An optimization approach to signal extraction from noisy multivariate data.

We consider a problem of blind signal extraction from noisy multivariate data, in which each datum represents a system's response, observed under a particular experimental condition. Our prototype example is multipixel functional images of brain activity in response to a set of prescribed experimental stimuli. We present a novel multivariate analysis technique, which identifies the different activity patterns (signals) that are attributable to specific experimental conditions, without a priori knowledge about the signal or the noise characteristics. The extracted signals, which we term the generalized indicator functions, are optimal in the sense that they maximize a weighted difference between the signal variance and the noise variance. With an appropriate choice of the weighting parameter, the method returns a set of images whose signal-to-noise ratios satisfy some user-defined level of significance. We demonstrate the performance of our method in optical intrinsic signal imaging of cat cortical area 17. We find that the method performs effectively and robustly in all tested data, which include both real experimental data and numerically simulated data. The method of generalized indicator functions is related to canonical variate analysis, a multivariate analysis technique that directly solves for the maxima of the signal-to-noise ratio, but important theoretical and practical differences exist, which can make our method more appropriate in certain situations.

The approach of a neuron population firing rate to a new equilibrium: an exact theoretical result.

The response of a noninteracting population of identical neurons to a step change in steady synaptic input can be analytically calculated exactly from the dynamical equation that describes the population's evolution in time. Here, for model integrate-and-fire neurons that undergo a fixed finite upward shift in voltage in response to each synaptic event, we compare the theoretical prediction with the result of a direct simulation of 90,000 model neurons. The degree of agreement supports the applicability of the population dynamics equation. The theoretical prediction is in the form of a series. Convergence is rapid, so that the full result is well approximated by a few terms.

On the simulation of large populations of neurons.

The dynamics of large populations of interacting neurons is investigated. Redundancy present in subpopulations of cortical networks is exploited through the introduction of a probabilistic description. A derivation of the kinetic equations for such subpopulations, under general transmembrane dynamics, is presented. The particular case of integrate-and-fire membrane dynamics is considered in detail. A variety of direct simulations of neuronal populations, under varying conditions and with as many as O(10(5)) neurons, is reported. Comparison is made with analogous kinetic equations under the same conditions. Excellent agreement, down to fine detail, is obtained. It is emphasized that no free parameters enter in the comparisons that are made.

Dynamics of encoding in neuron populations: some general mathematical features.

The use of a population dynamics approach promises efficient simulation of large assemblages of neurons. Depending on the issues addressed and the degree of realism incorporated in the simulated neurons, a wide range of different population dynamics formulations can be appropriate. Here we present a common mathematical structure that these various formulations share and that implies dynamical behaviors that they have in common. This underlying structure serves as a guide toward efficient means of simulation. As an example, we derive the general population firing-rate frequency-response and show how it may be used effectively to address a broad range of interacting-population response and stability problems. A few specific cases will be worked out. A summary of this work appears at the end, before the appendix.

The power ratio and the interval map: spiking models and extracellular recordings.

We describe a new, computationally simple method for analyzing the dynamics of neuronal spike trains driven by external stimuli. The goal of our method is to test the predictions of simple spike-generating models against extracellularly recorded neuronal responses. Through a new statistic called the power ratio, we distinguish between two broad classes of responses: (1) responses that can be completely characterized by a variable firing rate, (for example, modulated Poisson and gamma spike trains); and (2) responses for which firing rate variations alone are not sufficient to characterize response dynamics (for example, leaky integrate-and-fire spike trains as well as Poisson spike trains with long absolute refractory periods). We show that the responses of many visual neurons in the cat retinal ganglion, cat lateral geniculate nucleus, and macaque primary visual cortex fall into the second class, which implies that the pattern of spike times can carry significant information about visual stimuli. Our results also suggest that spike trains of X-type retinal ganglion cells, in particular, are very similar to spike trains generated by a leaky integrate-and-fire model with additive, stimulus-independent noise that could represent background synaptic activity.

Representation of spatial frequency and orientation in the visual cortex.

