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Nonlinear System Identiﬁcation of Neural Systems from Neurophysiological Signals

Fei He1, ∗and Yuan Yang2, 3, 4, †

1Centre for Data Science, Coventry University, Coventry CV1 2JH, UK.

2Stephenson School of Biomedical Engineering, The University of Oklahoma, Tulsa, Oklahoma-74135, USA.

3Department of Physical Therapy and Human Movement Sciences,

Feinberg School of Medicine, Northwestern University, Chicago, Illinois-60611, USA.

4Laureate Institute for Brain Research, Tulsa, Oklahoma-74136, USA.

The human nervous system is one of the most complicated systems in nature. Complex nonlinear

behaviours have been shown from the single neuron level to the system level. For decades, lin-

ear connectivity analysis methods, such as correlation, coherence and Granger causality, have been

extensively used to assess the neural connectivities and input-output interconnections in neural sys-

tems. Recent studies indicate that these linear methods can only capture a certain amount of neural

activities and functional relationships, and therefore cannot describe neural behaviours in a precise

or complete way. In this review, we highlight recent advances in nonlinear system identiﬁcation of

neural systems, corresponding time and frequency domain analysis, and novel neural connectivity

measures based on nonlinear system identiﬁcation techniques. We argue that nonlinear modelling

and analysis are necessary to study neuronal processing and signal transfer in neural systems quan-

titatively. These approaches can hopefully provide new insights to advance our understanding of

neurophysiological mechanisms underlying neural functions. These nonlinear approaches also have

the potential to produce sensitive biomarkers to facilitate the development of precision diagnostic

tools for evaluating neurological disorders and the eﬀects of targeted intervention.

I. INTRODUCTION

The human nervous system is a complicated network

comprised of more than 10 billion neurons, with tril-

lions of synapses connecting them. Neuronal information

processing is complex at diﬀerent levels, from the mi-

croscopic pre- and post-synaptic cellular interactions to

the macroscopic interactions between large populations

of neurons in, for instance, sensory processing and motor

response [1]. The behaviour of a single neuron is highly

nonlinear, showing a step-like ‘none-or-all’ ﬁring response

[2], while the behaviour of neurons in a population could

be relatively similar. Therefore, the nonlinear response of

each individual neuron may be smoothed out by the dis-

tribution of membrane thresholds across the population,

known as the pool eﬀect [3]. This eﬀect typically occurs

in a mono-synaptic neural system such as the cortico-

spinal tract where the supraspinal motor command is

linearly transferred to the motor output due to the pool

eﬀect of motor units [4]. However, multi-synaptic neural

systems, such as the somatosensory system, have been

reported highly nonlinear, showing harmonic responses

to periodic stimuli [5–8]. Cross-frequency coupling in

the corticothalamic interactions has also been reported

when characterising essential tremor [9]. Nonlinear be-

haviours in neural systems are thought to be associated

with various neural functions, including neuronal encod-

ing, neural processing of synaptic inputs, communication

between diﬀerent neuronal populations and functional in-

tegration [10–13].

∗fei.he@coventry.ac.uk

†yuan.yang-2@ou.edu

Various functional and eﬀective connectivity measures

have been developed [14] to characterise such linear

and nonlinear functional integration in neural networks,

from large-scale neurophysiological signals. These sig-

nals, such as electroencephalogram (EEG) and magne-

toencephalogram (MEG) from the brain, electromyo-

gram (EMG) from muscles, measure neural activities

from macro-scale neuronal populations. While functional

connectivity measures, e.g. correlation, coherence, mu-

tual information, only quantify the undirected statistical

dependencies among signals from diﬀerent areas, eﬀective

connectivity attempts to quantify the directed causal in-

ﬂuences of one neural system over another, either at a

synaptic or population level [15]. Mostly, eﬀective con-

nectivity measures are based on models of neural in-

teractions or coupling (although there exist model-free

measures like transfer entropy [16]) and is often time-

dependent (dynamic). Therefore, eﬀective connectivity

has a strong link with dynamic modelling, also known as

system identiﬁcation in control systems theory [17–19],

and corresponding model-based causality analysis.

Nonlinear system identiﬁcation techniques have been

formally applied to study neuronal information process-

ing and neural systems since the 1970s. Some pio-

neering work includes: the nonlinear dynamic mod-

elling of the retinal neuron chains in receptive-ﬁeld re-

sponses [20, 21], the identiﬁcation of nonlinear synaptic

interactions [22], the identiﬁcation of neural systems us-

ing stimulus-response and white-noise approach [23–25],

the development of nonlinear systems analytic approach

based on functional power (or Volterra/Wiener) series to

study central nervous system function and hippocampal

formation [26, 27], and nonlinear identiﬁcation of stretch

reﬂex dynamics [28]. Until now, many linear and non-

linear system identiﬁcation methods have then been pro-

2

posed and developed in the neuroscience context. Nev-

ertheless, recent studies indicate that linear methods can

only capture a certain amount of neural activities and

functional relationships, and therefore cannot describe

neural behaviours in a precise or complete way [29, 30].

Nonlinear approaches provide us with useful tools to ex-

plore the nonlinear nature of neural systems [10]. In this

review article, we highlight the need and recent advances

in nonlinear system identiﬁcation of neural systems, as

well as novel neural connectivity analysis methods based

on nonlinear system identiﬁcation techniques. A diagram

that summarises the linear and nonlinear functional and

eﬀective connectivity measures and their links with sys-

tem identiﬁcation is provided in Figure 1.

II. NONLINEARITY IN THE NEURONAL

LEVEL AND NEURAL SYSTEMS

At a single neuron level, the action potential spike is

the principal basis of information encoding, which al-

lows signal transmission across diﬀerent neuronal pop-

ulations [13]. The spike timing is thought to be associ-

ated with the coding scheme in neural systems [31]. The

nonlinear nature of the neuronal process of synaptic in-

put inﬂuences the temporal ﬁring behaviour of individual

neurons. Diﬀerent types of neurons have their own reper-

toire of ion channels that are responsible for their char-

acteristic nonlinear ﬁring patterns and associated neural

functions. For example, persistent inward currents me-

diated by their voltage-gated sodium and calcium chan-

nels are an important source of the nonlinear behaviour

of spinal motoneurons. They are instrumental in gen-

erating the sustained force outputs required for postural

control [32]. Activation of the L-type calcium channels in

nigral dopaminergic neurons results in intrinsic bursting

behaviour [33], exhibiting low-dimensional determinism

and likely encodes meaningful information in the awake

state of the brain [34]. The nonlinearity of the neuronal

transfer function mediated by its component ion channels

can generate various types of nonlinear output patterns

such as harmonic, subharmonic and/or intermodulation

of input patterns.

