Identification of directed interactions in networks.

Multichannel data collection in the neurosciences is routine and has necessitated the development of methods to identify the direction of interactions among processes. The most widely used approach for detecting these interactions in such data is based on autoregressive models of stochastic processes, although some work has raised the possibility of serious difficulties with this approach. This article demonstrates that these difficulties are present and that they are intrinsic features of the autoregressive method. Here, we introduce a new method taking into account unobserved processes and based on coherence. Two examples of three-process networks are used to demonstrate that although coherence measures are intrinsically non-directional, a particular network configuration will be associated with a particular set of coherences. These coherences may not specify the network uniquely, but in principle will specify all network configurations consistent with their values and will also specify the relationships among the unobserved processes. Moreover, when new information becomes available, the values of the measures of association already in place do not change, but the relationships among the unobserved processes may become further resolved.

Combining frequency and time domain approaches to systems with multiple spike train input and output.

A frequency domain approach and a time domain approach have been combined in an investigation of the behaviour of the primary and secondary endings of an isolated muscle spindle in response to the activity of two static fusimotor axons when the parent muscle is held at a fixed length and when it is subjected to random length changes. The frequency domain analysis has an associated error process which provides a measure of how well the input processes can be used to predict the output processes and is also used to specify how the interactions between the recorded processes contribute to this error. Without assuming stationarity of the input, the time domain approach uses a sequence of probability models of increasing complexity in which the number of input processes to the model is progressively increased. This feature of the time domain approach was used to identify a preferred direction of interaction between the processes underlying the generation of the activity of the primary and secondary endings. In the presence of fusimotor activity and dynamic length changes imposed on the muscle, it was shown that the activity of the primary and secondary endings carried different information about the effects of the inputs imposed on the muscle spindle. The results presented in this work emphasise that the analysis of the behaviour of complex systems benefits from a combination of frequency and time domain methods.

Neural Spike Train Synchronisation Indices: Definitions, Interpretations and Applications.

A comparison of previously defined spike train syncrhonization indices is undertaken within a stochastic point process framework. The second order cumulant density (covariance density) is shown to be common to all the indices. Simulation studies were used to investigate the sampling variability of a single index based on the second order cumulant. The simulations used a paired motoneurone model and a paired regular spiking cortical neurone model. The sampling variability of spike trains generated under identical conditions from the paired motoneurone model varied from 50% { 160% of the estimated value. On theoretical grounds, and on the basis of simulated data a rate dependence is present in all synchronization indices. The application of coherence and pooled coherence estimates to the issue of synchronization indices is considered. This alternative frequency domain approach allows an arbitrary number of spike train pairs to be evaluated for statistically significant differences, and combined into a single population measure. The pooled coherence framework allows pooled time domain measures to be derived, application of this to the simulated data is illustrated. Data from the cortical neurone model is generated over a wide range of firing rates (1 - 250 spikes/sec). The pooled coherence framework correctly characterizes the sampling variability as not significant over this wide operating range. The broader applicability of this approach to multi electrode array data is briefly discussed.

Estimating space and time constants for active neuronal models from measurements of conduction speed.

Space constants and time constants characterize the spatial and temporal behavior of the membrane potential of a neuronal membrane with constant conductance. However, more realistic models of membrane potential assume that membrane conductance depends on the membrane potential and its history, and therefore, it is not clear that space and time constants can be defined for membranes with this property. However, through a consideration of the properties of trains of action potentials treated as traveling waves, space and time constants for the total membrane current during a propagated action potential can be derived and estimated. We show that the formal definitions of the space and time constants for the membranes with constant conductance can be extended to membranes with voltage-dependent conductance in which the behavior of each type of membrane is distinguished through the choice of the representation of the membrane conductance used in these definitions. In the case of a membrane with voltage-dependent conductance, the conductance to be used in the definitions of the space and time constants can be estimated from the extracellularly determined conduction speed of the propagated action potential.

The effects of static magnetic field on action potential propagation and excitation recovery in nerve.

