0006-Ch06.pdf

The Use of Correlational

Methods to Investigate

the Organization of

Spinal Networks for

Pattern Generation

Thomas M. Hamm, Martha L. McCurdy,

Tamara V. Trank, and Vladimir V. Turkin

CONTENTS

6.1 Introduction

6.2 The Time Domain: Correlation as an Indicator of Neuronal

Connections

6.2.1 Basic Concepts and Terminology

6.2.2 Cross-Correlation as an Indicator of Neuronal

Connections

6.2.3 Identifying the Organization of Synaptic Inputs to

Motoneurons Based on Cross-Correlation Histograms

6.3 The Frequency Domain: Coherence as an Indicator of Neuronal

Connections

6.3.1 Basic Terminology and Concepts

6.3.2 Previous Studies Using Coherence

6.3.3 What Information Does the Coherence Function Provide?

6.3.4 Partial Coherence Functions

6.3.5 Limitations of Coherence and Partial Coherence Functions

6.4 Detecting Correlations, Coherence in Composite Signals

6.4.1 The Use of Neurograms and EMGs for Composite Recordings

6.4.2 The Effects of Rectiﬁcation on Composite Recordings

6.4.3 Comparison of Coherence between Unitary and

Composite Recordings

6.5 Using Correlation to Study Spinal Pattern Generation

6.5.1 Basic Concepts in Spinal Pattern Generation

6.5.2 Correlations during Fictive Locomotion

6.5.2.1 Spike-Triggered Averages

6.5.2.2 Neurogram Correlations

6.5.3 Coherence Functions during Fictive Locomotion

and Scratching

6.5.3.1 Coherence Functions between Rectiﬁed

Neurograms

6.5.3.1.1 Comparison with Results from

Time-Domain Correlations

6.5.3.1.2 Coherence Functions for Different

Functional Groups of Motor Nuclei

6.5.3.1.3 Comparison of Coherence Functions

to Phase Relations among Motor

Pool Activities

6.5.3.2 Coherence Functions between LDPs and

Neurograms

6.5.3.2.1 Comparison of LDP-Ng to Ng-Ng

Coherence Functions

6.5.3.2.2 Frequency Content of LDPs

6.5.3.2.3 Support for Module Concept

6.6 Summary and Conclusions

References

6.1 INTRODUCTION

Correlation has been used to determine the organization and connections of neurons

that are otherwise inaccessible or difﬁcult to record. It is, therefore, useful for

determining the organization of complex neural circuits and networks in the verte-

brate nervous system. In this chapter we focus on the application of correlational

techniques, primarily in the frequency domain, to the investigation of spinal circuits

involved in the generation of rhythmic motor patterns, such as locomotion and

scratching. Studies on the use of correlation to investigate neural systems and studies

of central pattern generation are extensive, so citations to these ﬁelds are necessarily

quite selective. Our intent is to establish a basic framework for the application of

correlation to study neural systems, discuss the implementation of correlational

techniques in the frequency domain, and brieﬂy consider the use of composite

recordings in correlation. Finally, we present examples of these applications from

our studies on ﬁctive locomotion and scratching.

6.2 THE TIME DOMAIN: CORRELATION AS AN

INDICATOR OF NEURONAL CONNECTIONS

A variety of studies have explored the use of correlation to examine neuron orga-

nization, including direct connections between neurons, common synaptic input to

neurons from branched presynaptic axons, or synchronized presynaptic inputs to

neurons. Most studies have used correlations in the time domain; many reviews are

available on these methods and their application. 1–2

6.2.1 B ASIC C ONCEPTS AND T ERMINOLOGY

Correlation functions describe the dependence of one variable on a second variable

at earlier and later times. The variables are the same in the case of the auto-correlation

function:

() = ( ) ∫

T xtxt dt

()

( )

−

(6.1)

. This information is related to the frequency content of

the signal, or its bandwidth, and its tendency to include periodicities. The cross-

correlation function expresses the dependence between two signals, x ( t ) and y ( t ):

() = ( ) ∫

T y t x t

()

( )

−

(6.2)

Using a ﬁnite period of integration, T , in practical applications provides estimates

of these functions whose reliability increases with longer integration intervals, pro-

vided the statistical properties of the signals remain constant. If x ( t ) and y ( t ) are

causally related, the cross-correlation function contains information on the process

that relates them, in addition to information contained in the auto-correlation function

of each signal.