Despite plenty of knowledge of the nonlinear behaviour

of a single neuron, the input-output relation at the neural

system level is yet to be understood entirely. The system-

level neural response is a composite output of collective

neuronal activities from a large number of neurons. In

a neuronal population, the pool eﬀect can reduce the

nonlinearity generated from each individual neuron, by

smoothing the neuronal dynamics from a scale of millisec-

onds (spikes) to 10 milliseconds (local ﬁeld potentials)

or to 100 milliseconds (large-scale neurophysiological ac-

tivities or signals such as EEG) [35]. Such eﬀects have

been previously demonstrated through both a computa-

tional model and an in vivo study in the human motor

system, where the motor command can be transmitted

linearly via the mono-synaptic corticospinal tract when

more than ﬁve motoneurons are activated [4]. However,

a small amount of nonlinearity may still be present [36].

A recent study simulated nonlinear neuronal dynam-

ics on a large-scale neural network that captured the

inter-regional connections of neocortex in the macaque.

The authors applied information-theoretic measures to

identify functional networks and characterized structure-

function relations at multiple temporal scales [37]. The

nonlinearity in each neuronal population can cumula-

tively increase if the system involves multiple synaptic

connections [38]. A recent study in hemiparetic stroke

shows that the nonlinearity in the motor system increases

due to an increased usage of multi-synaptic indirect mo-

tor pathways, e.g. cortico-reticulospinal tract [39], fol-

lowing damage to the mono-synaptic corticospinal tract

[40].

Assessing the input-output relation in neural systems,

e.g. sensory, motor and cognitive processes, is essen-

tial to a better understanding of the nervous system.

For instance, it could help to gain a better insight into

the normal and pathological neural functions. It is well

known that a linear system generates only iso-frequency

interactions between an input and the output, e.g. the

coupling of neuronal oscillations at a speciﬁc frequency

band [41]. For decades, correlation and coherence mea-

sures have been used to identify the linear interaction in

neural systems. More recently, various studies indicate

the input-output neural interactions can cross diﬀerent

frequency components or bands, which is named cross-

frequency coupling [42–45] and is a distinctive feature of

a nonlinear system. In the following sections, we review

both the linear and nonlinear approaches for identifying

neural systems and associating neural connectivity, espe-

cially from a system identiﬁcation perspective.

III. LINEAR CONNECTIVITY AND SYSTEM

IDENTIFICATION

The nervous system is a highly cooperative network

composed of diﬀerent groups of neurons. Neural connec-

tivity, i.e., the synchronization of neural activity across

these groups, is crucial to the coordination among dis-

tant, but functionally related, neuronal groups [46]. Lin-

ear neural connectivity can be assessed by determining

the signal correlation or causality between the recorded

neural signals. This section reviews commonly used lin-

ear connectivity, system identiﬁcation methods and their

interconnections in studying neural systems.

A. Correlation and coherence

The most widely used measure of interdependence be-

tween two time series in the time domain is the cross-

correlation function [14], which measures the linear cor-

relation between two signals or stochastic processes X

and Ywith discrete observations x(t) and y(t), at t=

3

Funconal Connecvity

Linear: correlaon, coherence

Nonlinear: nonlinear correlaon, mutual informaon

higher-order spectrum & coherence

Eﬀecve Connecvity

Modeling & System Idenﬁcaon Causality

i) Linear:

Time: AR/ARX

Frequency: transfer funcon or FRF

Time: linear GCA

Frequency: PDC, DTF, DCOH, Spectral GC

ii) Nonlinear:

Time: Volterra Series, NARX or NARMAX

ANN, DNN, Gaussian process, state-space

Frequency: GFRFs, OFRF

Time: nonparametric, RBF, kernel, local linear,

DCM, transfer entropy, NARX-based causality

Frequency: NPDC, nonlinear Spectral GC

iii) Nonlinear me-varying:

Time: TV-NA(MA)RX, me-varying

Volterra series, sliding-window method

Frequency: TV-GFRFs, TV-OFRF

FIG. 1: Overview of the linear and nonlinear functional and eﬀective connectivity (causality) measures and their links with

system identiﬁcation methods. The linear functional connectivity, linear system identiﬁcation and linear causality measures

are ﬁrst reviewed in Section III. The nonlinear and nonlinear time-varying system identiﬁcation approaches (in both time and

frequency domains) are then investigated in Section IV. The recently proposed nonlinear function connectivity measures and

nonlinear causality measures (based on nonlinear system identiﬁcation) are introduced in Section V. The abbreviations in the

diagram are deﬁned as: autoregressive (AR), autoregressive with exogenous input (ARX), frequency response function (FRF),

nonlinear autoregressive (moving average) model with exogenous inputs (NAR(MA)X), artiﬁcial neural networks (ANN),

deep neural networks (DNN), generalised frequency response function (GFRF), output frequency response function (OFRF),

time-varying (TV), Granger causality analysis (GCA), partial directed coherence (PDC), directed transfer function (DTF),

spectral Granger causality and directed coherence (DCOH), Granger causality (GC), dynamic causal modelling (DCM),

nonlinear partial directed coherence (NPDC).

1,2, ..., N , as a function of their delay time:

Cxy(τ) = 1

N−τ

N−τ

X

k=1

x(k+τ)y(k) (1)

where Nis the number of samples and τthe time lag

between two signals. This function ranges from -1 (com-

plete linear inverse correlation) to 1 (complete linear di-

rect correlation). The value of τthat maximizes this

function can be used to estimate the linearly related de-

lay between signals. The well-known Pearson correlation

coeﬃcient is equal to Cxy(τ) when τ= 0. The linear

dependence between two signals in the frequency domain

is usually measured by the spectral coherence. The co-

herence between two signals at frequency fis deﬁned as:

Cxy(f) = |Sxy(f)|2

Sxx(f)Syy (f)(2)

where Sxy(f) is the cross-spectral density between x

and y, and Sxx(f) and Syy(f) the auto-spectral density

of xand yrespectively. The cross-spectral and auto-

spectral densities are the Fourier transforms of the cross-

correlation and auto-correlation functions of the two sig-

nals. Values of coherence are always between 0 and 1.

The correlation and coherence measures have been widely

applied to EEG, MEG or EMG signals to characterise the

neuronal interactions, from the ﬁring of cortical neuron

spike trains to complicated neural systems (for reviews

see [14, 47]).

B. System identiﬁcation and causality

Unlike functional connectivity, eﬀective connectivity

emphasises on the directional causal inﬂuences between

neural areas or signals. Here, we ﬁrst introduce the clas-

sical Granger causality and its link with the time-domain

linear system identiﬁcation, i.e. regression models of

bivariate time series. The frequency-domain causality

measures can then be linked with the frequency response

function of linear systems.

Considering two signals or variables Xand Y, the in-

teractions of the signals can be described by bivariate

linear autoregressive with exogenous input (ARX) mod-

4

els jointly,

x(t) =

q

X

k=1

a11,kx(t−k) +

p

X

k=1

a12,ky(t−k) + ex(t)

y(t) =

q

X

k=1

a21,kx(t−k) +

p

X

k=1

a22,ky(t−k) + ey(t)

(3)

where pand qare the model order of yand xregressors;

ex(t) and ey(t) are the uncorrelated model prediction er-

rors over time. A linear causal inﬂuence from Xto Y

deﬁned by Granger can be expressed as a log ratio of the

prediction error variances of the corresponding restricted

(AR) and unrestricted (ARX) models:

FX→Y= ln var(y|y−)

var(y|y−, x−)= ln Σy−

22

Σ22

(4)

where x−and y−denotes contributions from lagged input

and output terms, respectively; y−22 denotes the vari-

ance of eywhen there are only regression terms of Y.