Calculations using the Hodgkin-Huxley and one-dimensional cable equations have been performed to determine the expected sensitivity of conduction and refractoriness to changes in the time constant of sodium channel deactivation at negative potentials, as reported experimentally by Rosen (Bioelectromagnetics 24 (2003) 517) when voltage-gated sodium channels are exposed to a 125 mT static magnetic field. The predicted changes in speed of conduction and refractory period are very small.

Neural spike train synchronization indices: definitions, interpretations, and applications.

A comparison of previously defined spike train syncrhonization indices is undertaken within a stochastic point process framework. The second-order cumulant density (covariance density) is shown to be common to all the indices. Simulation studies were used to investigate the sampling variability of a single index based on the second-order cumulant. The simulations used a paired motoneurone model and a paired regular spiking cortical neurone model. The sampling variability of spike trains generated under identical conditions from the paired motoneurone model varied from 50% to 160% of the estimated value. On theoretical grounds, and on the basis of simulated data a rate dependence is present in all synchronization indices. The application of coherence and pooled coherence estimates to the issue of synchronization indices is considered. This alternative frequency domain approach allows an arbitrary number of spike train pairs to be evaluated for statistically significant differences, and combined into a single population measure. The pooled coherence framework allows pooled time domain measures to be derived, application of this to the simulated data is illustrated. Data from the cortical neurone model is generated over a wide range of firing rates (1-250 spikes/s). The pooled coherence framework correctly characterizes the sampling variability as not significant over this wide operating range. The broader applicability of this approach to multielectrode array data is briefly discussed.

From Maxwell's equations to the cable equation and beyond.

Maxwell's equations are taken as the starting point for the development of a mathematical model of a dendrite. The three-dimensional model of the evolution of the dendritic membrane potential based on these equations gives rise to a hierarchy of one-dimensional membrane equations. Under sufficiently strong assumptions, the first membrane equation is identical to the conventional cable equation. The second membrane equation explicitly includes the influence of dendritic taper and non-axial gradients in the intra-cellular potential. The procedure of starting from a three-dimensional model and extracting from it a one-dimensional approximation provides a prescription of how to incorporate three-dimensional properties of a dendrite in a one-dimensional representation, by contrast with an approach which aims to modify the traditional cable equation to take account of three-dimensional structure. Finite element methods are used to solve the membrane equations. An example based on a simple model of a tapered dendrite with differently placed distributions of synaptic input suggests that the effect of taper on the spike train output from the model is more important for distal synapses than those closer to the soma.

An investigation into the influence of boundary condition specification in finite difference methods on the behaviour of passive and active neuronal models.

Neuronal models provide a major aid to understanding the behaviour of individual neurons and networks of neurons. The solution of the model equations by finite difference methods is widespread because of the inherent simplicity of the technique. Error in the finite difference approach due to spatial and temporal discretisation is shown to be equivalent to a mis-specification of membrane current density. The effect of this mis-specification on the accuracy of the solution to the model equations is shown to depend on the structure of the model and its input, as well as the size of the discretisation intervals themselves. Through a theoretical analysis, illustrated by a number of examples on passive and active dendrites, this article demonstrates that the accuracy with which core current is implemented numerically at segment end-points in elementary models influences the behaviour of the numerical solution of these models, and consequently any physiological conclusions drawn from them.

On trees as equivalent cables.

By means of a suitable transformation, any passive dendritic tree may be reduced to an equivalent, possibly non-uniform cable. Under certain conditions the equivalent cable has disjoint sections of which only one communicates with the soma. Inputs that map on to the disconnected sections cannot be seen by the soma. Ralls's equivalent cylinder and its generalizations emerge naturally as the simplest cases of this behaviour. Even where, as is more usual, decomposition does not occur exactly the equivalent cable together with the input mapping from the tree to the cable provides a readily visualisable and intuitively appealing description of quite subtle relationships on the tree. The structure of the equivalent cable is dominated by approximate geometric symmetries of the tree. These symmetries cause well-defined subspaces of the total space of synaptic inputs to arrive at the soma at different times, thus allowing them, in principle, to be reflected, for example in the temporal statistics of the neurons' spike output.