Application of these equations to studies of neural function requires their adap-

tation for use with spike trains in place of continuous processes. By representing

spike trains by values of 1 at times of spike occurrence and 0 at other times, the

products in eqs. 6.1 and 6.2 are only non-zero when spikes occur in both trains at

an interval of

, yielding

the familiar auto- and cross-correlation histograms. Dividing histogram counts by

the number of reference spikes (those of train x ( t )) and the bin width

∆τ

yields

probability density functions in units of frequency. Alternatives include the cross-

interval density histogram, which shows the time of occurrence only of spikes in

train y ( t ) that immediately precede or follow each spike in the reference train x ( t ).

∆τ

6.2.2 C ROSS -C ORRELATION AS AN I NDICATOR

OF N EURONAL C ONNECTIONS

Dependencies between spike trains can be produced by fundamental physiological

processes, such as monosynaptic or oligosynaptic projections between neurons or

common synaptic inputs from branched, presynaptic axons. Recognition of this fact

led several investigators to consider applying correlational methods to determine the

probable connections between neurons. Perkel et al. 3 discussed the uses of cross-

correlation histograms and interval histograms to determine whether two spike trains

which indicates how the value of a signal, x ( t ), depends statistically on its values

displaced by the interval

. The integral eqs. 6.1 and 6.2 are then replaced by simple summations

of the numbers of spikes in the two trains separated by interval

were statistically dependent and to form guidelines for interpreting these histograms

if dependence was demonstrated. On the basis of simulation, they suggested that (1)

common synaptic input to neurons from branched presynaptic neurons is more

difﬁcult to detect than monosynaptic or oligosynaptic projections between neurons;

(2) oligosynaptic connections are more difﬁcult to detect than monosynaptic pro-

jections; and (3) several conﬁgurations can lead to the same form of correlation. A

subsequent study described characteristic features of cross-correlation histograms

produced by several simple neuronal circuits, based on simulations and recordings

in Aplysia . 4 This study concluded that “fairly precise statements about the waveform,

amplitude and polarity of the intracellular synaptic potential which couples the

correlated cells” can be made using cross-correlation histograms based on spike

trains. Furthermore, the existence of other neurons present in the network, like a

common source of input, can be inferred from these histograms, even though the

activity of these neurons is not observed directly. Cross-correlation and cross-interval

histograms can be supplemented with other methods 5 to determine the conﬁguration

of neural circuits.

One limitation of this work was that rather large PSPs were used, while most

synaptic inputs, in the absence of composite stimulation, are quite small. 6 Never-

theless, the net effect of many small PSPs can have a clear and measurable effect

on neuron discharge. Calvin and Stevens 7 showed that background synaptic noise

is sufﬁcient to produce the variability observed in motoneuron discharge by causing

threshold crossings sooner or later than the mean interspike interval. Sears and Stag 8

hypothesized that EPSPs produced by many presynaptic ﬁbers shared by motoneu-

rons would increase their probability of discharge during the rising phase of the

EPSPs. Cross-correlation histograms between the discharge of pairs or small groups

of intercostal motoneurons contained short-term synchronization, central peaks with

widths of

6.2.3 I DENTIFYING THE O RGANIZATION OF S YNAPTIC

I NPUTS TO M OTONEURONS B ASED

ON C ROSS -C ORRELATION H ISTOGRAMS

Identifying probable neuronal circuits based on correlation functions is discussed in

several of the publications cited above. We conﬁne our remarks to the issue of

identifying the organization of synaptic inputs to motoneuron pools using correla-

tions. Several correlating mechanisms can be distinguished in cross-correlation his-

tograms between motoneurons. For example, Kirkwood et al. 10 show that cross-

correlation histograms between respiratory motoneurons can exhibit short-term syn-

chronization associated with common input, broad peaks which appear to be pro-

duced by correlated interneuron inputs, and correlation peaks associated with high-

frequency oscillations originating in medullary neurons (see 6.3.2).

