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.

Linear and quadratic models of point process systems: contributions of patterned input to output.

In the 1880's Volterra characterised a nonlinear system using a functional series connecting continuous input and continuous output. Norbert Wiener, in the 1940's, circumvented problems associated with the application of Volterra series to physical problems by deriving from it a new series of terms that are mutually uncorrelated with respect to Gaussian processes. Subsequently, Brillinger, in the 1970's, introduced a point-process analogue of Volterra's series connecting point-process inputs to the instantaneous rate of point-process output. We derive here a new series from this analogue in which its terms are mutually uncorrelated with respect to Poisson processes. This new series expresses how patterned input in a spike train, represented by third-order cross-cumulants, is converted into the instantaneous rate of an output point-process. Given experimental records of suitable duration, the contribution of arbitrary patterned input to an output process can, in principle, be determined. Solutions for linear and quadratic point-process models with one and two inputs and a single output are investigated. Our theoretical results are applied to isolated muscle spindle data in which the spike trains from the primary and secondary endings from the same muscle spindle are recorded in response to stimulation of one and then two static fusimotor axons in the absence and presence of a random length change imposed on the parent muscle. For a fixed mean rate of input spikes, the analysis of the experimental data makes explicit which patterns of two input spikes contribute to an output spike.

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.

A periodogram-based test for weak stationarity and consistency between sections in time series.

In one approach to spectral estimation, a sample record is broken into a number of disjoint sections, or data is collected over a number of discrete trials. Spectral parameters are formed by averaging periodograms across these discrete sections or trials. A key assumption in this approach is that of weak stationarity. This paper describes a simple test that checks if periodogram ordinates are consistent across sections as a means of assessing weak stationarity. The test is called the Periodogram Coefficient of Variation (PCOV) test, and is a frequency domain test based on a technique of spectral analysis. Application of the test is illustrated to both simulated and experimental data (EMG, physiological tremor, EEG). An additional role for the test as a useful tool in exploratory analysis of time series is highlighted.

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.

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.

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.

Increased computational accuracy in multi-compartmental cable models by a novel approach for precise point process localization.

Compartmental models of dendrites are the most widely used tool for investigating their electrical behaviour. Traditional models assign a single potential to a compartment. This potential is associated with the membrane potential at the centre of the segment represented by the compartment. All input to that segment, independent of its location on the segment, is assumed to act at the centre of the segment with the potential of the compartment. By contrast, the compartmental model introduced in this article assigns a potential to each end of a segment, and takes into account the location of input to a segment on the model solution by partitioning the effect of this input between the axial currents at the proximal and distal boundaries of segments. For a given neuron, the new and traditional approaches to compartmental modelling use the same number of locations at which the membrane potential is to be determined, and lead to ordinary differential equations that are structurally identical. However, the solution achieved by the new approach gives an order of magnitude better accuracy and precision than that achieved by the latter in the presence of point process input.

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.

A note on the discrepancy between the predicted and observed speed of the propagated action potential in the squid giant axon.

The Hodgkin-Huxley model for the ionic currents in the membrane of the squid giant axon has become the standard model for the electrophysiological behaviour of many excitable cells. A strong test of the model predicted a travelling wave speed of 18.76 m/s for the propagated action potential in an axon with a reported speed of 21.2 m/s. This discrepancy between prediction and observation was considered satisfactory when the model was proposed 50 years ago, appears not to have been re-evaluated, but is unsatisfactory for a mature and important model. The separate and combined influences of measurement error and biological variability on the discrepancy between prediction and observation are quantified, as is the effect of using of a one-dimensional model to represent a three-dimensional axon. The main tool in this investigation is the use of simulation to study the behaviour of the Hodgkin-Huxley membrane model. These studies show that measurement error in combination with biological variability cannot account for the discrepancy between prediction and observation. Also, calculation shows that the one-dimensional description of the behaviour of the axon is adequate. Further calculation shows that the travelling wave description of the propagated action potential is valid only for sufficiently long axons. In shorter axons the propagated action potential is predicted to travel faster than the travelling wave; consequently under suitable experimental conditions the discrepancy between prediction and observation may be negligible.

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.

Analytical and numerical construction of equivalent cables.

The mathematical complexity experienced when applying cable theory to arbitrarily branched dendrites has lead to the development of a simple representation of any branched dendrite called the equivalent cable. The equivalent cable is an unbranched model of a dendrite and a one-to-one mapping of potentials and currents on the branched model to those on the unbranched model, and vice versa. The piecewise uniform cable, with a symmetrised tri-diagonal system matrix, is shown to represent the canonical form for an equivalent cable. Through a novel application of the Laplace transform it is demonstrated that an arbitrary branched model of a dendrite can be transformed to the canonical form of an equivalent cable. The characteristic properties of the equivalent cable are extracted from the matrix for the transformed branched model. The one-to-one mapping follows automatically from the construction of the equivalent cable. The equivalent cable is used to provide a new procedure for characterising the location of synaptic contacts on spinal interneurons.