The linear ARX models (3) can be re-written in matrix

form and mapped to the frequency domain by Fourier

transformation:

A11(f)A12 (f)

A21(f)A22 (f)X(f)

Y(f)=Ex(f)

Ey(f)(5)

where the components of the coeﬃcient matrix A(f) are

Alm(f) = δlm −Pp(or q )

k=1 alm,ke−j2π kf /fswith fsthe sam-

pling frequency and δlm the Kronecker delta function.

We can re-write the above equation by inverting the co-

eﬃcient matrix G(f) = A−1(f) and moving the so-called

transfer function matrix G(f) to the right-hand-side the

equation. Diﬀerent frequency-domain Granger causality

measures, such as partial directed coherence (PDC), di-

rected transfer function (DTF), spectral Granger causal-

ity and directed coherence (DCOH) [48–50], can then be

expressed as a function of the elements of either the coef-

ﬁcient matrix A(f) or the transfer function matrix G(f)

(Baccala and Sameshima, 2001; Chicharro, 2011). By

dividing both sides of (5) with the corresponding diago-

nal elements in the coeﬃcient matrix A, the oﬀ-diagonal

elements in the transformed coeﬃcient matrix are actu-

ally related to the negative frequency response functions

(FRFs) of linear ARX systems, if one signal is treated as

the input while the other is treated as the output. For

instance,

A21(f)

A22(f)=−HX→Y(f) (6)

the FRF, HX→Y(f), describes the input-output relation-

ship, i.e., with input Xand output Y, of the (noise-free)

‘system’ in the frequency domain. It is also known as the

‘transfer function’ in linear system theory. Frequency-

domain Granger causality measure, e.g. PDC, can be

expressed as a function of the FRFs of the corresponding

linear ARX and AR models:

P DCX→Y(f) = −HX→Y(f)

p|A11(f)/A21(f)|2+|HX→Y(f)|2

(7)

Establishing such a link between the causality measures

and linear system identiﬁcation, in both time and fre-

quency domains, is crucial to the further development of

nonlinear model-based causality measures via nonlinear

system identiﬁcation, which will be investigated in Sec-

tion IV and V.

C. Limitation of linear approaches on identifying

neural system

Linear connectivity and system identiﬁcation allow the

assessment of communication between neuronal popu-

lations at the same oscillatory frequency band or sim-

ilar neuronal ﬁring patterns. The applications of lin-

ear approaches have been thoroughly reviewed previ-

ously [51, 52]. However, it is not clear how much informa-

tion is missing when using the linear approach since the

behaviour of various neural systems can be highly non-

linear [10, 11]. When one uses a linear measure to inves-

tigate a neural system, the nonlinear neural interaction

is ignored, especially between the neuronal populations

which have very diﬀerent mean ﬁring rates such as the

central nervous system and the periphery. A recent study

reported that in the human somatosensory system over

80% of the cortical response to wrist joint sensory input

comes from nonlinear interactions, where a linear model

explains only 10% of the cortical response [29]. There-

fore, nonlinear connectivity and modelling approaches

are needed to investigate neural systems in a complete

way.

IV. NONLINEAR SYSTEM IDENTIFICATION

OF NEURAL SYSTEMS

It is often impossible to derive a mechanistic model of

a neural system, due to the complexity of the underly-

ing biological process and many unobservable state vari-

ables. In this section, we focus on the generic nonlinear

model representations of a single-input and single-output

(SISO) neural dynamic system, its identiﬁcation process

in the time domain, and corresponding frequency-domain

analysis. We ﬁrst investigate the identiﬁcation of nonlin-

ear time-invariant systems, and then time-varying non-

linear systems.

5

A. Time-domain nonlinear system identiﬁcation

1. Volterra series

The Volterra series model is a direct generalisation of

the linear convolution integral and provides an intuitive

representation for a nonlinear input-output system. The

output y(t) of a SISO nonlinear system can be expressed

as a Volterra functional of the input signal u(t):

y(t) =

M

X

n=1

y(n)(t) + ey(t)

y(n)(t) = Z+∞

−∞ ·· · Z+∞

−∞

hn(τ1,·· · , τn)

n

Y

i=1

u(t−τi)dτi

(8)

where y(n)(t) is the nth-order output and Mis the max-

imum order of the system’s nonlinearity; hn(τ1,··· , τn)

is the nth-order impulse response function or Volterra

kernel, which describes nonlinear interactions among n

copies of input and generalises the linear convolution in-

tegral to deal with nonlinear systems. Neurobiologically,

Volterra series can be directly interpreted as the eﬀec-

tive connectivity - ‘the inﬂuence that one neural system

exerts over another, either at a synaptic or population

level’ [53]. The ﬁrst-order kernel describes the linear

‘driving’ eﬃcacy or linear synchronous interactions, and

the second- or higher-order kernels represent the ‘modu-

latory’ inﬂuence or asynchronous interactions [10]. The

Fourier transform of the ﬁrst-order kernel is the FRF

(or transfer function) and describe the interactions in

the same frequencies, while the frequency-domain coun-

terparts of the higher-order kernels are the GFRFs (to

be discussed in Section IV B) which quantify the cross-

frequency interactions.

Practically, to deal with a large number of Volterra

series coeﬃcients, a regularization strategy is often em-

ployed in the estimation procedure [54]. Volterra model

has been widely used in physiological systems, including

neural systems, modelling. Some recent examples include

the study of nonlinear interactions in the hippocampal-

cortical neurons [55], in the spectrotemporal receptive

ﬁelds of the primary auditory cortex [56], in the sensory

mechanoreceptor system [29], in the human somatosen-

sory system (i.e. the cortical response to the wrist

joint sensory input) indicating the dominance of nonlin-

ear response [30], in multiple-input and multiple-output

(MIMO) spiking neural circuits [57] and hippocampal

memory prostheses [58]. The Volterra model also has

a strong theoretical link with the NARMAX model [59]

and the dynamic causal modelling [60] to be discussed

next.

