circuit basis of such flexible behavior, we have developed a biophysically based decision-making network model of spiking neurons for arbitrary sensorimotor mapping. The model quantitatively reproduces behavioral and prefrontal single-cell data from an experiment in which monkeys learn visuomotor associations that are reversed unpredictably from time to time. We show that when synaptic modifications occur on multiple timescales, the model behavior becomes flexible only when needed: slow components of learning usually dominate the decision process. However, if behavioral contexts change frequently enough, fast components of plasticity take over, and the behavior exhibits a quick forget-and-learn pattern. This model prediction is confirmed by monkey data. Therefore, our work reveals a scenario for conditional associative learning that is distinct from instant switching between sets of well-established sensorimotor associations.

We recently introduced likelihood-based methods for fitting stochastic integrate-and-fire models to spike train data. The key component of this method involves the likelihood that the model will emit a spike at a given time t. Computing this likelihood is equivalent to computing a Markov first passage time density (the probability that the model voltage crosses threshold for the first time at time t). Here we detail an improved method for computing this likelihood, based on solving a certain integral equation. This integral equation method has several advantages over the techniques discussed in our previous work: in particular, the new method has fewer free parameters and is easily differentiable (for gradient computations). The new method is also easily adaptable for the case in which the model conductance, not just the input current, is time-varying. Finally, we describe how to incorporate large deviations approximations to very small likelihoods.

Both in vivo and in vitro recordings indicate that neuronal mem- brane potentials can make spontaneous transitions between distinct up and down states. At the network level, populations of neurons have been observed to make these transitions synchronously. Although synap- tic activity and intrinsic neuron properties play an important role, the precise nature of the processes responsible for these phenomena is not known. Using a computational model we explore the interplay between intrinsic neuronal properties and synaptic fluctuations. Model neurons of the integrate-and-fire type were extended by adding a nonlinear mem- brane current. Networks of these neurons exhibit large amplitude syn- chronous spontaneous fluctuations that make the neurons jump between up and down states, thereby producing bimodal membrane potential distributions. The effect of sensory stimulation on network responses depends on whether the stimulus is applied during an up state or deeply inside a down state. External noise can be varied to modulate the net- work continuously between two extreme regimes in which it remains permanently in either the up or the down state.

Recent developments in multi-electrode recordings enable the simultaneous measurement of the spiking activity of many neurons. Analysis of such multineuronal data is one of the key challenges in computational neuroscience today. In this work, we develop a multivariate point-process model in which the observed activity of a network of neurons depends on three terms: 1) the experimentally-controlled stimulus; 2) the spiking history of the observed neurons; and 3) a latent noise source that corresponds, for example, to ``common input'' from an unobserved population of neurons that is presynaptic to two or more cells in the observed population. We develop an expectation-maximization algorithm for fitting the model parameters; here the expectation step is based on a continuous-time implementation of the extended Kalman smoother, and the maximization step involves two concave maximization problems which may be solved in parallel. The techniques developed allow us to solve a variety of inference problems in a straightforward, computationally efficient fashion; for example, we may use the model to predict network activity given an arbitrary stimulus, infer a neuron's firing rate given the stimulus and the activity of the other observed neurons, and perform optimal stimulus decoding and prediction. We present several detailed simulation studies which explore the strengths and limitations of our approach.

Memories maintained in patterns of synaptic connectivity are rapidly over-written and destroyed by ongoing plasticity due to the storage of new memories. Short memory lifetimes arise from the bound that must be imposed on synaptic efficacy in any realistic model. We explore whether memory performance can be improved by allowing synapses to traverse a large number of states before reaching their bounds, or by changing the way these bounds are imposed. In the case of hard bounds memory lifetimes grow proportional to the square of the number synaptic states, but only if potentiation and depression are precisely balanced. Improved performance can be obtained without fine tuning by imposing soft bounds, but this improvement is only linear in the number of synaptic states. We explore a number of other possibilities and conclude that improving memory performance requires a more radical modification of the standard model of memory storage.

Circadian rhythms are daily oscillations in behaviors that persist in constant light/dark conditions with periods close to 24 hours. A striking feature of these rhythms is that their periods remain fairly constant over a wide range of physiological temperatures, a feature called temperature compensa- tion. Although circadian rhythms have been associated with periodic oscillations in mRNA and protein levels, the question of how to construct a network of chemical reactions that is temperature compensated remains unanswered. We discuss a general framework for building such a network.

