U.S. patent application number 14/419072 was filed with the patent office on 2015-07-23 for method and computing system for modelling a primate brain.
The applicant listed for this patent is Baycrest Centre for Geriatric Care, Max-Planck-Gesellschaft Zur Foderung Der Wissenschaften e.V.. Invention is credited to Viktor Jirsa, Anthony Randal Mclntosh, Jochen Mersmann, Petra Ritter.
Application Number | 20150206051 14/419072 |
Document ID | / |
Family ID | 49681062 |
Filed Date | 2015-07-23 |
United States Patent
Application |
20150206051 |
Kind Code |
A1 |
Mclntosh; Anthony Randal ;
et al. |
July 23, 2015 |
METHOD AND COMPUTING SYSTEM FOR MODELLING A PRIMATE BRAIN
Abstract
In one aspect the application relates to a computing system for
providing data for modelling a human brain comprises a database
including a plurality of datasets (or allow access to a plurality
of datasets), each dataset including at least a dynamical model of
the brain including at least one node and a neurodataset of a
neuroimaging modality input. The at least one node include a
representation of a local dynamic model and a parameter set of the
local dynamic model.
Inventors: |
Mclntosh; Anthony Randal;
(Toronto, CA) ; Mersmann; Jochen; (Stuttgart,
DE) ; Jirsa; Viktor; (Aubagne, FR) ; Ritter;
Petra; (Berlin, DE) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Max-Planck-Gesellschaft Zur Foderung Der Wissenschaften e.V.
Baycrest Centre for Geriatric Care |
Munchen
Toronto |
|
DE
CA |
|
|
Family ID: |
49681062 |
Appl. No.: |
14/419072 |
Filed: |
August 2, 2013 |
PCT Filed: |
August 2, 2013 |
PCT NO: |
PCT/IB2013/001707 |
371 Date: |
February 2, 2015 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
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61678950 |
Aug 2, 2012 |
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Current U.S.
Class: |
706/15 |
Current CPC
Class: |
G06N 3/049 20130101;
G06N 3/10 20130101; G06N 3/04 20130101 |
International
Class: |
G06N 3/10 20060101
G06N003/10; G06N 3/04 20060101 G06N003/04 |
Claims
1. A computing system for standardized modelling of a primate
brain, the computing system comprising: a plurality of dynamical
models of the brain; an interface for including at least one
neuroimaging modality input; and, a decision engine for comparing
results of the dynamical model to the at least one neuroimaging
modality input.
2. The system of claim 1 wherein the dynamical model comprises a
plurality of nodes, wherein at least a first subset of the
plurality of nodes is connected to a second subset of the plurality
of nodes based on an anatomical founded connectivity structure and
wherein at least one of the plurality of nodes includes a
representation of a local dynamic model and a parameter set of the
local dynamic model.
3. The system of claim 1 wherein the dynamical model is configured
to incorporate an anatomically founded connectivity structure and
information from the neuroimaging modality input.
4. The system of claim 1 wherein the decision engine comprises a
forward transformation model for mapping activity of the plurality
of nodes into a space corresponding to the neuroimaging modality
input.
5. The system of claim 1 wherein the decision engine comprises an
entity for fitting at least one parameter set of the at least one
of the plurality of nodes by comparing an output of the dynamical
model of the brain and the imaging modality input in a space
corresponding to the neuroimaging modality input.
6. The system of claim 2 wherein the decision engine comprises an
inverse model for transformation of the imaging modality input into
a source space corresponding to the plurality of nodes.
7. The system of claim 5 wherein the entity for fitting the at
least one parameter set is configured to simulate an output of at
least a first mapping subset of the plurality of nodes by a mapped
neuroimaging modality input, which is transformed into source
space, and to fit the parameter set of at least one of the
plurality of nodes, the at least one node not belonging to the
first mapping subset.
8. The system of claim 5, wherein the entity for fitting the at
least one parameter set of the at least one of the plurality of
nodes is configured to fit the at least one parameter set based on
at least one time segment of the neuroimaging modality input.
9. The system of claim 1 including neuroimaging modality input from
at least two different neuroimaging methods.
10. A computing system for modelling a human brain, the computing
system comprising: a database including a plurality of datasets,
each dataset including: at least a dynamical model of the brain
including at least one node including a representation of a local
dynamic model and a parameter set of the local dynamic model; a set
of values for the parameter set of the local dynamic model; and a
neurodataset of a neuroimaging modality input, wherein the set of
values represents a fit between the dataset and the parameter set
in a predetermined space.
11. The computing system of claim 10 wherein the computing system
further comprises a search entity configured to receive input of a
property of a neurodataset as a search request and to forward an
output of the plurality of datasets, the output including one or
more parameter sets and the corresponding representation of a local
dynamic model, the neurodataset corresponding to the one or more
parameter sets matching the search request.
12. The computing system of claim 11 wherein the search request
includes signal properties of the neurodatasets and the search
entity is configured to analyze the neurodatasets of the database
using signal processing methods to match neurodatasets to the
signal properties of the search request.
13. The computing system of any of claim 10 wherein the
neurodatasets include at least one time snippet or epoch of
experimental data.
14. The computing system of any of claim 10 wherein a subunit of
the dynamical model includes a plurality of nodes, the plurality of
nodes connected to each other based on an anatomically founded
connectivity structure and each node includes a representation of a
local dynamic model and a parameter set of the local dynamic
model.
15. The computing system of claim 14 wherein at least one dataset
of the database further includes information of at least one of the
plurality of nodes, the parameter set of which is fit to the
neurodataset.
16. The computing system of any of claim 10 wherein the
predetermined space includes at least one of a space corresponding
to the neuroimaging modality input and a source space corresponding
to the at least one node.
17. A computing system for distributing and receiving data on
primate brain modelling, the computing system comprising: a
back-end system configured to connect to a web interface and
configured for accessing neuroimaging modality input data, the
back-end system including a simulation component with a simulation
core and a simulation controller, wherein the simulation core is
configured to receive a set of parameters and configured to
retrieve neuroimaging modality input data; and the simulation
controller is configured to control a work-flow of the simulation
core; and a storage system configured for storing: at least one
neurodataset of neuroimaging modality input data; a plurality of
dynamical models, each dynamical model including a plurality of
nodes; anatomical connectivity data; a plurality of local dynamic
models representing a node; at least one forward transformation
model for mapping an activity of the plurality of nodes into a
space corresponding to the neuroimaging modality input; at least
one inverse transformation model for mapping the imaging modality
input into a source space corresponding to the plurality of nodes;
and a decision engine including at least one entity for fitting at
least one node of the plurality of nodes to the at least one
neurodataset.
18. A computing system for standardized modelling of a primate
brain by a dynamical model, which comprises a plurality of nodes,
wherein at least a first subset of the plurality of nodes is
connected to a second subset of the plurality of nodes based on an
anatomical founded connectivity structure and/or functional
connectivity data and wherein at least one of the plurality of
nodes includes a representation of a local dynamic model and a
parameter set of the local dynamic model.
Description
PRIORITY CLAIM
[0001] This application claims priority to US provisional
application U.S. 61/678,950, which is fully incorporated herein by
reference.
FIELD OF THE INVENTION
[0002] This invention relates to a method and apparatus for
simulating brain function. In particular, this invention relates to
a system that simulates brain function by integrating brain models
with measured neuroimaging data.
BACKGROUND OF THE INVENTION
[0003] Brain function is thought to emerge from the interactions of
large numbers of neurons. One aspect of difficulty in the research
of brain function is the inability to directly observe brain
dynamics. It would be useful to provide an abstract mathematical
model of modelled brain dynamics that exhibit a lower complexity
than actual brain dynamics. Such a model may be limited to only a
few selected aspects of brain dynamics that are of interest.
Further model complexity or detail may be added to the model to
alter a different range of modelled brain dynamics. Clinical
assessment and research may accordingly be conducted on the model
to both investigate conditions and test theories.
[0004] Besides insights into mechanisms of how brain dynamics
emerge, such models allow for the exploration of the consequences
of pathological changes in the system, enabling identification of
pathological changes and subsequent treatment.
[0005] In an aspect, there is a need for a method and apparatus
that assists in modelling brain dynamics.
[0006] In an aspect, there is a need for a method and
apparatus--that enables model-based inference of neurophysiological
mechanisms on different brain scales that underlie the generation
of macroscopic neuroimaging signals.
SUMMARY
[0007] The application discloses a computing system for modelling a
human, or more generally a primate brain comprising a dynamical
model of the brain, at least one neuroimaging modality input and a
decision engine for comparing results of the dynamical model to the
at least one neuroimaging modality input. In some examples, the
neuroimaging modality input refers to single or multimodal
neuroimaging data, i.e. data from one or more different types of
neuroimaging data, such as electroencephalography (EEG),
magnetoencephalography (MEG), magnetic resonance imaging (MRI) or
positron emission tomography (PET). The system or its components
may be configured to be implemented as and/or processed by software
running on a single computer, a computer cluster or via
cloud-computing. Cloud computing is a synonym for distributed
computing over a network and includes the ability to run a program
on many connected computers at the same time. This includes the
provisioning of application services that run client-server
software on a remote location. Furthermore, the system or its
components may be configured to be implemented as functional
modules of a software or modular software, or a computer based
database. Throughout this application a computer will be understood
to include an entity including a real or virtual central processing
unit to receive, compute and output commands from a software or a
functional module as described above.
[0008] In a further aspect, a computing system for providing data
for modelling a human brain comprises a database including a
plurality of datasets (or allow access to a plurality of datasets),
each dataset including at least a dynamical model of the brain
including at least one node and a neurodataset of a neuroimaging
modality input. The at least one node include a representation of a
local dynamic model and a parameter set of the local dynamic model.