2. NARMAX model

Although Volterra series can provide an intuitive rep-

resentation for nonlinear systems, there are several crit-

ical limitations including i) it cannot represent severely

nonlinear systems; ii) the order of the Volterra series ex-

pansion can be very high in order to achieve a good ap-

proximation accuracy; however iii) the estimation of high

order Volterra kernel requires a large number of data and

can be computationally very expensive. Nonlinear Au-

toregressive Moving Average Model with Exogenous In-

puts (NARMAX) model [59, 61] has therefore been devel-

oped as an alternative to the Volterra series. NARMAX

model normally contains a much smaller number of terms

due to the inclusion of output delay terms, and its iden-

tiﬁcation process is computationally more eﬃcient. Sim-

ilar to the Volterra series, a polynomial Nonlinear Au-

toregressive Model with Exogenous Inputs, NARX (the

simplest NARMAX) model, can be expressed as a sum-

mation of a series of output terms with diﬀerent orders

of nonlinearity:

y(n)(t) =

n

X

p=0

K

X

k1,kp+q=1

cp,q(k1, ..., kp+q)×

p

Y

i=1

y(t−ki)

p+q

Y

i=p+1

x(t−ki) (9)

where p+q=n, ki= 1, ..., K, and PK

k1,kp+q=1 ≡

PK

k1·· · PK

kp+q=1. The number of model terms depends

on the order of input and output (qand p) and the max-

imum lags (K). The NARX model structure and param-

eters are typically identiﬁed based on the forward regres-

sion with the orthogonal least squares (FROLS) method

[62]. In cases where the system under study is stochas-

tic with unknown coloured noise, noise moving average

(MA) models should be employed to form a full NAR-

MAX model. The identiﬁed model can be statistically

validated using nonlinear correlation tests [63, 64].

A wide range of nonlinear systems can be represented

by NARMAX method, including systems with exotic

nonlinear behaviours such as subharmonics, bifurcations,

and chaos, as observed in the human nervous system [65].

Until now, NARMAX methodology has been employed to

develop dynamic models for nonlinear sensory process-

ing circuit from spiking neuron data [66] as an improve-

ment to the previous Volterra model-based studies [57],

to investigate the somatosensory aﬀerent pathways from

muscles to the brain [67, 68]; as well as to study the cor-

ticothalamic nonlinear interactions during tremor active

and resting states [9]. Apart from eﬃcient time-domain

predictive modelling, NARMAX also provides an essen-

tial base for the nonlinear frequency-domain analysis,

nonlinear time-varying modelling, and nonlinear causal-

ity analysis to be discussed in the following sections.

3. Dynamic causal modelling

Most of the eﬀective connectivity models, e.g. linear

and nonlinear autoregressive models, are directly identi-

6

ﬁed from functional neurophysiological signals. However,

sometimes it would be more accurate and meaningful to

identify the causal interactions of the underlying neu-

ronal activities at the level of neuronal dynamics [69].

The aim of dynamic causal modelling (DCM) [60, 70]

is to infer such connectivity among brain regions (or

sources) under diﬀerent experimental factors or inputs.

A DCM comprises typically two parts: a neuronal part

that describes dynamics among neuronal sources and a

measurement part that describes how the source dynam-

ics generate measurable observations, e.g. EEG or MEG

[71, 72]. Therefore, DCM can be naturally expressed as a

nonlinear state-space model with hidden states denoting

unobserved neuronal dynamics and the observation equa-

tion (e.g. the lead-ﬁeld) assumed linearly in the states.

The eﬀective connectivity among those sources can be

identiﬁed via Bayesian model selection and Bayesian in-

ference of the neuronal model parameters. One strength

of DCM is its biophysical and neuronal interpretation

of how the neurophysiological signals are generated from

the underlying neuronal system, through the generative

or forward (state-space) modelling. Due to the complex-

ity and computational cost of Bayesian model selection,

DCM is more suitable to investigate the connectivity

among pre-deﬁned regions of interest, rather than ex-

ploratory analysis of relatively large brain or neural net-

works [73]. Compared to the hypothesis-driven DCM,

the NARMAX or Volterra models are more ﬂexible in

terms of model structure identiﬁcation and their direct

frequency-domain mapping (to be discussed) is a power-

ful tool to study the nonlinear cross-frequency interac-

tions between neurological regions.

4. Other black-box neural nonlinear system identiﬁcation

methods

Apart from the aforementioned three important

generic nonlinear model representations, other black-box

modelling approaches have also been applied in the ‘neu-

ral system identiﬁcation’ context. For example, artiﬁcial

neural networks (ANNs), e.g. recurrent, multilayer per-

ceptron, fuzzy, probabilistic neural networks, have often

been used as an alternative to classical system identiﬁca-

tion models. ANNs have been applied to predict neural

responses in visual cortex [74, 75], and to improve the

prediction of synaptic motor neuron responses [76]. More

recently, deep neural networks (DNNs), such as convolu-

tional neural network (CNN) or recurrent neural network

(RNN), are employed to model sensory neural responses,

to understand neural computations and to learn feature

spaces for neural system identiﬁcation [77–81]. Never-

theless, in the current neuroscience literature, ANNs and

DNNs are applied more towards automatic feature ex-

traction and classiﬁcation problems rather than tradi-

tional ‘system identiﬁcation’. For instance, automatic de-

tection and diagnosis of neurological disorders via a com-

bination of ANN with other nonlinear feature extraction

techniques such as approximate entropy and wavelet [82–

85], or direct implementation of DNNs [86, 87]. Nonpara-

metric Bayesian approaches like Gaussian process (GP)

is closely related to ANN. GP has recently been used

for system identiﬁcation purpose [88–90] and applied to

analyse neurophysiological signals [91], such as the use of

GP modelling for EEG-based seizure detection and pre-

diction [92] and heteroscedastic modelling of noisy high-

dimensional MEG data [93]. Compared with ANN, GP

can be applied to model datasets with small sample size

and it has a relatively small number of hyperparameters.

Additionally, due to its Bayesian nature, GP can incor-

porate prior knowledge and speciﬁcations into the mod-

elling and can directly capture the model uncertainty.

Another well-known system identiﬁcation paradigm is

the nonlinear state-space model [94, 95]. Its strength in

dynamic (latent) state estimation and sequential infer-

ence process makes it a suitable candidate in the identiﬁ-

cation of certain neural systems. The state-space models

have been applied to infer neural spiking activity induced

by an implicit stimulus observed through point processes

[96], to perform optimal decoding given multi-neuronal

spike train data and tracking nonstationary neuron tun-

ing properties (for a review, see [97]), and to perform

source localization from neurophysiological signals like

MEG and EEG [98, 99]. All of those black-box modelling

approaches are usually ﬂexible and accurate in quantify-

ing complex and long-range nonlinear interactions. In

comparison, the advantages of NARMAX and Volterra

models are their modelling simplicity, interpretability of

nonlinear interactions in the time-domain (e.g. the or-

der of nonlinearity, phase delay), and frequency-domain

mapping and analysis (e.g. energy transfer, intermodu-

lations).

