Traveling waves and breathers in an excitatory-inhibitory neural field

We study existence and stability of traveling activity bump solutions in an excitatory-inhibitory (E-I) neural field with Heaviside firing rate functions by deriving existence conditions for traveling bumps and an Evans function to analyze their spectral stability. Subsequently, we show that these existence and stability results reduce, in the limit of wave speed $c\to 0$, to the equivalent conditions developed for the stationary bump case. Using the results for the stationary bump case, we show that drift bifurcations of stationary bumps serve as a mechanism for generating traveling bump solutions in the E-I neural field as parameters are varied. Furthermore, we explore the interrelations between stationary and traveling types of bumps and breathers (time-periodic oscillatory bumps) by bridging together analytical and simulation results for stationary and traveling bumps and their bifurcations in a region of parameter space. Interestingly, we find evidence for a codimension-2 drift-Hopf bifurcation occurring as two parameters, inhibitory time constant $\tau$ and I-to-I synaptic connection strength ${\overline{w}}_{ii}$, are varied and show that the codimension-2 point serves as an organizing center for the dynamics of these four types of spatially localized solutions. Additionally, we describe a case involving subcritical bifurcations that lead to traveling waves and breathers as $\tau$ is varied.

Figure 1

Traveling bump wave profiles $V_e\left(\xi \right),V_i\left(\xi \right)$ in traveling wave coordinates for different time constants of inhibition $\tau$. In this parameter regime, as $\tau$ increases from (a) $\tau =1.2$ to (b) $\tau =1.5$ to (c) $\tau =2.0$, the wave speed increases from $c=0.77$ to $c=1.1$ to $c=1.3$, accordingly. Waves in this parameter regime are determined to be stable using the Evans function (Sec. 3) and numerical simulations. Other parameters are ${\overline{w}}_{ee}=1,{\overline{w}}_{ei}=0.7,{\overline{w}}_{ie}=0.55,{\overline{w}}_{ii}=0.24,{\sigma }_{ee}={\sigma }_{ei}={\sigma }_{ii}=1,{\sigma }_{ie}=1.3,{\theta }_e={\theta }_i=0.15,I_{\circ }=0.$

Figure 2

Graphs of the zero sets of the real (dark) and imaginary (light) parts of the Evans function $\mathcal{E}\left(\lambda \right)$ for $\text{Re}\lambda >-1/\tau$ in the complex plane where (a) ${\overline{w}}_{ii}=0.24$ and (b) ${\overline{w}}_{ii}=0.16$. Eigenvalues in the region identify as intersections of the zero sets of the real and imaginary parts of the Evans function. In addition to the persistent 0 eigenvalue, panels (a) and (b) show a pair of complex conjugate eigenvalues crossing the imaginary axis into the open right half plane as ${\overline{w}}_{ii}$ is decreased, destabilizing the traveling bump in a Hopf bifurcation at ${\overline{w}}_{ii}\approx 0.172$. Other parameters are as in Fig. 1.

Figure 3

Graph of the width of a propagating activity bump vs time $t$ (in units of the excitatory time constant) as it evolves towards a stable traveling bump in a simulation of EI neural field (1) for parameters in Fig. 2 with ${\overline{w}}_{ii}=0.185$ (only E population shown). At this point, a pair of complex eigenvalues with negative real part is positioned near the imaginary axis. The imaginary part is 1.221 which closely matches the oscillation frequency in the simulation (crest-to-crest time interval length is approximately 5.15). At the bifurcation point ${\overline{w}}_{ii}\approx 0.172$, the Hopf frequency is ${\omega }_{\mathcal{H}}=1.355$.

Figure 4

A curve of drift bifurcations of stationary bumps, denoted by ${\mathcal{D}}^{\ominus }$, and a curve of Hopf bifurcations of stationary bumps, denoted by ${\mathcal{H}}^{\oplus }$, in the $\tau {\overline{w}}_{ii}$ plane. Other (fixed) parameters are ${\overline{w}}_{ee}=1,{\overline{w}}_{ei}=0.84,{\overline{w}}_{ie}=0.8,{\sigma }_{ee}={\sigma }_{ei}={\sigma }_{ii}=1,{\sigma }_{ie}=1.3,{\theta }_e=0.16,{\theta }_i=0.24,I_{e\circ }=I_{i\circ }=0.$

