Open & accessible for everyone. Coming to your browser.

Network nodes in TVB: neural populations with local mesoscopic dynamics

Mesoscopic dynamics describe the mean field activity of populations of neurons organized as cortical columns or subcortical nuclei. Common assumptions in mean-field modeling 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 behavior is only weakly sensitive to the details of individual neuron behaviour (Breakspear Jirsa 2007).

Basic mean field models capture changes of the mean firing rate (Brunel Wang 2003), where as more sophisticated mean field models account for parameter dispersion in the neurons and the subsequent richer behaviour of the mean field dynamics (Assisi et al 2005; Stefanescu Jirsa 2008, 2011; Jirsa Stefanescu 2010).

These approaches demonstrate the relatively new concept from statistical physics that macroscopic physical systems obey laws that are independent of the details of the microscopic constituents they are built of (Haken 1983).

These and related ideas have been exploited in neurosciences (Kelso 1995; Buzsaki 2006).In TVB, our main interest lies in deriving the mesoscopic laws that drive the observed dynamical processes at the macroscopic large brain scale in a systematic manner.

Various mean-field models are available in TVB reproducing typical features of mesoscopic population dynamics.

For each node of the large-scale network, a neural population model describes the local dynamics. The neural population models in TVB are well-established models derived from the ensemble dynamics of single neurons (Wilson Cowan 1972; Jansen Rit 1995; Larter 1999; Brunel Wang 2003; Stefanescu Jirsa 2008).

TVB offers also a generic two-dimensional oscillator model for the use at a network node capable of generating a wide range of phenomena as observed in neuronal population dynamics such as multistability, coexistence of oscillatory and non-oscillatory behaviors, various behaviors displaying multiple time scales, etc. just to name a few.

The generic large-scale brain network equation in TVB

When traversing the scale to the large-scale network, then each network node is governed by its own intrinsic dynamics in interaction with the dynamics of all other network nodes.

This interaction happens through the connectivity matrix via specific connection weights and time delays due to signal transmission delays. The following (generic) evolution equation (Jirsa 2009) captures all the above features and underlies the emergence of the spatiotemporal network dynamics in TVB:

:math:`{\dot\Psi(x,t)} = N(\Psi(x,t)) +`

:math:`\int_{\Gamma}g_{local}(x,x\prime)S(\Psi(x\prime,t))dx\prime +`

:math:`\int_{\Gamma}g_{global}S(\Psi(x\prime,t - \frac{|x-x\prime|}{\nu}))dx\prime + I(x,t) + \xi (x,t)`

The equation describes the stochastic differential equation of a network of connected neural populations.

:math:`\Psi(x,t)` is the neural population activity vector at the location x in 3D physical space and time point t. It has as many state variables as are defined by the neural population model, which is specified by :math:`N(\Psi(x,t))`.

The connectivity distinguishes local and global connections, which are captured separately in two expressions.

The local network connectivity :math:`g_{local}(x,x\prime)` is described by connection weights between x and x’, whereas global connectivity is defined by :math:`g_{global}(x,x\prime)`.

The critical difference between the two types of connectivity is threefold:

• 1. Local connectivity is short range (order of cm) and global connectivity is long range (order of 10cm).
• 2. Signal transmission via local connections is instantaneous, but via global connections undergoes a time delay dependent on the distance :math:`|x-x\prime|` and the transmission speed v.
• 3. Local connectivity is typically spatially invariant (of course with variations from area to area, but generally it falls off with distance), global connectivity is highly heterogeneous.

Stimuli of any form, such as perceptual, cognitive or behavioral perturbations, are introduced into the Virtual Brain via the expression I(x,t) and are defined over a location x with a particular time course.

Noise plays a crucial role for the brain dynamics and hence for brain function (see McIntosh et al 2010). In TVB it is introduced via the expression :math:`\xi (x,t)` where the type of noise and its spatial and temporal correlations can be specified independently.

The TVB Simulator

Various numerical algorithms are available in TVB and can be coarsely categorized into deterministic (no noise) and stochastic (with noise). They include the Heun algorithm, Runge Kutta of various orders, Euler Maruyama, and others.

EEG-MEG forward solution in TVB

Noninvasive 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 inverse problem.

As a consequence, the EEG and MEG backward solution is underdetermined (Helmholtz 1853). 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.

The forward problem of the EEG and MEG is the calculation of the electric potential V(x,t) on the skull and the magnetic field B(x,t) outside the head from a given primary current distribution D(x,t). The sources of the electric and magnetic fields are both, primary and return currents.

The situation is complicated by the fact that the present conductivities such as the brain tissue and the skull differ by the order of 100.

In TVB, three compartment volume conductor models are constructed from structural MRI data using the MNI brain; 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 it is assumed 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 (~85 % of all neurons).

So far subcortical regions are not considered in the forward solution. We also neglect dipole contributions from inhibitory neurons since they are only present in a low number (~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. Then the primary current distribution D(x,t) is obtained as the set of all normal vectors perpendicular to the vertices at locations x of the cortical surface multiplied by the relevant state variable in the population vector :math:`\Psi(x,t)`.

fMRI-Bold contrast in TVB

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 (Petzold, Albeanu et al. 2008; Giaume, Koulakoff et al. 2010).

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.

In order to estimate the BOLD signal, the mean-field amplitude time course of a neural source may be convolved with a canonical hemodynamic response function as included in the SPM software package or the “Balloon-Windkessel” model of (Friston, Harrison et al. 2003) may be employed; cf. (Bojak, Oostendorp et al. 2010) for some more technical details.