B. Frequency-domain nonlinear system analysis:

nonlinear frequency response functions

Many nonlinear eﬀects, such as harmonics, intermod-

ulations and energy transfer, can only be accurately and

intuitively characterised in the frequency domain. Thus,

it is important to ‘map’ the identiﬁed time-domain non-

linear models to the frequency domain for further analy-

sis. A multidimensional Fourier transform of the nth-

order Volterra kernel in (8) yields the so-called nth-

order generalised frequency response function (GFRF),

Hn(f1,·· · , fn), which is a natural extension of the lin-

ear frequency response function to the nonlinear case

[59]. The output spectrum Y(f) of a nonlinear system

can then be expressed as a function of the input spec-

trum X(f) and GFRF, known as the output frequency

response function (OFRF) [59, 100]:

Y(f) =

M

X

n=1 1

√nZf1+···+fn=f

Hn(f1,·· · , fn)

n

Y

i=1

X(fi)df

(10)

Compared with the Volterra series, the GFRFs can

7

A B C

FIG. 2: The GFRFs of an exemplar nonlinear system. (A)

The linear 1st-order GFRF, H1(f), shows a ‘resonance’ peak

at f= 0.9Hz; (B) and (C) the 3-D and contour plots of the

2nd-order GFRF, H2(f1, f2). It shows a peak at

f1= 0.9Hz , f2= 0.9Hz, which indicates harmonics at

2f=f1+f2= 1.8Hz can be introduced in the output

spectrum if input contains a 0.9Hz component.

be more eﬃciently computed from the identiﬁed time-

domain NARMAX model (9) and corresponding model

parameters [101]. As shown in Figure 2, the peaks in 1st-

order GFRF indicate the well-known ‘resonance frequen-

cies’ of the linear part of the system; and the peaks (or

ridges) in the 2nd-order GFRF would indicate nonlinear

harmonics (f1+f2when f1=f2) or inter-modulation ef-

fects (f1±f2when f16=f2) in the output spectrum, and

so on. Since Y(f) = Pm

n=1 Yn(f), the nth OFRF Yn(f)

represents the nth-order (linear or nonlinear) contribu-

tion from the input to the output spectrum. Practically,

by comparing the OFRF with the spectrum of the output

signal, obtained from a classical nonparametric estima-

tion such as fast Fourier transform, one can also ‘validate’

the accuracy of the time-domain modelling in addition to

the aforementioned NARMAX model validation [9].

NARMAX-based frequency-domain analysis method

has been applied to quantify the dynamic characteristics

of nonlinear sensory processing circuit models from spik-

ing neuron data [66], the cross-frequency interactions in

the corticothalamic loops with respect to tremor [9], and

the characterisation of epileptic brain states [64]. More

details will be discussed in the Section VI.

C. Time-varying nonlinear system identiﬁcation

Many neurological subsystems are inherently nonsta-

tionary, since the brain is a dissipative and adaptive dy-

namical system [102, 103]. Modelling and identiﬁcation

approaches of nonstationary processes have been well de-

veloped for linear systems, i.e. linear time-varying (LTV)

systems. They are primarily based on adaptive recur-

sive methods, such as recursive least squares, least mean

squares, and the Kalman ﬁlter [104], or based on a ﬁnite

basis function expansion of the time-varying coeﬃcients

[105–108]. The identiﬁcation of a nonlinear time-varying

system is more sophisticated. The primary diﬃculty is

how to eﬀectively and simultaneously resolve the nonlin-

ear model structure selection and the time-varying co-

eﬃcient estimation. Approaches based on time-varying

Volterra series combining artiﬁcial neural networks [109],

principal dynamic modes [110], or sliding-window strat-

egy [111], have been proposed. However, the model struc-

ture selection is still an unsolved issue, and the identiﬁca-

tion and frequency-domain analysis are computationally

costly.

A better strategy is to extend the basis function ex-

pansion approach, originally proposed for LTV identiﬁ-

cation, to nonlinear time-varying cases [112]. The time-

varying (TV) parameters in TV-NARX models are ﬁrst

expanded using multi-wavelet basis functions, and TV

nonlinear model is transformed into an expanded time-

invariant model; the challenging TV model selection and

parameter estimation problem can then be solved using

the computational eﬃcient FROLS algorithm. To accom-

modate the stochastic perturbations or additive coloured

noise, this procedure can also be extended to more gen-

eral TV-NARMAX models by introducing a modiﬁed

extended least squares (ELS) algorithm [113]. Several

modiﬁcations to the TV-NARX model has recently been

proposed using diﬀerent basis functions or model selec-

tion procedure [114, 115]. The corresponding frequency-

domain analysis for nonlinear time-varying systems has

also been developed based on the identiﬁed time-domain

TV-NARX or TV-NARMAX model and the TV-GFRF

concepts [113, 116]. By ﬁtting TV-NARX models to two

fragments of intracranial EEG recordings measured from

epileptic patients, the corresponding frequency-domain

analysis (i.e. TV-GFRF and TV-OFRF) shows the non-

linear energy transfer eﬀect – the underlying neural sys-

tem transfers the energy from lower frequencies to higher

frequencies as the seizure spreading from the left to the

right brain regions over time [116, 117].

An overview of the NARMAX model-based system

identiﬁcation framework, including both time-invariant

(NARMAX) and time-varying (TV-NARX) modelling

along with corresponding frequency-domain analysis to

neurophysiological signal analysis, is summarised in Fig-

ure 3.

V. NONLINEAR NEURAL CONNECTIVITY

ANALYSIS

The communication between diﬀerent neuronal popu-

lations which have very diﬀerent ﬁring behaviours can

result in nonlinear neural connectivity, showing neural

coupling across two or more diﬀerent frequencies. To

quantitatively study such a ‘cross-frequency coupling’,

this section reviews recent advances in nonlinear neural

functional and eﬀective connectivity analysis.

A. High-order spectrum and nonlinear coherence

The power spectra and coherence discussed in Sec-

tion III A are Fourier transforms of the auto- and

cross-correlations of signals, hence they are only linear

frequency-domain measures. Practically these measures

8

60 65 70 75

Time (s)

-20

-10

0

10

20

30

Freq (Hz)

-40

-30

-20

-10

0

10

20

60 65 70 75

Time (s)

-30

-20

-10

0

10

20

30

Freq (Hz)

-60

-50

-40

-30

-20

-10

0

H1(f, t)

H1(f)

H2(f1, f2)

H2(f, t),

f=f1+f2

X(f)

NARMAX

input spectrum

TV-NARX

TV-OFRF

time-frequency spectrum

TV-AR based input

input signal: X

output signal: Y

58 60 62 64 66 68 70 72 74 76

Time (s)

-200

-150

-100

-50

0

50

100

V

58 60 62 64 66 68 70 72 74 76

Time (s)

-400

-200

0

200

400

V

0 10 20 30 40 50 60

Freq (Hz)

-30

-20

-10

0

10

Relative PSD (dB)

0 10 20 30 40 50 60

Freq (Hz)

-20

-15

-10

-5

0

5

10

15

Relative PSD (dB)

GFRFs

TV-GFRFs

OFRF

Y(f) = Y1(f) + Y2(f)

FIG. 3: Analysing neurophysiological signals uses nonlinear time-invariant and time-varying system identiﬁcation and

corresponding frequency-domain analysis methods. The upper part of the diagram illustrates the nonlinear time-invariant

modelling: ﬁrst a NARMAX model is identiﬁed from the input and output neurophysiological signals (e.g. EEG, EMG,

MEG, LFP); this time-domain model is then mapped to the frequency-domain with GFRFs (i.e. H1(f), H2(f), ...), and the

OFRF (Y(f) = Y1(f) + Y2(f), ...) can be computed from the input spectrum and GFRFs. The lower part of the diagram

shows the nonlinear time-varying system identiﬁcation using a TV-NARX model, and the identiﬁed time-varying model can

then be mapped to the time-frequency domain with (averaged) TV-GFRFs (i.e. H1(f , t), H2(f, t), ...). The TV-OFRF can

therefore be computed from a combination of the input time-frequency spectrum and the TV-GFRFs.

cannot detect certain nonlinear eﬀects such as quadratic

moments in or between signals that have a zero mean

[59]. Higher-order spectral analysis has been developed

to detect nonlinear coupling between spectral compo-

nents [118]. For example, the most widely used bispectral

analysis is the frequency-domain mapping of the third-

order statistics. It can be used to quantify the quadratic

nonlinear interactions, i.e. quadratic phase coupling.