Furthermore, each of the datasets includes a set of values for the
parameter set of the local dynamic model, which were generated by
fitting at least parts of the dynamical model to the neurodataset
in a predetermined space. In some examples, the predetermined space
may be a space corresponding to the neuroimaging modality input or
a source space corresponding to the dynamical model of the brain
including the at least one node.
[0009] In a further aspect, a computing system for distributing and
receiving on human brain modelling includes a back-end system
configured for accessing neuroimaging modality input data. The
back-end system includes a simulation component with at least a
simulation core and a simulation controller. The back-end system
may be implemented as a server running on a computer or computer
cluster or in a cloud-computing environment. The simulation core is
a functional module, such as a software module of the back-end
system configured to receive at least a set of parameters from an
input, such as a human user input into a computer or an uploaded
file including data as input. Furthermore, the simulation core is
configured to retrieve neuroimaging modality input from an input,
such as an uploaded file including the neuroimaging data or a
storage system coupled to the stimulation component. A workflow of
the simulation core is controlled by the stimulation controller. In
some examples, it is configured to analyse and optionally correct
the input, to schedule resources for the stimulation core. In some
examples, it is configured to distribute tasks to be performed by
the stimulation core to different cluster nodes of a computer
cluster or different computers over a network. A storage system may
be configured to store data of at least one neurodataset of
neuroimaging modality input. One neurodataset is a dataset of
neuroimaging modality input data. The different wording is chosen
to distinguish it from dataset of the above-mentioned database.
Furthermore, the storage system may include a plurality of
dynamical models, each dynamical model including a plurality of
nodes, anatomical connectivity data, which may be derived from
individuals by methods such as diffusion tensor imaging (DTI) or by
brain cartography data such as data from the Montreal Neurological
Institute (MNI). Furthermore, the storage system may store at least
one forward and or inverse transformation model to map the output
of the dynamical model to a space corresponding to the neurodataset
or to map the neurodataset to a source space corresponding to the
plurality of nodes and their output. Additionally, the storage
system may provide a decision engine including an entity to fit at
least one of the plurality of nodes to the at least one
neurodataset.
[0010] Further aspects of the computing system will become apparent
by the detailed description.
DETAILED DESCRIPTION OF THE INVENTION
[0011] This application incorporates by reference Ritter et. al.,
"The Virtual Brain Integrates Computational Modelling and
Multimodal Neuroimaging" in Brain Connect. 2013; 3(2): pages 121-45
and Sanz Leon et. al., "The Virtual Brain: a simulator of primate
brain network dynamics." in Front Neuroinform. 2013 Jun. 11; pages
7:10. Furthermore, this application incorporates by reference the
source code of the virtual brain app, obtainable from
"http://www.thevirtualbrain.org/register/". The current
implementation of the virtual brain app represents a working mode
of aspects of the invention implementing some of the features
described hereafter.
[0012] The problem is to create a standardized simulation framework
for a full primate brain model that allows a) comparison of
different modelling strategies/model specifications b) simulating
brain activity of individual primates.
[0013] WHY A COMUPTATIONAL MODEL? Brain function--i.e. cognition
and behaviour--are generated by complex neuronal interactions.
Brain imaging methods allow us to measure brain dynamics with good
spatial and temporal resolution. In order to understand and
replicate the underlying generative mechanisms we need a
computational model of the interacting neurons.
[0014] WHY A FULL BRAIN MODEL? An individual function does not
emerge from a localized neuronal population--rather it is the
result of orchestrated neuronal activities involving the entire
brain. Hence in order to understand and replicate brain function
the entire brain web needs to be considered. We need a full brain
computational model.
[0015] WHY A STANDARDIZED SIMULATION FRAMEWORK? A model reduces
complexity. It simulates brain dynamics that exhibit a lower
complexity than the actual brain dynamics. Such a model may be
limited to only a few selected aspects of brain dynamics that are
of interest. Different models are capturing different aspects of
brain activity. Further model complexity or detail may be added to
a model to alter a different range of modelled brain dynamics. The
ultimate goal is to build an increasingly general model that
accounts for many aspects of brain dynamics. The creation of a
comprehensive model requires an iterative process where models are
tested, modified and merged. Such a systematic model creation is
only feasible in a standardized framework where the model outcomes
and model specifications are well tractable. A standardized
framework does not exist for full brain modelling and scientific
community struggles with the fact that models are not reproducible
and hence can't be utilized in an iterative model generalization
process.
[0016] WHY INDIVIDUAL VIRTUAL BRAINS? Each brain is different. In
order to be able to use the brain model for individual predictions
for example in a clinical setting, we need to be able to account
for inter-individual differences.
[0017] WHY IMAGING RESOLUTION? A full brain model that simulates
activity at the same temporal and spatial resolution as brain
imaging methods on the one hand can be validated against these
signals and on the other hand integrates the information of
multimodal data and reveals the underlying--identical and
differential--origins. A full brain model that simulates brain
activity at imaging resolution is optimal for adding detail to
those noninvasively obtained functional brain maps by revealing
hidden states and parameters.
[0018] Some of the key challenges overcome by the system and
methods described within this application are 1) Building the first
fully standardized framework that is easily accessible and usable
by the research community around the globe. 2) Fitting a full brain
model at imaging resolution to empirical functional data of
individuals and hence creating individual virtual brains. 3)
Creating a database of dynamical regimes that stores all model
insights and is accessible for creation of new hypotheses/theories,
model refinement and real-life TVB applications.
BRIEF DESCRIPTION OF THE DRAWINGS
[0019] FIG. 1 represents a schematic overview of a node, and a
plurality of nodes connected by anatomical connectivity structure
information;
[0020] FIG. 2 illustrates a dynamic model comprising a plurality of
nodes;
[0021] FIG. 3 shows an example of a computing system;
[0022] FIG. 4 discloses an example of a computing system including
a database;
[0023] FIG. 5 illustrates an architecture of a computing
system;
[0024] FIG. 6 illustrates some optional refinements of the
computing system of FIG. 5;
[0025] FIG. 7 shows additional optional features of the computing
system of FIGS. 5 and 6;
[0026] Non-invasive neuroimaging modalities like
electroencephalography (EEG), magnetoencephalography, functional
magnetic resonance imaging (fMRI) and positron emission tomography
provide a macroscopic picture of neuronal dynamics, but fall short
of identifying mesoscopic or microscopic brain dynamics such as
population oscillations or neuronal spike trains. However,
macroscopic signal features are thought to emerge from the
interaction of neuronal populations, while the explicit
mechanism--the neural code--still remains unclear. Instead of
attempting to understand neuronal interactions in every detail, it
is proposed to reproduce functionally relevant brain dynamics using
a large-scale network model that may span one or more salient brain
regions. The one or more brain regions may be selected based upon
either a particular brain dynamic being evaluated, or a particular
brain function being analysed.
[0027] The inventors have realized that for providing a computing
system for standardized modelling of a primate brain, the computing
system may include at least one dynamical model or may provide a
choice of a multitude of dynamical models of the brain.
[0028] A dynamical model in the sense of this application includes
at least a plurality of nodes, wherein at least one, preferably
more than one, preferably each node includes a representation of a
local dynamic model and a parameter set of the local dynamic model.
The dynamic model further includes a connectivity structure based
on anatomical principles.
[0029] Each node of the dynamical model represents a local subunit
of the brain. The number of subunits included in the dynamical
model may depend on the spatial resolution of the neuroimaging
modality chosen as an input of the computing system. For example,
if the chosen neuroimaging modality input is fMRI, then the number
of subunits may resemble the number of voxels of the fMRI data.
However, if the neuroimaging modality input is EEG, the number of
subunits can be chosen to correspond to the number of dipoles,
where dipole locations are assumed to be identical to source
locations (ROIs) used for DTI-tractography, for example. The number
of subunits may also correspond to the number of defined brain
regions in which case the data of the neuroimaging modality are
getting aggregated for each brain region.
[0030] The connectivity structure of the dynamical model is used to
connect the plurality of nodes with each other. For example, a
first subset of nodes is connected to a second subset of nodes
based on the anatomical connections between different nodes, or
subunits. This connectivity structure can be chosen from generic
data, such as standard brain atlases, for example the NIH
Connectome database, or can be based on one individual by taking
DTI tractography data from this one individual and connecting the
nodes based on this data. Furthermore, the connectivity structure
may also include directionality data, such as the data provided by
the CoCoMac database (www.cocomac.org) or
(http://cocomac.g-node.org/drupal/?).
[0031] The nodes include a representation of a local dynamic model,
which comprises a parameter set for adapting the local dynamic
model. This parameter set governs the local dynamics of the local
dynamic model, but may also govern the interactions of a specific
node with a connected node. A local dynamic model represents a
population of biological neurons, such as a column of the cortex.
However, it could also represent a smaller number of biological
neurons. Several examples of a local dynamic model include two
coupled and modified Hindmarsh-Rose models as presented later on.
However, different local dynamic models may also be chosen, such as
coupled Wilson-Cowan oscillators, a single oscillator (phase
oscillator, Wilson-Cowan oscillator, Fitz Hugh Nagumo oscillator)
or others known in the art, such as oscillator models,
biophysically founded population models, artificial neuronal
networks for example. While in some examples, each node of the
dynamic model may be modelled after the some local dynamic model,
different examples may include different subset of the plurality of
nodes to be based on a first type of local dynamic model, for
example the above-mentioned Hindmarsh-Rose models, while other
subset of the plurality of nodes are based on different types of
dynamic models. For example, the choice of the dynamic model may be
based on the physiological behaviour of the population of the brain
to be modelled by a node, for example, whether the population
generally shows bursting behaviour or high firing rates or low
firing rates.
[0032] FIG. 1 represents a schematic overview of a node, and a
plurality of nodes connected by anatomical connectivity structure
information. A representative node 1 can be mathematically
represented by a function f(x(t))=x'(t), where x(t) in this example
represents a membrane potential and x(t) is also considered to be
the output of the node. In a computing system, the node may be
represented by a software-implemented module or class which
receives as input a choice of a local dynamic model for this node
and a parameter set. As will become apparent later, this parameter
set is varied and fitted to mimic the behaviour of the data of the
neuroimaging modality input.
