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, 55 (1), 39-48

Neural Fields, Spectral Responses and Lateral Connections

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Neural Fields, Spectral Responses and Lateral Connections

D A Pinotsis et al. Neuroimage.

Abstract

This paper describes a neural field model for local (mesoscopic) dynamics on the cortical surface. Our focus is on sparse intrinsic connections that are characteristic of real cortical microcircuits. This sparsity is modelled with radial connectivity functions or kernels with non-central peaks. The ensuing analysis allows one to generate or predict spectral responses to known exogenous input or random fluctuations. Here, we characterise the effect of different connectivity architectures (the range, dispersion and propagation speed of intrinsic or lateral connections) and synaptic gains on spatiotemporal dynamics. Specifically, we look at spectral responses to random fluctuations and examine the ability of synaptic gain and connectivity parameters to induce Turing instabilities. We find that although the spatial deployment and speed of lateral connections can have a profound affect on the behaviour of spatial modes over different scales, only synaptic gain is capable of producing phase-transitions. We discuss the implications of these findings for the use of neural fields as generative models in dynamic causal modeling (DCM).

Figures

Fig. 1
Fig. 1
Connectivity kernel. Connectivity kernel describing the strength of intrinsic (lateral) connections within a neuronal field mode; see Eq. (12). The insert was modified from www.ini.uzh.ch/node/23776.
Fig. 2
Fig. 2
The spectrum obtained using the connectivity kernel (12). The spectrum ω of predicted steady-state responses using the bimodal connectivity scheme in Eq. (12). The spectrum is shown in the complex plane, where the imaginary part (vertical axis) determines the frequency of the response and the real part (horizontal axis) reflects the time course over which responses to perturbations decay (cf, the amplitude under white noise perturbations). The blue dots are the origin of each semi-branch at k = 0 (i.e. at the lowest spatial frequency). The semi-branches B1 and B2 are first order branches of multi-branched spectra associated with the neural field equation (for more details on spectra with an infinity of branches, see (Grindrod and Pinotsis)). Also, RS denotes the finite positive real part of the spectrum.
Fig. 3
Fig. 3
Real and imaginary parts of the spectrum. The blue and green curves are the real and imaginary parts respectively of a spectrum (the blue semi-branch in the previous figure) as a function of spatial wave-number (frequency).
Fig. 4
Fig. 4
Log-power spectra for different values of connectivity range. Log-power spectra as a function of frequency, based on the solution to Eq. (14). Red curve for a connectivity range of a = 1 and blue curve for a = 1.3.
Fig. 5
Fig. 5
Log-power spectra for different values of conduction velocity. As for Fig. 4, but changing the conduction velocity from low (red curve: ε = 5) to higher (blue curve: ε = 2) values.
Fig. 6
Fig. 6
Log-power spectra for different values of dispersion. As for Fig. 4, but changing the spatial extent (dispersion) of afferent populations from c = 1 (red curve) to c = 2 (blue curve).
Fig. 7
Fig. 7
Complex spectra for various values of connectivity range. The spectrum ω of predicted steady-state responses using the bimodal connectivity scheme in Eq. (12). The spectrum is shown in the complex plane, where the imaginary and real parts of ω are depicted on the vertical and horizontal axes respectively (see also Fig. 2).Here c = 2, ε= 20 and we vary a between 0.6 and 1.4: Each coloured line corresponds to a first order branch for various values of a; the branch corresponding to a = 1 is depicted in red. Each blue dot depicts the corresponding origin of each branch.
Fig. 8
Fig. 8
Complex spectra for various values of velocity. As for Fig. 7, but choosing c = 2, a= 1 and varying ε between one fourth and four times its original value ε= 20. The panels show a first order branch as the conduction velocity increases (above) or decreases (below); the branch corresponding to ε= 20 is depicted in red.
Fig. 9
Fig. 9
Complex spectra for various values of dispersion. As for Fig. 7, but choosing ε= 20, a= 1 and varying c between one fourth and four times its original value c = 2.
Fig. 10
Fig. 10
Complex spectra for various values of gain. As for Fig. 7, but choosing ε= 20, a= 1, c = 2 and varying g between one fourth and eight times its original value g = 4. The right and left panels show a first order branch as the gain decreases (above) or increases (below). The branch corresponding to g = 4 is depicted in red. A Hopf bifurcation occurs for g = 16 and temporal frequency w = 0.36 (red point in the right panel).
Fig. 11
Fig. 11
Log-power spectrum involving a peak. Log-power spectrum as a function of frequency (w), based on the solution to Eq. (14) with c = 2, ε= 5, a = 1 and g = 16. Note the peak at w = 1.7.

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