The bispectrum or bicoherence (and the bivariate cross-

bispectrum or cross-bicoherence) analysis is well-known

in engineering signal processing, whereas it has only rel-

atively recently been applied to study the nonlinear in-

teractions in neurophysiological signals [119–121]. For

example, bispectral measures were used to detect long-

range nonlinear coupling and synchronization in healthy

subjects from human EEG [120, 122], to characterise and

predict epileptic seizures [123], and to study the nonlin-

ear interactions between diﬀerent frequency components

related to Parkinson’s disease and tremor [9, 124, 125].

However, bispectrum or bicoherence cannot detect

nonlinearity beyond second order, such as the higher-

order harmonics and intermodulation eﬀects, or the sub-

harmonic coupling. A generalised nonlinear coherence

analysis framework has therefore been proposed, based

on two diﬀerent nonlinear mappings from the input to

the output of an ‘open-loop’ system in the frequency do-

main [6]:

1) n:m Mapping: to measure harmonic or subhar-

monic coupling related to individual input frequencies.

Ym(fY) = H(n:m)Xn(fX), Xn≡XX ·· · X

| {z }

n

, where

the output frequencies (fY) are related to the input fre-

quencies (fX) by the ratio n/m (nand mare co-prime

positive integers), and H(n:m) is the n:m mapping

function. The n:m mapping can generate cross-frequency

(e.g. harmonic m= 1 or subharmonic m > 1) coupling

between the input and the output.

2) Integer Multiplication Mapping: to quantify inter-

9

modulation coupling among multiple (≥2) input fre-

quencies.

Y(fY) = H(f1, f2,·· · , fN;a1, a2,· · · , aN;fY)M

N

Y

i=1

Xai(fi)

(11)

where fY=a1f1+a2f2+·· · +aNfN. The

Mis the corresponding multinomial coeﬃcient, and

H(f1, f2,·· · , fN;a1, a2,· · · , aN;fY) indicates amplitude

scaling and phase shift from the input to the output.

According to these two diﬀerent types of nonlinear

mapping, Yang and colleagues proposed two basic met-

rics for quantifying nonlinear coherence: (i) n:m coher-

ence and (ii) multi-spectral coherence [6].

1. n:m coherence

The n:m coherence is a generalized coherence mea-

sure for quantifying nonlinear coherence between two fre-

quency components of the input X(f) and the output

Y(f) [6]:

Cnm(fX, fY) = |SXY (fX, fY)|

pSn

XX (fX)Sm

Y Y (fY)(12)

where n:m=fY:fX.SXY (fX, fY) =<

Xn(fX)(Ym(fY))∗>is the n:m cross-spectrum,

with <·>represents the averaging over repe-

titions. Sn

XX (fX) =< X n(fX)(Xn(fX))∗>=<

|X(fX)X(fX)·· · X(fX)

| {z }

n

|2>is the nth-order auto-

spectra. According to Cauchy-Schwarz-inequality, we

have:

|< Xn(fX)(Ym(fY))∗>|

≤<|Xn(fX)|2>1/2·<|Yn(fY)|2>1/2(13)

Thus, n:m coherence is bounded by 0 and 1, where one

indicates that two signals are perfectly coupled for the

given frequency pair (fX,fY).

A simpliﬁed version of n:m coherence that considers

only the phase relation between the input and the out-

put is known as n:m phase synchronization index [126].

The n:m coherence and n:m phase synchronization index

has been widely applied to neuroscience research to inves-

tigate nonlinear functional connectivity in diﬀerent brain

regions [127, 128], as well as the nonlinear connectivity

between the brain and muscles [36].

2. Multi-spectral coherence

Multi-spectral coherence measures the multi-frequency

nonlinear coupling generated by the integer multiplica-

tion mapping [6]. It is deﬁned as:

CMS (f1,· · · , fN;a1,· · · , aN;fY)

=|SXY (f1,· · · , fN;a1,··· , aN;fY)|

q(Qn

i=1 S|ai|

XX (fi))SY Y (fY)

(14)

where fY=a1f1+·· · +aNfN;

SXY (f1,· · · , fN;a1,··· , aN;fY) is the high-order

cross-spectrum between Xand Y, and equal to

<Qn

i=1 X|ai|(fi))Y∗(fY)>. Here ‘*’ indicates the

complex conjugate. When there are only two input

frequencies, the multi-spectral coherence is degraded

to the bicoherence [129]. The multi-spectral coherence

or bicoherence has been applied to study the nonlinear

behaviours in visual [130], auditory [131] and somatosen-

sory systems [6], which are thought to be associated

with neural coding and functional integration of various

sensory inputs [132].

A simpliﬁed version of multi-spectral coherence that

considers only the phase relation between the input and

the output is known as multi-spectral phase coherence

[133]. Similarly, there is a degraded measure, named bi-

phase locking value [134], for the case involving only two

input frequencies. The advantage of multi-spectral phase

coherence or bi-phase locking value is that it allows pre-

cise estimation of time delay in the nervous system based

on the relative phase relationship between the input and

output [133, 135]. The multi-spectral phase coherence or

bi-phase locking value has been previously used to de-

termine neural transmission delays in the human visual

system [130] and the stretch reﬂex loops [36].

B. Nonlinear causality analysis: system

identiﬁcation based approaches

1. Time-domain analysis

In terms of eﬀective connectivity, classical linear

Granger causality analysis (GCA) (as discussed in Sec-

tion III B) may provide misleading results when used to

analyse EEG/MEG or EMG signals, as the possible non-

linear interactions within a neural system are not mod-

elled explicitly by simply using linear regression models.

The Granger causality deﬁnition has been extended to

nonlinear cases in the time domain, based on nonpara-

metric methods [136, 137], radial basis functions [138],

kernel methods [139], local linear models [140]. DCM

[60] (see Section IV A 3) was developed to accommodate

both linear and nonlinear causal eﬀects using a dynamic

state-space model, and the eﬀective connectivity among

hidden states (unobserved neuronal dynamics) can be

identiﬁed via Bayesian inference. Information-theoretical

eﬀective connectivity measures have also been proposed,

such as the bivariate transfer entropy (TE) [141, 142] and

phase transfer entropy (PTE). TE is a model-free mea-

sure, which compares two conditional probabilities using

10

the Kullback-Leibler divergence - the amount of uncer-

tainty in the future of target signal Yconditioned only

on the target’s past and the future of Yconditioned on

the past of both its own Yand the source X, in a concep-

tually similar way as the GCA. A more recent work [143]

generalised the TE method by using multivariate trans-

fer entropy, which can overcome the problems of inferring

spurious or redundant causality and missing synergistic

interactions between multiple sources and target.