[0033] FIG. 2 illustrates a dynamic model 10 built up from a
plurality of nodes 1, which are interconnected by connections w_ij.
The information, which nodes are to be connected to which other
nodes is extracted from connectivity structure information. The
connections w_ij have strengths and time delays due to the spatial
distance and the conductivity speed in the brain. Furthermore, each
node has a geometric position representing its position in a brain.
In the dynamic model, each node has an output. It is apparent that
each dynamic model may be individualized by choosing the
appropriate connectivity structure and by choosing the appropriate
number of nodes and by choosing the local dynamic model of each
node and the accompanying parameter set. Later on, a dynamic model
is explained in further detail.
[0034] In further embodiments, the dynamical model can be further
individualized or adapted to an individual by fitting the parameter
set of one or more, in some embodiments each node of the plurality
of nodes, to neurodatasets obtained from an individual brain. These
embodiments not only incorporate information of an (anatomical)
connectivity structure, but also functional connectivity
information from the neuroimaging modality input, i.e.
neurodatasets. The integration of the dynamical model with the
neurodataset will be explained in more detail below.
[0035] In some embodiments, the computing system includes a
plurality of dynamic models. This plurality of dynamic models arise
from a plurality of local dynamic model for each node and/or from a
plurality of different anatomical connectivity structures
connecting the plurality of nodes of the dynamic model.
Furthermore, the location of the nodes in the dynamical model, the
number of nodes which may depend on the neuroimaging modality input
used for the computing system as, for example, described above.
[0036] A dynamical model may be provided by a class of software
representing the computing system. The computing system allows a
user to choose the number of nodes, the local dynamical model for
each node, and the connectivity structure of the nodes amongst
others. Thus, the computing system may be configured to uses
different instances or classes of each of the number of node, the
local dynamical models for a node and the connectivity structure to
construct a user-decided dynamical model.
[0037] In some further embodiments, the computing system is further
configured to provide a standardized framework of making different
user-decided or predetermined dynamical model comparable to each
other. In these embodiments, pre-processing steps are performed to
ensure a comparability of the different dynamical models.
[0038] The creation of a comprehensive dynamical model requires an
iterative process where models may be tested, modified and merged.
Such a systematic model creation is often feasible only in a
standardized framework where the model outcomes and model
specifications are well tractable. A standardized framework does
not exist for full brain modelling and scientific community
struggles with the fact that models are not reproducible and hence
can't be utilized in an iterative model generalization process.
[0039] Individual full brain models are based on information
extracted from different data modalities, i.e. structural
information derived from MRI T1-weighted and DTI images are
incorporated along with functional data from BOLD-fMRI and EEG.
Preparing empirical raw data for implementation in the model
involves several pre-processing steps. In order to compare
subject-specific models all pre-processing steps are standardized
and anatomical locations are identified using atlas based
anatomical labelling. Since we are modelling brain signals up to
imaging resolution and standard brain atlases are typically
region-based (e.g. gyrus-based) their parcellation is too coarsely
grained requiring finer subdivisions as well as a precise
co-registration of all data modalities is crucial. Apart from
anatomical labelling, normalization of structural and functional
MRI images to MNI templates is another option for establishing
spatial correspondence between brains.
[0040] In some embodiments the user can choose between
normalization and anatomical labelling to standardize the dynamical
models with each other. Thus, in embodiments in which the computing
system is configured to store a dynamical model set-up by a user,
this dynamical model is stored and labelled to be comparable to
other dynamical models stored in the computing system.
[0041] The computing system includes a decision engine to compare
results of the dynamical model to the at least neuroimaging
modality input. The decision engine may be implemented as a
software module or a routine to compare the data of the
neuroimaging modality to the output of the dynamical model. In many
cases, the neuroimaging modality input or neurodatasets are
recorded in a space different from the output of the dynamical
model. To compare the out put of the dynamical model, the output
can be transformed applying a forward transformation model
corresponding to the neuroimaging modality input to a space of the
neuroimaging modality input. For example, when EEG is the
neuroimaging modality input, an EEG forward model may be applied to
the output of the dynamical model to represent the output in the
space of the EEG, i.e. the output is transformed to represent a
simulated EEG signal. This simulated EEG signal can be compared to
the experimentally obtained EEG signal. Other forward models exist
for different neuroimaging modalities such as the BOLD signal in
case of fMRI.
[0042] In further embodiments, the neurodatasets may be transformed
to a source space of the output of the dynamical model, where in
some embodiments each source may be represented by one of the
plurality of nodes. An example of such an inverse transforming
method is the EEG lead-field inversion method. Further examples of
inverse transforming methods are spatiotemporal kernels taken for
example from the dictionary of dynamical regimes converting the
fMRI BOLD maps in maps of electric activity.
[0043] The forward and inverse transform methods may be implemented
as software modules or classes. Several standard transform methods
have been implemented in the prior art and are referred to
below.
[0044] In some further embodiments, the decision engine includes a
fitting method to fit the parameter set of a node of the plurality
of nodes or several of the plurality of nodes to the neurodatasets.
Standard mathematical fitting methods are known in the art. In some
specific embodiments, the fitting of the dynamical model to the
neurodatasets is performed by transforming the neurodataset into
the source space of the dynamical model by an inverse transform
method. Instead of fitting the parameter sets for all nodes, a
subset of the plurality of nodes is replaced by the inversely
transformed neurodataset corresponding to the subset. The remaining
nodes are then fitted using standard fitting methods. In a further
specific embodiment, the output of all but one node is replaced by
the inversely transformed neurodataset. The remaining node, i.e.
the parameter set of the remaining node is then fitted to the
inversely transformed neurodataset corresponding to this node. This
greatly reduces the necessary computational power necessary to
handle the fitting process. This process can be repeated for every
node, wherein each previously fitted node can be either used or
replaced by its corresponding inversely transformed
neurodataset.
[0045] FIG. 3 illustrates a schematic representation of a computing
system 11 including a plurality of dynamical models 12 provided by
a library 13 of dynamical models. The library may be implemented as
a database and a dynamical model may be implemented as a set of
coupled differential equations. The dynamical model 12 includes a
plurality of nodes as is exemplified in FIG. 2. After a user has
chosen a configuration of the dynamical model 12, i.e. its
connectivity structure, the initial values of the parameter sets
amongst others, the dynamical model can be forwarded to a decision
engine 14. In order to standardize the dynamical model and make it
comparable to the individualized models of other subjects or
patients, spatial transformation rules are implemented that map the
individualized model to a standard brain space. An example of how
standardization may be implemented is fount in the Virtual brain
project, found at http://www.thevirtualbrain.org
[0046] Additionally, the computing system provides access to
neurodatasets 15, which may be added by the user via an
upload/download module or can be chosen by the user from a set of
neurodatasets provided by the computing system. The plurality of
neurodatasets may be pre-processed by the decision engine or may be
pre-processed by a further module of the computing system, removing
artefacts from the neuroimaging data. The chosen and pre-processed
neurodataset 16 is also forwarded to the decision engine.
[0047] The decision engine 14 may be implemented as part of a
simulation component, which may be part of several computing
systems. The decision engine prepares the neurodataset and the
dynamical model to be comparable. In our present example, the
neurodataset is a simultaneously measured fMRI/EEG dataset, the
dynamical model has 96 nodes mapped to different regions as
discussed in Ritter et. al.
[0048] The decision engine 14 has access to different forward
transformation models 17, for example an EEG forward model to
project or map the dynamical model output from its source space to
the space corresponding to the EEG dataset and a BOLD model to map
the output into a space corresponding to the fMRI dataset.
[0049] Additionally, the decision engine in this example includes
an inverse transformation model 18 to project or transform the
neurodataset into a source space represented by the locations of
the nodes of the dynamical model. For example, this inverse
transform model can be based on the lead-field matrix of the EEG,
which is available from several standard toolboxes.
[0050] The decision engine may schedule both, the dynamical model
and the neurodataset to be copied and transformed into each others
space. By doing this operation it can be seen whether the simulated
output on different levels, e.g. the output of each node, or the
input from other nodes to a specific node, or the output of the
dynamical model as a whole or the forwardly transformed output of
the dynamical output in the space of the neurodataset resembles the
experimental neurodataset on all or some of the different
levels.
[0051] The decision engine 14 may also include a plurality of
fitting methods 19 to fit the simulated output to the experimental
output of the neurodatasets on the different levels. Depending on
the size of the parameter space to be fitted, all parameters may be
fitted simultaneously. However, in many instances the parameter
space is too large and amore sensible approach may be helpful. The
transforms and the fitting may be performed by the decision engine
itself, or may be forwarded by the decision engine to a processing
unit, such as a real or virtual central processing unit.
[0052] One example of a fitting method provided by the decision
engine is fitting each node sequentially and independently of each
other. In further examples, several nodes may be fitted
simultaneously of the remaining nodes of the dynamical model. In
the present example, in which each node is fitted sequentially,
i.e. one node after the other, the fitting method tunes this node
that receives local and global inputs from all connected nodes.
However, the activity of all connected nodes is represented by the
empirical data transformed into source space. Hence the activities
of all the connected nodes need not to get fitted while tuning the
single node or group of nodes of interest. Subsequently the next
node is tuned following the same principle. This is repeated until
each of the nodes of interest has been estimated. With this
approach the fitting method fundamentally downsizes the number of
parameter combinations to be tested. This fitting approach allows
disentangling three constituents of local brain activity: 1) local
intrinsic activity, 2) local network inputs and 3) global network
inputs. This allows for systematically exploring how the
computations of local populations operate under the influence of
global context such as resting state activity.
