INFERENCE ABOUT MULTIPLE PATHWAYS IN MOTOR

CONTROL LIMB IN LOCUST

C. D. Maciel

, D. M. Simpson

and P. L. Newland

Electrical Eng. Dept. - EESC, University of S

ao Paulo, Av. Trabalhador S

aocarlense 400, S

ao Carlos, Brazil

Signal Processing and Control Group, University of Southampton

University Road Highﬁeld, S017 1BJ, Southampton, U.K.

School of Biological Science, University of Southampton, University Road Highﬁeld, S017 1BJ, Southampton, U.K.

Keywords:

Neuronal Analysis, Information Theory, Causal Inference.

Abstract:

In locust local circuits that control limb movements, the neural signals are processed by both spiking and

nonspiking interneurons that operate in parallel to process sensory information. These interneurons receive

sensory inputs from leg mechanoreceptors and together project to leg motor neuron pools. The main feature

of the nonspiking interneurons is their ability to communicate with other neurons without the intervention

of nerve impulses, or spikes, so that they exert graded control over their postsynaptic motor neurons, while

spiking local interneurons communicate by means of action potentials and are involved in the integration

of sensory signals. Our work presents an investigation from different classes of neurons driven by random

Gaussian excitatory movements to a proprioceptor at the knee joint. The underlying aim of this work was to

use information theory in understanding connectivity in the neural network.

1 INTRODUCTION

To improve the performance of robots many stud-

ies have focused on understanding how animals such

as insects (Fourtner, 1976) perform complex move-

ments using relatively simple neuromuscular reﬂex

control systems (Delcomyn, 2004). To produce a

reﬂex movement of a limb, the neuromuscular con-

trol system must transform an external stimulus into

a limb movement and in doing so must generate a

movement driven by appropriate neuronal patterns

(Burrows, 1996).

In many arthropods, and locusts in particular, it

is possible to perform measurements of the underly-

ing control signals (patterns of a neural activity) at

the level of the relatively few neurons responsible for

controlling movements of the legs. Locusts have a

distributed nervous system with a brain and a series

of segmental ganglion. Those in the thorax are re-

sponsible for generating the local movements of the

three pairs of legs (Burrows, 1996). The movements

of the tibia relative to the femur of a hing leg are mon-

itored and encoded by a sensory structure containing

approximately 90 sensory cells, the femoral chordo-

tonal organ (FeCO) that converts a mechanical stim-

ulus into electrical neuronal signals (Kondoh et al.,

1995). These sensory signals are processed in neu-

ronal networks containing different types of interneu-

rons that use both digital and analogue signalling to

control the activation of eleven excitatory motor neu-

rons (9 causing ﬂexion and 2 causing extension) that

activate the tibial limb muscles to generate movement

of the tibia (Newland and Kondoh, 1997).

In a number of studies (Schreiber and Schmitz,

2000; Bialek et al., 2001; Ebeling, 2002) the idea that

a natural approach to analyse non-linear and stochas-

tic signals should be based on information theory has

been suggested. Information theory (Shannon, 1948;

Cover and Thomas, 2006) quantiﬁes statistical un-

certainty in random processes and statistical depen-

dencies between multiple random processes (Cover

and Thomas, 2006). Entropy is a measure of uncer-

tainty of a random vector with a probability distribu-

tion function (pdf) p

(x) (Cover and Thomas, 2006;

Shannon, 1948). In addition, divergence is a gener-

alization of variance to processes with non-Gaussian

distributions (Cover and Thomas, 2006) while mutual

information is a measure of statistical dependency and

may be interpreted as a generalization of correlation

to arbitrary non-linear relationships between multiple

processes with arbitrary probability distributions (Er-

dogmus and Principe, 2006; Li, 1990).

D. Maciel C., M. Simpson D. and L. Newland P..

INFERENCE ABOUT MULTIPLE PATHWAYS IN MOTOR CONTROL LIMB IN LOCUST.

DOI: 10.5220/0003782200690075

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing (BIOSIGNALS-2012), pages 69-75

ISBN: 978-989-8425-89-8

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

The main limitation to mutual information is due

to its inability to differentiate the direction of associ-

ation between two signals (Schreiber, 2000). To over-

come this limitation (Schreiber, 2000) introduced a

quantity called information transfer that shares some

of the desired properties of mutual information but

takes into account the dynamics of information trans-

port. These quantities quickly spread to different

areas; from econometrics (Marschinski and Kantz,

2002) to biomedical engineering (Ward and Maza-

heri, 2008).