Another strategy to implement nonlinear granger

causality under a system identiﬁcation framework is to

use NARX models [144, 145], by calculating the relative

predictability improvement obtained from the NARX

model compared to the corresponding NAR model. More

importantly, compared to other nonlinear causality mea-

sures (e.g. nonparametric or information-theoretic mea-

sures), the advantage of using NARX-based causal infer-

ence [145] is that one can easily separate the linear and

nonlinear causal inﬂuence, for example from an input X

to an output Y. After ﬁtting a polynomial NARX model

with the form (9), the linear causality can still be cal-

culated from (4) based on the linear part of the NARX

model, while the nonlinear causal inﬂuence from Xto Y

can be deﬁned as:

Fn

X→Y= ln var(Y|Y−

l, Y −

n, X−

l)

var(Y|Y−

l, Y −

n, X−

l, X−

n,(XY )−

n)

or ln var(Y|X−

l)

var(X−

l, X−

n,(XY )−

n)(15)

Here Y−

n, X−

nand (XY )−

ndenote the sets of all nonlinear

delayed terms of Y,Xand nonlinear product terms XY .

This nonlinear causality measure can also be generalised

to nonlinear time-varying systems, by computing similar

linear and nonlinear causality indices based on the iden-

tiﬁed TV-NARX models (as described in Section IV C),

as proposed in [145, 146].

2. Frequency-domain analysis

In the frequency domain, linear Granger causality mea-

sures, such as PDC, DTF and spectral Granger causality,

can all be expressed as a function of the elements in the

coeﬃcient matrix or its inverse the transfer function ma-

trix of the corresponding linear ARX models (3). By

identifying the link between the PDC and the FRFs of

the corresponding linear ARX models (as described in

the Section III B), a new nonlinear PDC (NPDC) mea-

sure has been proposed [64] by generalising the spectrum

decomposition with respect to a nonlinear NARX model

in a similar way as the linear case. The NPDC from X

to Ycan then be expressed as a direct generalization of

the linear PDC:

N P DCX→Y(f) = −HX→Y(f)

q|He

y(f)/He

x(f)|2+|HX→Y(f)|2

(16)

Here, the HX→Y(f) is the ‘nonlinear FRF’ which re-

places the FRF in the linear PDC (7), and He

y(f) and

He

x(f) are the error-driven GFRFs correspond to the re-

stricted NAR models with respect to Yand Xas dis-

cussed in [64]. The NPDC measures both linear and non-

linear causal inﬂuences from Xto Y. The linear causal

eﬀects can be interpreted as a special case of (16) by

only considering the 1st-order nonlinear FRFs of NARX

(i.e. H1,X→Y(f)) and NAR (i.e. He

1,x(f) and He

1,y(f))

models.

This new NPDC measure has recently been applied to

predict epileptic seizures from EEG data [147] by advanc-

ing the construction of functional brain networks, nonlin-

ear feature selection and classiﬁcation. This new nonlin-

ear causality measure helps to provide better prediction

accuracy compared to other standard graph theory or

nonlinear classiﬁcation based methods. A nonlinear gen-

eralization of Geweke’s spectral Granger causality has

also been proposed [117] using the NARX methodology.

VI. NEUROLOGICAL AND CLINICAL

APPLICATIONS

Movement, sensation and cognition arise from the cu-

mulative activity of neurons within neural circuits and

across distant, macroscale networks in the nervous sys-

tem. Although the behaviour of an individual neuron

has been investigated and well understood for decades,

the mechanisms underlying neural communications be-

tween macroscale neural networks are still yet to better

understand. Newly developed nonlinear system identiﬁ-

cation approaches allow us to investigate neural commu-

nications from large-scale neural activities measured by

EEG, MEG and EMG, with the most recent application

examples discussed below.

A. Nonlinear cortical response to somatosensory

inputs

The human somatosensory system is highly nonlin-

ear [11]. Previous studies applied periodic sinusoid tactile

stimulations to the index ﬁnger and measured the cortical

response, where they found harmonic and subharmonic

patterns in the response [5, 148]. Several recent studies

used sum-of-sinusoid stimulations to the wrist joint and

found not only harmonics and subharmonics but also in-

termodulation patterns [6, 29]. The majority of inter-

modulation responses presented the second-order nonlin-

earity, which is the sum or the diﬀerence between input

frequencies [8]. These ﬁndings indicate that the nonlin-

earities in the somatosensory system allow the functional

integration of input signals at diﬀerent frequencies, and

they can be transmitted in diﬀerent somatosensory as-

cending pathways.

Yang and colleagues recently built a hierarchical neural

network based on known neuroanatomical connections

11

and corresponding transmission delays in neural path-

ways to model the cortical response to somatosensory

input [67]. The proposed computational model contains

a neural layer at the thalamus that integrates the inputs

from diﬀerent ascending pathways, including Ia and II af-

ferents. The computational model well captured the ma-

jority of the cortical response to the given somatosensory

inputs, indicating the functional integration of diﬀerent

somatosensory input signals may occur at the thalamus

and is then transmitted to the cortex via the thalamo-

cortical radiation.

B. Tremor: nonlinearity in the thalamocortical

loop

Essential tremor is a common neurological movement

disorder widely considered to have a centrally-driven ori-

gin. There is both neurophysiological and clinical evi-

dence of thalamic involvement in the central oscillatory

network generating essential tremor [149–151]. Local

ﬁeld potential (LFP) recordings of thalamic ventralis in-

termedius (Vim) nucleus show a strong linear correlation

with the contralateral EMG during tremor [150]. Some

studies using EEG and MEG suggest that the sensorimo-

tor cortex is also part of the central tremor-related oscil-

latory network, with signiﬁcant coupling in some cases

with the contralateral tremorgenic EMG [152–154]. De-

spite a well-established reciprocal anatomical connection

between the thalamus and cortex, the functional associ-

ation between the two structures during ‘tremor-on’ pe-

riods had not been extensively deﬁned.