[0053] By successively tuning each node, a set of values for each
parameter set is fitted which resembles the experimentally measured
activity in the neurodataset.
[0054] In further examples the parameter set may be fitted for
succeeding epochs or time snippets of the neurodataset to reflect
different output patterns of the dynamical model due to different
experimentally recorded signals. The set of values for the
parameter set may then be derived for different epochs. The set of
values for the parameter set thereby can be considered to be
time-dependent. In case it is known that certain parameters of a
node are more likely to stay invariable over several epochs (i.e.
the range of a minute, or half an hour or more for example) it can
also be considered to fix this parameter and only allow a variation
of parameters known to change even on shorter time-scales such as
seconds
[0055] To further simplify the fitting process or to account for
temporal trajectories of the dynamical model, the neurodataset may
be separated into neurodatasets containing a smaller time window
than the full neurodataset. For example, the fitting can be
performed on small time snippets or epochs of the neurodataset
containing only seconds (0.1 seconds to 10, 20, 40 or more seconds)
of the original dataset. In other embodiments, the time length of
the entire neurodataset may be used.
[0056] The time course of the neurodataset encodes a functional
connectivity of the dynamical model, which is different from the
connectivity structure mentioned before. As an analogy, the
connectivity structure or structural connectivity provides the
pathways of how different nodes of the dynamical model are
connected with each other similar a road network. The functional
connectivity encoded in the neurodataset resembles the traffic over
time in this "road" (or in this case node) network. Thus, the
parameter sets of different nodes may be fitted to represent the
(neural information) traffic through the network.
[0057] According to another aspect, the computing system is
configured for providing data modelling a primate or human brain
comprising a database including a plurality of datasets. An example
of the database is a dictionary of dynamical regimes, a specific
example of which is explained below.
[0058] It is noted that the computing system, in some examples,
need not necessarily contain one of the plurality of datasets, but
that the database is configured to store, retrieve or modify
datasets as described in the following paragraph. A database may be
implemented as software or a server environment and may include
interfaces and/or routines for retrieving, modifying, downloading
and/or uploading datasets from the database. The database may be a
SQL variant or other well-known software architectures.
[0059] Each dataset of the database includes a container for a
dynamical model of the brain. The dynamical model, as has been
discussed above, may include information of the number of nodes,
the local dynamic model type chosen for each node, and a set of
values for the parameter set of a local dynamic model of at least
one node, in further examples all nodes of the model. Furthermore
the container may contain a structure to save the connectivity
structure of the dynamic model, for example the tractography for a
node of interest with the respective fibre tract length.
Additionally, the dataset may contain the output of the dynamic
model based on the set of values of the parameter set of the nodes
of the models.
[0060] The neuroimaging dataset, i.e. neurodataset is also part of
the dataset of the database. In some examples, the neurodataset has
been pre-processed and freed of artefacts, such as heartbeat or
eye-movement based activity in an EEG. The neurodataset may be
stored as a matrix, wherein the rows (or columns, respectively)
represent the different recording channels and the columns (or
rows, respectively) represent time. For example, in a 64-channel
EEG recording, the matrix may have 64 rows, one for each channel of
the EEG and 10000 columns for a length of 100s at a resolution of
10 ms. When the neurodataset includes or consists of an fMRI
recording the matrix may have a number of rows (or columns,
respectively) corresponding to the number of voxels of the fMRI and
a number of columns (or rows, respectively) for each time step.
Including neurodatasets from different neuroimaging methods results
in a corresponding number of transform matrices. The neurodataset
may further include an inversely transformed matrix, for example, a
convolution of the lead-field matrix of EEG and the corresponding
EEG matrix to show the source activities for all dipoles of the
inverse EEG model times the time steps, i.e. a representation of
the EEG in source space. The neurodataset may also comprise signal
analysis results of aforementioned imaging data such as functional
connectivity matrices (that are matrices describing the degree of
similarity/mutual information of the temporal behaviour of pairs of
nodes) or graph theoretical measures that equally may serve as
reference for parameter tuning.
[0061] The set of values of the parameter set is derived by fitting
the dynamic model to an instance of the neurodataset, for example,
a time window of the duration of 100 seconds as an epoch to the
representation of an EEG in source space.
[0062] In other examples, the set of values may be derived by using
a forward model and fitting the set of values for the parameter set
so that the forwardly transformed simulated activity of the dynamic
model in a space corresponding to the neuroimaging modality input
reflects the actual neurodataset data in this space.
[0063] The set of values may also be stored in a matrix type
structure, wherein each entry of the matrix is a vector containing
the fitted or prior set of values and index of the matrix entry
indicates the position of the node. Additionally, the matrix entry
may also include the position of the node with respect to source
space of the neurodataset.
[0064] In further example, a dataset may include additional
information, such as information about the methods applied during
the pre-processing of the data, the length of the neurodataset, in
case the neurodataset is separated in shorter neurodatasets a
labelling of the different neurodatasets according to their
occurrence, the condition of the subject from which the
neurodataset was obtained, i.e. a healthy or sick subject, which
disease, the age of the subject and other types of meta information
such as the experimental/recording condition, e.g. as resting state
or a specific task.
[0065] The computing system may include a search entity configured
to receive input from a user via an interface (for example a web
interface) indicating a property of a neurodataset as a search
request. For example, the property may be to look for a dataset
including EEG data including an alpha-rhythm or another
prototypical rhythm known to occur in an EEG. The search entity
processes the search request and analyses the datasets. If some of
the datasets include the desired property these datasets are marked
and displayed for the user, including the particular time frame
during which the particular property occurred. The user may then
choose to inspect the datasets manually to see whether one of the
datasets is helpful to the user. For example, the fitting of the
parameter sets of the dynamic model of this user may be performed
faster of the prior values for the parameter set are chosen to
resemble the particular searched property, which is also found in
the neurodataset of the user. In other words, in some examples, the
database may help the user to find suitable prior values for a
parameter set. In further embodiments, the search request may also
include the used or preferred local dynamic model used by the user
to exclude datasets using other local dynamic models. The search
request may further include additional information to reduce the
number of dataset which are suitable for the user.
[0066] The search entity, thus, may scan the datasets and forward
datasets including suitable dynamical or local dynamic models to a
signal-processing unit, which in return analyzes the neurodataset
of the dataset for a property indicated by the search request.
[0067] In other examples, a computing system may automatically
search a database for suitable sets of values for parameter sets
before fitting a yet unfitted dynamical model to a current
neurodataset. In these examples, the computing system may analyze
the current neurodataset to be used for the fitting of the yet
unfitted dynamical model and search the database for datasets
including neurodatasets showing similar behaviour than the current
neurodataset. If the datasets also include comparable dynamical
models, the set of values of the dynamical model may be used as a
prior for the fitting of the yet unfitted dynamical model.
[0068] In the previous examples, the search entity initiated a
search of the neurodataset, in particular the part of the
neurodataset in the space of the neuroimaging modality. However, in
further examples, the search entity may also initiate a search of
the neurodatasets in a source space. If, for example, the
neurodataset includes information dipole source activity derived
from an EEG, this data may also be searched for properties. The
search entity may not only provide a mask to enter the properties
to be searched, but also be functionally connected to a
signal-processing unit to analyze the data contained in each
dataset to search for the property.
[0069] In addition or alternatively to searching and analyzing the
neurodatasets, the search entity may also search or forward the
data corresponding to the dynamical models. For example, if a user
is interested in dynamical models that exhibit a certain type of
global behaviour (for example an alpha rhythm), while the
corresponding nodes exhibit bursting behaviour the search entity
may forward the dataset to the signal-processing unit and receive
and analyze a result from the signal-processing unit whether a
dataset has the desired properties. In this sense the search entity
can perform searches of behaviour on the node level, the dynamical
model level, or a property of a pattern of incoming input into a
node (which reflects the output of nodes connected to said node).
Many other properties can be envisioned to be searched by the
search entity.
[0070] In some examples the database may also be configured to
extend a dataset. For example, if a certain dataset is found in
connection with a particular property by man different search
requests, this property may be added to the dataset. Over time this
adding of properties may also result in a ranking of dataset with
respect to specific properties. By performing a ranking based on
number of the dataset being found, datasets can identified to show
prototypical behaviour. Further rankings can be performed by the
database, for example, by noting whether a set of values for a
parameter set of a node when used as a prior for other datasets to
be analysed results in a faster convergence of the fitting.
[0071] The database may be configured to add further datasets
outputted by computing system as discussed in this application
which include a dynamical model of the brain, a neuroimaging
modality input and a decision engine to compare and or fit the
results of the dynamical model and the neuroimaging modality
input.
[0072] The database may also be used to infer knowledge. Based on
simulations of empirically fitted full-brain models we perform
state-space explorations. Identified state-space motifs are
associated with physiological or pathological behaviour and stored
in the dictionary. A state space is spanned for example by the
state variables of local population models, global input and local
input. We investigate possible flows on manifolds that are
associated with resting state networks of functional networks in
general. Such a manifold is a definable subspace. A flow lies on
such a manifold as a laminar vector field. In other words we now
identify generalizable geometric objects in the state space on
which the entire neural behaviour takes place. Different objects
characterize different functional network/resting state network
(RSM) states/properties. Initially we calculate the state space
trajectory and identify clearly dissolvable sub spaces with laminar
flows. In other words we identify state-space motifs that are
associated with RSN activity. For example, consider a node in the
visual cortex. At this site the BOLD signal time course is almost
identical to the time course of the visual RSN. Or put it
differently the visual RSN determines strongly the activity of that
node. With TVB we are able to simulate the electric and fMRI
activity for this node--i.e. visual RSN activity. We now are able
to see how global, local input and local computation behave at
times of high or low activity of the visual RSN or at transition
points.
