Transient neocortical gamma oscillations induced by neuronal response modulation

Abstract

In this paper a mean field model of spatio-temporal electroencephalographic activity in the neocortex is used to computationally study the emergence of neocortical gamma oscillations as a result of neuronal response modulation. It is shown using a numerical bifurcation analysis that gamma oscillations emerge robustly in the solutions of the model and transition to beta oscillations through coordinated modulation of the responsiveness of inhibitory and excitatory neuronal populations. The spatio-temporal pattern of the propagation of these oscillations across the neocortex is illustrated by solving the equations of the model using a finite element software package. Thereby, it is shown that the gamma oscillations remain localized to the regions of neuronal modulation. Moreover, it is discussed that the inherent spatial averaging effect of commonly used electrocortical measurement techniques can significantly alter the amplitude and pattern of fast oscillations in neocortical recordings, and hence can potentially affect physiological interpretations of these recordings.

Appendix: Bifurcation analysis results for a different set of parameter values

To show the robustness of the results of Sections 3 and 4 against changes in parameter values, the bifurcation analysis of these sections is repeated in this appendix for a different set of parameter values as given in Table 3. These parameter values are generated by randomly perturbing the values used in Sections 3 and 4 by a magnitude of 10 to 100 percent. The codimension-one bifurcation diagrams and the frequency of the limit cycles are shown in Figs. 13 and 14. These diagrams resemble the diagrams shown in Figs. 2 and 3, respectively, and equivalently imply the emergence of gamma oscillations as a result of a Hopf bifurcation. The result of two-parameter continuation of the bifurcation points detected in Fig. 13a for neuronal sensitivity modulation is shown in Fig. 15. The bifurcation diagrams for the nominal value of F_e given in Table 3, as well as the diagram for a modulated lower value of F_e = 13 [s^− 1] are shown in Fig. 16, which illustrates the coordinated transition from gamma oscillations to the initial resting state. The results are highly comparable to the results obtained in Section 4. Similarly, two-parameter analysis of excitability (μ_e and μ_i) and saliency (σ_e and σ_i) modulations leads to results very close to those obtained in Section 4, and hence are not included here.

Fig. 13

Codimension-one bifurcation diagrams of the ODE system described in Section 3 for the nominal parameter values given in Table 3. The same description as given in Fig. 2 holds for the curves of equilibria and limit cycles, as well as the detected bifurcation points

Fig. 14

Frequency of the limit cycles in the bifurcation diagrams shown in Fig. 13

Fig. 15

Two-parameter continuation of the bifurcation points shown in Fig. 13a. The starting bifurcation points are indicated by dots with the same colors as they appear in Fig. 13a. The curve shown in gray is the result of the continuation of a fold bifurcation of limit cycles that is not made visible in Fig. 13a. The detected codimension-two zero-Hopf bifurcation point is indicated by ZH. The detected codimension-two generalized Hopf bifurcation point is indicated by GH

Fig. 16

Mechanism of transient emergence of gamma oscillations through coordinated modulation of inhibitory and excitatory neuronal sensitivity. The diagrams are obtained using the parameter values described in the Appendix. The description of the modulatory actions and phase transitions in Steps 1 to 4 follows the same description as given in Fig. 8

A note on using MatCont for the numerical bifurcation analysis of this paper can be of interest for future works on this model. The MatCont versions 6.6 and 7.1 were used to perform the computations. To achieve convergence and obtain the full extent of the curves presented in the bifurcation diagrams, the continuation parameters are usually needed to be re-adjusted manually for each diagram. These re-adjustments are especially needed when continuing the curves of limit cycles and their fold bifurcations. The convergence error typically observed is ‘current step size too small’. The direction of the continuation of limit cycles may occasionally be reversed during the continuation, especially near the fold bifurcations. No general rule was observed for adjusting the continuation parameters so that these problems are resolved. However, setting the number of mesh points equal to 6 and the number of collocation points equal to 5 usually results in smoother continuation of the limit cycles. Adjustment of the initial amplitude can affect the convergence of the continuations. A value between 0.1 to 5 was typically chosen for this parameter. Finally, adjustment of the maximum step size MaxStepsize option for continuation of the limit cycles is frequently needed to resolve convergence errors or to avoid reversals in the direction of continuation. Values as large as 50000 or larger for MaxStepsize are needed in some cases.