The word-frequency effect (WFE) in recognition memory refers to the finding that more rare words are better recognized than more common words. We demonstrate that a familiarity-discrimination model operating on data from a semantic word-association space yields a robust WFE in data on both hit rates and false-alarm rates. Our modeling results suggest that word frequency is encoded in the semantic structure of language, and that this encoding contributes to the WFE observed in item-recognition experiments.

The dynamics of neural networks is influenced strongly by the spectrum of eigenvalues of the matrix describing their synaptic connectivity. In large networks, elements of the synaptic connectivity matrix can be chosen randomly from appropriate distributions, making results from random matrix theory highly relevant. Unfortunately, classic results on the eigenvalue spectra of random matrices do not apply to synaptic connectivity matrices because of the constraint that individual neurons are either excitatory or inhibitory. Therefore, we compute eigenvalue spectra of large random matrices with excitatory and inhibitory columns drawn from distributions with different means and equal or different variances.

We compute the exact spike-triggered average (STA) of the voltage for the nonleaky IF cell in continuous time, driven by Gaussian white noise. The computation is based on techniques from the theory of renewal processes and continuous-time hidden Markov processes (e.g., the backward and forward Fokker-Planck partial differential equations associated with first-passage time densities). From the STA voltage it is straightforward to derive the STA input current. The theory also gives an explicit asymptotic approximation for the STA of the leaky IF cell, valid in the low-noise regime $\sigma \to 0$. We consider both the STA and the conditional average voltage given an observed spike ``doublet'' event, i.e. two spikes separated by some fixed period of silence. In each case, we find that the STA as a function of time-preceding-spike, $\tau$, has a square-root singularity as $\tau$ approaches zero from below, and scales linearly with the scale of injected noise current. We close by briefly examining the discrete-time case, where similar phenomena are observed.

Biophysically accurate multi-compartmental models of individual neurones have signi cantly advanced our understanding of the input-output function of single cells. These models depend on a large number of parameters which are dif cult to estimate. In practise, they are often hand tuned to match measured physiological behaviors, thus raising questions of identifiability and interpretability. We propose a statistical approach to the automatic estimation of various biologically relevant parameters, including 1) the distribution of channel densities; 2) the spatiotemporal pattern of synaptic input; and 3) axial resistances across extended dendrites. Recent experimental advances, notably in voltage-sensitive imaging, motivate us to assume access to: a) the spatiotemporal voltage signal in the dendrite, and b) an approximate description of the channel kinetics of interest. We show here that, given a) and b), the parameters 1)-3) can be inferred simultaneously by nonnegative linear regression; that this optimization problem possesses a unique solution and is guaranteed to converge despite the large number of parameters and their complex nonlinear interaction; and that standard optimization algorithms ef ciently reach this optimum with modest computational and data requirements. We demonstrate that the method leads to accurate estimations on a wide variety of challenging model data sets that include up to on the order of 10,000 parameters (roughly two orders of magnitude more than previously feasible), and describe how the method gives insights into the functional interaction of groups of channels.

Reward-based learning in neural systems is challenging because a large number of parameters that affect network function must be optimized solely on the basis of a reward signal that indicates improved performance. Searching the parameter space for an optimal solution is particularly difficult if the network is large. We show that Hebbian forms of synaptic plasticity applied to synapses between a supervisor circuit and the network it is controlling can effectively reduce the dimension of the space of parameters being searched to support efficient reinforcement-based learning in large networks. The critical element is that the connections between the supervisor units and the network must be reciprocal. Once the appropriate connections have been set up by Hebbian plasticity, a reinforcement-based learning procedure leads to rapid learning in a function approximation task. Hebbian plasticity within the network being supervised ultimately allows the network to perform the task without input from the supervisor.