[0073] FIG. 4 discloses a schematic overview of an example of a
computing system including a database as described above and
includes some workflow aspects handled by the system. Computing
system 20 includes a database 30, which either stores itself or
accesses a data storage system 40 storing datasets as described in
one of the previous examples. The computing system 20 includes a
web interface 50 by which a user may access the database 30 through
a search entity 60. In other examples, the web interface 50 may
also allow direct access to the database 30. Furthermore, the
computing system includes an upload/download module 70 allowing
other components to upload or download data from the database 30.
The other components may include a human user or a component of the
computing system such as a server or another computing system.
[0074] The search entity 60 may be a search engine, configured to
process search requests input into a web interface. Other
interfaces may also be possible, for example by parsing files for
search terms. Other interfaces my provide access for or to a
computing system to fit parameter sets of nodes of a dynamical
model to neurodatasets.
[0075] In the present example, a user enters input 51 into the web
interface. The input 51 is formatted and forwarded as a search
request 61 to the search engine 60. The search request 61 includes
properties of neurodatasets which the user seeks. As an output the
user requests a set of values for a parameter set of a specific
node in his or her dynamical model of a brain. The node is
represented by a local dynamic system of an excitatory population
represented by a Hindmarsh Rose mean-field model coupled with an
inhibitory sparsely firing integrate-and-fire model and is
connected to other nodes. The connectivity structure suggests that
the node is located in a region of the visual cortex
[0076] The search entity starts a query 31 with the database and
searches the database 30 for datasets which include a node with a
similar local dynamical model. The database 30 returns several
neurodatasets 32 of datasets including such a node. The search
entity 60 forwards each neurodataset 32 with a command 81 to a
signal-processing unit 80. The signal-processing unit may be a
software module or a toolbox which can be included in a search
entity. The command 81 indicates to the signal-processing unit 80
which property is desired in the neurodataset and analyzes the
different neurodatasets. Upon success, a message 82 is returned to
the search entity indicating the neurodatasets which fulfil the
different desired properties. The search entity then may request
from the database to allow the download module 70 to access the
database (71) and download the parameter set 72 of the specific
node desired by the user. The download module 70 then outputs a
file 73 to the user. In the meantime the search entity 60 has
forwarded a result 62 of the search request 61 to the web
interface, which outputs that the search has found a number of hits
and file 73 may be accessed.
[0077] FIG. 5 illustrates a computing system 100 including a
back-end system 110, a storage system 120, a data connector 130 and
a web interface which is accessible by a web browser 140. The
computing system 100 may run on a standalone computer, but may also
be distributed over several computers. For example, the web browser
140 may be located in a location of the user, while the other
components are implemented in a cloud for cloud computing. A system
which is distributed over computers, may be configured with a
minimum set of installed components on the client side, and uses
the cluster infrastructure for all operations, including data
storage.
[0078] In the present example, a set of multiple interaction
interfaces is provided, to accommodate support for different user
typologies and deployment alternatives. For example, some users may
access the web interface with a graphical user interface, while
other users may interact with a scripting interface, uploading more
detailed scripts.
[0079] A system which is distributed over computers, may be
configured with a minimum set of installed components on the client
side, and uses the cluster infrastructure for all operations,
including data storage.
[0080] The back-end system 110 may summarize the logic and
applications, which enable datasets to be fed into the system, to
be processed according to a configurable parameter-set as described
above and to be visualized.
[0081] In this example, the back-end system 110 handles the
different interfaces, such as for example the shown web interface.
Regardless of the chosen interface for accessing the back-end
system 110, the back-end system handles and distributes data in a
unified manner.
[0082] The back-end system 110 has dynamic components packed,
dependent on the chosen interface and deployment procedure. An
automatic packaging tool will make sure the correct components are
packed. For example, with the web interface.
[0083] The storage system 120 is configured to process meta-data,
which should be searchable and, in some examples, on big datasets
(which are also structured, but do not benefit from decomposing and
feeding into a standard database). The meta-data may be kept in
database, for example a relational database, or an XML-file or
another suitable searchable file.
[0084] Furthermore, the storage system 120 may be configured to
store meta-data regarding one or more of the following: at least
one neurodataset of neuroimaging modality input data, a plurality
of dynamical models, each dynamical model including a plurality of
nodes, connectivity structure data, a plurality of local dynamic
models representing a node, at least one forward transformation
model for mapping an activity of the plurality of nodes into a
space corresponding to the neuroimaging modality input, at least
one inverse transformation model for mapping the imaging modality
input into a source space corresponding to the plurality of nodes
and a decision engine including at least one entity for fitting at
least one node of the plurality of nodes to the at least one
neurodataset. The data itself may be stored elsewhere, for example
in a data warehouse component of the computing system attaches to
the storage system. However, the storage system may also store the
above mentioned data structures by itself.
[0085] The data connector 130 is configured to allow access to
external data provider system. Big datasets of medical information
already exist in multiple external data storages (used for other
purposes, too). The data connector may be configured to serve as an
interface for accessing external data storage systems, such that
data from this external system can be imported. The data connector
130 may also be configured to forward data toward some External
Data Storages. Thus, the Output Data requires to be processed and
exported from an internal form into an accepted external format.
Some possible access protocols, such as ssh, http(s), ftp, soac or
dbc are listed in FIG. 4. A DataConnect command can be initiated by
the Browser interface or the Console Interface, be passed to the
back-end system, and then be forwarded to the data connector. The
Data Connector may be configured to receive a Job Descriptor, which
includes information about the External. Data Source, credentials
to access the system and operation target (i.e. what to do with the
data). The meaning of the data to be imported/exported can come
from the External Data Source itself, in an accepted format
(metadata), or from the Back-End System.
[0086] An alternative to the data connector is to have all the
input data "uploaded" from the user interface. While this approach
simplifies the architecture of the computing system, the size such
data can have, and the frequency of the operation often allows a
data connector to improve the overall performance of the system.
Nevertheless an upload from the browser interface can be
established in addition to the data connector.
[0087] In FIG. 6 further details of the computing system of FIG. 5
are illustrated.
[0088] The web server application 111 of the back-end system
controls an HTML Interface and receives data from it. For example
it can be used to serve as the point to input properties to be
forwarded to the storage system 120, which searches a dictionary of
dynamical regimes like database (for a description please see
above).
[0089] Also at this application 111, visualization parameters and
input data will be prepared, for usage at the HTML Interface level,
and for passing it to the back-end components, respectively.
[0090] The back-end system includes a simulation controller 112 for
controlling the workflow between different Application Interfaces
and the actual business logic in the Back-End. Thus this component
needs to be abstract and may not have a connection with the
interaction method (script, html or stand-alone interface).
[0091] The Simulation Component 113 is responsible for all the
scientific computation related to brain models and data. The
Simulation Component may receive controlling messages from the
simulation controller and will know how to retrieve required
Brain-Input data from the Storage System, i.e. the desired local
dynamic models, the connectivity structure data, the resulting
dynamical model of the brain, the number of nodes etc. based on
that input. Furthermore, it may be used to integrate neurodatasets
into the analysis. The simulation component may also include or may
be functionally connected to the decision engine or compare and/or
fit a dynamical model to a neurodataset. This Brain Input data and
its information flow between the storage system 120 and the
stimulation component is illustrated by data 115.
[0092] A post-processor 114 operates on simulation results, takes a
configurable set of parameters to adjust the applied algorithms
(applied by the simulation core) and may return results wither to
the simulation component or the web server. Additionally, it might
include a visualization component to help illustrate results in a
more easy to understand fashion. The visualization algorithms may
include one or more of the following: cortical surface data
visualization, volume based data (fMRI, lesion site), visualization
as a movie of selected sources with the brain as a background,
visualization of connectivity matrices (structural and functional
connectivity), visualization for the Connectome proposed data
formats (NIFTI, GIFTI, GraphML, TrackVis), display of multiple time
series with possibility to let it shift through (as known from EEG)
for arbitrary sources, nodes, electrodes, display of wave let time
series with possibility to let it shift through (as known from EEG)
for arbitrary sources, nodes, electrodes, power spectrum for time
series of arbitrary sources, nodes, electrodes, frequency-time plot
of arbitrary sources, nodes, electrodes, coherence matrix of
arbitrary sources, nodes, electrodes.
[0093] Sub-components for the Storage system are Database(s) and
File System(s). One of these two elements could be used
exclusively, but a complementary solution is considered better.
Different distributions (Cluster or Stand-Alone) might have
differences in the way data is stored.
[0094] In FIG. 7 several additional optional refinements of
components of the computing system are illustrated. In particular,
it is illustrated that the simulation core 116 handles the
simulation algorithms, such as the differential equation solvers,
the transform models, the access to mathematical libraries to
perform pre-processing of data, and the fitting of parameter sets
of the nodes of the dynamical model, for example.
[0095] Furthermore, it is illustrated how the brain information
including metadata and data such as the number of nodes, the local
dynamical model chosen for a node, the anatomically founded
connectivity structure of the nodes to arrive at a global model and
other data such as the simulation results, i.e. the set of fitted
values of the parameter sets and the neurodataset used to fit the
parameter sets are stored in the storage system as a dataset which
may be accessed by the web browser to find statistical prior
information for future simulations, i.e. which may form a dataset
of a database of dynamical regimes or a dictionary of dynamical
regimes.
[0096] In the following, several aspects of the computing system,
particularly relevant to the derivation of the dynamical model,
pre-processing of neurodatasets the comparison of the output of the
dynamical model and the integration of the dynamical model and the
neurodatasets to arrive at a fitted set of values for a parameter
set of one or more of the plurality of nodes are described. These
aspects are only optional aspects.
[0097] In an aspect, a dynamical mathematical model that is able to
capture relevant features of brain activity at the mesoscopic
scale, i.e., the scale of cortical columns, nuclei and populations
comprising up to several hundreds of neurons, may be provided for
modelling the one or more brain regions. In an aspect, at least
some of the one or more brain regions may be modelled by a
region-specific dynamical mathematical model.
