nips nips2001 nips2001-48 nips2001-48-reference knowledge-graph by maker-knowledge-mining
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Author: Odelia Schwartz, E. J. Chichilnisky, Eero P. Simoncelli
Abstract: Spike-triggered averaging techniques are effective for linear characterization of neural responses. But neurons exhibit important nonlinear behaviors, such as gain control, that are not captured by such analyses. We describe a spike-triggered covariance method for retrieving suppressive components of the gain control signal in a neuron. We demonstrate the method in simulation and on retinal ganglion cell data. Analysis of physiological data reveals significant suppressive axes and explains neural nonlinearities. This method should be applicable to other sensory areas and modalities. White noise analysis has emerged as a powerful technique for characterizing response properties of spiking neurons. A sequence of stimuli are drawn randomly from an ensemble and presented in rapid succession, and one examines the subset that elicit action potentials. This “spike-triggered” stimulus ensemble can provide information about the neuron’s response characteristics. In the most widely used form of this analysis, one estimates an excitatory linear kernel by computing the spike-triggered average (STA); that is, the mean stimulus that elicited a spike [e.g., 1, 2]. Under the assumption that spikes are generated by a Poisson process with instantaneous rate determined by linear projection onto a kernel followed by a static nonlinearity, the STA provides an unbiased estimate of this kernel [3]. Recently, a number of authors have developed interesting extensions of white noise analysis. Some have examined spike-triggered averages in a reduced linear subspace of input stimuli [e.g., 4]. Others have recovered excitatory subspaces, by computing the spiketriggered covariance (STC), followed by an eigenvector analysis to determine the subspace axes [e.g., 5, 6]. Sensory neurons exhibit striking nonlinear behaviors that are not explained by fundamentally linear mechanisms. For example, the response of a neuron typically saturates for large amplitude stimuli; the response to the optimal stimulus is often suppressed by the presence of a non-optimal mask [e.g., 7]; and the kernel recovered from STA analysis may change shape as a function of stimulus amplitude [e.g., 8, 9]. A variety of these nonlinear behaviors can be attributed to gain control [e.g., 8, 10, 11, 12, 13, 14], in which neural responses are suppressively modulated by a gain signal derived from the stimulus. Although the underlying mechanisms and time scales associated with such gain control are current topics of research, the basic functional properties appear to be ubiquitous, occurring throughout the nervous system. a b 0 k0 0 Figure 1: Geometric depiction of spike-triggered analyses. a, Spike-triggered averaging with two-dimensional stimuli. Black points indicate raw stimuli. White points indicate stimuli eliciting a spike, and the STA (black vector), which provides an estimate of , corresponds to their center of mass. b, Spike-triggered covariance analysis of suppressive axes. Shown are a set of stimuli lying on a plane perpendicular to the excitatory kernel, . Within the plane, stimuli eliciting a spike are concentrated in an elliptical region. The minor axis of the ellipse corresponds to a suppressive stimulus direction: stimuli with a significant component along this axis are less likely to elicit spikes. The stimulus component along the major axis of the ellipse has no influence on spiking. ¢ £ ¡ ¢ ¡ Here we develop a white noise methodology for characterizing a neuron with gain control. We show that a set of suppressive kernels may be recovered by finding the eigenvectors of the spike-triggered covariance matrix associated with smallest variance. We apply the technique to electrophysiological data obtained from ganglion cells in salamander and macaque retina, and recover a set of axes that are shown to reduce responses in the neuron. Moreover, when we fit a gain control model to the data using a maximum likelihood procedure within this subspace, the model accounts for changes in the STA as a function of contrast. 1 Characterizing suppressive axes ¤¥ As in all white noise approaches, we assume that stimuli correspond to vectors, , in some finite-dimensional space (e.g., a neighborhood of pixels or an interval of time samples). We assume a gain control model in which the probability of a stimulus eliciting a spike grows monotonically with the halfwave-rectified projection onto an excitatory linear kernel, , and is suppressively modulated by the fullwave-rectified projection onto a set of . linear kernels, ¨ ¤§ ¤¥ ©¤ § ¨ ¤ ¥ ©¤ § ¦ First, we recover the excitatory kernel, . This is achieved by presenting spherically symmetric input stimuli (e.g., Gaussian white noise) to the neuron and computing the STA (Fig. 1a). STA correctly recovers the excitatory kernel, under the assumption that each of the gain control kernels are orthogonal (or equal) to the excitatory kernel. The proof is essentially the same as that given for recovering the kernel of a linear model followed by a monotonic nonlinearity [3]. In particular, any stimulus can be decomposed into a component in the direction of the excitatory kernel, and a component in a perpendicular direction. This can be paired with another stimulus that is identical, except that its component in the perpendicular direction is negated. The two stimuli are equally likely to occur in a spherically Gaussian stimulus set (since they are equidistant from the origin), and they are equally likely to elicit a spike (since their excitatory components are equal, and their rectified perpendicular components are equal). Their vector average lies in the direction of the excitatory kernel. Thus, the STA (which is an average over all such stimuli, or all such stimulus pairs) must also lie in that direction. In a subsequent section we explain how to Model: Retrieved: Excitatory: Excitatory: Eigenvalues: Suppressive: Suppressive: Weights Variance (eigenvalue) { 1.5 { 2{ 2.5 { 3{ 1 1 Arbitrary 0 Axis number 350 Figure 2: Estimation of kernels from a simulated model (equation 2). Left: Model kernels. Right: Sorted eigenvalues of covariance matrix of stimuli eliciting spikes (STC). Five eigenvalues fall significantly below the others. Middle: STA (excitatory kernel) and eigenvectors (suppressive kernels) associated with the lowest eigenvalues. recover the excitatory kernel when it is not orthogonal to the suppressive kernels. Next, we recover the suppressive subspace, assuming the excitatory kernel is known. Consider the stimuli lying on a plane perpendicular to this kernel. These stimuli all elicit the same response in the excitatory kernel, but they may produce different amounts of suppression. Figure 1b illustrates the behavior in a three-dimensional stimulus space, in which one axis is assumed to be suppressive. The distribution of raw stimuli on the plane is spherically symmetric about the origin. But the distribution of stimuli eliciting a spike is narrower along the suppressive direction: these stimuli have a component along the suppressive axis and are therefore less likely to elicit a spike. This behavior is easily generalized from this plane to the entire stimulus space. If we assume that the suppressive axes are fixed, then we expect to see reductions in variance in the same directions for any level of numerator excitation. Given this behavior of the spike-triggered stimulus ensemble, we can recover the suppressive subspace using principal component analysis. We construct the sample covariance matrix of the stimuli eliciting a spike: £ §¥
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