It has been shown that motion aftereffect (MAE) not only is present at the adapted location but also partially transfers to nearby non- adapted locations. However, it is not clear whether MAE transfers across the fixation point. Since cells in area MSTd have receptive fields that cover both sides of the fixation point and since many MSTd cells, but not cells in earlier visual areas, prefer complex motion patterns such as expansion, we tested cross-fixation transfer of MAE induced by expanding random-dots stimuli. We also used rightward translational motion for comparison. Subjects adapted to motion patterns on a fixed side of the fixation point. Dynamic MAE was then measured with a nulling procedure at both the adapted site and the mirror site across the fixation point. SubjectsÕ eye fixation during stimulus presentation was monitored with an infrared eye tracker. At the adapted site, both the expansion and the translation patterns generated strong MAEs, as expected. However, only the expansion pattern, but not translation pattern, generated significant MAE at the mirror site. This remained true even after we adjusted stimulus parameters to equate the strengths of the expansion MAE and translation MAE at the adapted site. We conclude that there is cross-fixation transfer of MAE for expansion motion but not for translational motion.

Neurons recorded from electrode arrays show a remarkable scaling property in their bursts of spontaneous activity, referred to as "avalanches' (Beggs and Plentz, 2003 & 2004). Such scaling suggests a critical property in the coupling of these circuits. We show that similar scaling laws can arise in a simple model for the growth of neuronal processes. In the model (Van Ooyen and Van Pelt, 1994 & 1996), the spatial range of the processes extending from each neuron is represented by a circle that grows or shrinks as a function of the average intracellular calcium concentration. Neurons interact when the circles corresponding to their processes intersect, with a strength proportional to the area of overlap.

Given the extensive attenuation that can occur along dendritic cables, location within the dendritic tree might appear to be a dominant factor in determining the impact of a synapse on the postsynaptic response. By this reasoning, distal syn- apses should have a smaller effect than proximal ones. However, experimental evidence from several types of neurons, such as CA1 pyramidal cells, indicates that a compensatory strengthening of syn- apses counteracts the effect of location on synaptic efficacy. A form of spike-timing-dependent plasticity (STDP), called anti-STDP, com- bined with non-Hebbian activity-dependent plasticity can account for the equalization of synaptic efficacies. This result, obtained originally in models with unbranched passive cables, also arises in multi- compartment models with branched and active dendrites that feature backpropagating action potentials, including models with CA1 py- ramidal morphologies. Additionally, when dendrites support the local generation of action potentials, anti-STDP prevents runaway dendritic spiking and locally balances the numbers of dendritic and backpropa- gating action potentials. Thus in multiple ways, anti-STDP eliminates the location dependence of synapses and allows Hebbian plasticity to operate in a more "democratic" manner.

Several models exist for explaining primary visual cortex (V1) orientation tuning. The modified feedforward model (MFM) and the recurrent model (RM) are major examples. We have implemented these two models, at the same level of detail, alongside a few newer variations, and thoroughly compared their receptive-field structures. We found that antiphase inhibition in the MFM enhances both spatial phase information and orientation tuning, producing well-tuned simple cells. This remains true for a newer version of the MFM that incorporates untuned complex-cell inhibition. In contrast, when the recurrent connections in the RM are strong enough to produce typical V1 orientation tuning, they also eliminate spatial phase information, making the cells com- plex. Introducing phase specificity into the connections of the RM (as done in an original version of the RM) can make the cells phase sensitive, but the cells show an incorrect 90¡ peak shift of orientation tuning under opposite contrast signs. An inhibition-dominant version of the RM can generate well-tuned cells across the simpleÐ complex spectrum, but it predicts that the net effect of cortical interactions is to suppress feedforward excitation across all orientations in simple cells. Finally, adding antiphase inhibition used in the MFM into the RM produces a most general model. We call this new model the modified recurrent model (MRM) and show that this model can also produce well-tuned cells throughout the simpleÐ complex spectrum. Unlike the inhibition-dominant RM, the MRM is consistent with data from cat V1, suggesting that the net effect of cortical interactions is to boost simple cell responses at the preferred orientation. These results sug- gest that the MFM is well suited for explaining orientation tuning in simple cells, whereas the standard RM is for complex cells. The assignment of the RM to complex cells also avoids conflicts between the RM and the experiments of cortical inactivation (done on simple cells) and the spatial-frequency dependency of orientation tuning (found in simple cells). Because orientation-tuned V1 cells show a continuum of simple- to complex-cell behavior, the MRM provides the best description of V1 data.