[0098] This approach accounts for two fundamental principles of
brain organization: segregation and integration. In both the brain
and existing computational brain models, these principles are
realized by a small-world architecture in which signals from
functionally specialized and densely connected local groups are
integrated by long-range connectivity Multi-modal neuroimaging
captures different aspects of brain dynamics that can be related to
cognitive states.
[0099] Providing a system that integrates modelled brain dynamics
with input of one or more neuroimaging modalities provides a tool
to enable the research and assessment biophysically relevant
parameter changes as they occur in changing brain states or as a
result of pathology. Accordingly, the tool may be used for clinical
assessment to identify and counteract unfavourable processes in the
brain and to promote beneficial processes. The tool may further be
used in research to identify (i) fundamental neuronal mechanisms
that give rise to typical features of brain dynamics and (ii) how
inter-subject variability in brain structure results in
differential brain dynamics.
[0100] In an aspect, a computational model may be provided that
integrates multimodal neuroimaging data together with existing
knowledge about brain functioning and physiology. In an aspect, the
computational model may model basic bottom-up interaction between
elementary brain processing units, to describe high-level brain
function. Instead of modelling the interaction between abstract
entities or concepts, the building blocks of the model are neurons,
respectively neural populations. The use of model building blocks
that correspond to real neurophysiological entities are useful for
modelling the generation of simulated signals that show similar
dynamics to actual neuroimaging signals.
[0101] Large-scale neural network models can be built in a way that
enables the incorporation of all prior knowledge about brain
physiology and processing that is necessary at the desired level of
abstraction. These models can be combined with forward models that
allow the simulation of neuroimaging modalities. This approach
promotes the direct formulation of hypotheses on how cognitive
processes are generated by neuronal interaction. Those hypotheses
can take the form of specific features of the model network
structure, i.e. connectivity strength and distance between neural
elements, or biophysical properties like resting potentials,
membrane permeability, plasticity effects, etc. The data generated
by these models can be directly compared to signals from the
respective imaging modality.
[0102] Comparing different imaging signal modalities to a single
underlying model of neural generators can be exploited in several
ways: the exploration of model weaknesses that lead to residuals
between data and model output can be traced either to shortcomings
in the model network structure or to suboptimal parameter settings.
The iterative refinement of model network structure and
optimization of parameter values leads to systematic improvements
in model validity and thereby in knowledge about brain physiology
and cognitive processing.
[0103] Models that have been proven to accurately reproduce resting
state data can be fitted with data from different experimental
conditions. Cognitive and behavioural experiments can then be
interpreted in the light of model behaviour that directly points to
the underlying neuronal processes. For example, analysing the
variation of parameter settings in response to experimental
conditions can deliver further insights on the role of the
associated structural or dynamical aspect in a certain cognitive
function. Furthermore, the relevance of neurobiological features
for the emergence of self-organized criticality and neuronal
information processing can be inferred.
[0104] In summary, the model-based nexus of experimental and
theoretical evidence allows the systematic inference of
determinants of the generation of neuronal dynamics. In the
following, we first describe the modelling environment and the data
and then outline two complementary approaches for the model based
integration of subject specific physiological priors and recorded
neuroimaging data from different modalities. We conclude with some
statements concerning feasibility and possible subsequent analyses
upon successful integration of model and data.
Large-Scale Brain Models with Local Mesoscopic Dynamics
[0105] The brain contains about 10.sup.11 neurons linked by
10.sup.15 connections, with each neuron having inputs in the order
of 10.sup.5. The complex and highly-nonlinear neuronal interaction
patterns are only poorly understood and the number of degrees of
freedom of a microscopic model attempting to describe every neuron,
every connection and every interaction is astronomically large and
therefore too high to directly fit with recorded macroscopic data.
The gap between the microscopic sources of scalp potentials at cell
membranes and the recorded macroscopic potentials can be bridged by
an intermediate mesoscopic description. Mesoscopic dynamics
describe the activity of populations of neurons organized as
cortical columns or subcortical nuclei. Several features of
mesoscopic and macroscopic electric behaviour, e.g., dynamic
patterns like synchrony of oscillations or evoked potentials, show
good correspondence to certain cognitive functions, e.g.,
resting-state activity, sleep patterns or event related
activity.
[0106] Common assumptions in mean-field modelling are that explicit
structural features or temporal details of neuronal networks (e.g.
spiking dynamics of single neurons) are irrelevant for the analysis
of complex mesoscopic dynamics and the emergent collective
behaviour is only weakly sensitive to the details of individual
neuron behaviour. Another common assumption is that neurons in a
population that constitutes a functional cluster exhibit similar
behaviour. This accounts for the relatively new concept from
statistical physics that macroscopic physical systems obey laws
that are independent of the details of the microscopic constituents
from which they are built. Thus, our main interest lies in deriving
the mesoscopic laws that drive the observed dynamical processes at
the macroscopic scale in a systematic manner.
[0107] Non-invasive neuroimaging signals constitute the
superimposed representations of the activity of many sources
leading to high ambiguity in the mapping between internal states
and observable signals, i.e., the pairing between internal states
of the neural network and observed neuroimaging signals is highly
surjective. In EEG and MEG this surjectivity arises from the
underdetermined nature of the backward solution. Therefore, a
crucial step towards the outlined goals is the correct
synchronization of model and data, that is, the alignment of model
states with internal--but often unobservable--states of the system.
Hence strategies for synchronizing brain models with experimental
neuroimaging data are required.
[0108] In the framework of the model, a biologically realistic
large scale coupling of neural populations is provided at salient
brain regions that is mediated by long-range neural fibre tracts as
identified with diffusion tensor imaging (DTI) based tractography
together with mean-field models that are able to reproduce typical
features of mesoscopic population dynamics such as the reduced
forms of the Hindmarsh-Rose and the FitzHugh-Nagumo models.
[0109] The Stefanescu-Jirsa reduced Hindmarch-Rose neural network
model provides a low dimensional description of complex neural
population dynamics including synchronization, multiclustered
solutions in phase space, and oscillator death. The conditions
under which this behaviour occurs are shaped by specific parameter
settings, including connectivity strengths and neuronal membrane
excitability.
[0110] Each network node is governed by its own intrinsic dynamics
superimposed with the dynamics of all other network nodes that are
each connected via specific connection weights and time delays
yielding the following evolution equation for the time course t=1,
. . . , T of the network mean-field potential {x.sub.i(t)} at node
i:
X i ( t + 1 ) = X i ( t ) + f ( X i ( t ) ) t + c j = 1 N W ij X j
( t - .DELTA. t ij ) + .eta. ( t ) . Eq . 1 ##EQU00001##
[0111] The equation describes the numerical integration of a
network of N connected neural populations i=1.quadrature. N. The
large-scale network is described by connection weights w.sub.ij
where index j indicates the weight of node j exerting an influence
on the node indexed by i. The time delays for information
transmission .DELTA.t.sub.ij=d.sub.ij/v depend on a distance matrix
d.sub.ij and a constant conduction speed v. Weights are scaled by a
constant c. Additive noise is introduced by the term .eta.(t).
[0112] For each node, a neural population model f(x.sub.i(t))={dot
over (x)}.sub.i(t) describes the local dynamic at each of the nodes
of the large-scale network. The six coupled first-order
differential equations are a reduced representation of the
mean-field dynamics of populations of fully connected neurons that
are clustered into excitatory and inhibitory pools. Since the
reduced system is described by three modes each variable and
parameter is either a column vector with 3 rows or a 3.times.3
square matrix:
{dot over
(x)}=y-p.sub.1x.sup.3+p.sub.2x.sup.2-z+K.sub.11(Ax-x)-K.sub.21(Bw-x)+IE.s-
ub.i
{dot over (y)}=p.sub.3-p.sub.4x.sup.2-y
=p.sub.5p.sub.6x-p.sub.5z-p.sub.7
{dot over
(w)}=v-p.sub.8w.sup.3+p.sub.9w.sup.2-u+K.sub.12(cx-x)+II.sub.i
{dot over (v)}=p.sub.10-p.sub.11w.sup.2-v
{dot over (u)}=p.sub.5p.sub.6w-p.sub.5z-p.sub.12 Eq. 2
[0113] This dynamical system describes the state evolution of two
coupled populations of excitatory (variables x, y and z) and
inhibitory neurons (variables w, v and u). In its original single
neuron formulation--that is known for its good reproduction of
burst and spike activity and other empirically observed
patterns--the variable x(t) encodes neuron membrane potential at
time t, while y(t) and z(t) account for the transport of ions
across the membrane through ion channels. The spiking variable y(t)
accounts for the flux of sodium and potassium through fast
channels, while z(t), called bursting variable, accounts for the
inward current through slow ion channels (Hindmarsh and Rose 1984,
incorporated herein by reference).
[0114] Parameters of the mean-field model are fitted with short
epochs of simultaneously acquired EEG-fMRI data recorded from
subjects. The coupling of the large-scale network is constrained by
individual DTI tractography data combined with directionality data
from the CoCoMac database http://cocomac.org. We aim for parameter
adaptation strategies that efficiently enable the brain model to
reproduce the recorded EEG-fMRI time-series with the long-term
target to analyse resulting parameter dynamics for biological
relevance.
[0115] EEG waveforms recorded on the scalp are due to a linear
superposition of micro-current sources. However, the mechanisms of
source interaction from which dynamic signals emerge remain mostly
unknown. It has been shown that time delays of signal transmission
between large-scale brain regions that emerge from the specific
underlying large-scale connectivity structure due to finite
transmission speeds can have a profound impact on the dynamic
properties of the system.
[0116] Ghosh and colleagues demonstrate that in large-scale models,
besides realistic long-range connectivity, the addition of noise
and time delays enables the emergence of fast neuroelectric rhythms
in the 1-100 Hz range and slow hemodynamic oscillations in the
ultraslow regimes <0.1 Hz. Hence the model, includes the
connectivity structure derived from tractography results and
investigate the effects of variation in coupling on emerging brain
dynamics by comparing modelling results across different
subjects.