In general information theory approaches makes

few assumptions about the relationship among the

system variables, but does require strict stationarity

(Nichols, 2006). Time-delayed mutual information

quantiﬁes co-dependence between variables by look-

ing at shared previous information content as a func-

tion of time lag (information ﬂow) (Palu

s et al., 2001;

Vastano and Harry, 1988).

Our study addresses the question of delayed

mutual information estimated from a large multi-

variate biological data set based on data used in

(Kondoh et al., 1995) and (Newland and Kondoh,

1997). Knowing the time-delay and information ﬂow

strength between two signals will inform understand-

ing of the neural network function underlying limb

motor control in locusts. These insights and method-

ological developments are part of our wider interest in

neuromuscular (dys)function and bioinspired sensing

and control. The key challenge is how to combine the

recordings made in different animals into one map of

neuronal interconnections. In this experimental set-

ting (as is also often the case in related work), it is

not possible to record simultaneously from more than

perhaps two neurones, due to the physical size of the

preparation.

This investigation uses mechanical excitation by

random Gaussian displacements of a joint movement

sensor (the FeCO) to analysis of possible pathways

leading to electrical signal collected at different points

within the neural network, Fig. 1(C). In addition,

a surrogate test procedure (Schreiber and Schmitz,

1996; Schreiber and Schmitz, 2000) was used to infer

if any relationships were signiﬁcantly above the noise

level. The ﬁnal result represents a functional connec-

tivity map displayed using a graph tools to show the

time differences in neurophysiological measurements

along the processing chain, combining measurements

from many different recordings.

2 THEORY

Consider X = {x

} a discrete random signal. The en-

tropy H

is deﬁned by (Cover and Thomas, 2006)

as H

= −

∑

x∈χ

(x)log

(x) where χ represents

the set of symbols used in this codiﬁcation, p

(x) the

probability density function (pdf ) of event x, and H

quantiﬁes the mean number of bits (when using base

two in log) that can optimally code random variables.

Entropy is a measure of disorder or more precisely un-

predictability. For example, systems with equiproba-

ble states have maximum entropy, since there is no

way to predict what will come next. A system that

could have many states but is held in just one particu-

lar state has zero entropy. Most data collections in the

real world lie somewhere between such extremes.

If p

(.) is the pdf of variable Y , then

] =

∑

(i)log

(i)/p

(i) is the

Kullback-Leibler divergence (Cover and Thomas,

2006) quantifying the difference between p

and p

It can be observed from the previous equations that

if p

= p

then the D

= 0. It should point out

that D

] 6= D

] and thus it is not

symmetric.

The mutual information from two random vari-

ables X and Y , I(X;Y ), quantiﬁes the average shared

information or how the knowledge from one time

series informs about another (Cover and Thomas,

2006). This equation has been derived in many

references (Cover and Thomas, 2006; Shannon,

1948; Palu

s and Vejmelka, 2007) I(X;Y ) = H(Y ) −

H(Y |X) = H(X) + H(Y) − H(X,Y) and expressed

from probabilities is:

I(X;Y) =

∑

x∈χ

∑

y∈φ

(x,y)log

(x,y)

(x)p

(y)

(1)

where p

(x,y) is the joint probabilities distribution

regarding X and Y , H(Y |X ) is the conditional en-

tropy of Y given X. Since H(Y ) ≥ H(Y|X) then

0 ≤ I(X,Y ) < ∞ (Dionisio et al., 2004), the equal-

ity is archived only when X and Y are indepen-

dent. From Eq. 1 it can easily be seen that

I(X ;Y ) = D

[p(x,y)kp(x)p(y)] is the “distance”

from p

(x,y) to p

(x)p

(y), i.e. the assumption that

X and Y are independent.