Knowledge of the response of the primary visual cortex to the various spatial frequencies and orientations in the visual scene should help us understand the principles by which the brain recognizes patterns. Current information about the cortical layout of spatial frequency response is still incomplete because of difficulties in recording and interpreting adequate data. Here, we report results from a study of the cat primary visual cortex in which we employed a new image-analysis method that allows improved separation of signal from noise and that we used to examine the neurooptical response of the primary visual cortex to drifting sine gratings over a range of orientations and spatial frequencies. We found that (i) the optical responses to all orientations and spatial frequencies were well approximated by weighted sums of only two pairs of basis pictures, one pair for orientation and a different pair for spatial frequency; (ii) the weightings of the two pictures in each pair were approximately in quadrature (1/4 cycle apart); and (iii) our spatial frequency data revealed a cortical map that continuously assigns different optimal spatial frequency responses to different cortical locations over the entire spatial frequency range.

Response variability and timing precision of neuronal spike trains in vivo.

We report that neuronal spike trains can exhibit high, stimulus-dependent temporal precision even while the trial-to-trial response variability, measured in several traditional ways, remains substantially independent of the stimulus. We show that retinal ganglion cells and neurons in the lateral geniculate nucleus (LGN) of cats in vivo display both these aspects of firing behavior, which have previously been reported to be contradictory. We develop a simple model that treats neurons as "leaky" integrate-and-fire devices and show that it, too, can exhibit both behaviors. We consider the implications of our findings for the problem of neural coding.

Separating spatially distributed response to stimulation from background. I. Optical imaging.

We consider the problem of estimating a small stimulus-induced response to stimulation that is masked by a fluctuating background when measurements of the background in the absence of stimulation are available, as is common in optical imaging of the cortex and in many other experimental situations. Two related methods based on the Karhunen-Loève procedure are discussed. One seeks the function, an indicator function, that is most parallel to the response data and most orthogonal to the background data. The second removes the subspace spanned by the background from the response. Numerical investigations on simulated optical imaging data show that the first method is generally superior. Connections between the two methods and techniques for assessing the quality of the result are discussed.

Contrast gain control in the primate retina: P cells are not X-like, some M cells are.

Primate retinal ganglion cells that project to the magnocellular layers of the lateral geniculate nucleus (M) are much more sensitive to luminance contrast than those ganglion cells projecting to the parvocellular layers (P). We now report that increasing contrast modifies the temporal-frequency response of M cells, but not of P cells. With rising contrast, the M cells' responses to sinusoidal stimuli show an increasing attenuation at low temporal frequencies while the P cells' responses scale uniformly. The characteristic features of M-cell dynamics are well described by a model originally developed for the X and Y cells of the cat, where the hypothesized nonlinear feedback mechanism responsible for this behavior has been termed the contrast gain control (Shapley & Victor, 1978, 1981; Victor, 1987, 1988). These data provide further physiological evidence that the M-cell pathway differs from the P-cell pathway with regard to the functional elements in the retina. Furthermore, the similarity in dynamics between primate M cells and cat X and Y retinal ganglion cells suggests the possibility that P cells, being different from either group, are a primate specialization not found in the retinae of lower mammals.

Response transfer functions of Limulus ventral photoreceptors: interpretation in terms of transduction mechanisms.

In recent years, our knowledge of the biochemical mechanisms underlying the transduction process in photoreceptors has expanded rapidly. However, a full picture of the temporal dynamics of these mechanisms remains elusive. To study the dynamics in the Limulus ventral photoreceptor, we measure its light-evoked transfer function under voltage clamp. Comparison of this transfer function to biochemically realistic theoretical models of transduction provides insights into the photoreceptor dynamics. This comparison supports the suggestion that the low-frequency behaviour of the Limulus photoreceptor, corresponding to light and dark adaptation, is that of a nonlinear negative feedback loop. The main reactions of this loop have time constants between about 1 and 40 s. Such a feedback loop does not account, however, for the high-frequency behaviour of the responses, which implies the existence of a further, fast-acting, mechanism.

The amplitudes of unit events in Limulus photoreceptors are modulated from an input that resembles the overall response.

