Independent component analysis (ICA) 8283 has proven to be a powerful exploratory tool for data-driven analysis of fMRI recordings. It is used to blindly estimate distributed spatial patterns in the brain, jointly with hidden source processes in the observed data, under the assumptions of statistically independent patterns and non-Gaussianity of the source components 8485. In this setting, one typically represents the 4-D fMRI dataset as a p × n matrix X, where the n columns consist of an enumeration of the n voxels that covers the imaged 3-D object of interest (i.e. the brain), and the rows consist of the recorded signals in these voxels at p different time points. Furthermore, one assumes that a matrix decomposition exists such that X = AS, where S is a p × n matrix of source signals, and A is a p × pmixing matrix. This expresses that the ICA model is a generative model that describes how the observed data are generated by a mixing process of the components (i.e. row vectors) in S, and that the independent components are latent variables that cannot be directly observed. Both A and S must be estimated from X under some general statistical assumptions (independence and non-Gaussianity), and the ICA method is thus a type of blind source separation (BSS).

During ICA estimation, the matrix S is optimized to contain, in its rows, statistically independent areas in the brain (spatial maps, or components), each with an internally consistent time course (the corresponding column vector in A).The observed signal intensity x_ij in time sample i of voxel j is thus a weighted sum (linear mixing) of the unobserved source signals in the voxel, i.e. . With iterative optimising of an unmixing matrixW = A^-1, we obtain S = WX with mutually independent rows using e.g. the InfoMax algorithm 8486or the FastICA algorithm 87. No noise is included in the model. Typically, the spatial maps are transformed to have zero mean and unit variance (Z-scores), thresholded at some level (e.g. |Z| > 2.0), and mapped back to 3-D image space, color-coded, and superimposed on a coregistered anatomical image for visual assessment of the ICA component.

It can be shown that the independent components are identifiable up to a permutation and scaling of the sources under the assumption that at most one of the sources s_k = [s_k1,..., s_kn] (k = 1,... ,p) is Gaussian, and that the mixing matrix A is full rank. The assumption that the mixing matrix is square i.e. the complete case with as many mixtures (M) as sources (K), can be relaxed 888990. Important challenges are to reduce large fMRI data sets for ICA 91 and to estimate the number of independent components in the data 92. Calhoun and coworkers 93 has also introduced an approach (GIFT) for drawing group inferences using ICA of fMRI data from many subjects, and Beckmann et al. 94 have constructed a "tensor probabilistic ICA" for multisubject fMRI analysis. These extensions have widened the scope and popularity of ICA for fMRI studies, especially group studies of resting state networks and the DMN (see 95 for a comparative framework with these approaches). Furthermore, ICA methods has also been applied to EEG recordings and combined fMRI-EEG data 96, with a large potential for integrated analysis of electrophysiological recordings (with high temporal, low spatial resolution) and BOLD image acquisitions (with high spatial, low temporal resolution). Recently, it has been argued that independence is not the right mathematical framework for blind source separation in fMRI and that representations in which the fMRI signal is spatially sparse are more promising 97, and in their simulation studies Daubechies et al. demonstrated that the ICA algorithms InfoMax and FastICA select for such sparsity of components rather than spatial independence.

Stochastic resonance (SR) is "a mechanism by which a system embedded in a noisy environment acquires an enhanced sensitivity towards small external time-dependent forcings, when the noise intensity reaches some finite level" 102. This SR effect introduces a new role of noise processes, not only as a detrimental nuisance, but as a constructive component in a broad range of natural and man-made systems. The mechanism was first observed and reported in climate research by Benzi et al. 103 in the early 1980's to explain the almost periodic recurrences of ice ages on the earth.