The conditional entropy (Palu

s and Vejmelka,

2007), H(Y |X), quantiﬁes the remaining uncertainty

(entropy) of a random variable Y given another ran-

dom variable, X, and it is evaluated as:

H(Y |X) =

∑

x∈χ

∑

y∈φ

(x,y)log

Y |X

(y|x). (2)

The conditional mutual information I(Y ; X |Z)

characterizes the dependence between X and Y with-

out the possible inﬂuence of another variable Z

(Cover and Thomas, 2006; Palu

s and Vejmelka,

2007). This measure assesses the interaction X → Y

BIOSIGNALS 2012 - International Conference on Bio-inspired Systems and Signal Processing

Figure 1: Diagram of the recording locations in the

metathoracic ganglion of the locust. The spiking local in-

terneurons were recorded from their somata on the ven-

tral midline of the metathoracic ganglion, whereas nonspik-

ing local interneurones were recorded in their neuropil pro-

cesses withing the area indicated by cross-hatching. Sen-

sory neurones were recorded in their axons in the ante-

rior half of nerve 5, while identiﬁed motor neurons were

recorded from their somata, the locations of which are

shown.

Table 1: Description of main neurones described in Fig. 1,

and used in signal acquisition.

Movement about knee:

ASFlTi Anterior Slow Flexor tibia motor neurone

AFFlTi Anterior Fast Flexor tibia motor neurone

AIFlTi Anterior Intermediate Flexor tibia motor neurone

PSFlTi Posterior Slow Flexor tibia motor neurone

PFFlTi Posterior Fast Flexor tibia motor neurone

PIFlTi Posterior Intermediate Flexor tibia motor neurone

SETi Slow Extensor Tibia motor neurone

FETi Fast Extensor Tibia motor neurone

Movement about ankle:

SleTa Slow Levator Tarsus

FleTa Fast Levator Tarsus

SDTa Slow Depressor Tarsus

FDTa Fast Depressor Tarsus

Movement of claw:

SRU Slow Retractor unguis motor neurone

FRU Fast Retractor unguis motor neurone

Movement about shoulder:

FLeTr Fast Levator Trochanter motor neurone

FDTr Fast Depressor Trochanter motor neurone

that is not only due to the parent random variable Z,

i.e. Z → X and Z → Y (Pearl, 2009).

Like mutual information, conditional

mutual information can be expressed as a

Kullback-Leibler divergence I(X;Y |Z) =

[p(x,y,z)kp(x|z)p(y|z)p(z)] or as an ex-

pected value of simpler Kullback-Leibler

divergences: I(X ;Y |Z) =

∑

z∈Z

p(Z =

z)D

[p(x,y|z)kp(x|z)p(y|z)].

(Nichols, 2006; Alonso et al., 2007) and others

have examined the structural dynamics of different

systems (the ﬁrst study was on interactions from mus-

cle activity in pathological patient conditions and the

second used simulated mechanical system) based on

mutual information and conditional mutual informa-

tion.

Time delayed mutual information quantiﬁes the

dependencies between dynamical variables by look-

ing at shared information content as a function of time

lags τ between X and Y . The delayed mutual informa-

tion that is (Nichols, 2006)

I(X,Y

) =

∑

x∈χ

∑

y∈φ

i+τ

)log

i+τ

)

i+τ

)

. (3)

If the information present in signal X at a discrete

time i is also present in signal Y at discrete time i + τ

there will be a peak in the curve I(X,Y (τ)) at τ

> 0

as the joint probability density increases. If the peak

occurs at τ

< 0, that implies that the information is

being transported from Y to X (reverse order). At this

point it is assumed that X and Y are stationary and the

joint probabilities will depend only on the time lag τ

(Nichols, 2006).

The evaluation of interaction between random

variables X and Y has been presented in many articles

(e.g. (Schreiber, 2000; Palu

s et al., 2001; Palu

s and

Vejmelka, 2007; Palu

s and Stefanovska, 2003)) and

is expressed using conditional entropies I(Y ; X

|X)

(Nichols, 2006). Due to peculiarities of system topol-

ogy, Fig 1 (B,C) the approach adopted to analyse this

system is I(Y ; X

|Z) where Z is the recorded mechani-

cal driver signal and τ varies from −T → T to include

analysis in both directions.

The data were also analysed to determine whether

they expressed any signiﬁcant underlying dynamics

(Kugiumtzis, 2002). The validation procedure was

based on null hypotheses statistical tests for the ob-

served data (Urbach, 2000). To this end surrogate

data analysis was used (Schreiber and Schmitz, 1996;

Schreiber and Schmitz, 2000; Kugiumtzis, 2002).

