J.R. Rosenberg

The Fourier approach to the identification of functional coupling between neuronal spike trains

I. I N T R O D U C T I O N The study of the behaviour of small networks of neurones frequently requires the determination of measures of the strength of association between component neurones, an assessment of their timing relations, and the identification of which neurones may interact directly or are influenced by common inputs. In many of these studies the principal quantities available for analysis are the sequences of extracellularly recorded action potentials (neuronal spike trains). The subsequent analytical work is then based entirely on the relations between the times of occurrence of the action potentials recorded from different neurones. In these circumstances neuronal spike trains are frequently represented as the mathematical entity known as a stochastic point process. These processes are described by providing a probability law for a set of ordered times

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.

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.

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 Parkinsons 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.

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.

The origin of ocular microtremor in man

Abstract A novel technique for the study of human eye movements was used to investigate the frequency components of ocular drift and microtremor in both eyes simultaneously. The tangential components of horizontal eye accelerations were recorded in seven healthy subjects using light-weight accelerometers mounted on scleral contact lenses during smooth pursuit movements, vestibulo-ocular reflexes and eccentric gaze with and without fixation. Spectral peaks were observed at low (up to 25 Hz) and high (60–90 Hz) frequencies. A multivariate analysis based on partial coherence analysis was used to correct for head movement. After correction, the signals were found to be coherent between the eyes over both low- and high-frequency ranges, irrespective of task, convergence or fixation. It is concluded that the frequency content of ocular drift and microtremor reflects the patterning of low-level drives to the extra-ocular muscle motor units.

On the application, estimation and interpretation of coherence and pooled coherence.

Coherence and related functions are powerful tools in the study of neurophysiological data. Amjad et al. (1997) introduced two measures which allow an arbitrary number of independent coherence functions to be evaluated for statistically significant differences and combined into a single population measure. This latter measure they termed ‘pooled coherence’. Baker (2000) presents an example of the application of pooled coherence to a population of simulated data which does not describe accurately the correlation structure within the data. The estimation of pooled coherence, as defined by Eq. (2.11) in Amjad et al. (1997) and Eq. (1) in Baker (2000), is equivalent to joining the separate records into a single long record and estimating the coherence for this single pooled record using a periodogram approach (e.g. Halliday et al., 1995). Thus, once the context of an analysis has been determined, the terms coherence and pooled coherence are interchangeable. Asymptotic confidence limits allow coherence estimates to be assessed for the presence of significant correlation at each Fourier frequency. In the case of periodogram estimates formed from disjoint sections this confidence limit can be estimated from a simple expression which depends on the number of sections averaged (Amjad et al., 1997). This approach is based on the extensively developed theory for the properties of Fourier transforms of stationary stochastic data (e.g. Brillinger, 1981). Why should a coherence estimate and confidence limit which are appropriate for single records fail to describe the correlation structure in a single record formed from 100 separate sections (Baker 2000)? Consider a plot of the two simulated signals constructed in Baker (2000), each consisting of 100 sections, with the amplitude of each section scaled by a normally distributed factor of 1098 (mean92 SD). The fine detail will be lost, but the two traces will exhibit step changes in their envelope at transition points between sections. In statistical terms the data exhibits a nonstationarity, in fact nonstationarity is built into the construction of such data. Second order stationarity (invariance of second order moments) is one of the assumptions underlying the application of coherence analysis (e.g. Brillinger, 1981). As described below, it is the violation of the assumption of stationarity which underlies the result reported in Baker (2000). The analogy between spectrum analysis and analysis of variance has long been recognized (e.g. Tukey, 1961). The spectrum of a signal can be interpreted as providing a measure of the fraction of the total variance (power) at a Fourier frequency. Scaling the amplitude of different sections of data by a constant is equivalent to scaling the variance of each signal resulting in an increase or decrease in the relative power at each frequency. Following this line of reasoning we can see that a pooled coherence estimate (calculated from pooled cross and auto spectra) will be dominated by the sections with the largest variance, and the sampling fluctuations will be determined mostly by the subset of sections with the largest variance. A confidence limit * Corresponding author. Present address: Department of Electronics, University of York, Heslington, York YO10 5DD, UK. Tel.: +44-1904-432345; fax: +44-1904-432335. E-mail address: dh20@ohm.york.ac.uk (D.M. Halliday).

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.

Time and Frequency Domain Analysis of Spike Train and Time Series Data

The concept of a spike triggered average will be familiar to many neurophysiologists. The first application in neurophysiology by Mendell and Henneman (1968, 1971) was used to examine the magnitude of monosynaptic excitatory postsynaptic potentials (EPSP) from muscle spindle la afferents onto homonymous motoneurons, which provided a major piece of evidence in the development of the size principle for motoneuron recruitment (see Henneman and Mendell, 1981). The technique has gained widespread acceptance, and been widely used to investigate the strength of synaptic connections in the mammalian central nervous system (e.g. Watt et al., 1976; Stauffer et al., 1976; Kirkwood and Sears, 1980; Cope et al., 1987), leading to new insights and an increased understanding of basic neurophysiological mechanisms.

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

BackgroundPredictions 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.ResultsIn 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.ConclusionsThese 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.