He and co-authors [9] investigated the functional con-

nectivity among cortical EEG, thalamic (Vim) LFPs and

contralateral EMG signals over both ‘tremor-on’ and

‘tremor-oﬀ’ states, using linear coherence and nonlin-

ear bispectral analysis methods. In addition to expected

strong coherence between EMG and thalamic LFP, non-

linear interactions (i.e. quadratic phase coupling) at dif-

ferent frequencies, i.e. low frequency during tremor on

and higher frequency during tremor oﬀ, in LFPs have

been reported. More importantly, by using the NARX-

based nonlinear system identiﬁcation and frequency-

domain analysis (as described in Section ‘IV B’), two dis-

tinct and non-overlapping frequency ‘channels’ of com-

munication between thalamic Vim and the ipsilateral

motor cortex were identiﬁed, which robustly deﬁned the

‘tremor-on’ versus ‘tremor-oﬀ’ states. Longer corticotha-

lamic nonlinear phase lags in the tremor active state were

also uncovered, suggesting the possible involvement of an

indirect multi-synaptic loop. This work demonstrates,

for the ﬁrst time, the importance of cross-frequency non-

linear interactions between the cortex and the thalamus

in characterising the essential tremor.

C. Nonlinear analysis for determining motor

impairment in stroke

After a stroke, damage to the brain increases the re-

liance on indirect motor pathways resulting in motor im-

pairments and changes in neural connectivity between

the brain and muscles. A hallmark of impairments post-

stroke is a loss of independent joint control that leads

to abnormal co-activation between shoulder, arm and

hand muscles, known as the upper limb synergy [155].

The upper limb synergy is thought to be caused by pro-

gressive recruitment of indirect motor pathways via the

brainstem following a stroke-induced loss of corticospinal

projections [156]. Thus, a neural connectivity measure

that quantiﬁes the recruitment of these indirect motor

pathways would be crucial to evaluate post-stroke motor

impairments. Recent model-based simulation and clini-

cal studies indicate that the increased usage of indirect

motor pathways enhances nonlinear distortion of motor

command transmissions, which leads to stronger nonlin-

ear interaction between the brain and muscles [38, 40].

The ratio of nonlinear interaction over linear interaction,

known as the nonlinear-over-linear index (N-L Index),

has been reported to be associated with the relative ratio

of the recruitment of indirect versus direct motor path-

ways [40]. This new measure may facilitate the future

determination of the eﬀect of new therapeutic interven-

tions that aim to optimise the usage of motor pathways,

and thus facilitate the stroke recovery.

D. Epilepsy

It has been widely recognised that epileptic seizures

are highly nonlinear phenomena, due to low dimen-

sional chaos during epileptic seizure or transitions be-

tween ordered and disordered stats [157]. Currently, the

treatment mainly relies on long-term medication with

antiepileptic drugs or neurosurgery, which can cause cog-

nitive or other neurological deﬁcits. New treatment

strategies such as on-demand therapies during the pre-

seizure (preictal) state or electrical stimulation are there-

fore needed. A vital part of this new on-demand strategy

is the accurate and timely detection of the preictal state,

even seconds before seizure onset [158]. A range of uni-

variate, bivariate and multivariate linear and nonlinear

measures have been developed for the characterisation

and detection or prediction of epileptic brain states and

achieving a better understanding of the spatial and tem-

poral dynamics of the epileptic process [158, 159]. There

is a comprehensive review of using diﬀerent parametric

and nonparametric nonlinear features (in time, frequency

and time-frequency domains) for the automated epilepsy

stage detection and classiﬁcation [160].

Given the current challenges in epilepsy detection and

diagnostics [158, 161], e.g. to improve the understanding

of brain dynamics and mechanisms during the seizure-

free interval and seizure initiation and termination,

12

there is a great need to develop more accurate nonlin-

ear methods to improve the detectability of directional

interactions in underlying functional and anatomical net-

works. Developing new nonlinear system identiﬁcation

and nonlinear causality measures are therefore crucial.

A nonlinear causality measure, partial transfer entropy

[162], has been applied to analyse the EEG of epileptic

patients during preictal, ictal and postictal states. It can

provide better detection of causality strength variations

compared to linear PDC. An adaptive nonlinear Granger

causality measure was also proposed [163] and applied to

LFP data (intracranial EEG in cortex and thalamus) in

rats. It was reported to provide more sensitive detection

of changes in the dynamics of network properties com-

pared to linear Granger causality. The recently proposed

nonlinear frequency-domain causality measure NPDC

[64] (as reviewed in Section ‘V B 2’) has been applied to

analyse EEG recordings of two bipolar channels from a

patient with childhood absence epilepsy. It shows this

nonlinear measure can detect extra frequency-domain

causal interactions compared to standard linear PDC.

VII. CONCLUSIONS AND PERSPECTIVES

The complexity and nonlinearity of neural systems re-

quire advanced system identiﬁcation techniques to un-

derstand their properties and mechanisms better. This

review investigated the links between connectivity anal-

ysis and system identiﬁcation, as well as recent progress

of nonlinear system identiﬁcation of neural systems.

With the state-of-the-art examples of clinical applica-

tions, we argued that nonlinear dynamic modelling and

corresponding connectivity analysis allows new insights

into the underlying neural functions and neuropatho-

logical mechanisms of the abnormality caused by vari-

ous neurological disorders. These novel approaches may

well facilitate the development of new precision diagnos-

tic tools and brain-computer interface (BCI) techniques

[164–166], and therefore improve the diagnosis and treat-

ment of neurological disorders as well as restore commu-

nication and movement for people with motor disabilities.

Compared to the linear system identiﬁcation and

iso-frequency connectivity analysis, nonlinear dynamic

modelling and cross-frequency analysis are much more

complicated. Such complexity brings challenges but

also research opportunities. Potential future work in-

cludes: i) further developing multivariate system identi-

ﬁcation techniques and corresponding multivariate non-

linear frequency-domain analysis and causality analysis

measures. Most existing nonlinear system identiﬁcation

based (time or frequency-domain) analysis or causality

analysis are primarily bivariate, which limits nonlinear

analysis to the only pre-deﬁned local brain or neural re-

gions. New multivariate system identiﬁcation (e.g. mul-

tivariate nonlinear regression modelling) or inference ap-

proaches would generalise the existing nonlinear connec-

tivity analysis to larger neural networks, although devel-

oping eﬃcient model selection and reducing the compu-

tational cost would be challenging tasks; ii) many neu-

ronal systems or interactions are in nature nonstationary

and nonlinear, how to automatically distinguish the non-

linearity and time-varying eﬀects (nonstationarity) via

novel system identiﬁcation technique is still an open and

important research topic, although signiﬁcant progress

has been made so far (as reviewed in Section IV C); iii)

machine learning and deep learning techniques have re-

cently been applied to Granger causality analysis [167–

169], an interesting future work is to further explore and

combine the advantages of deep learning, e.g. accurate

quantiﬁcation of complex and long-range nonlinear inter-

actions, and nonlinear system identiﬁcation approaches

to study the nonlinear causal interactions in complex

neural networks; iv) using nonlinear system identiﬁcation

techniques to extract new nonlinear features for the BCI;

v) apart from those clinical applications described in Sec-

tion VI, the importance of nonlinearity in understand-

ing and characterising other important neurological dis-

orders, e.g. Parkinson’s disease and Alzheimer’s disease

[170–173], has been reported recently. Therefore nonlin-

ear system identiﬁcation approaches will have great po-

tential in developing new diagnostic tools for those pri-

mary neurological disorders that aﬀect a large population

worldwide.

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