Desarrollo de las actividades

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The cortex can be approximated by a two-dimensional system of interacting neural masses if it is assumed that the vertical direction is redundant, as is the case when considering the local activity of neural masses (Wilson and Cowan, 1973; Mount- castle, 1997). This induces an additional spatial degree of freedom compared to the one-dimensional case, which must be investigated in model systems arranged

Figure 3.3: Effect of changes in excitatory a) and inhibitory b) gain on the threshold required to propagate activity to the edge of the 21 compartment system. In a), R=60, B=22. In b), R=60, A=3.25. Vertical lines indicate previously used default values of these parameters (Jansen and Rit (1995)).

in two-dimensions. Here, we examined these systems in a hexagonally arranged, nearest neighbour coupled scheme. The central compartments have 6 neighbours each, whereas the peripheral compartments have 3 neighbours.

We start with a model of one central compartment surrounded by 6 symmet- rically coupled compartments, which represents the simplest such 2 dimensional configuration (c.f. the arrangement of grey hexagons in Figure 3.8). The dynam- ics of this model of 7 coupled compartments are characterised by the bifurcation diagram in Figure 3.4 (a). It can be seen that these dynamics follow a similar struc- ture to those of the one dimensional chain, in that the fixed points are interrupted by a region of oscillations. However, the critical value for onset of oscillations is lower (compare Figure 3.2 (a)) due to each compartment communicating with more neighbours and therefore receiving an increased net contribution for equivalent R.

For R ≤ 32 the fixed point background solution persists. At R=32.5, the system enters a region of bistability between fixed point and oscillatory dynamics. The oscillatory activity in this region is around 8 Hz, with phases distributed across the 7 compartments. Figure 3.4 (b) shows that whilst the amplitude of the peripheral compartment can be relatively stable, the amplitudes in other compartments (here, the central compartment) may vary, leading to sub-harmonic components in the Fourier spectrum. Between R=36 and R=40, the system is in a region of bistability between large amplitude spiking activity, occurring at around 1 spike per second, and

oscillations of smaller amplitude at around 8Hz, with phases distributed across the 7 compartments. The dynamics of the former are shown in Figure 3.4 (c), and the smaller amplitude oscillations are shown in Figure 3.4 (d). The effect of increasing R beyond 40 was i) to bring into closer alignment the phases of the oscillations in the peripheral compartments, and ii) to produce an offset between the central compartment, which oscillated at a higher mean value, and the rest of the system, which oscillated around a lower mean.

Figure 3.4: Dynamics of 7 compartments with hexagonal nearest neighbour coupling. a) bifurcation diagram for changing R. Maxima and minima are plotted for a peripheral compartment. The forward scan is plotted in black and the backward scan is plotted in red. b) mixed oscillations at R=34, c) spiking solution at R=38, , d) oscillatory solution at R=38. In b-d, time series for the central compartment and an outer compartment are plotted with an offset along the vertical axis.

A larger region of cortex was modelled by an extended two-dimensional sheet of connected model compartments. At the default value of B=22, low values of R in large two dimensional systems typically lead to sustained mixed oscillations. For example a 15 × 15 grid with R=20 outputs heterogeneous oscillations with apparent

frequency components ranging from 5 to 10 per second and heterogeneous ampli- tudes and phases, somewhat reminiscent of mixed oscillations in the one dimensional case. As in the one dimensional case, these oscillations can be replaced by the fixed point solution if B is increased for a given coupling strength. In such a case, a stimulus applied to the system in the background fixed point can evoke a travelling wave, as shown in Figures 3.5 (a) and (b). A movie of this response is given in supplementary file “SFigure5a.mp4”, which can be obtained from Goodfellow et al. (2011b).

In contrast, at different values of coupling strength R, more complex responses can be observed (Figures 3.5 (c) and (d)). Here, the system is identical to that in Figures 3.5 (a) and (b), but R = 35, i.e. coupling is stronger. The initially induced wave activity evolves with a tail of higher amplitude compared to that observed for R=20, and causes the evolution of two subsequent travelling fronts. Collision of wavefronts then eliminates further activity leading to a total duration of around 1 second for this transient. A movie of this response is given in supplementary file “SFigure5c.mp4”, which can be obtained from Goodfellow et al. (2011b).

In addition to travelling waves, it is expected that a two dimensional excitable system can propagate spiral activity (see e.g. Winfree (2001)). Indeed, our two dimensional, excitable system is capable of generating complex spiral waveforms, as shown in Figure 3.6 (a) and (b), for a N =421 square arrangement of hexagonally coupled compartments with B=35 and R=50. A movie of these dynamics is given in supplementary file “SFigure6.mp4”, which can be obtained from Goodfellow et al. (2011b).

The effect of changes in inhibitory gain, B, and connection strength, R, on the dynamics of a cortical patch modelled by the 421 compartment system were explored with random initial conditions over different combinations of these two parameters. Figure 3.7 (a) maps the presence of fixed point versus oscillatory dynamics and therefore indicates the region appropriate for examining stimulus induced transitions from the lower steady state. It can be seen that fixed point activity is generally found for extremely low or high B. The fixed point region to the left of the oscillatory band is saturated in a higher state, though peripheral compartments still oscillate for smaller R. The band to the right of the oscillatory region resides at the lower steady state and is therefore the starting point for the investigation of responses to stimuli.

The response to stimuli in the region of the lower steady state solution (white region to the right of Figure 3.7 (a)) was found to vary depending upon the value of R and B, the effects of which are explored in Figure 3.7 (b). Figure 3.7 (b) is colour coded by length of activity evoked by stimulation, with darker colours indicating longer transients, and displays approximately three regions of differing activity. In the very light grey region at the top right (approximately R ≤ 18 and

Figure 3.5: The effect of perturbation to a central compartment in the two dimen- sional, homogeneous system. a) snapshots of wave propagation due to stimulus, R=20 and B=28, at t=10.1, 10.3 and 10.5 seconds The black line indicates a one dimensional projection for the time series plotted in b), which are plotted with an offset on the vertical axis. c) more complex transient due to stimulus, snapshots shown at t=10.1, 10.5 and 10.8 seconds. R=35 and B=28, time series for compart- ments along the black line in c) are plotted in d) with an offset on the vertical axis. The duration of these responses can be seen in supplementary files “SFigure5a.mp4” and “SFigure5c.mp4”, respectively, which can be obtained from Goodfellow et al. (2011b).

B ≥ 22) a stimulus evokes no travelling response and therefore the duration of the perturbation to model output is of the order of the length of stimulation. In the darker grey region (approximately 18 ≤ R ≤ 40 and B ≥ 25), the stimulus evokes at least one travelling wave. In general, for constant R, the responses are more complex for smaller B, i.e. near the boundary of the oscillatory domain.

The darker squares within this region, including those coloured black, elicit more than one travelling wave or more complex responses. We note that the number of compartments stimulated, as well as their position in the system, affects activity

Figure 3.6: Example of complex spiral waves in a system with N =421, B=35, R=50. Images in (a) represent t=12.1, 12.35 and 12.6 seconds. Time series for the compartments along the black line in (a) are plotted in (b) with an offset along the vertical axis. A movie of these dynamics is provided in supplementary file “SFigure6.mp4”, which can be obtained from Goodfellow et al. (2011b).

post-stimulus. For instance, if stimulation of a single compartment does not elicit a response, the stimulation of multiple compartments occasionally can. The response of the system to stimulus at the centre and at the edge of the system is often not equivalent.