Spike-frequency and h-current based adaptation are dynamically equivalent in a Wilson-Cowan field model

TL;DR

The dynamical systems are mathematically equivalent under a compensatory external input, which depends on the adaptation strength, leading to a shift in state space of the otherwise equivalent bifurcation structure.

Abstract

During slow-wave sleep, the brain produces traveling waves of slow oscillations (SOs; $\leq 2$ Hz), characterized by the propagation of alternating high- and low-activity states. The question of internal mechanisms that modulate traveling waves of SOs is still unanswered although it is established that it is an adaptation mechanism that mediates them. One mechanism investigated is spike-frequency adaptation, a hyperpolarizing feedback current that is activated during periods of high-activity. An alternative mechanism is based on hyperpolarization-activated currents, which are positive feedback currents that are activated in low-activity states. Both adaptation mechanisms were shown to feature SO-like dynamics in neuronal populations, and the inclusion of a spatial domain seems to enhance observable differences in their effects. To investigate this in detail, we examine a spatially extended two-population Wilson-Cowan model with local spatial coupling and the excitatory populations equipped with either one of the two adaptation mechanisms. We describe them with the same dynamical equation and include the inverse mode of action by changing the signs of adaptation strength and gain. We show that the dynamical systems are mathematically equivalent under a compensatory external input, which depends on the adaptation strength, leading to a shift in state space of the otherwise equivalent bifurcation structure. Strong enough adaptation is required to induce traveling waves. Additionally, adaptation modulates the properties of the spatio-temporal activity patterns, such as temporal and spatial frequencies, and the speed of the traveling waves, all of which increase with increasing strength. Though being dynamically equivalent, our results also explain why location-dependent variations in feedback strength cause differences in the propagation of traveling waves between both adaptation mechanisms.

Figures (11)