The basic idea from the surrogate data analysis was

to compute the statistics of interest for the original

data set and for each of the ensemble of surrogate

data sets with equivalent amplitude distribution and

power spectra but forming independent from X, Y and

Z. If the computed statistics for the original data set

are signiﬁcantly different from the values obtained for

the surrogate sets it is possible to infer that the output

data are related by the input signal.

For univariate time series the most common

method is the use of Fourier Transformation of the

data, randomising the phase and inverting the trans-

form. This removes any correlation between signals.

The surrogate data will have the same power spectrum

of the original and considering the Wiener-Khintchin

INFERENCE ABOUT MULTIPLE PATHWAYS IN MOTOR CONTROL LIMB IN LOCUST

theorem the same autocorrelation function. The liter-

ature presents many improvements on this algorithm

(Venema et al., 2006) in particular with reference to

amplitude distribution.

3 MATERIALS & METHODS

Adult male and female desert locusts, Schistocerca

gregaria (Forsk

al) were used for all experiments. Lo-

custs were mounted ventral-side-uppermost in mod-

elling clay and the apodeme of the FeCO exposed by

opening a small window of cuticle in the distal ante-

rior femur (Kondoh et al., 1995), grasped between the

tips of ﬁne forceps attached to a vibrator and cut dis-

tal to the forceps. The metathoracic ganglion was ex-

posed by making a small window in the ventral thorax

and removing air sacs and connective tissue. Micro-

electrodes with DC resistances of 50 − 80 MΩ were

driven through the sheath and into the neuropilar pro-

ceses of the spiking and nonspiking interneurons, Fig.

The forceps holding the FeCO were moved with a

Gaussian white noise (GWN) signal produced by ﬁl-

tering a pseudorandom binary sequence band-limited

to 27 Hz with a fourth order Chebyshev low-pass ﬁl-

ter. Stimulus and evoked responses of the interneu-

rons were stored on magnetic tape using a PCM-DAT

data recorder. Subsequently, all signals were sampled

at a rate of 10 kHz ofﬂine to a PC for later analysis.

3.1 The Algorithms

The algorithms were developed in plain Python 2.67

running on a Linux (Ubuntu 10.10) i7 computer

with 12 GB of RAM. During the execution process

2 computers were used with the same speciﬁcation

and communications base on SSH. The code used

mainly the resources of parallel programming from

the iphython (P

erez and Granger, 2007) environment.

The main libraries used in this work were: numpy,

scipy and matplotlib.

The data set consist of signals from 20 different

nonspiking local interneurons, 35 spiking local in-

terneurons, 20 sensory neurons and 34 identiﬁed mo-

tor neurons. All neuron pathways were stimulated for

at least 40s. We chose to analyse approximately 25

percent of each dataset where signals were approxi-

mately stationary (grey area in Fig. 2(a). The main

consequence of this long acquisition period is that

each analysed signal (discounting the transients and

zero input) has from 200.000 - 400.000 samples. For

regular signal analysis it took over 20 min to analy-

ses each ﬁle. Moreover, 35 repetitions of surrogate

(a)

(b)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-150 -100 -50 0 50 100 150

200 Hz -> 27 Hz

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-150 -100 -50 0 50 100 150

27 Hz -> SETi

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-150 -100 -50 0 50 100 150

27 Hz -> FETi

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-150 -100 -50 0 50 100 150

SETi -> FETi

(c)

Figure 2: (A) Simultaneous intracellular recordings from

two motor neurons PIFlTi, SETI, and a nonspiker during

stimulation applied to the FeCO. The grey areas represents

the signal selection used in this analysis and the horizontal

axis is time in seconds. (B) Delayed Conditional Mutual

Information from combinations of signals shown in (A) and

the horizontal axis is delay between signals in milliseconds.

mation (line associate with the peak) between signal indi-

cated at the legend and delayed conditional mutual informa-

tion using surrogate date with 35 repetitions (baseline). The

vertical axis, left, has a normalized values and each mark

correspond to a step of 0.1.

data were also performed. The full analysis of this

data took approximately 35 days in a continuous par-

allel environment and at least one extra month was

required to analyse these initial results.

The estimation of probability density used a his-

BIOSIGNALS 2012 - International Conference on Bio-inspired Systems and Signal Processing

togram approach, while for joint probability distribu-

tions bidimensional histograms were used. In both

cases the number of bins was ﬁxed at 128. The de-

layed mutual information and conditional mutual in-

formation were implemented as a function of a lag.

Each lag was executed in a different core of the com-

puters.