3 msec, consistent with this hypothesis. This work was further supported

by the demonstration of an EPSP-like depolarization, the average common excitatory

(ACE) potential, revealed by averaging the intracellular potentials in one motoneuron

while using the spike discharge of another as a trigger. 9 A theory relating the waveform

of the EPSP to an increased probability of discharge (see 6.2.3) successfully accounted

for both the shape of the ACE potential and short-term synchronization.

As Kirkwood and Sears 9 suggested originally, the form of the cross-correlation

histogram for two motoneurons whose discharge is correlated by common input can

be predicted from the convolution of their primary correlation kernels (PCK), in the

terminology of Knox, 11 which describe the increased probability of neuron discharge

produced by the PSP. Kirkwood and Sears found that a PCK expressed by a linear

combination of the EPSP and its derivative was sufﬁcient to account for the shape

of both the cross-correlation peak and the ACE potential. Application of this theory

to human motor unit recordings 12 suggests that synchrony in the discharge of human

motor units occurs as a result of common inputs, probably from branched corticospi-

nal ﬁbers. 13 The dependence of this short-term synchronization on various physio-

logical factors has been investigated in several studies. 14–16

Clearly, the relationship between EPSP characteristics and the change in the

probability of discharge is critical in the interpretation of cross-correlation histo-

grams. The increased probability of discharge produced by an EPSP has been

described as most directly related to the proﬁle of the EPSP, 4 the derivative of the

EPSP, 11 or a combination of the EPSP proﬁle and its derivative. 9 Several studies

have examined the relationship between the EPSP (or IPSP) received by a moto-

neuron and its effect on discharge probability. 17–19 These studies show that this

relationship depends on PSP size in relation to the level of synaptic noise. 20 More-

over, the increased discharge probability produced by an EPSP is determined by its

amplitude 19 or the amplitude of the EPSP and the derivative of its rising phase. 18

Demonstration that even the EPSPs produced by single afferents can increase the

probability of motoneuron discharge 19 indicates that cross-correlation may be sen-

sitive enough to detect dependencies between spike trains produced by neuronal

circuits with a variety of connection patterns.

A central problem in this area is distinguishing the correlation peak produced

by common, branched input ﬁbers from other forms of correlating input, such as

input ﬁbers with correlated activity (presynaptic synchronization). Making this dis-

tinction can be difﬁcult, as discussed by Kirkwood and Sears. 21 A central correlation

peak produced by presynaptic synchronization should be wider than if produced by

common input. But how much wider should it be? If the presynaptic neurons are

synchronized by common input and project monosynaptically to motoneurons, then

the peak should by widened by an additional convolution with the PCK of the

presynaptic neurons. Any dispersion in the arrival time of action potentials in the

set of synchronized presynaptic neurons would widen the peak as well. However,

this increase in peak width may be rather slight, particularly if presynaptic synchro-

nization occurs in neurons close to the motoneuron pools, as might be expected for

interneuronal inputs. Kirkwood and Sears 21 suggest that correlation peaks with half-

widths of 2.1 msec or less can be assumed to indicate the presence of common input,

as opposed to presynaptic synchronization or some combination of the two. This

decision must be based on the particular conditions of the system under study. A

recent, careful application of this approach is given in Vaughn and Kirkwood. 22 We

consider this issue with respect to interpreting correlations between motor pool

activities during ﬁctive locomotion in 6.5.2.2.

The precision with which the characteristics of PSPs and the organization of

presynaptic inputs can be identiﬁed has not been fully resolved. As indicated above,

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