Mathematically, the SR phenomenon can be captured in a bistable nonlinear dynamical system where a one-variable formulation could be 104:

where the state variablex(t), interpreted as e.g. the position of a Brownian particle in a bi-stable potential U : ℝ → ℝ, is subject to both a weak periodic forcing parameterized with amplitude A, frequency ω and initial phase ϕ, and a noise process ξ(t). In our example U(x) can be the reflection-symmetric quartic potential , with minima x_m located at ±1 and with height of the potential barrier between the two stable minima equal to . The noise term ξ(t) in the stochastic differential equation (1) is typically a zero-mean Gaussian where the SR mechanism is related to its autocorrelation function 〈ξ(t), ξ(0)〉, which could be on the form where τ is correlation time and D is intensity of the colored noise. The effect of SR can be enhanced by introducing an ensemble of stochastic resonators. In this case, a weak periodic signal can induce large-scale stochastic synchronization and self-organization (Ch. 3 in 34).

More recently, SR has been introduced to explain behavior in biological systems (see 105, 106 for reviews), including single neurons and large-scale brain systems (e.g. 107). Neural synchrony, noise, and SR has also been applied to the modeling of attention and consciousness 108, and how age-related neuromodulatory deficiencies may contribute to increased neuronal noise leading to less robust information processing in aging neurocognitive systems 109110. Most interestingly, the stochastic resonance mechanisms has recently been proposed to explain resting state results in connectivity data from BOLD fMRI experiments 43.

As an illustrative example of stochastic resonance in basic neurophysiology, we will briefly present a simulation study of the spiking behavior in a neuron with Hodgkin-Huxley (HH) dynamics 111, where the firing of action potentials is driven by a sinusoidal external force and Gaussian noise. The simulation code we are using is modified from the exposition on nonlinear neurodynamics by H.R. Wilson 112 (HHnoise.m), using a fourth order Runge-Kutta solver with constant step size to solve the Rinzel approximation to the HH equations:

For the simulations we used time increment Δt = 0.02, and a = 17.81, b = 47.71, c = 32.63, d = 26. The equilibrium potentials were E_Na = 0.55 and E_K = –0.92. The neural capacitance was C = 0.8, and the recovery time constant τ_R = 1.9 ms. The stimulating oscillating current was given as I(t) = A sin(ωt + ϕ) with amplitude A = 0.035 and initial phase ϕ = 0, were the ω was fixed to give an angular input frequency of 200 Hz. We adjusted zero mean noise level η with standard deviations (SD) in steps of 0.01 in the range [0.0, 0.5] to observe the effect of noise level of the system and the occurrence of stochastic resonance. The results from this simulation study are given in Fig. 4, where Figs. 4c &4e depict SR with clustering of spike intervals corresponding to integral multiples of the principal inter-spike interval of about 5 ms.

The theory of dynamical systems has been applied to fMRI in different settings 113114115116117118. To explore a particularly interesting link between fMRI and neurodynamical modeling in more detail, we adopt the approach taken by Deco et al. 43. Here, the dynamics of each 'node' (to be interpreted below) is captured by a mean-field type rate model, expressing the coupling between excitatory and inhibitory nodes in the network. This is very similar to the classical attractor neural network (ANN) model stated by Amit 37 and others.

More specifically a Wilson-Cowan 119 type neural network dynamics is introduced, following 120121

where u_i denotes the activity of the i-th 'neuron' (neural mass, or enumerated spatial localization i = 1,..., n in the brain); α is inverse relaxation time; W_ij is the 'synaptic' or connection strength between locations i and j (i.e. the connectivity matrix); I_i is the external input to location i; and ξ_i is random noise related to i. The nonlinear (e.g. sigmoidal) function Θ(x) limits the growth of x, accounting for saturation of 'firing rates' or local neural mass activity. In the 'resting state', external inputs to the network nodes vanish, i.e. I_i ≡ 0, where we also assume the noise terms ξ_i are Gaussian N(0, σ_i), and that saturation can be ignored, i.e. Θ(x) ≡ x. With these assumptions, the coupled stochastic ODE's (2) can be discretized in time, with time step Δt, to obtain

which can be written on matrix form

u(t + Δt) = Au(t)+ξ(t) (4)