4 RESULTS

For evaluation of the algorithms we used simulated

signals (with a linear and non-linear system with

Gaussian random input signals) and real experimen-

tal signals. In many signal records there were both

200 Hz and 27 Hz data. The time delays from 200 Hz

→ 27 Hz, Fig.2(c) ﬁrst plot, agreed with time-delay

speciﬁcations of the low-pass ﬁlter used and when re-

versing input and output, 27 Hz → 200 Hz, the correct

negative lag were obtained.

Simultaneous recordings from interneurons and

motor neurons showed that they received either exci-

tatory or inhibitory inputs during stimulation applied

to the FeCO, Fig.2(a). This conclusion is based on

extensive previous work (Kondoh et al., 1995; New-

land and Kondoh, 1997; Vidal-Gadea et al., 2010) and

parallel work not further reported here.

We evaluated the time delay from recording sites,

1, considering the peaks of delayed mutual informa-

tion and delayed conditional mutual information. Dif-

ferent combinations of these sites were evaluated and

distinct peaks with relatively short delays were clearly

apparent (all graphics in Fig.2(b)). In Fig.2(b) (top

to down, three ﬁrst graphics) the analyses was per-

formed with delayed mutual information and last plot

from delayed conditional mutual information where

the conditional was 27 Hz signal. Using this approach

we analysed the delayed (conditional) mutual infor-

mation from responses of sensory neurons, spiking

and nonspiking local interneurons and motor neurons

resulting from stimulation of the FeCO.

The results of our analysis are summarised in

Fig.3 in which the presumed interconnections based

on delayed (conditional) mutual information are

shown. The numbers on the arrows give the mean

time-delay (ms).

Some of recorded signal were collected at the

same place in neuronal ganglia from different ani-

mals. In this case, data were average and in few cases

outliers were removed from mean evaluation. In one

case (Nonspiker1 and Nonspiker2 in Fig.3) there were

enough data to cluster into two different sets, with dif-

ferent mean delay value and the graph was rerouted to

accommodate this split state. Also, in part of the data

there were signals with two peaks with similar am-

plitude time delay mutual information that were sta-

tistically signiﬁcant. In these cases the peaks were

considered as representing different pathways.

The results reveal a number of important points.

First, that neurons sharing similar functions and that

are closely located in the network give similar time-

delays. For example, all identiﬁed ﬂexor motor neu-

rons recorded from anterior and posterior groups, that

generate ﬂexion movements of the tibia, have only

short delays of only a few milliseconds relative to

each other. Similarly, the two extensor motor neurons

(the slow, SETi, and fast, FETi) that produce exten-

sion movements of the tibia share information with

similar short delays.

Secondly, nonspiking interneurons can be parti-

tioned into two distinct groups (marked as nonspiker

1 and 2) with short and long time delays with respect

to the stimulus input, reﬂecting their potential roles

in the neural pathways controlling limb movements.

Third, while the dataset on which this analysis was

based was very large, it should be considered as a

small sample probing network characteristics. It is

impossible to record from all combinations of neu-

rons and so many potential interconnections are miss-

ing or have few repetitions. Additional recordings are

clearly needed for a more complete analysis of the

network.

Clearly it would be desirable to conﬁrm the anal-

ysis with detailed anatomical studies of the neuronal

connections of the actual cells from which the data

was collected. However, this would be challenging

and was not carried out in this data set. In addition,

the connections with longer time-lags are unlikely to

be mono-synaptic, and such connectivities would be

difﬁcult to disentangle from anatomical studies. Thus

while agreement between anatomy and the current

study would provide supportive evidence for the ap-

proach taken, disagreement would not necessarily im-

ply that the proposed signal processing method failed.

It is a strength of the current method that connec-

tions are identiﬁed based on information transmis-

sion, rather than direct neuronal connections.

It should also be pointed out that the current

method based on the simultaneous recordings from

only one or two neurons cannot clearly indicate if

connections are direct or indirect.

