Id

Fig. 37. Left: Neural connectivity has local intracortical symmetric components (homogeneous) and patchy corticocortical components (heterogeneous). Right: Approximation of the real local and global connectivity by a symmetric connectivity function with an average path length

Fig. 38. Realistic connectivities are characterized by translationally invariant local connections and translationally variant global connections. The basic model for the study of the interplay of local and global interactions is the embedded two-point connection in a locally connected neural network as shown in the figure

Fig. 38. Realistic connectivities are characterized by translationally invariant local connections and translationally variant global connections. The basic model for the study of the interplay of local and global interactions is the embedded two-point connection in a locally connected neural network as shown in the figure connectivity identifies the basic toy model for the study of local and global interactions. The change in the connection topology destabilizes the initial stationary dynamics and the system undergoes a transition to a new stationary state via a Hopf bifurcation. Detailed bifurcation diagrams are given in (Jirsa & Kelso 2000) in which the spatiotemporal reorganization is characterized as a function of the length of the two point connection.

Minor changes in the location of the terminals or the system parameters, such as the homogeneous or heterogeneous transmission speeds, may result in qualitatively different global neural field dynamics. As an example, in Fig. 39 a stimulus is introduced in the neighborhood of a terminal of a two-point

Fig. 39. A neural field following Jirsa and Haken (1996) with an embedded two-point connection at x=10 and x=40 is established. In the neighborhood of x=10, a brief stimulus excites the neural sheet locally and the neural field reorganizes globally in a large scale transient wave which damps out after a sufficiently long time (not shown here)

Fig. 39. A neural field following Jirsa and Haken (1996) with an embedded two-point connection at x=10 and x=40 is established. In the neighborhood of x=10, a brief stimulus excites the neural sheet locally and the neural field reorganizes globally in a large scale transient wave which damps out after a sufficiently long time (not shown here)

connection (based at x=10 and x=40), then the excitation of the neural field travels through the continuous sheet, but also transmits a signal via the heterogeneous pathway. A transient wave dynamics is observed on the global system scale and damps out after a sufficiently long enough time. With no heterogeneous connection, only a local excitation at x=10 would have been observed. Similarly, with no heterogeneous connection and with two stimuli at terminal sites x=10 and x=40, only two local excitations would have been observed, but no large scale organization as observed in Fig. 39.

5 Conclusion

The neurosciences have historically leaned strongly towards empiricism - a tradition which continues today. However, mathematical formalisms of dynamical phenomena have provided extraordinary explanatory and unifying insights in the physical sciences. The emerging advances in computational neurosciences, particularly with respect to brain connectivity, suggest that they will also come to play an important role in the brain sciences. The cross-fertilization of dynamical systems theory (see also the Chapters by Campbell, Horowitz & Husain and Stephan & Friston), graph theory (Sporns & Tononi), basic physics (Ferree & Nunez), and methodological advances in neuroimag-ing (Darvas & Leahy, Fuchs) will hopefully underpin advances which do not merely reduce problems in neuroscience to problems already solved in other fields, but instead allow those properties of the brain that are unique to inform novel and specific discoveries. We see this blending of universality and specificity as absolutely crucial. Too much of the former will yield simplifications that lose what is required of a system in order to look (and function) like a brain. Conversely, too much specific detail yields volumes of descriptive data that adds little to our understanding of the underlying principles of brain function.

In this chapter, we have overviewed developments in the field of dynamical neural modeling across several scales of magnitude - from the microscopic conductance models of bifurcating neurons, (briefly) through systems of coupled chaotic oscillators at the mesoscopic scale to models of large scale neural networks whose behavior generates the electroencephalographic and neuroimag-ing data that is acquired non-invasively from human subjects. Evidence of computationally significant processes has been documented in data sets from across this spectrum of scales - i.e. from the single cell to the whole brain. An open and important question then is the relationship between activity at different temporal and spatial scales (Churchland and Sejnowski 1992). A possible answer could be that the macroscopic dynamics is an epiphenomenon -that is, a summed output of dynamics that can only truly be modeled at the neuronal level. However, this approach cannot be reconciled with the successes of large-scale models, which engage the brain at macroscopic scales only, to provide descriptive explanations of neuroscience data. That is, as discussed in

Sect. 4 of the present chapter, a mean field reduction of the present state of the system is able to predict its future states. This suggests that synchronizing processes are able to enslave many of the (small scale) degrees of freedom into dynamical structures at larger scales whose behavior is then - to some degree - determined by the state of the system at that scale. Whilst the large scale processes that are sustained by such processes inevitably influence the dynamics of the small scale units, it also remains possible that small scale events - such as critical sensory inputs - are able to rapidly influence the behavior of the system as a whole.

Whilst such considerations preclude a purely reductionist approach, an adequate explanatory framework remains elusive. The situation may be analogous to a heated magnet that it is close to the Curie temperature (above which it loses the ability to be magnetized): The magnetic fields are purely an outcome of the dipoles of spinning electrons. Yet the spinning electrons are also strongly influenced by the larger-scale magnetic fields. Below the Curie temperature, the fields are sufficiently strong to overcome stochastic fluctuations of individual spin directions. Above the Curie temperature the emergent fields are insufficient in strength to enslave the electron dipoles and the metal cannot hold a macroscopic field. However, at the Curie temperature, there is just a sufficient degree of coherence at any given scale to overcome the stochastic fluctuations at the next smaller scale. However, fluctuations at a small scale are able to transiently cascade to larger scales, a phenomena known as criticality and exhibit scale free fluctuations. Or perhaps even more attractive are the spin glass systems where - in addition to these processes - there exist disordered structures embedded in the system which preclude a perfectly ordered system even at low temperatures. Many of such spatiotemporal pattern formation phenomena and their underlying mechanisms have been understood in the framework of Synergetics, a field pioneered by Hermann Haken (1983, 1999).

Whilst such arguments have an attractive appeal, we should bear in mind our own warning that the brain is not just another complex physical system - such as a heated metal - even one with embedded impurities! There exist additional complexities that are surely important to brain function. One such critical difference is that there do exist structures across spatial and temporal scales prior to the emergence of dynamically driven scale-free (and scale-specific) fluctuations. Is it possible that the interaction between scale-specific processes across a hierarchy of scales is somehow optimal? Fusi et al. (2005) have shown how a hierarchy of synaptic processes - each with characteristic time scales - can interact in order to optimize memory retention (upgrading new memories) and storage (maintaining selected memory for long periods of time). Breakspear & Stam (2005) modeled the interaction between scale-free dynamics and multiscale spatial architectures by defining dynamical systems on different wavelet subspaces, and with cross-scale coupling between subspaces. This would potentially allow for a recursive relationship between small and large-scale dynamics.

Such observations hopefully reflect the challenge of fusing the universal with the specific as an emerging frontier in neuroscience research.

Acknowledgements

The authors wish to acknowledge helpful comments by SA Knock, PL Nunez, PA Robinson and E Tognoli and funding by ATIP (CNRS), Brain NRG JSMF22002082, NH&MRC, Australia and ARC Australia.

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