Activity-dependent modification of synaptic strengths due to spike-timing-dependent plasticity (STDP) is sensitive to correlations between pre- and postsynaptic firing over timescales of tens of milliseconds. Temporal associations typically encountered in behavioral tasks involve times on the order of seconds. To relate the learning of such temporal associations to STDP, we must account for this large discrepancy in timescales. We show that the gap between synaptic and behavioral timescales can be bridged if the stimuli being associated generate sustained responses that vary appropriately in time. Synapses between neurons that fire this way can be modified by STDP in a manner that depends on the temporal ordering of events separated by several seconds even though the underlying plasticity has a much smaller temporal window.

Many biological systems exhibit complex temporal behavior that cannot be adequately characterized by a single time constant. This dynamics, observed from single channels up to the level of human psychophysics, is often better described by power-law rather than exponential dependences on time. We develop and study the properties of neural models with scale-invariant, power-law adaptation and contrast them with the more commonly studied exponential case. Responses of an adapting firing-rate model to constant, pulsed and oscillating inputs in both the power-law and exponential cases are considered. We construct a spiking model with power-law adaptation based on a nested cascade of processes and show that it can be "programmed" to produce a wide range of time delays. Finally, within a network model, we use power-law adaptation to reproduce long-term features of the tilt aftereffect.

Movement duration is an integral com- ponent of motor control, but nearly all extant optimization models of motor planning prefix duration instead of explaining it. Here we propose a new optimization principle that predicts movement dura- tion. The model assumes that the brain attempts to minimize move- ment duration under the constraint of meeting an accuracy criterion. The criterion is task and context dependent but is fixed for a given task and context. The model determines a unique duration as a trade-off between speed (time optimality) and accuracy (acceptable endpoint scatter). We analyzed the model for a linear motor plant, and obtained a closed-form equation for determining movement duration. By solv- ing the equation numerically with specific plant parameters for the eye and arm, we found that the model can reproduce saccade duration as a function of amplitude (the main sequence), and arm-movement duration as a function of the ratio of target distance to size (FittsÕs law). In addition, it explains the dependency of peak saccadic speed on amplitude and the dependency of saccadic duration on initial eye position. Furthermore, for arm movements, the model predicts a scaling relationship between peak velocity and distance and a reduc- tion in movement duration with a moderate increase in viscosity. Finally, for a linear plant, our model predicts a neural control signal identical to that of the minimum-variance model set to the same movement duration. This control signal is a smooth function of time (except at the endpoint), in contrast to the discontinuous bangÐ bang control found in the time-optimal control literature. We suggest that one aspect of movement planning, as revealed by movement duration, may be to assign an endpoint accuracy criterion for a given task and context.

Sensory neuroscience seeks to understand how the brain encodes natural environments. However, neural coding has largely been studied using simplified stimuli. In order to assess whether the brain's coding strategy depends on the stimulus ensemble, we apply a new information-theoretic method that allows unbiased calculation of neural filters (receptive fields) from responses to natural scenes or other complex signals with strong multipoint correlations. In the cat primary visual cortex we compare responses to natural inputs with those to noise inputs matched for luminance and contrast. We find that neural filters adaptively change with the input ensemble so as to increase the information carried by the neural response about the filtered stimulus. Adaptation affects the spatial frequency composition of the filter, enhancing sensitivity to under-represented frequencies in agreement with optimal encoding arguments. Adaptation occurs over 40 s to many minutes, longer than most previously reported forms of adaptation.

Controlled responses differ from reflexes because they can be turned off and on. This is a critical part of what distinguishes animals from automatons. How does the nervous system gate the flow of information so that a sensory stimulus that elicits a strong response on some occasions, evokes no response on others? A related question concerns how the flow of sensory information is altered when we pay close attention to something as opposed to when we ignore it. Most research in neuroscience focuses on circuits that directly respond to stimuli or generate motor output. But what of the circuits and mechanisms that control these direct responses, that modulate them and turn them off and on?
Self-regulated switching is vital to the operation of complex machines such as computers. The essential building block of a computer is a voltage-gated switch, the transistor, that is turned off and on by the same sorts of currents that it controls. By analogy, the question of my title refers to neural pathways that not only carry the action potentials that arise from neural activity, but are switched off and on by neural activity as well. By what biophysical mechanisms could this occur?
In the spirit of this volume, the point of this contribution is to raise a question, not to answer it. I will discuss three possible mechanisms— neuromodulation, inhibition, and gain modulation—and assess the merits and short-comings of each of them. I have my prejudices, which will become obvious, but I do not want to rule out any of these as candidates, nor do I want to leave the impression that the list is complete or that the problem is in any sense solved.

