Dynamics of a neural system with a multiscale architecture

Abstract

The architecture of the brain is characterized by a modular organization repeated across a hierarchy of spatial scales-neurons, minicolumns, cortical columns, functional brain regions, and so on. It is important to consider that the processes governing neural dynamics at any given scale are not only determined by the behaviour of other neural structures at that scale, but also by the emergent behaviour of smaller scales, and the constraining influence of activity at larger scales. In this paper, we introduce a theoretical framework for neural systems in which the dynamics are nested within a multiscale architecture. In essence, the dynamics at each scale are determined by a coupled ensemble of nonlinear oscillators, which embody the principle scale-specific neurobiological processes. The dynamics at larger scales are 'slaved' to the emergent behaviour of smaller scales through a coupling function that depends on a multiscale wavelet decomposition. The approach is first explicated mathematically. Numerical examples are then given to illustrate phenomena such as between-scale bifurcations, and how synchronization in small-scale structures influences the dynamics in larger structures in an intuitive manner that cannot be captured by existing modelling approaches. A framework for relating the dynamical behaviour of the system to measured observables is presented and further extensions to capture wave phenomena and mode coupling are suggested.

Figure 1

Wavelet representation of a chirp, f(x). The chirp is shown as its centre frequency increases with time, showing a helpful time–frequency decomposition. Left column shows the projection of f on to a hierarchy of approximation subspaces through Π, and the right column shows the projection of f on to a hierarchy of wavelet subspaces through Λ. Note that the increasing frequency component (scales) is located progressively towards the end of the signal. The unmasked components represent an exemplar decomposition up to scale J=4.

Figure 2

Contour map of a single slice BOLD signal from a human subject (1.5 T). (a) Raw (unfiltered) data; (b) Gaussian smoothed data with confluence from all scales; and (c)–(f) four wavelet subspace projections, with scale-specific signal component.

Figure 3

Exemplar attractors and time-series for the neural system employed in this study for equation (3.1). Complex periodic (panels a and b) and chaotic (panels c and d) are shown. Panels (a) and (c) show the orbits and attractors in the system's phase space. Panels (b) and (d) show the corresponding time-series of the pyramidal cell potentials, V.

Figure 4

Dependence of the intrascale ‘synaptic footprint’ H on four choices of K, equation (3.4): K=4 (solid); K=1 (dotted); K=0.1 (dot-dashed); and K=0.01 (dashed).

Figure 5

Uncoupled array, c=0. (a) Cross-section through the system's evolution at an arbitrary time point; (b) contour plot of the system's spatio-temporal evolution; and (c) time-series of the mean field of the ensemble.

Figure 6

Dynamics with a broad synaptic footprint (K=0.01) and weak coupling (c=0.03). (a) Cross-section through the system's evolution at an arbitrary time point; (b) contour plot of the system's spatio-temporal evolution; and (c) time-series of the mean field of the ensemble.

Figure 7

Local and scale-free structures with a narrow synaptic footprint (K=1) and c=0.54. (a) Cross-section through the system's evolution at an arbitrary time point; (b) contour plot of the system's spatio-temporal evolution; and (c) time-series of the mean field of the ensemble.

Figure 8

Complete synchronization, K=0.01, c=0.06. (a) Cross-section through the system's evolution at an arbitrary time point; (b) contour plot of the system's spatio-temporal evolution; and (c) time-series of the mean field of the ensemble.

Figure 9

Travelling waves, K=16, c=3.54. The wave in (a) is moving from left to right. (a) Cross-section through the system's evolution at an arbitrary time point; (b) contour plot of the system's spatio-temporal evolution; and (c) time-series of the mean field of the ensemble.

Figure 10

Spatial spectra of four coupled systems. (1) No coupling, (2) weak coupling c=0.03 and K=1, (3) moderate coupling c=0.54, K=1, (4) travelling wave c=3.54, n=16.