where A = (1 – αΔt)1 + WΔt;ξ = N(0,σ)(t)Δt, and N(0,σ)(t) denotes the noise column vector at time point t. The temporal averaged covariance matrix of the resting state time courses is C = 〈u(t) • u(t)^T〉, and the eigenvectors v_k of the real-valued symmetric matrix C are the principal components or the dominant patterns or modes of the resting state activity, i.e. . In 120 it is shown that the covariance matrix of the resting state activity is determined by the covariance matrix of the intrinsic noise and the eigenvalues of the connectivity matrix W. As the overall weight of each mode in the resting state activity is given by the magnitude of the associated eigenvalue λ_k, there is link between the dominant patterns and the concept of attractor in neural dynamics 120. Morover, Galán 120 provides arguments that the resting state neural activity 'fluctuates most of the time around the basin of attraction ( 37122) of the dominant pattern', and that 'the amount of time spent around the basin of attraction of the remaining principal components is proportional to the magnitude of their eigenvalues'.

In their simulation study of resting brain fluctuations, Deco and co-workers 43, using the above Wilson-Cowan model with 38 cortical and 2 subcortical thalamic nodes and a realistic neuroanatomical connectivity matrix and signal transmission delay between nodes in the network, were able to reproduce typically occurring resting state fMRI brain dynamics. In their synchronization analysis (employing the Kuramoto synchronization index 123) of simulated neuroelectric activity in each of two individual community clusters (the occipital-temporal-prefrontal community, and the sensorimotor-premotor community), using internode communication velocity of v ~ 1.65 m/s and global coupling strength of α ~ 0.007, a global slow 0.1 Hz) oscillation at rest could emerge from a network built up with simple fast oscillators in the γ-band of 40 Hz. This underscores the role of neural synchronization as a mechanism for the emergence of ultraslow fluctuations in e.g. the default mode network as observed in resting state BOLD fMRI. Moreover, Deco et al. also observed a stochastic resonance effect for the same level of fluctuations that revealed optimal emergence of 0.1 Hz global slow oscillations, with the occurrence of anti-correlated spatiotemporal patterns (as reported in fMRI) in both communities – without the use of (unrealistic) long-range inhibition. From their work, they conclude that "the particular dynamics of the intrinsic properties of the brain are useful for keeping the system in a high competition state between the different subnetworks that later are used during different tasks. ... In this way, a relatively small external stimulation is able to stabilize one or the other subnetwork giving rise to the respective evoked activity. So, the anticorrelated fluctuating structure of the subnetwork patterns characteristic of the resting state is particularly convenient for that."

The field of neurodynamics has also had impact on the theory of consciousness 3841124125126127 (so has stochastic resonance 32108128). A prominent example is the 'dynamic core hypothesis' presented in a seminal paper by Tononi and Edelman 129, and later refined and extended 127130131. Based on modeling concepts from statistical physics and information theory, together with accumulated empirical findings in the history of clinical and experimental neuroscience, they propose the following:

• A group of neurons can contribute directly to conscious experience only if it is part of a distributed functional cluster that achieves high integration in hundreds of milliseconds.

• To sustain conscious experience, it is essential that this functional cluster be highly differentiated, as indicated by high values of complexity.

• A large cluster of neuronal groups exist that together constitute, on a time scale of hundreds of milliseconds, a unified neural process of high complexity – termed the dynamic core, where

– its participating neuronal groups are much more strongly interactive among themselves than with the rest of the brain

– its global activity patterns must be selected within less than a second out of a very large repertoire

– deliberately does not refer to a unique, invariant set of brain areas (e.g. prefrontal, extrastriate, or striate cortex) and,

– can transcend traditional anatomical boundaries and may change in composition over time

An other interesting neurodynamics theory of consciousness is the COrollary Discharge of Attention Movement (CODAM) model of J. G. Taylor and co-workers 132133, based on application of engineering control theory and artificial neural networks. The CODAM framework is claimed to have considerable experimental support for certain of its modules. Support that comes from both brain imaging, ERP studies, and single cell experiments, as well as transcranial magnetic stimulation (TMS) that enables non-invasive, temporarily 'knock out' of brain circuitry in operator-defined cortical regions. In this model, the so-called 'working memory corollary discharge (WMcd) buffer' is the most important element in the circuitry to create consciousness. The framework consists of an input module (for pre-processing in low-level visual cortex), the object map (where object codes are stored), the inverse model controller (IMC) which is a generator of the signal to move the focus of attention in lower cortices, the corollary discharge module where a copy of the attention movement signal is stored temporarily, the working memory (WM) holding an estimate of the attended target representation, and the monitor producing an error signal given by the difference of the required goal signal and that produced by the corollary discharge module as a predictor of the attended next state or of the working memory module activation (cf. 133). So far, very little work has been done to put fMRI experiments into the CODAM framework, which might have been due to difficulties of interpreting fMRI data in terms of timing and inhibition ( 134).

