Bruce W. Knight

Voltage noise in Limulus visual cells.

Intracellular recordings from Limulus eccentric cells suggest that the generator potential arises from the superposition of numerous discrete fluctuations in membrane conductance. If this is so, a relation between frequency response to flickering light and noise characteristics under steady light may be predicted. This prediction is verified experimentally. If a discrete fluctuation model is assumed, the data indicate that increased light has two major effects: (i) the discrete events are strongly light-adapted to smaller size, and (ii) the time course of each event becomes briefer.

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(105) 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.

A population study of integrate-and-fire-or-burst neurons

Any realistic model of the neuronal pathway from the retina to the visual cortex (V1) must account for the burstingbehavior of neurons in the lateral geniculate nucleus (LGN). A robust but minimal model, the integrate- and-fire-or-burst (IFB) model, has recently been proposed for individual LGN neurons. Based on this, we derive a dynamic population model and study a population of such LGN cells. This population model, the first simulation of its kind evolving in a two-dimensional phase space, is used to study the behavior of bursting populations in response to diverse stimulus conditions.

Dynamics of neuronal populations: the equilibrium solution

The behavior of an aggregate of neurons is followed by means of a population equation which describes the probability density of neurons as a function of membrane potential. The model is based on integrate-and-fire membrane dynamics and a synaptic dynamics which produce a fixed potential jump in response to stimulation. In spite of the simplicity of the model, it gives rise to a rich variety of behaviors. Here only the equilibrium problem is considered in detail. Expressions for the population density and firing rate over a range of parameters are obtained and compared with like forms obtained from the diffusion approximation. The introduction of the jump response to stimulation produces a delay term in the equations, which in turn leads to analytical challenges. A variety of asymptotic techniques render the problem solvable. The asymptotic resultsshow excellent agreement with direct numerical simulations.

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 systems 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 populations 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.

Causality calculations in the time domain: An efficient alternative to the Kramers–Kronig method*

The consequences of causality and analyticity are commonly invoked in procedures for determining optical constants from reflectance data using the Kramers–Kronig relation. Here an entirely elementary argument is advanced, which exploits only the parity of Fourier transforms and the vanishing of the impulse response for negative times, and which avoids the concept of analyticity. This leads to a simply understood algorithm for such computations. The new procedure shows large gains of computational efficiency over the classical Kramers–Kronig approach. The method is applied first to model data and compared with exact results; it is then applied to real data and compared with the result obtained by the standard method. Excellent agreement is obtained in all cases.

Modeling the functional organization of the visual cortex

Abstract While many models of the dynamics and interactions of single neurons are extant, analogous constructs which attempt to describe large-scale (≥O(108)) neuronal activity are few and far between. Optical imaging of the visual cortex makes such macroscopic neuronal activity accessible. Symmetries latent in the cortical architecture are used here to develop a scheme for analyzing such images. In this way, intrinsic modes of cortical response can be uncovered, using minimal assumptions. Some of these modes correspond to already-familiar features of the functional architecture, and it is highly likely that others hold physiological relevance as well. Finally, the number of such modes that would be required in a more fully developed model (incorporating cortical dynamics) is approximated.

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

Abstract. 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.