Dynamics of neural fields with exponential temporal kernel

TL;DR

A dynamic bifurcation analysis of the standard neural field equation with an exponential temporal kernel gives explicit bifurcation conditions and shows that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations.

Abstract

We consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatiotemporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations. However, the exponential temporal kernel possesses the important property that it takes into account the finite memory of past activities of neurons, which Green's function does not. Through a dynamic bifurcation analysis, we give explicit bifurcation conditions. Hopf bifurcations lead to temporally non-constant, but spatially constant solutions, but Turing-Hopf bifurcations generate spatially and temporally non-constant solutions, in particular, traveling waves. Bifurcation parameters are the coefficient of the exponential temporal kernel, the transmission speed of neural signals, the time delay rate of synapses, and the ratio of excitatory to inhibitory synaptic weights.

Figures (8)

• Figure 1: The solid curves represent the quantity $D$ from Theorem \ref{['theo1']} plotted against the bifurcation parameters: $\alpha$ in (a) with $\tau=0.7$, $r=0.5$; $r$ in (b) with $\alpha=2.0$, $\tau=0.7$; and $\tau$ in (c) with $\alpha=2.0$, $r=0.5$. The intervals of $\alpha\in(0.0,1.4)$, $r\in(0.60,1.84)$, and $\tau\in(0.0,0.51)$ in which the solid curves are below the dashed horizontal line fulfills the sufficient condition of asymptotic stability of the equilibrium solution $v_0 = \tau E$, following Theorem \ref{['theo1']}. Other parameters are fixed at $c=15.0$, $E=0.275$, $a_{e} = 10.0$, $a_{i} = a_{e}/r$. The nonlinear dependence on $\tau$, as opposed to the linear dependence on $\alpha$, arises because in $\beta=\alpha c \tau F'(v_0)$, we get an additional dependence since $v_0=\tau E$.
• Figure 2: Panels (a) and (b) show the interval of $\alpha\in(0.0,8.0)$ and $\tau\in(0.0,2.0)$ for which \ref{['eq3.4b7']} is satisfied. In the panels (c)-(e), the blue curves represent the solutions of \ref{['eq3.4b8']} in the parameter spaces $\alpha$-$\nu$, $\tau$-$\nu$, and $r$-$\nu$, respectively. The gray areas in the panels represent the region of the parameter spaces where \ref{['eq3.4b7']} holds, i.e., the oscillatory region. The parts of blue curves from \ref{['eq3.4b8']} that lie in the gray region represent the values of the parameters for which oscillations exist, as these values satisfy \ref{['eq3.4b7']}. In (a)$\tau=0.75$, $r=5.0$; in (b)$\alpha=6.0$, $r=5.0$; in (c)$\tau=0.75$, $r=5.0$; in (d)$\alpha=6.0$, $r=5.0$; and in (e)$\alpha=6.0$, $\tau=0.75$. In (a)-(e), the other parameter values are: $c=15.0$, $E=0.275$, $a_{e}=10.0$, $v_0=\tau E$, $\beta=\alpha c\tau F'(v_0)$, and in (e)$a_{i}=a_e/r$.
• Figure 3: Panels (a)--(c) show color coded space-time patterns of the membrane potential $v(x,t)$ emerging from Hopf instability. In (a), $\alpha=7.0$, $\nu=0.16$, i.e., below the Hopf bifurcation (blue) curve. In (b), $\alpha=7.0$, $\nu=1.83$, i.e., on the Hopf bifurcation curve. In (c), $\alpha=7.0$, $\nu=2.5$, i.e., above the Hopf bifurcation curve. In all cases, we have periodic oscillations of spatially constant solutions. The solutions are obtained for the Gaussian connectivity kernel in the panels, with initial conditions chosen randomly from a uniform distribution on $[v_0-0.1, v_0+0.1]$. Other parameters are fixed at $\tau=0.75$, $a_e=10.0$, $a_i=2.0$, $r=a_e/a_i=5.0$, $c=15.0$, and $k=0$.
• Figure 4: A dispersion relation of the neural field satisfying the Turing-Hopf bifurcation equation given by \ref{['Turing1']} for a fixed set of parameters values: $\alpha=5.0$, $\nu=1.0$, $\tau=0.75$, $r=5.0$, $a_{e}=10.0$, $a_i=a_e/r=2.0$.
• Figure 5: The curves in panels (a)-(d) represent the Turing-Hopf bifurcation curves in \ref{['Turing1']} in the ($\alpha$-$\omega$), ($\nu$-$\omega$), ($\tau$-$\omega$), and ($r$-$\omega$) planes, respectively. Parameter values are: (a)$k=25.0$, $\nu=1.0$, $\tau=0.75$, $r=5.0$; (b)$k=1.0$, $\alpha=5.0$, $\tau=0.75$, $r=5.0$; (c)$k=1.0$, $\alpha=5.0$, $\nu=1.0$, $r=5.0$; (d)$k=1.0$, $\alpha=5.0$, $\nu=1.0$, $\tau=0.75$. The remaining parameters are fixed at $a_{e}=10.0$, $a_i=a_e/r$.

Theorems & Definitions (6)

• theorem 1
• lemma 1
• theorem 2
• proof
• theorem 3
• proof