An analysis of the dependence of saccadic latency on target position and target characteristics in human subjects.

BACKGROUND: Predictions from conduction velocity data for primate retinal ganglion cell axons indicate that the conduction time to the lateral geniculate nucleus for stimulation of peripheral retina should be no longer than for stimulation of central retina. On this basis, the latency of saccadic eye movements should not increase for more peripherally located targets. However, previous studies have reported relatively very large increases, which has the implication of a very considerable increase in central processing time for the saccade-generating system. RESULTS: In order to resolve this paradox, we have undertaken an extended series of experiments in which saccadic eye movements were recorded by electro-oculography in response to targets presented in the horizontal meridian in normal young subjects. For stationary or moving targets of either normal beam intensity or reduced red intensity, with the direction of gaze either straight ahead with respect to the head or directed eccentrically, the saccadic latency was shown to remain invariant with respect to a wide range of target angular displacements. CONCLUSIONS: These results indicate that, irrespective of the angular displacement of the target, the direction of gaze or the target intensity, the saccade-generating system operates with a constant generation time.

A framework for the analysis of neuronal networks.

The object of this work is to consider the application of some methods of spike train analysis that are not widely known, and are concerned with the description of the interaction between spike trains and the determination of causal connections between them. The notation and terminology follow conventions established in the statistical literature. The examples given are based on in-continuity recordings of the spontaneous activity of single Ia afferents from the soleus muscle and single motor units from the same muscle. Cumulant densities are shown to be simple extensions of the traditional cross-correlation methods, and are useful in characterizing the pattern of activity in one spike train that influences that in another, and to reveal interactions between spike trains that would not be apparent from the correlation histogram alone. Parameters based on the Fourier transforms of the spike trains are shown to be useful in determining timing relations between them, and in inferring patterns of connectivity not possible by correlation methods alone.

The interaction between membrane kinetics and membrane geometry in the transmission of action potentials in non-uniform excitable fibres: a finite element approach.

By solving the partial differential equations for an axonal segment using a finite element method, the interaction between membrane kinetics and axonal inhomogeneities, measured by their influence on propagated action potentials and stochastic spike trains, is investigated for Morris-Lecar and Hodgkin-Huxley membrane models. To facilitate comparisons of both kinetic models, parameter values are matched to give approximately the same speed for propagated action potentials. In all cases examined, the Morris-Lecar membrane model is more sensitive to geometric inhomogeneities than the comparable Hodgkin-Huxley membrane model. This difference in sensitivity can, in part, be attributed to significant differences in the membrane current supplied by each kinetic model ahead of the action potential. Also, the Morris-Lecar membrane model did not generate reflected action potentials whereas these were observed over a narrow range of geometric parameters for the comparable Hodgkin-Huxley membrane model. Simulations using stochastic spike train input showed that the presence of a sharp flare could significantly modify the statistical characteristics of the spike train output. The behaviour of action potentials governed by Morris-Lecar kinetics were more sensitive to changes in axonal geometry than those generated by comparable Hodgkin-Huxley kinetics. As a consequence of the fine balance between membrane kinetics and axon geometry, local changes in membrane properties, such as those caused by synaptic activity, can be expected to have a strong influence on the behaviour of stochastic spike trains at regions of changing axonal geometry.

Identification of patterns of neuronal connectivity--partial spectra, partial coherence, and neuronal interactions.

The cross-correlation histogram has provided the primary tool for inferring the structure of common inputs to pairs of neurones. While this technique has produced useful results it not clear how it may be extended to complex networks. In this report we introduce a linear model for point process systems. The finite Fourier transform of this model leads to a regression type analysis of the relations between spike trains. An advantage of this approach is that the full range of techniques for multivariate regression analyses becomes available for spike train analysis. The two main parameters used for the identification of neural networks are the coherence and partial coherences. The coherence defines a bounded measure of association between two spike trains and plays the role of a squared correlation coefficient defined at each frequency lambda. The partial coherences, analogous to the partial correlations of multiple regression analysis, allow an assessment of how any number of putative input processes may influence the relation between any two output processes. In many cases analytic solutions may be found for coherences and partial coherences for simple neural networks, and in combination with simulations may be used to test hypotheses concerning proposed networks inferred from spike train analyses.