Light adaptation is a gain-control process that endows photoreceptors with large dynamic range. In invertebrates, this process appears to be mediated by a negative feedback that sets the amplitude of the isolated photon responses (bumps) by modulating an enzyme's rate of catalysis. This paper reports measurements of the feedback dynamics of Limulus from the responses to small modulations in light intensity. The responses show a noise that apparently arises from the random arrival of photons. We use a dynamic noise-analysis technique to extract the cell's frequency-response transfer function for bump amplitude. Its ratio to the transfer function for the summed response of the cell has a simple form at low frequencies. This indicates that the origin of the feedback responsible for the adaptation is at a stage temporally close to the final conductance response. Moreover, the form of the transfer function suggests feedback by a chemical agent which is removed by a single enzymatic-like stage at low light intensity and by several such stages in parallel but with a spread of time constants at high intensity.

The quantal source of area supralinearity of flash responses in Limulus photoreceptors.

The time integrals of the responses of dark-adapted Limulus ventral photoreceptors to flashes exhibit a supralinear dependence on intensity at intermediate intensities. By decomposing the responses into their elementary single-photon components ("bumps"), we are able to calculate the overall quantum efficiency and to display the time courses of the bump amplitude and rate of appearance. Since the time course of the flash response is not slow compared with that of the bump, it was necessary, in order to carry out the decomposition, to develop a new technique for noise analysis of dynamic signals. This new technique should have wide applications. Our main finding is that the supralinearity of the flash responses corresponds to an increase in bump amplitude, with little change in bump duration or quantum efficiency. The time courses of the bump rate and of the change in bump amplitude are peaked and have widths similar to that of the response itself. The peaks of the time courses of the bump rate and amplitude displayed against the starting times of the bumps do not coincide and occur approximately 80 and approximately 40 ms, respectively, before the peak of the response. The time from the start of a bump to its centroid is approximately 70 ms, which means that the time at which the bump centroid reaches its maximum follows the response peak by 30 ms. These results impose constraints on possible mechanisms for the amplitude enhancement.

Source geometry and dynamics of the visual evoked potential.

A method is developed for the analysis of the steady-state visual evoked potential (VEP) recorded at an array of electrodes. The VEP is assumed to result from the sum of a number of independent mechanisms. Each mechanism proves to have a fixed intracerebral source (independent of stimulus paradigm) and dynamics which depend on the stimulus. The steps in the analysis are: Fourier analysis of the responses and retention of the first 4 even harmonics; factor analysis in the frequency domain; an invariant representation ('the invariant fingerprint') of the geometric information contained in the factor analysis; and interpretation of the invariant fingerprint in terms of a simple biophysical model. This analysis is applied to responses elicited by a contrast-reversing checkerboard. The dependence of the midline VEP wave form on reversal frequency, check size, and area of stimulation (upper, lower, and full-field) reflects a change in the dynamics of the generators, rather than a change in their geometry.

Characteristic patterns of an inhomogeneous imaging system with an application to vision.

We examine consequences of image-forming inhomogeneity in the form of a point-spread function that changes with position on the image plane. The familiar self-replicating sinusoids, which a homogeneous system simply multiplies by its spatial modulation-transfer function, generalize to eigenfunctions, which the system multiplies by eigenvalues. We give a way to calculate the eigenfunctions and eigenvalues from the variable point-spread function. We illustrate this with data from the visual system and show that these lead to a discrete set of most-sensitive eigenfunctions, which we construct.

The eigenfunction problem in higher dimensions: Exact results.

A hermitian integral kernel in N-space may be mapped to a corresponding Hamiltonian in 2N-space by the Wigner transformation. Linear simplectic transformation on the phase space of the Hamiltonian yields a new kernel whose spectrum is unchanged and whose eigenfunctions follow from an explicit unitary transformation. If an integral kernel has a Wigner transform whose surfaces of constant value are concentric ellipsoids, then the Wigner transform yields exact results to the eigenfunction problem. Such behavior is asymptotically generic near extrema of the Wigner transform, from which follow simple and robust asymptotic results for the ends of the eigenvalue spectrum and for the corresponding eigenfunctions.

The eigenfunction problem in higher dimensions: Asymptotic theory.

The spectral problem for linear operators on fully infinite domains is considered. A transformation first introduced by Wigner is used to show a number of asymptotic results. The method leads to a WKB (Wentzel-Kromers-Brillouin) theory for operators in more than one dimension. This includes practical tools for the approximate evaluation of spectra and eigenfunctions. Several general examples are developed.

Adapting bump model for ventral photoreceptors of Limulus.