Figure 11

Schema of the multi-scale model of equation (3.5). Two-tiered system with coarse-grained (J) and fine-grained (j) scales. Local ensemble at position x_i in the coarse-grained system is governed by F^(J). Within-scale coupling is introduced through H. The scale-congruent ‘harmonic’ of the dynamics at scale j are introduced through G, acting on the wavelet projection Λ^(jJ)V^(j) evaluated at x_i.

Figure 12

Impact of uncoupled fine-grained dynamics on spatio-temporal structures within a coarse-grained scale c^(J)=0. Contour plots of coarse-grained system with (a) no inter- or intrascale coupling; (b) strong input C=0.5 from an uncoupled fine-grained system; and (c) weak interscale coupling C=0.075 and strong intrascale coupling c^(J)=0.6.

Figure 13

Surface maps of entropy in coarse-grained system S^(J) as a function of intra- (c) and inter- (C) scale coupling. (a) Input from an uncoupled fine-grained system; (b) input from a synchronized system; and (c) difference in entropy when input is from a fine-grained system with spatio-temporal structures (c^(j)=0.03, k=0.01) compared with input from an uncoupled system. Note that for visualization, plot (b) is flipped in the {C, c}plane compared with (a) and (b).

Figure 14

Between-scales bifurcation. Panels (a), (c) and (e) show contour plots of synchronized fine-grained system. c^(j)=0.6. Impact of this system on a coarse-grained system with c^(J)=0.35 and (b) C=0, (d) C=0.025, (f) C=0.075. Panel (g) shows a cross-section through the surface plot of figure 13b at c^(J)=0.35.

Figure 15

Results of leaving t^(j)=1 but setting t^(J)=2. Panels (a), (c) and (e) show the mean-field oscillations of the fine- (dashed) and coarse- (solid) grained systems. Panels (b), (d) and (f) show the dynamics of the coarse-grained ensemble. In panels (a) and (b), c^(j)=0.6 and c^(J)=0.35 but C=0. Introducing between-scale coupling C=0.025, panels (c) and (d). Increasing the within-scale coupling of the coarse-grained system c^(J)=0.75, panels (e) and (f).

Figure 16

Injection of travelling waves from fine-grained system. Panel (a) strong interscale input (C=0.4) with weak intrascale coupling in the coarse-grained system (c^(J)=0.05); panel (b) increasing the strength of the intrascale coupling (c^(J)=0.3); and panel (c) increasing c^(J)=0.6.

Figure 17

Injection of spatio-temporal structures from fine-grained system. Both systems have internal connectivity (c^(j)=0.54, c^(J)=0.1). The fine-grained system is overlaid in red on the coarse-grained system. Panel (a) no interscale coupling C=0; and panel (b) interscale coupling, C=0.4, causes coincident spatio-temporal structures (arrows).

Figure 18

Three-tiered system with the injection of travelling waves from a fine-grained system j₁ with n=64 into a coarse-scaled system J with n=16 and additional influence of a very fine-grained system. Intrascale (c^(J)=0.2) and interscale (C=0.35) coupling were chosen to yield coarse-grained travelling waves in the absence of the third scale. Panel (a) no coupling in the very fine system c^(j2)=0, with G_j1=0.33 and G_j2=0.67; (b) injection of a very fine-grained system with internal coupling (c^(j2)=0.54) and G_j1=0.33, G_j2=0.67; and (c) effect of a completely synchronized very fine-grained system (c^(j2)=0.8) and G_j1=0.33, G_j2=0.67.

Figure 19

Three-tiered system with different internal time-scales. Panel (a) contour maps with parameters as shown; (b) mean-field oscillations of very fine system (dashed), fine-grained system (dotted) and coarse-grained system (solid); and panels (c) and (d) the same system but G_j2j1 has been increased from 0.05 to 0.11.

Figure 19

Three-tiered system with different internal time-scales. Panel (a) contour maps with parameters as shown; (b) mean-field oscillations of very fine system (dashed), fine-grained system (dotted) and coarse-grained system (solid); and panels (c) and (d) the same system but G_j2j1 has been increased from 0.05 to 0.11.