5 CONCLUSIONS

Previous work (Kondoh et al., 1995; Newland and

Kondoh, 1997) has analysed the anatomical connec-

tions between neurones, but such methods are not able

INFERENCE ABOUT MULTIPLE PATHWAYS IN MOTOR CONTROL LIMB IN LOCUST

Nonspiker

Nonspiker1

Nonspiker2

13.0

SETi

18.7

PSFlTi

20.5

AFFlTi

2.9 20.6

PFFlTi

3.1

ASFlTi

0.3

PIFlTi

2.2

2.1

FETi

2.5

2.7

0.31.7

2.8

2.7 2.5

4.6

1.7

2.6

MechInput

sensory

11.9

3.9

19.9 6.7

7.8

Spiking

6.0

Figure 3: Graph showing the presumed interconnections

based on delayed conditional mutual information analyses

with edge labels showing the mean time delays through

nodes in ms. The red edge labels indicate presumed con-

nections with incomplete data. Solid lines represent path-

ways that were evaluated directly from each dataset. Dotted

lines were inferred under the conditions that FeCO move-

ment activates sensory neurons whose time delays were not

included in the analyses. Dashed lines represent the parti-

tioning of nonspiking interneurons into two nodes.

to identify the activation of these connections during

the GWN stimulation used in the current study and

related work. The functional connections identiﬁed

in the current study broadly match those known from

the anatomical studies.

In previous work on this data we have used lin-

ear and non-linear system identiﬁcation approaches

to study connectivity. However, this form of analy-

sis is limited by the class of models chosen (Volterra-

Wiener using polynomial non-linearities) whereas

the current information theoretic approach does not

impose such a constraints. Mutual information is thus

a better suited tool for mapping the connectivity in the

network. However, mutual information only shows

strength of connections (this will be considered in fur-

ther studies) and delays (considered here), but not de-

tailed input-output relationships.

The analysis of this data was realized using a com-

puter with multiple cores and with no downsampling

of the data. The approach was computationally inten-

sive and without a multiple core computer and a high

performance computational environment could not be

complete within a reasonable time. A key feature of

our analysis is that it can be used to understand the

interconnections between neurons in neural networks

composed of both spiking (digital action potentials)

and nonspiking (analogue synaptic) signals.

We found that neurons that share similar func-

tions, for example ﬂexor or extensor motor neurons

that drive muscle activity (Newland and Kondoh,

1997) share similar time delays with respect each

group indicating their combined activity in control-

ling movements. Knowledge such as this can help

further understand the structure of neural pathways

within local circuits and can help inform further neu-

rophysiological analysis

(Burrows, 1996) suggested that the interactions

between nonspiking interneurons could lead to them

acting in many different ways to process the signals

from the FeCO. We found that a subset (4 and 6 mea-

surements of 14) of a population of nonspiking local

interneurons showed two distinct delayed mutual in-

formation time delays. Also, the last 4 measures from

nonspinking had 2 peaks with similar delays from 2

sets described before. This revealed itself in distinct

peaks in their responses indicating the presence of two

main pathways to the same neurone and points to dif-

ferences in the function of the two types of interneu-

ron in local networks, a feature not documented from

previous neurophysiological analyses.

The next steps in our study will be to include more

experimental samples in data analysis. The data anal-

ysis will also include an analysis of transfer entropy

and a Bayesian model to group partial graph informa-

tion.

ACKNOWLEDGEMENTS

The ﬁrst author is indebted to the Research Founda-

tion of Brazil (CNPq) for support this collaboration.

We are also grateful to the BBSRC (UK) and EPSRC

(UK) for ﬁnancial support.

REFERENCES

Alonso, J. F., Mananas, M. A., Topor, D. H. Z. L., and

Bruce, E. N. (2007). Evaluation of respiratory mus-

cles activity by means of cross mutual information

function at different levels of ventilatory effort. IEEE

Trans. on Biomedical Eng., 54(9):1573–1582.

Bialek, W., Nemenman, I., and Tishby, N. (2001). Pre-

dictability, complexity and learning. Neural Comp,

(13):2409 – 2463.

Burrows, M. (1996). The Neurobiology of an Insect Brain.

Oxford University Press, USA.

Cover, T. M. and Thomas, J. A. (2006). Elements of Infor-

mation Theory (Wiley Series in Telecommunications

and Signal Processing). Wiley-Blackwell, second edi-

tion.

Delcomyn, F. (2004). Insect walking and robots. Annu. Rev.

Entomol., 49(3):51–70.

Dionisio, A., Menezes, R., and Mendes, D. A. (2004). Mu-

tual information: a measure of dependency for nonlin-

ear time series. Physica A, 344:326–329.