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.

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.

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.

Dendritic subunits determined by dendritic morphology.

A theoretical framework is presented in which arbitrarily branched dendritic structures with nonhomogeneous membrane properties and nonuniform geometry can be transformed into an equivalent unbranched structure (equivalent cable). Rall's equivalent cylinder is seen to be one part of the equivalent cable in the special case of dendrites satisfying the Rall criteria. The relation between the branched dendrite and its equivalent unbranched representation is uniquely defined by an invertible mapping that connects configurations of inputs on the branched structure with those on the unbranched structure, and conversely. This mapping provides a new definition of dendritic subunit and provides a mechanism for characterizing local and nonlocal signal processing within dendritic structures.

Equivalence transformations for dendritic Y-junctions: a new definition of dendritic sub-unit.

A sequence of equivalence transformations is used to represent the mathematical model of a simply branched neuron with non-homogeneous membrane properties and non-uniform geometry by an entirely equivalent model of an unbranched structure. The analysis indicates how neuronal morphology, in combination with its biophysical properties, shapes neuronal output in response to current input. The equivalence transformations described here reveal the types of operations that are likely to feature in the analysis of complex multi-branched structures, neuronal or otherwise. These transformations provide a new definition of dendritic sub-unit and a basis of a mechanism for characterising local and non-local signal processing within dendritic structures. It is anticipated that the capacity to transform biological neurons into an equivalent unbranched structure will make an important contribution to the understanding of the functional role of neuron geometry as well as to the construction of silicon neurons with realistic biological properties.

Coherence between low-frequency activation of the motor cortex and tremor in patients with essential tremor.

BACKGROUND: In healthy people, rhythmic activation of the motor cortex in the 15-30 Hz frequency range accompanies and contributes to voluntarily-generated postural contractions of contralateral muscle. In patients with Parkinson's disease, an abnormal low-frequency activation of the motor areas of the cortex occurs and has been directly linked to the characteristic 3-6 Hz rest tremor of this disease. We therefore investigated whether the motor cortex is involved in the transmission of the rhythmic motor drive responsible for generating essential tremor. METHODS: Non-invasive recordings of activity from the hand area of the motor cortex were made from six patients with essential tremor by magnetoencephalography. The recordings were made simultaneously with the electromyogram recorded from contralateral finger muscles during periods of postural tremor. A statistical spectral analysis was done to determine at which frequencies the two signals were correlated. FINDINGS: Spectral analysis of the electromyogram signals showed a significant low-frequency component at the frequency of the tremor bursts. However, there was no coherence between magnetoencephalogram and electromyogram recordings at the tremor frequency, indicating that no correlation existed between the tremor signal and low-frequency activity recorded from the primary motor cortex in individuals with essential tremor. Coherence at frequencies higher than the tremor frequency was similar to that in healthy individuals performing voluntary postural contractions. INTERPRETATION: The absence of significant coherence between the magnetoencephalogram and electromyogram at tremor frequencies suggests that in essential tremor the tremor is imposed on the active muscle through descending pathways other than those originating in the primary motor cortex. These findings challenge the model widely used to explain the efficacy of neurosurgical treatment of essential tremor, are in contrast to those of previous studies of parkinsonian rest tremor, and highlight an important difference in the pathophysiology of essential and parkinsonian tremor.

The unilateral and bilateral control of motor unit pairs in the first dorsal interosseous and paraspinal muscles in man.

1. The discharges of two motor units were identified in an intrinsic hand muscle (first dorsal interosseous, FDI) or an axial muscle (lumbar paraspinals, PSP) in ten healthy subjects. Each motor unit was situated in the homologous muscle on either side of the body (bilateral condition) or in the same muscle (ipsilateral condition). The relationship between the times of discharge of the two units was determined using coherence analysis. 2. Motor unit pairs in the ipsilateral FDI showed significant coherence over the frequency bands 1-10 Hz and 12-40 Hz. Motor units in the ipsilateral PSP were significantly coherent below 5 Hz. In contrast there was no significant coherence at any frequency up to 100 Hz in the bilateral FDI condition and only a small but significant band of coherence below 2 Hz in the bilateral PSP condition. 3. Common drive to motor units at frequencies of < 4 Hz was assessed by cross-correlation of the instantaneous frequencies of the motor units. A significantly higher coefficient was found in the ipsilateral FDI, ipsi- and bilateral PSP compared with shifted, unrelated data sets. This was not the case for the bilateral FDI condition. 4. The presence of higher frequency coherence ( > 10 Hz) in the ipsilateral FDI condition and its absence in ipsilateral PSP is consistent with a more direct and influential cortical supply to the intrinsic hand muscles compared with the axial musculature. The presence of low frequency drives (< 4 Hz) in the bilateral PSP condition and its absence in the bilateral FDI condition is consistent with a bilateral drive to axial, but not distal, musculature by the motor pathways responsible for this oscillatory input.