• Figure 1: Examples for static (top row) and dynamic (bottom row) Turing bifurcations in a Hopf- (top row) and Turing-unstable state (bottom row) for the system with h-currents. Panels from left to right show (cf. Eqs. \ref{['eq:cs']}): Coefficients $c_{1}(k)$ (solid line) and $c_2(k)$ (dashed line), coefficient $c_0(k)$ (solid line) and condition $c_1(k)c_2(k)-c_0(k)$ (dashed line), real ($\alpha_{0,1,2}(k)$, solid lines) and imaginary ($\omega_{0,1,2}(k)$, dashed lines) parts of the eigenvalues of the linearization matrix, and the corresponding activity pattern $u_e(x,t)$. Circles denote the wavenumbers $k_0$ (red) and $k_{max}$ (pink) (see text). Insets below each plot show close-ups of the thin boxes. Values for the external input currents and the adaptation parameter were $(I_e, I_i) = (-0.7, -0.6),\ b=-0.5$ (top row) and $(I_e, I_i) = (-0.3, -0.3),\ b=-0.5$ (bottom row). All other parameters are given in Table \ref{['tab:default-params']}.
• Figure 2: State spaces and activity patterns for the Wilson-Cowan model with h-currents. A Results of the stability analyses for a slice of parameter space spanned by the external input currents $(I_e, I_i)\in[-2.5,2]\times[-3.5,1]$ for increasing adaptation strength $|b|\in\{0, 0.5, 1.0\}$ (left to right). Colors denote the different dynamical regimes (see legend). Pentagon markers identify the positions in state space corresponding to the activity traces shown in B. Resolution of state space is 181 x 181. The black dot in the state space for $b=0$ denotes center of point symmetry (see Section \ref{['sec:equivalency-transformation']}). B Activity traces $u_e(x,t)$ for different locations in state space (see Table \ref{['tab:state-space-input-pairs']} in Appendix \ref{['appendix:state-space-activity-pairs']}), denoted by pentagons in A. Two types of bistability are exemplified in panels 3, 3(*) and 5, 5(*) (see text). Lighter green denotes low, darker green high activity values (see color bar). All simulations were initialized around their corresponding fixed points, except for the activity traces marked with (*) which were initialized close to zero (see Appendix \ref{['subsec:initialisation-methods']}).
• Figure 3: Comparison of state spaces and activity patterns for both adaptation mechanisms. A Slices of state space for SFA (left) and h-currents (right) for $|b|=0.5$ spanned by the external input currents $I_e$ and $I_i$. Colors denote the different dynamical regimes (see legend). Colored pentagon markers identify the positions in state space corresponding to the activity traces shown in B. The external input currents are $(I_e, I_i)=(0.4, -0.3)$ for SFA and $(I_e, I_i)=(-0.1, -0.3)$ for h-currents. B Top row: Activity traces $u_e(x,t)$ for the locations in state space, marked in A. Middle row: Corresponding traces of the adaptation variable $m$. Bottom row: Activity traces and traces of the corresponding adaptation variable for a fixed spatial location for the traveling waves shown above for SFA (left panel) and h-currents (middle panel). All traces are superimposed in the right panel.
• Figure 4: Eigenvalue spectrum for small values of $|b|$. The panels show the real, $\alpha_{0,1,2}(k)$, (solid lines) and imaginary parts, $\omega_{0,1,2}(k)$, (dashed lines) of the eigenvalues of the linearization matrix in Eq. \ref{['eq:spatial-adaps-jacobian']}. Top row: Hopf-unstable state at $(I_e, I_i) = (-0.35, -1.225)$. Center row: Turing-unstable down-state at $(I_e, I_i) = (-0.35, -0.55)$. Bottom row: same Turing-unstable down-state as in the center row but with weaker adaptation. Adaptation strength $|b|$ decreases from left to right. Lower panels show close-ups around the zero crossing of the real parts. The imaginary parts vanish for $b=0$. Arrows in panels below the first two plots of the upper row indicate zero crossing of a static (s) and dynamic (d) Turing bifurcation.
• Figure 5: Properties of the spatio-temporal patterns emerging in Hopf- and Turing-unstable states for increasing h-current strength. A$k_{max}$ values (top row), dominant spatial frequency $f_x$ (middle row), and dominant temporal frequency $f_t$ (bottom row) for each location in state space. The figure shows slices of state space spanned by the external input $(I_e, I_e)\in[-2.5,2]\times[-3.5,1]$ for $b\in\{0,-0.5, -1.0\}$. Purple colors denote the corresponding feature values. Darker colors indicate higher values, see color bar. Yellow denotes states of no spatio-temporal activity, light green denotes states with irregular patterns with a standard deviation $r$ of the Kuramoto-order parameter of $r>10^{-2}$. Thin lines denote the boundaries of the dynamical regimes shown in Fig. \ref{['fig:equivalency-results-with-activity']}A. Green boxes correspond to the panels shown in D. B Mean (dots) and variance (bars) of $k_{max},\ k_{0},\ f_t,$ and $f_x$ for $b\in\{-0.25, -0.5, -0.75, -1\}$ for the states located in the Hopf-unstable, Turing-unstable up-, and Turing-unstable down-state regimes, for regular spatio-temporal activity for which the standard deviation of the Kuramoto-order parameter over time and space was $r\leq 10^{-2}$ (see Appendix \ref{['subsec:regularity-methods']}). C Histograms and kernel density estimates (KDE; thin lines) of the numerically calculated spatial frequency ($f_x$; top row), the spatial frequency corresponding to $k_0$ ($\frac{L\cdot k_0}{2\pi}$; middle row), and the spatial frequency corresponding to $k_{max}$ ($\frac{L\cdot k_{max}}{2\pi}$; bottom row) in the regimes of Hopf instability (left column), Turing instability in down-states (middle column), and Turing instability in up-states (right column) for $b=-0.5$. Dashed vertical lines with numbers denote the location of the peak of the KDE. D Close-ups of state space (see green boxes in A) with values for $k_{max}$ (lower right), $f_x$ (upper right), and $f_t$ (upper left) in the Turing-unstable up- (left panels) and down-states (right panels) in bi. Yellow denotes states of no spatio-temporal activity or with irregular patterns with a standard deviation $r$ of the Kuramoto-order parameter of $r>10^{-2}$. The other colors denote feature values (see color bars). Wavenumbers were acquired as described in Section \ref{['sec:3d-stability-analysis-methods']}; spatial and temporal frequencies as described in the Appendices \ref{['subsec:initialisation-methods']} and \ref{['subsec:frequency-methods']}. Numerical simulations were conducted with initialization close to the corresponding unstable fixed points.