BIOSIGNALS 2012 - International Conference on Bio-inspired Systems and Signal Processing

Ebeling, W. (2002). Entropies and predictability of nonlin-

ear processes and time series. In Sloot, P., Hoekstra,

A., Tan, C., and Dongarra, J., editors, Computational

Science ICCS 2002, volume 2331 of Lecture Notes in

Computer Science, pages 1209–1217. Springer Berlin

/ Heidelberg.

Erdogmus, D. and Principe, J. C. (2006). From linear

adaptative ﬁltering to nonlinear information process-

ing. IEEE Signal Processing Magazine, 14.

Fourtner, C. (1976). Central nervous control of cockroach

walking, pages 401–418. Plenum, New York.

Kondoh, Y., Okuma, J., and Newland, P. L. (1995). Dy-

namics of neurons controlling movements of a locust

hind leg: Wiener kernel analysis of the responses of

proprioceptive afferents. Journal of Neurophysiology,

73(5):332–340.

Kugiumtzis, D. (2002). Surrogate Data Test on Time Se-

ries, chapter 12, pages 267–282. Kluwer Academic

Publishers.

Li, W. (1990). Mutual information functions versus corre-

lation functions. Journal of Statistical Physics, 60(5-

6):823 – 837.

Marschinski, R. and Kantz, H. (2002). Analysing the infor-

mation ﬂow between ﬁnancial time series. Eur. Phys.

J. B, 30(2):275–281.

Newland, P. L. and Kondoh, Y. (1997). Dynamics of neu-

rons controlling movements of a locust hind leg ii.

ﬂexor tibiae motor neurons. Journal of neurophysi-

ology, 77(3):1731–1746.

Nichols, J. (2006). Examining structural dynamics using in-

formation ﬂow. Probabilistic Engineering Mechanics,

21(4):420–433.

Palu

s, M., Kom

arek, V., Hrn

r, Z., and

erbov

a, K. (2001).

Synchronization as adjustment of information rates:

Detection from bivariate time series. Phys. Rev. E,

63(4):046211.

Palu

s, M. and Stefanovska, A. (2003). Direction

of coupling from phases of interacting oscillators:

An information-theoretic approach. Phys. Rev. E,

67(5):055201.

Palu

s, M. and Vejmelka, M. (2007). Directionality of cou-

pling from bivariate time series: How to avoid false

causalities and missed connections. Phys. Rev. E,

75(5):056211.

Pearl, J. (2009). Causal inference in statistics: An overview.

Statistics Surveys, (3):96–146.

erez, F. and Granger, B. E. (2007). IPython: a System for

Interactive Scientiﬁc Computing. Comput. Sci. Eng.,

9(3):21–29.

Schreiber, T. (2000). Measuring information transfer. Phys.

Rev. Lett., 85(2):461–464.

Schreiber, T. and Schmitz, A. (1996). Improved surrogate

data for nonlinearity tests. Phys. Rev. Lett., 77:635.

Schreiber, T. and Schmitz, A. (2000). Surrogate time series.

Physica D, 142:346–382.

Shannon, C. E. (1948). A mathematical theory of communi-

cation. Bell System Technical Journal, 27(3):379–423.

Urbach, R. M. A. (2000). Footprints of Chaos in the Mar-

kets: Analyzing Non-linear Time Series in Financial

Markets and other Real Systems. Prentice Hall Pub-

lishing, USA.

Vastano, J. A. and Harry, S. L. (1988). Information trans-

port in spatiotemporal systems. Phys. Rev. Lett.,

60(18):1773–1776.

Venema, V. K. C., Ament, F., and Simmer, C. (2006). A

stochastic iterative amplitude adjusted fourier trans-

form algorithm with improved accuracy. Nonlinear

Processes in Geophysics, 13(3):247–363.

Vidal-Gadea, A., Jing, X. J., Simpson, D. M., Dewhirst, O.,

Kondoh, Y., Allen, R., and Newland, P. (2010). Cod-

ing characteristics of spiking local interneurons dur-

ing imposed limb movements in the locust. Journal of

Neurophysiology, 103:603 – 615.

Ward, B. D. and Mazaheri, Y. (2008). Information transfer

rate in fmri experiments measured using mutual in-

formation theory. Journal of Neuroscience Methods,

167(1):22–30.

INFERENCE ABOUT MULTIPLE PATHWAYS IN MOTOR CONTROL LIMB IN LOCUST