As resting state fMRI networks revealed by e.g. independent component analysis might represent neural mass activity related to the state of consciousness ( 17), further empirical investigations and numerical simulations of distinct network components and their time courses, IC₁(t), ..., IC_n(t), should be performed in the framework of Wilson-Cowan type neural network dynamics ( 43121). Moreover, measuring fMRI resting state functional connectivity (rsFC) and structural connectivity (SC), using DTI or diffusion spectrum imaging tractography at high resolution in the same individuals, it has been shown that the organizations of SC and of rsFC are strongly interrelated 121135 135. In 121 the authors applied Eq. (4) to model macroscopic cortical dynamics within segmented cortical parcellations (n=998), where the inter-nodal interaction efficacy matrix W in the generalized coupling matrix A = (1 – αΔt)1 + WΔt was the resampled fiber strengths obtained from diffusion spectrum imaging tractography. Such approaches might open up a new opportunity to combine mutlimodal MRI and neurodynamics in the study of consciousness.

In the following experiment, we illustrate the first steps of deriving time course information from resting state fMRI data that can be further analyzed in such modeling frameworks.

While the subject was lying comfortably in the scanner with eyes closed, head fixed with noise-protecting ear-pads, and asked not to fall asleep, we used an EPI gradient echo pulse sequence (TR / TE / FA = 2 s / 50 ms / 90⁰); 64 × 64 acquisition matrix, 25 slices, 256 time frames, with voxelsize = 3.75 × 3.75 × 5.5 mm³ and time resolution of 2 s to obtain "resting state" data. In addition, the imaging protocol consisted of an alternating left and right hand finger-tapping fMRI paradigm, two anatomical 3D T1-weighted scans for brain segmentation with FreeSurfer (http://surfer.nmr.mgh.harvard.edu), and DTI acquisitions (TR / TE / FA = 7900 / 104.8 / 90; 25 b=1000, 5 b=0; 25 slices; voxelsize = 1.88 × 1.88 × 4.0 mm³) for assessment of white matter integrity and fibertracking with TrackVis (http://trackvis.org).

Image processing and analysis of the "resting state fMRI" data was carried out using Probabilistic Independent Component Analysis 85 as implemented in MELODIC (Multivariate Exploratory Linear Decomposition into Independent Components) Version 3.09, part of FSL (FMRIB's Software Library, http://www.fmrib.ox.ac.uk/fsl) 29. The following data pre-processing was applied to the input data: masking of non-brain voxels; voxel-wise de-meaning of the data; normalization of the voxel-wise variance; Pre-processed data were whitened and projected into a 60-dimensional subspace using probabilistic Principal Component Analysis where the number of dimensions was estimated using the Laplace approximation to the Bayesian evidence of the model order 85140. The whitened observations were decomposed into sets of vectors which describe signal variation across the temporal domain (time-courses) and across the spatial domain (maps) by optimizing for non-Gaussian spatial source distributions using a fixed-point iteration technique 87. Estimated Component maps were divided by the standard deviation of the residual noise and thresholded by fitting a mixture model to the histogram of intensity values 85. Figure 5 shows recorded images and a voxel time course from the subject, while Fig. 6 depicts MELODIC component no. 45, representing the default mode network.

A more advanced processing scheme of the multimodal MRI acquisitions from the same subject (651), combining both structural connectivity information from DTI recordings and functional connectivity from resting state fMRI recordings, is shown in Fig. 7. [The original acquisitions Anatomyi.nii, Anatomy2.nii, DWI.nii, bvec.dat, bval.dat, Resting.nii, Fingertapping.nii are available in NIFTI-1 format from the author; see also http://sites.google.com/site/hufy372 ]