Figure 5

Subbranches of curves of drift (${\mathcal{D}}_{\pm }^{\ominus }$) and Hopf (${\mathcal{H}}_{\pm }^{\oplus }$) bifurcations of stationary bumps. Subscripts ${}_{-}$ (${}_{+}$) denote where the opposite spatial eigenmode has eigenvalues with negative (positive) real part. Roman numerals denote regions: region I: $\text{Re}{\lambda }_{\pm }^{\oplus },{\lambda }_{-}^{\ominus }<0$; region IIa: ${\lambda }_{-}^{\ominus }>0$, $\text{Re}{\lambda }_{\pm }^{\oplus }<0$; region IIb: $\text{Re}{\lambda }_{\pm }^{\oplus }>0$, ${\lambda }_{-}^{\ominus }<0$; region III: $\text{Re}{\lambda }_{\pm }^{\oplus },{\lambda }_{-}^{\ominus }>0$.

Figure 6

Curves of Hopf bifurcations ${\mathcal{H}}_{1,2}^{\mathcal{T}\mathcal{W}}$ of traveling bumps indicated in blue. Other curves as described in Fig. 5.

Figure 7

Wave speed $c$ vs $\tau$ along curve ${\mathcal{H}}_1^{\mathcal{T}\mathcal{W}}$.

Figure 8

Space-time plots of $u_e(x,t)$ illustrating the four solution types: (a) stationary bump, (b) traveling bump, (c) stationary breather, (d) traveling breather; $[u_i(x,t)$ not shown]. Parameter values are those listed in Fig. 4 and (a) $(\tau ,{\overline{w}}_{ii})=(0.84,0.31)$, (b) $(\tau ,{\overline{w}}_{ii})=(0.96,0.28)$, (c) $(\tau ,{\overline{w}}_{ii})=(0.83,0.265)$, (d) $(\tau ,{\overline{w}}_{ii})=(0.96,0.243)$ which are points encircling point $\mathcal{X}$ in the $(\tau ,{\overline{w}}_{ii})$ plane in Fig. 9.

Figure 9

Bifurcation and simulation results: (a) Curves of drift (${\mathcal{D}}_{-}^{\ominus }$) and Hopf (${\mathcal{H}}_{-}^{\oplus }$) bifurcations of stationary bumps and the curves of Hopf bifurcations (${\mathcal{H}}_{1,2}^{\mathcal{T}\mathcal{W}}$) of traveling bumps in a subregion of the $\tau {\overline{w}}_{ii}$ plane (depicted in Fig. 6) that was studied extensively in numerical simulations of E-I neural field (1). (b) Different regions indicate the stable attractors that occur in numerical simulations evolving from the offset (allotopic) and centered (syntopic) activity bump initial conditions.

Figure 10

(a) Schematic bifurcation diagrams along lines ${\ell }_1$, ${\ell }_2$, ${\ell }_3$ as denoted in (b) the $\tau {\overline{w}}_{ii}$ plane in a subregion of Fig. 9. In the bifurcation diagrams, solid black lines denote stable solutions, gray dashed lines denote unstable solutions, and black dots denote bifurcation points. Initials denote different solution branches: $\mathcal{S}\mathcal{W}$, stationary bumps; $\mathcal{T}\mathcal{W}$, traveling bumps; $\mathcal{T}\mathcal{B}$, traveling breathers. The drift bifurcation ${\mathcal{D}}_{-}^{\ominus }$ from a $\mathcal{S}\mathcal{W}$ to a $\mathcal{T}\mathcal{W}$ is supercritical. Numerical simulations suggest that Hopf bifurcation ${\mathcal{H}}_2^{\mathcal{T}\mathcal{W}}$ from a $\mathcal{T}\mathcal{W}$ to a $\mathcal{T}\mathcal{B}$ is supercritical along line ${\ell }_1$ but subcritical along lines ${\ell }_2$ and ${\ell }_3$ with a stable-unstable pair of $\mathcal{T}\mathcal{B}$ instead annihilating in a (presumed) saddle-node bifurcation of traveling breathers (limit cycles) corresponding to the red curve.

Figure 11

A hypothetical codimension-2 bifurcation with a mode interaction occurring between the odd ${\lambda }_{\pm }^{\ominus }$ eigenmode (drift instability) and the even ${\lambda }_{\pm }^{\oplus }$ eigenmode (Hopf instability) consistent with results in Fig. 9. Curves are denoted as follows: ${\mathcal{H}}_{-}^{\oplus }$, curve of Hopf bifurcations of stationary bumps; ${\mathcal{D}}_{-}^{\ominus }$, curve of drift bifurcations of stationary bumps; ${\mathcal{H}}^{\mathcal{T}\mathcal{W}}$, curve of Hopf bifurcations of traveling bumps; and ${\mathcal{D}}^{\mathcal{S}\mathcal{B}}$, curve of drift bifurcations of stationary breathers. ${\mathcal{H}}_{-}^{\oplus }$, ${\mathcal{D}}_{-}^{\ominus }$, and ${\mathcal{H}}^{\mathcal{T}\mathcal{W}}$ are calculated from analytical results. Curve ${\mathcal{D}}^{\mathcal{S}\mathcal{B}}$ is a hypothetical curve of drift bifurcations of stationary breathers consistent with the transition from stationary to traveling breathers (from speed 0) in simulations. Each bifurcation is supercritical with each region denoting the stable attractor.