Light-evoked current fluctuations have been recorded from ventral photoreceptors of Limulus for light intensity from threshold up to 10(5) times threshold. These data are analyzed in terms of the adapting bump noise model, which postulates that (a) the response to light is a summation of bumps; and (b) the average size of bump decreases with light intensity, and this is the major mechanism of light adaptation. It is shown here that this model can account for the data well. Furthermore, the model provides a convenient framework to characterize, in terms of bump parameters, the effects of calcium ions, which are known to affect photoreceptor functions. From responses to very dim light, it is found that the average impulse response (average of a large number of responses to dim flashes) can be predicted from knowledge of both the noise characteristics under steady light and the dispersion of latencies of individual bumps. Over the range of light intensities studied, it is shown that (a) the bump rate increases in strict proportionality to light intensity, up to approximately 10(5) bumps per second; and (b) the bump height decreases approximately as the -0.7 power of light intensity; at rates greater than 10(5) bumps per second, the conductance change associated with the single bump seems to reach a minimum value of approximately 10(-11) reciprocal ohms; (c) from the lowest to the highest light intensity, the bump duration decreases approximately by a factor of 2, and the time scale of the dispersion of latencies of individual bumps decreases approximately by a factor of 3; (d) removal of calcium ions from the bath lengthens the latency process and causes an increase in bump height but appears to have no effect on either the bump rate or the bump duration.

Dispersion of latencies in photoreceptors of Limulus and the adapting-bump model.

To light stimuli of very low intensity, Limulus photoreceptors give a voltage response with a fluctuating delay. This phenomenon has been called "latency dispersion." If the generator potential is the superposition of discrete voltage events ("bumps"), and if the effect of light upon bump size is negligible, then the latency dispersion and the bump shape completely characterize the frequency response to sinusoidal flicker. For very low light intensities, the latency dispersion of the bumps, the bump shape, and the frequency response are measured. It is found that for data obtained at 20 degrees C, the frequency response can be accounted for completely by the latency dispersion and by the bump shape derived from steady-state noise characteristics. At 10 degrees C, the time scale of the response of the photoreceptor is lengthened. The dispersion of latencies and the bump shape are found not to have the same temperature dependence. However, just as those measured at 20 degrees C, the bump shape and the dispersion of latencies measured at 10 degrees C can predict the frequency response measured under the same conditions. These results strongly suggest that the major mechanisms involved in the generator potential are the latency process and the bump process. At high light intensities, the time scale of the generator potential shortens. The decrease in time scale of the generator potential can be attributed to the decreases in time scales of the bumps and of the latency dispersion process.

Adapting-bump model for eccentric cells of Limulus.

Light-evoked intracellular voltage noise records have been obtained from Limulus eccentric cells, from threshold light intensity to an intensity .10(5) times threshold. These data are analyzed in terms of a simple "adapting-bump" noise model. It is shown how the model yields a data reduction procedure that slightly generalizes the familiar use of Campbell's theorem for Poisson shot noise: the correlative effect of adaptation amends Campbell's theorem by a single multiplicative factor, which may be estimated directly from the power spectrum of the noise data. The model also permits direct estimation of the bump shape from the power spectrum. The bump shape estimated from noise at dim light is in excellent agreement with the average shape of bumps observed directly in the dark. The data yield a bump rate that is linear with light up through about 50 times threshold intensity but that falls short of linearity by a factor of 35 at the brightest light. The bump height decreases as the -0.4 power of light intensity across the entire range. Bump duration decreases by a factor of 2 across the entire range, and the adaptation correlation factor descends from unity to about one-third. The modest change of the adaptation correlation shows that naive application of Campbell's theorem to such data is adequate for rough estimation of the model's physiological parameters. This simple accounting for all the data gives support to the adapting-bump model.

Effect of boundaries on the response of a neural network.

The effect an abrupt boundary has upon the dynamical response of a neural network is investigated. The retina of the Limulus eye is used as a model system for studying this effect. A theoretical technique is presented for the quantitative prediction of the manner in which this neural network responds in the vicinity of its boundary. Corresponding experimental measurements of the response to moving stimuli by single optic neurons located near retinal boundaries are presented. Theory and experiment show detailed quantitative agreement.