Forward Model
[0117] Neural source activity time courses are projected into EEG
respectively BOLD (blood oxygen level dependent contrast) space
using a forward model. In the following, we will briefly outline
simulation pipelines for the forward estimation of BOLD and EEG
signals and the backward estimation of source time courses.
Forward Model 1: Computing the EEG Signal
[0118] Three compartment volume conductor models are constructed
from structural MRI data; surfaces for the interfaces between grey
matter, cerebrospinal fluid and white matter are approximated with
triangular meshes. For EEG predictions, volume conduction models
for skull and scalp surfaces are incorporated. Here we assume that
electric source activity can be well approximated by the
fluctuation of equivalent current dipoles generated by excitatory
neurons that have dendritic trees oriented roughly perpendicular to
the cortical surface and that constitute the majority of neuronal
cells (.sup..about.85% of all neurons). We neglect dipole
contributions from inhibitory neurons since they are only present
in a low number (.sup..about.15%) and their dendrites fan out
spherically. Therefore, dipole strength can be assumed to be
roughly proportional to the average membrane potential of the
excitatory population.
[0119] Besides amplitude, every dipole has six additional degrees
of freedom necessary for describing its position and orientation
within the cortical tissue. Dipole locations are assumed to be
identical to source locations (ROIs) used for DTI-tractography,
while orientations are inferred from segmented anatomical MRI
resting on the assumption that cortical dendrites are perpendicular
to gyri surfaces.
[0120] Under these assumptions, the transfer matrix A can be
calculated using, e.g., a boundary element method, yielding a
transformation from source dipole activity D.sub.j(t) at a triangle
corner vertex j to the potential .phi..sub.i(t) of electrode i at
time t:
.phi..sub.i(t)=[AD].sub.i=.SIGMA..sub.ja.sub.ijD.sub.j(t).
[0121] BEM models are based on meshes that form a closed
triangulation of the compartment surfaces from segmented
T1-weighted MRI surfaces. Finite Element Method models consist of
multiple tetrahedra allowing to model tissue anisotropy that is
physiologically more realistic at the cost of computational
resources.
Forward Model 2: Computing fMRI BOLD Contrast
[0122] Subsequent to the fitting of electric activity with the
model, we want to compare model predictions of BOLD contrast with
actual recorded fMRI time-series data in order to deduce and
integrate further constrains from and into the model and to perform
analyses on the coupling between neural activity and BOLD contrast
fluctuation.
[0123] The BOLD signal time course is approximated from the
mean-field time-course of excitatory populations accounting for the
assumption that BOLD contrast is primarily modulated by glutamate
release. Apart from these assumptions, there is relatively little
consensus about how exactly the neurovascular coupling is realized
and whether there is a general answer to this problem.
[0124] Up to now, we approximated neurovascular coupling with a
hemodynamic response function that is convoluted with the
excitatory mean-field potential in order to obtain BOLD time
courses. We may include to more sophisticated accounts like the
"Balloon-Windkessel" model of (Friston, Harrison et al. 2003) for
some more technical details.
Backward Solution: Source Imaging and Equivalent Current
Dipoles
[0125] As a first step towards the integration of recorded EEG
time-series with mean-field population models, the time courses of
cortical sources are estimated from recorded data using neural
source imaging methods.
[0126] The forward problem of EEG has a unique solution.
Conversely, the inverse problem of EEG, i.e., the estimation of a
tomography of neural sources from EEG channel data, is an
ill-conditioned problem lacking a unique solution. EEG constitutes
a convolved signal from multiple, simultaneously active and
regionally overlapping neural sources that are spatially mixed and
summed across the brain by volume conduction. Due to the
underdetermined nature of the inverse problem it may yield multiple
solutions that equivalently reproduce the observations.
[0127] We address this ill-posedness by the introduction of
aforementioned constraints, namely: realistic, subject specific
head models segmented from anatomical MRI images, physiological
priors and source space based regularization schemes and
constraints.
[0128] A commonly used prior is to restrict neural sources using
the popular equivalent current dipole model reducing the backward
problem to the estimation of one or a few dipole locations and
orientations. This approach is straightforward and fairly realistic
since the basis of our modelling approach rests on the assumption
that significant parts of a recorded EEG time-series are generated
by the interaction of our large-scale model sources. Consequently,
we can incorporate the location and orientation information of
these sources as priors, thereby alleviating the ill-posedness of
the backward solution for this special modelling scenario.
[0129] More general source imaging approaches that attempt to
estimate source activity over every point of cortex rest on more
realistic assumptions but need further constraining in order to be
computationally tractable, i.e., the degrees of freedom need to be
reduced by applying a regularization scheme that yields likely
regions for cortical activation. Nevertheless, current density
field maps are not necessarily "closer to reality" than dipole
source models, since the spatial spread is due to imprecision in
the source estimation method rather than a direct reconstruction of
the potential field of the actual.
[0130] In order to estimate the source waveforms for a given
recorded EEG time-series a current equivalent dipole approach can
be based on the inversion of the lead-field matrix. Source space
projection in order to derive source time-courses can be done with
commonly used software packages (e.g., the open-source Matlab
toolbox FieldTrip or BrainStorm or the commercial product
BrainVoyager or Besa) by computing the inverse of the lead-field
matrix on the basis of the given source dipole positions and
orientations and the volume conductor-model that can all be derived
from anatomical MRI data of the subjects. Furthermore, priors
derived from BOLD contrast fluctuation can be exploited to
alleviate source imaging.
Data-Model Integration
[0131] Synchronizing computational models that are able to
reproduce typical dynamical features of brain activity with actual
recorded neuroimaging data requires several methodological
considerations.
[0132] For models that describe broad-band neuronal population
activity comprising spiking and slow oscillations--as the reduced
Hindmarsh-Rose model--it is reasonable to fit simulated outputs
with EEG and BOLD signals simultaneously since different aspects of
neuronal activity are captured by the imaging modalities.
[0133] We do not attempt to fuse the mean-field output of the model
with recorded EEG in signal space but rather in source space by
applying a source reconstruction scheme to the EEG data. The
simultaneously acquired fMRI signal is compared to the mean-field
amplitude fluctuation that was convolved with a hemodynamic
response function. The reason for this is simple and
straightforward. Assuming an equivalent current dipole model, the
integration of model and data in channel space requires a forward
model that in turn requires an accurate representation of source
distribution in terms of location and orientation. Dipole locations
are known since source locations are known, they are obtained by
DTI tractography. Anatomical priors for dipole orientation are
derived from the cortical folding pattern of anatomical MRI data.
Dipole orientations are assumed to remain fixed and normal to the
cortical folding pattern.
[0134] A further advantage of source space integration arises from
the fact that we intend to fit short snippets of model output with
a long (20 minutes) time series of recorded data in order to obtain
relevant parameter settings for reproducing the ongoing fluctuation
of brain activity. Source time-course reconstruction enables us to
initialize model variables with actual cortical source time-courses
and to compare model outputs for different parameter settings with
the actual continuation of the source time-series.
[0135] This allows for efficient recalibration of model parameters.
Instead of computing a time-consuming forward solution for every
model output subsequent to each parameter variation, we only have
to calculate a source reconstruction once and are then able to
effectively compare it to model output in the light of parameter
variations. This approach integrates into the overall concept of
generating a "model library" that maps specific parameter settings
to dynamic model output (in terms of a succession of relevant
signal features, for example, brain dynamics characterized by a
trajectory of the relative power of frequency bands in each source
node).
[0136] The backward estimation of source potential time courses is
valid since only assumptions and parameter constraints that were
already implemented in the forward part of the model are used.
Therefore, weaknesses in the backward estimate directly relate to
model weaknesses or posed differently: the backward estimate as
accurate as the model.
[0137] Furthermore, it is highly unlikely that the neural
population activity we consider in our model is exactly congruent
with the actual sources generating the recorded EEG signal.
Therefore, we (i) restrict our model to the reconstruction of
source activity time-series obtained by backward inference and (ii)
try to improve the ill-posed backward estimate by deriving methods
that maximize the fit between source and image dynamics in an
iterative backward-forward estimation manner.
[0138] Parameter values are systematically inferred using an
estimation strategy that is guided by several different
principles:
[0139] Parameter ranges are constrained by biophysical and
graph-theoretic priors.
[0140] In an aspect, models are not fitted with time-series using
standard parameter optimization techniques, but dictionaries are
created that associate specific parameter settings with resulting
model dynamics. This provides a convenient way to fit the model
with experimental data without the need to re-optimize parameters
for every time-step. Aside from dictionaries that associate
parameter settings with mean-field amplitudes of large-scale nodes,
dictionaries may be built that associate typical
mean-field-configurations of nodes with resulting EEG and BOLD
topologies.
[0141] Unlikely parameter changes are penalized and parameters are
chosen such that implausible changes are avoided. Based on
bifurcation analyses, the likelihood of parameter settings is
estimated in order to drive the system into a state of
self-organized criticality.
[0142] A typical workflow implementing the discussed framework
starts with the aforementioned initial backward estimate of
underlying source activity that is in turn used as initial
condition for model simulation, i.e., source time-series are
estimated and a short time snippet that we obtain from a simulation
database is used as initial model source time series.
[0143] This initial estimate serves as a preliminary configuration
of parameter and source time-series states. The initially short
snippet is incrementally prolonged and parameters are refitted.
A Dictionary of Dynamical Regimes
[0144] Due to the high number of free parameters (number of free
parameters of all six equations for the excitatory and inhibitory
population times the number of sources) and the resulting high
model complexity, manual parameterization of the model is, of
course, infeasible.
