Nonlinear System Identification of Neural Systems from Neurophysiological Signals

Abstract

The human nervous system is one of the most complicated systems in nature. The complex nonlinear behaviours have been shown from the single neuron level to the system level. For decades, linear 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 systems. 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 on 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 quantitatively. These approaches can hopefully provide new insights to advance our understanding of neurophysiological mechanisms underlying neural functions. They 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.

Full text

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
Funconal Connecvity
Linear: correlaon, coherence
Nonlinear: nonlinear correlaon, mutual informaon
higher-order spectrum & coherence
Effecve Connecvity
Modeling & System Idenficaon Causality
i) Linear:
Time: AR/ARX
Frequency: transfer funcon 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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