Transmission of signals within the brain is essential for cognitive function, but it is not clear how neural circuits support reliable and accurate signal propagation over a sufficiently large dynamic range. Two modes of propagation have been studied: synfire chains, in which synchronous activity travels through feedforward layers of a neuronal network, and the propagation of fluctuations in firing rate across these layers. In both cases, a sufficient amount of noise, which was added to previous models from an external source, had to be included to support stable propagation. Sparse, randomly connected networks of spiking model neurons can generate chaotic patterns of activity. We investigate whether this activity, which is a more realistic noise source, is sufficient to allow for signal transmission. We find that, for rate-coded signals but not for synfire chains, such networks support robust and accurate signal reproduction through up to six layers if appropriate adjustments are made in synaptic strengths. We investigate the factors affecting transmission and show that multiple signals can propagate simultaneously along different pathways. Using this feature, we show how different types of logic gates can arise within the architecture of the random network through the strengthening of specific synapses.

Observers can better discriminate orientation or direction near the cardinal axes than near an oblique axis. We investigated whether this well-known oblique effect is determined by the physical or the perceived axis of the stimuli. Using the simultaneous tilt illusion, we generated perceptually different orientations for the same inner (target) grating by contrasting it with differently oriented outer gratings. Subjects compared the target orientation with a set of reference orientations. If orientation discrimina- bility was determined by the physical orientations, the psychometric curves for the same target grating would be identical. Instead, all subjects produced steeper curves when perceiving target gratings near vertically as opposed to more obliquely. This result of orientation discrimination was confirmed by using adaptation-generated tilt aftereffect to manipulate the perceived ori- entation of a given physical orientation. Moreover, we obtained the same result in direction discrimination by using motion repul- sion to alter the perceived direction of a given physical direction. We conclude that when the perceived orientation or direction differs from the physical orientation or direction, the oblique effect depends on perceived, rather than physical, orientation or direction. Finally, as a by-product of the study, we found that, around the vertical direction, motion repulsion is much stronger when the inducing direction is more clockwise to the test direction than when it is more counterclockwise.

Motion repulsion involves interaction between two directions of motion. Since attention is known to bias interactions among different stimuli, we investigated the effect of attentional tasks on motion repulsion. We used two overlapping sets of random dots moving in different directions. When subjects had to detect a small speed-change or luminance change for dots along one direction, the repulsive influence from the other direction was significantly reduced compared with the control case without attentional tasks. However, when the speed-change could occur to either direction such that subjects had to attend both directions to detect the change, motion repulsion was not different from the control. A further experiment showed that decreasing the difficulty of the atten- tional task resulted in the disappearance of the attentional effect in the case of attention to one direction. Finally, over a wide range of contrasts for the unattended direction, attention reduced repulsion measured with the attended direction. These results are con- sistent with the physiological finding that strong attention to one direction of motion reduces inhibitory effects from the other direction.

Stereo plaid stimuli were created to investigate whether depth perception is determined by an intersection of constraints (IOC) or vector average (VA) operation on the Fourier components, or by the second-order (non-Fourier) feature in a pattern. We first cre- ated stereo plaid stimuli where IOC predicted vertical disparity, VA predicted positive diagonal disparity and the second-order fea- ture predicted negative diagonal disparity. In a depth discrimination task, observers indicated whether they perceived the pattern as near or far relative to a zero-disparity aperture. Observers perception was consistent with the disparity predicted by VA, indicat- ing its dominance over IOC and the second-order feature in this condition. Additional stimuli in which VA predicted vertical dis- parity were created to investigate whether VA would dominate perception when it was a less reliable cue. In this case, observers performance was consistent with disparity predicted by IOC or the second-order feature, not VA. Finally, in order to determine whether the second-order feature contributes to depth perception, stimuli were created where IOC and VA predicted positive hor- izontal disparity while the second-order feature predicted negative horizontal disparity. When the component gratings were oriented near horizontal (±83 from vertical), depth perception corresponded to that predicted by the second-order feature. However, as the components moved away from horizontal (±75 and ±65 from vertical), depth perception was increasingly likely to be predicted by an IOC or VA operation. These experiments suggest that the visual system does not rely exclusively on a single method for com- puting pattern disparity. Instead, it favours the most reliable method for a given condition.