Figure 12

Bumps and breathers in an different bifurcation scenario involving subcritical bifurcations in a alternate parameter region described in Sec. 5f. (a) Curves of Hopf bifurcations ${\mathcal{H}}_{-}^{\oplus }$ and drift bifurcations ${\mathcal{D}}_{+}^{\ominus }$ of stationary bumps in the $(\tau ,{\overline{w}}_{ii})$ plane, with other parameters as in Figs. 1 and 2. Stationary bumps are stable (unstable) to the left (right) of curve ${\mathcal{H}}_{-}^{\oplus }$. (b) Bifurcation diagram for stationary bumps ($\mathcal{S}\mathcal{W}$), traveling bumps ($\mathcal{T}\mathcal{W}$), and traveling breathers ($\mathcal{T}\mathcal{B}$) where the vertical axis measures the traveling wave speed $c$ vs the bifurcation parameter $\tau$ for fixed $w_{ii}=0.24$ [average speed $\tilde{c}$ of the bump center is used in the case of a traveling breather ($\mathcal{T}\mathcal{B}$)]. Solid (dashed) curves denote the stable (unstable) solutions.

Figure 13

The oscillation amplitude $\delta {\xi }_e$ of the time-varying width of the traveling breather $\mathcal{T}\mathcal{B}$ (red curve) in Fig. 12 plotted as $\tau$ varies from 1.065 to $\tau \approx 1.06805$ (large dot) which coincides with the Hopf bifurcation point ${\mathcal{H}}^{\mathcal{T}\mathcal{W}}$ occurring on the branch of traveling bumps $\mathcal{T}\mathcal{W}$. This is consistent with a supercritical bifurcation giving rise to a branch of stable traveling breathers emerging at the bifurcation point.

Figure 14

Two hypothetical (schematic) bifurcation diagrams that may connect the structure of the stable attractors found the in analytical calculations and simulations in Fig. 12. Different solution types are denoted by colors and symbols: $\mathcal{S}\mathcal{W}$, stationary bump (wave); $\mathcal{T}\mathcal{W}$, traveling bump (wave); $\mathcal{S}\mathcal{B}$, stationary breather; and $\mathcal{T}\mathcal{B}$, traveling breather. The solid (dashed) curves indicate the stable (unstable) solutions. The vertical axis abstractly represents the amplitude of the bifurcating solutions from the stationary bump (amplitude 0) and, in the traveling breather case, represents a mix of wave speed and oscillation amplitude. Symbols $\mathcal{S}{\mathcal{N}}^{\mathcal{X}}$, ${\mathcal{H}}^{\mathcal{X}}$, ${\mathcal{D}}^{\mathcal{X}}$ denote saddle-node, Hopf, and drift bifurcation, respectively, occurring on a solution (or solution pair) of type $\mathcal{X}$. Points ${\mathcal{H}}_{-}^{\oplus }$ and ${\mathcal{D}}_{+}^{\ominus }$ along the branch of stationary bumps ($\mathcal{S}\mathcal{W}$) denote the points of Hopf and drift bifurcation as $\tau$ is varied. In simulations of stable traveling breathers (Figs. 12 and 13), the average wave speed $\tilde{c}$ increases as $\tau$ increases from $\mathcal{S}{\mathcal{N}}^{\mathcal{T}\mathcal{B}}$ to ${\mathcal{H}}^{\mathcal{T}\mathcal{W}}$ whereas the oscillation amplitude $\delta {\xi }_j$ (breathing) increases from amplitude 0 as $\tau$ decreases from ${\mathcal{H}}^{\mathcal{T}\mathcal{W}}$ to $\mathcal{S}{\mathcal{N}}^{\mathcal{T}\mathcal{B}}$. The hypothetical component is the connection from the curve of stable traveling breathers ($\mathcal{T}\mathcal{B}$) to the Hopf bifurcation ${\mathcal{H}}_{-}^{\oplus }$.