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Fig. 33. The characteristic connectivity of Wilson & Cowan's coupled population approach reflects local excitation and inhibition with various degrees of laterality the axons not to leave the gray matter and clearly limits the application of these neural fields to local area networks.

The Neural Field Model of Jirsa & Haken (1996) and Wave Equations

Based on first principles using pulse-wave and wave-pulse conversions, Jirsa & Haken (1996, 1997) developed a neural field approach (see for early accounts of neural field theories Griffith 1963, 1965) targeted specifically towards large scale phenomena as observed in EEG, MEG. Initially based on two locally coupled neural masses of excitatory and inhibitory neurons, the action of the inhibitory neural mass is absorbed into an effective excitatory neural mass action ^(x,t). This reduction is possible under the assumption that the in-trisic dynamics of the neural mass is negligible and relaxes instantly to its steady state, i.e. the neural mass action displays a fixed point dynamics. Then the network dynamics will exclusively be determined by the connectivity and its time delays and captured by an equation equivalent to (48), but with a scalar connectivity function h(x—x') (see Fig. 34). As in the Nunez model, the connectivity includes local intracortical connections and global corticocortical projections. As a first approximation, h(x—x') is assumed to be translationally invariant and follows an exponential decay as plotted in Fig. 32. Under these xi x2 x3 x, ; x-, xf, x- xy x

Fig. 34. The characteristic connectivity of the Jirsa-Haken wave equation emphasizes excitatory long range connectivity after elimination of the local inhibitory effects. The latter are captured in an effective neural mass action ^(x,t)

conditions, Jirsa & Haken showed that the integro-differential (48) is equivalent to the following partial differential equation in one physical dimension:

where w0 = v/a, v is the transmission speed along myelinated axons and a the mean fiber length. Early accounts of wave phenomena in EEG and their discussion in the context of wave equations can be found in (Nunez 1995).

The Jirsa-Haken wave equation (51) approximates various connectivity functions of large scale networks in the limit for long waves, or, in other words, large scale activity patterns. If the slope of the sigmoid function S increases beyond a threshold, then the rest state becomes unstable and undamped wave propagation occurs. Below the threshold damped wave propagation exists. Steven Coombes and colleagues (2003) discuss the effects of connectivity strengths which do not decrease with increasing distance, but rather remain constant within a finite regime. In this case, it is not sufficient to describe the spatiotemporal dynamics by a local partial differential equation as in (51), but non-local delayed terms arise (see Coombes 2005 for a review). Wright and colleagues introduced much physiological detail and were able to address issues of rhythm generation (Wright & Liley 1996), as well as clinical aspects such as hysteresis phenomena in anesthesia (Steyn-Ross et al. 1999). Robinson and colleagues introduced expressions for the corticothalamic loop into the Jirsa-Haken equation (see next section) and included dendritic dynamics while implementing detailed physiologically realistic parameter ranges (Robinson 1997, 2001). Frank and colleagues developed a Fokker-Planck approach to the Jirsa-Haken equation which captures the time evolution of the stochastic properties of the neural fields (Frank et al. 1999, 2000). Applications to encephalograpic data can be found in (Jirsa and Haken 1997; Jirsa et al. 1998, 2002; Fuchs et al. 2000; Liley et al. 2002; Jirsa 2004b; Robinson et al. 2004, 2005; Breakspear et al. 2006).

The Inclusion of the Thalamocortical Loop into Neural Fields (Robinson 2001)

In 1997 Robinson et al. presented an equivalent derivation of the Jirsa-Haken equation considering effects of dendritic dynamics and added the important extension of the thalamocortical loop in 2001 (see Fig. 35). The inclusion of the thalamocortical interactions proved to be crucial to reproduce the essential spectral properties observed in scalp topographies. Robinson and colleagues preserve the neural field as a vector field *(x,t) = (*i(x,t), *2(x,t)) of excitatory and inhibitory neural masses and write the following equations d 2*(x,t) d *(x,t) 2 d2*(x,t) 2l/ N

Fig. 35. Robinson et al. (2001) capture corticothalamic effects contributing to neural field dynamics. The effect of excitatory and inhibitory influences is collapsed into the upper row

Fig. 35. Robinson et al. (2001) capture corticothalamic effects contributing to neural field dynamics. The effect of excitatory and inhibitory influences is collapsed into the upper row for the dynamics of the neural field. The critical step is that the sigmoid function p (^ (x, t)) does not only depend on the neural fields ^i(x,t), ^2(x,t), but also receives time-delayed thalamic input ^th(t — t/2).

The thalamic action ^th(t) is governed by the following differential equation d^t(t) + ( + b) ^f^ + ab^th(t) = input(*(z, t — t/2)) (53)

where the cortical input to the thalamus also undergoes a delay t/2 via propagation resulting in a effective delay t of the total corticothalamic loop. Computer simulations of equations (52) and (53) provide representative EEG power spectra as shown in Fig. 36.

Frequency in Hz

Fig. 36. Power spectra from the Robinson model of corticothalamic activity in eyes closed (solid) and eyes open (dashed) resting states. The increase of low frequencies in the eyes open condition reflects increased corticocortical gain, whereas the increased alpha (10 Hz) peak in the eyes closed condition reflects increased corti-cothalamic gain

### Frequency in Hz

Fig. 36. Power spectra from the Robinson model of corticothalamic activity in eyes closed (solid) and eyes open (dashed) resting states. The increase of low frequencies in the eyes open condition reflects increased corticocortical gain, whereas the increased alpha (10 Hz) peak in the eyes closed condition reflects increased corti-cothalamic gain

### Extensions and Limitations of Large Scale Models

Large scale systems are characterized by an anatomical connectivity with massively parallel and serial, hierarchical structures, as well as time delays due to signal transmission. Such architecture produces an interareal connection topology, which is patchy as observed by Braitenberg & Schiiz (2001) and results in a heterogeneous connectivity. Yet it has been approximated in various attempts by a homogeneous connectivity with a larger extension (see Fig. 37). The approach uses a larger mean path length and hence effectively mixes functionally the intracortical and corticocortical fiber systems. Research of this kind has successfully reproduced various large scale characteristics of activity including the dispersive properties of the cortex (Nunez 1995) or global EEG power spectra (Robinson 2001); it also shows promise in situations of highly symmetric functional connectivity (Jirsa et al. 1997, 1998; Fuchs et al. 2000). However, to this date, it has not been shown rigorously under what conditions the homogeneous approximation holds.

Mallot and colleagues (1989, 1996) discussed in a series of papers a conceptual framework in which, rather than just mean fields, local networks communicate across distances. These local networks have an intrinsic fixed point dynamics, but exchange information via time-delayed pathways. Mallot and colleagues applied this approach to examples of the thalamocortical loop (Mallot et al. 1996) and for the geniculate-striate pathway of the visual system (Mallot et al. 1989). Similarly, discretely coupled local networks incorporate time delays in the connecting pathways and absorb all local dynamics within a set of coupled neural masses (Freeman 1975, 1992; David & Frison 2003). Jirsa & Kelso (2000) studied the neural field dynamics of the Jirsa-Haken equation in which a heterogeneous pathway is included (Fig. 38). Such a two-point pathway connects the neural masses at locations X2 and X8 which are embedded into a continuous sheet with local connections only. This

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