[0145] Model Inversion approaches (e.g. Dynamic Causal Modelling,
Bayesian Inversion or Bayesian Model Comparison based approaches)
build on the inversion of a generative model of only a few sources
(up to about ten). These approaches are intractable, in our case,
due to the high number of contributing sources and the resulting
combinatorial explosion of parameter combinations.
[0146] In the framework of control theory, the problem of parameter
estimation in complex dynamical systems is approached by the
implementation of state observers in order to provide estimates of
the internal states of a system given measurements of the input and
output of the system. Recently, several state observer techniques
have been developed and successfully applied to biological systems
identifying parameters by incorporating the specific structure of
the problem based on an expansion of the system and the
transformation of parameters into or by the use of extended and
unscented Kalman filtering methods.
[0147] Modern automatic parameter optimization methods like
evolutionary algorithms are often able to find acceptable solutions
in complex optimization problems or to fully generate the structure
of brain network models based on physiological constraints.
However, it is very likely that model parameters are subject to
ongoing variations throughout the course of even short time series
and it hence becomes necessary to re-fit parameters for every time
segment. Therefore, it is unreasonable to perform time-consuming
searches through very large parameter spaces (the number of model
instances is in the order of 10.sup.94 if we assume 10 possible
settings for each parameter and 94 source nodes) for every short
epoch of experimental data. Furthermore, it is unlikely that there
is a single correct model, but many parameter combinations might
yield the same result.
[0148] Instead, we build on a strategy that--under certain
assumptions--allows us to explicitly calculate exact parameter
values by algebraic solution, respectively least squares
estimation. Exploiting several features of our modelling setup, we
become able to calculate parameter values by simply solving a
system of linear equations.
[0149] A considerable reduction of parameter space can be achieved
by regularising model space from coupled dynamics to uncoupled
dynamics by disentangling all coupled interactions, i.e., all
incoming potentials from all other nodes are subtracted from each
time series. This can be consistently done inside our modelling
framework by inverting the forward formulation by the simple
subtraction of the coupling term
( c j = 1 N W ij X j ( t - .DELTA. t ij ) ) ##EQU00002##
from Eq.1. Thereby, the forward model is reversed in a well-defined
manner, since all coupling weights w.sub.ij and all time delays
t.sub.ij are given by tractography and the time series x.sub.j(t)
by source estimation results.
[0150] This operation and omitting the noise term yields us an
approximation of the uncoupled mean-field time-series x.sub.i(t)
for all source nodes i. Furthermore, all differentials
f(x.sub.i(t))dt can be easily obtained by subtracting consecutive
time-points.
[0151] The next step is to obtain an initial parameter and state
estimate that captures the prevalent dynamic regime over a short
time snippet (.sup.181-5 seconds) of the uncoupled source
time-series.
[0152] This issue can be addressed by extending the initial
parameter space mapping (with a database driven approach by the
construction of simulation databases. Instead of tuning parameters
for every short time segment, a dictionary is created that relates
specific parameter settings to resulting model dynamics.
[0153] Thus, we aim to generate a database that associates
parameter and variable settings to model dynamics. This dictionary
is used to decompose source time-courses into prototypic atoms of
mode time-series in order to derive initial coarse model state
priors for subsequent parameter estimation and refinement.
[0154] We start by laying a coarse grid over the parameter space
and simulate mean-field activity for grid points, similar to the
initial parameter mapping. Subsequently, the resolution of the grid
can be increased according to desired accuracy or space and runtime
constraints. Each resulting waveform is classified according to
several criteria that discern specific dynamical regimes.
[0155] In this first estimation step, only coupling parameters
(K.sub.11, K.sub.12, K.sub.21) and distribution-parameters of
membrane excitabilities (mean m and dispersion .sigma.) are
estimated since these are the main determinants of the resulting
dynamic regime of the mean-field over both populations. Parameters
that correspond to biophysical properties of the respective neuron
populations (p.sub.1-p.sub.12) are optimized in subsequent steps.
Along with initial parameter estimates, this first step also
provides us with initial estimates of time course snippets of the
second and third variable (y and z) and the time-courses of all
three variables of the inhibitory population (w, v, u), or more
specifically their corresponding mode time courses for all three
modes. These initial estimates will be used for the estimation of
all remaining parameter values that are left unspecified up to that
point. The matrices A, B and C are given and need not to be
inferred.
[0156] Analyses of model simulations under several different
parameter settings and initial conditions revealed good
discriminability of the dynamic properties of the three modes
during uncoupled population simulations in the presence of noise.
Long-term simulation runs under stable conditions show that while
the third mode contains most of the fast-varying dynamics, the two
other modes reveal only slowly varying dynamics and linear
trends.
[0157] Mode decomposition, component analysis, factor analysis and
signal separation techniques refer to the problem of representing a
set of mixed signals by a linear superposition of a set of
generating signals or sub-components.
[0158] In our case, dictionary entries are classified according to
prototypical dynamical model properties as inferred by model
simulations in the presence of additive noise. Therefore, we
classify snippets according to a variety of dynamical metrics that
have relevance for cognition. We disentangle estimated source
time-series into modes using a dictionary that resembles the
prototypical dynamics of model modes. Thereby, the problem of
uniqueness of the solution is addressed by the inclusion of
appropriate priors on the dynamical properties of the respective
time-series.
[0159] Therefore, we suppose our source signal to be a linear
superposition (with equal weights) of atoms taken from our dynamic
regime dictionary made up of short time-sequences and their
respective modes. In a two-step procedure, we first estimate a set
of mean-field atom snippets from the dictionary that show a high
similarity to the source snippet. Then, among these snippets, we
choose one snippet that resembles the constitutive components of
the source signal most closely. To start with, one could employ a
simple correlation based mode decomposition that estimates the
correlation between the previously estimated source time courses
are simply correlated with the full time-series first and the
decomposed time-series later. Later, it might be reasonable to use
more sophisticated signal separation techniques. Specifically, we
aim for the adaptation of existing non-approximate state observer
or Kalman filter approaches from control theory to our problem
setting.
Refining Source Parameters
[0160] As outlined above, we first subtract long-range coupling
terms from each of the estimated source time-courses using the
time-delays and coupling strengths specified in the large-scale
model in order to reconstruct uncoupled source dynamics. Then, we
convolve the uncoupled source time-courses with entries in the
"dynamic regime dictionary" yielding one mean-field waveform that
fits the short-term dynamics of source estimates best. Thereby, we
obtain a coarse estimate of coupling parameters and prototypical
time-series for each mode of the excitatory and inhibitory
populations as well as parameter and variable estimates of the
second and third differential equations of both populations. This
time-course snippet resembles prototypical model behaviour, i.e.,
typical trajectories in the state space of the model resulting from
random initial conditions and simulations in the presence of noise.
Several features of the simulated dynamics of the model are similar
to the dynamics observed in neuroimaging signals. However, noisy
conditions are, as mentioned earlier, critical for the model to
explore different dynamical regimes. In simulations without noise,
model variables are attracted to steady state within a few thousand
time-steps. In this part of the reconstruction scheme, we are
interested in expressing the observed dynamical regime as the
result of parameter fluctuations, instead of additive noise.
Therefore, we now estimate parameter trajectories that, when
inserted in the model, reconstruct the observed variable
trajectories. We assume that parameter settings stay fixed for at
least a short period of time, i.e., ten to one hundred time-steps,
respectively several picoseconds.
[0161] This assumption allows us to estimate parameter values by
constructing for each of the six dynamic equations and their
respective modes, a system of linear equations by reversing
parameters and variables, with parameters being then variables and
vice versa using the time-course variables over several consecutive
time-steps. After lumping all approximated parameter and source
value estimates of the six equations together into six terms
c.sub.1-c.sub.6 and exchanging model variables (x, y, z) with the
estimated initial values (referred to as k.sub.1-k.sub.4) the
dynamic equations 1-3 from (Eq. 2) yield:
c.sub.1=p.sub.lk.sub.i+p.sub.2k.sub.2
c.sub.2=p.sub.3-p.sub.4k.sub.2
c.sub.3=p.sub.5p.sub.6k.sub.3-p.sub.5k.sub.4-p.sub.7
[0162] and analogous for equations 4-6 and values (u, v, w). Note
that for the construction of this system of equations we assume the
coupling parameters (K.sub.11, K.sub.12, K.sub.21) to be fixed and
given from the "dynamic regime dictionary"; nevertheless, it would
also be possible to solve the system with those parameters being
unassigned. Since the first equation is a linear equation
containing only two unknowns, it can be solved using the values
from two consecutive time-steps.
[0163] Sweeping this estimation procedure over the whole snippet
yields parameter trajectories for the whole time-series and allows
us to express recorded neuroimaging time-series as parameter
fluctuations of a computational brain model.
Parameter-Space Analyses
[0164] Following the reconstruction of model dynamics from
time-series, i.e., the expression of time-series in terms of model
behaviour, several neurobiological questions can be addressed.
State and parameter space trajectories can be related to
experimental conditions and brain dynamics to cognition. Model
dynamics are matched with observed behaviour and model dynamics in
turn relate to biophysical properties of the system. Therefore, our
approach allows us to directly associate low-level neuronal
dynamics with top-level processes and to identify metrics that
quantify the functional relevance of dynamical features for
cognition and behaviour under normal and pathological conditions.
Parameter and state space dynamics can be analysed with regard to
biophysical relevance, e.g., parameter dynamics may be associated
with biological phenomena; parameter dynamics may be associated
with behaviour or pathological conditions; macroscopic, mesoscopic
or microscopic connectivity impact on resulting model dynamics may
be measured; potential relationships between the slow-varying
dynamics of the first and second modes of the model and the
localized hemodynamic responses at the respective fitting site of
the electrodynamic source may be evaluated; and, BOLD information
may be integrated into the model. Using a deconvolution model, the
behaviour of slowly varying dynamics in the model may be directly
associated with BOLD contrast dynamics.
* * * * *
References