A review of recent applications of cross-correlation methodologies to human motor unit recording.

This article reviews some recent applications of time and frequency domain cross-correlation techniques to human motor unit recording. These techniques may be used to examine the pre-synaptic mechanisms involved in control of motoneuron activity during on-going motor tasks in man without the need for imposed and artificial perturbations of the system. In this review we examine, through several examples, areas in which insights have been gained into the basic neurophysiological processes that bring about motoneuron firing in man and illustrate how these processes are affected by central nervous system pathology. We will demonstrate that synchronization and coherence may be revealed between human motor unit discharges and give examples that support the hypothesis that these phenomena are generated by activity in a focused common corticospinal input to spinal motoneurons. Disruption of central motor pathways due to diseases of the nervous system leads to pathophysiological alterations in the activity of these pre-synaptic motoneuron inputs that can be revealed by cross-correlation analysis of motor unit discharges. The significance of these studies and outstanding questions in this field are discussed.

An extended difference of coherence test for comparing and combining several independent coherence estimates: theory and application to the study of motor units and physiological tremor.

Recently there has been an increase in the use of spectral methods for the analysis of experimental data. These analytical methods allow the study of interactions between simultaneously recorded signals and are particularly suited to the study of systems displaying rhythmic behaviour. A useful parameter in this context is the coherence function which provides a bounded measure of linear association between two signals. In this report we introduce two new techniques for dealing with an arbitrary number of independent coherence estimates. The first technique provides a test to compare the coherence estimates for statistically significant differences. The second allows the original coherence estimates to be combined, or 'pooled' into a single representative estimate. These two measures, taken together, provide a powerful tool for characterising and summarising the correlations within data sets. Applications of the techniques are illustrated by analysing the interactions between single motor unit discharges and finger tremor, and between pairs of motor unit discharges in human subjects.

The participation of forelimb flexors in labyrinth and neck reflexes in the decerebrate cat.

The reflex behaviour of triceps and biceps brachii was assessed by EMG recordings during natural labyrinth or neck stimulation. Labyrinth reflexes resulting from changes in head position are antagonistic to those resulting from changes in neck position. Reflexes recorded from biceps were reciprocal to those recorded in ipsilateral triceps and symmetrical to those from contralateral triceps. A scheme of labyrinth and neck reflexes involving forelimb flexors is presented.

Labyrinth and neck reflex modification of the tonic vibration reflex in the decerebrate cat.

The interaction between tonic labyrinth or neck reflexes and the tonic vibration reflex acting on the medial head of triceps in the decerebrate cat is described. Medial triceps was isotonically loaded and reflex actions were measured as changes in muscle length. Natural stimulation of the receptors giving rise to tonic labyrinth or neck reflexes can either enhance or diminish the size of a pre-existing tonic vibration reflex. It is also shown that descending activity from either the labyrinth or neck reflex systems can completely suppress the tonic vibration reflex, whereas the tonic vibration reflex was never observed to suppress an established labyrinth or neck reflex.

Using electroencephalography to study functional coupling between cortical activity and electromyograms during voluntary contractions in humans.

Previous studies of neuronal oscillations in sensorimotor cortex in humans and primates have observed rhythmic 15-30 Hz activity, which is correlated with motor output. In humans, this work has been limited to magnetic recordings. In the present study we investigate if similar results can be obtained using electroencephalography (EEG). EEG recordings were made from over the sensorimotor cortex of five adult subjects who performed repeated periods of maintained wrist extension and flexion. Coherence analysis between EEG and electromyogram (EMG) recordings from these muscles revealed correlation in the 15-30 Hz range, with a synchronous correlation structure which matches that previously observed in humans and in paired cortical recordings from primates. We conclude that EEG is equally efficient at investigating functional aspects of these cortical rhythms during voluntary movement in humans.