Synaptic propagation in neuronal networks with finite-support space-dependent coupling

Traveling waves of electrical activity are ubiquitous in biological neuronal networks. Traveling waves in the brain are associated with sensory processing, phase coding, and sleep. The neuron and network parameters that determine traveling waves' evolution are the synaptic space constant, synaptic conductance, membrane time constant, and synaptic decay time constant. We used an abstract neuron model in a one-dimensional network to investigate the propagation characteristics of traveling wave activity. We formulate a set of evolution equations based on the network connectivity parameters. Using a combination of numerical and analytical approaches, we show that these traveling waves are stable to a series of perturbations with biological relevance.

Figure 1

Comparison of analytical solutions and numerical simulations. The finite support neuronal network of integrate-and-fire neurons with excitatory coupling is at rest; at $t=0$, the neurons at $x=0$–1 receive an additional current that drives them over the threshold. All shocked neurons spike synchronously because they receive their input at the same time. The wave evolution shows damped oscillations with an amplitude that decays exponentially. The self-propagating wave settles at the constant speed $c_{\mathrm{fast}}$. (a) Wave speed as a function of space. The red trace is the computer simulation and the black trace is the speed solution from Eq. (4). (b) Traveling wave acceleration as a function of space. The blue trace is the computer simulation and the green trace is the solution from Eq. (6). The parameters are $g_{\mathrm{syn}}=15, \sigma =1, {\tau }_1=1, {\tau }_2=2, V_T=1$, and $\delta =1\times {10}^{-3}$.

Figure 2

Stable and unstable fixed points of the wave propagation system. (a) Solutions to the consistency equation as intersections between $V=V\left(c\right)$ and the horizontal line at $V_T$. According to the consistency equation, the neuron membrane potential is a function of traveling wave speed. These numerical solutions allow one to compute both $c_{\mathrm{fast}}$ and $c_{\mathrm{slow}}$. Here $V_{\max }$ denotes the maximum achievable voltage as a function of the traveling wave speed. If this value is less than $V_T$ no traveling wave can exist. (b) Results from numerical simulations where the wave is initiated through shocking the region $(-\sigma ,0)$. The attractor graph of acceleration as a function of speed indicates that the traveling wave settles into a constant speed solution. The magnitude of the oscillations is consistent with the synaptic footprint space constant $\sigma$. The speed $c$ and acceleration $a$ of the traveling wave oscillate while approaching the intersection between $c=c_{\mathrm{fast}}$ and $a=0$. The parameters are $g_{\mathrm{syn}}=15, \sigma =1, {\tau }_1=1, {\tau }_2=2, V_T=1$, and $\delta =1\times {10}^{-3}$.

Figure 3

Traveling wave speed as a function of $g_{\mathrm{syn}}$. This graph represents $c_{\mathrm{fast}}$ and $c_{\mathrm{slow}}$ as a function of the global excitability parameter, in logarithmic scale. For this set of simulations, we systematically varied $g_{\mathrm{syn}}$ and kept all other parameters constant. Our results show there is a critical value $g_{\mathrm{crit}}$ where global excitation is so small that traveling wave activities are not sustainable. These curves also illustrate the dependence of the traveling wave dynamics on global excitability: As $g_{\mathrm{syn}}$ grows, $c_{\mathrm{fast}}$ increases while $c_{\mathrm{slow}}$ decreases, both asymptotically. The parameters are $\delta =1\times {10}^{-3}\sigma =1, {\tau }_1=1, {\tau }_2=2$, and $V_T=1$.

Figure 4

Traveling wave speed as a function of $V_T$. For this set of simulations, we systematically varied $V_T$ and kept all other parameters constant. Our results show that as $V_T$ increases, the wave speed decreases. The critical value $V_{\max \mathrm{critical}}$ is the critical value between activity propagation and failure, determined by the consistency equation (7) (see Fig. 2). Intuitively, when $V_{\max \mathrm{critical}}$ is found the voltage threshold is greater than the voltage as a function of speed $[V_T\ge V\left(c\right)]$. The parameters are $\delta =1\times {10}^{-3}\sigma =1, g_{\mathrm{syn}}=15, {\tau }_1=1, {\tau }_2=2, V_T=[1,6.125]$, and $V_{\max \mathrm{critical}}=6.125$.

Figure 5

Faster wave speed perturbation. This graph shows the stability test of the wave speed solution $c_{\mathrm{fast}}$. At $x=0$, there is no external drive and the wave is free to evolve due to the dynamics defined by network and neuron parameters. Removal of the external drive creates a delay between the spiking time of the last neuron (with external drive) and the first neuron (without external drive). We obtain the following series of estimates: delay from numerical simulation, $8.6253\times {10}^{-4}$; delay from Eq. (15), $8.9347\times {10}^{-4}$; initial speed after the break from simulation, 7.0253; initial speed from Eq. (16), 7.0225; initial acceleration from simulation, $-0.2914$; and initial acceleration from Eq. (17), $-0.3095$. $V_1=-0.928915$. (a) Traveling wave speed as a function of space. The red trace is the computer simulation and the black trace is the result from Eq. (16). (b) Traveling wave acceleration as a function of space. The blue trace is the computer simulation and the green trace is the solution from Eq. (17). Note that acceleration is initially negative (the wave slows down), but then becomes positive because of the oscillations. The parameters are $g_{\mathrm{syn}}=15, \sigma =1, {\tau }_1=1, {\tau }_2=2, V_T=1, \epsilon =0.012$, and $\delta =1\times {10}^{-3}$.

Figure 6

Accuracy of estimated delays as a function of $\epsilon$. For these graphs, we performed multiple simulations with varying magnitudes for $\epsilon$ and saved the delay, speed, and acceleration after the break. Here we summarize the numerical and theoretical values of relevant wave characteristics after removing perturbation: numerical and theoretical $t_0, c_0^{+}$, and $a_0^{+}$. These numerical and theoretical delays, speeds, and accelerations after perturbation removal show consistent trends dependent on the control parameter $\epsilon$. The range of relative error of $t_0$ [Eq. (15)] compared to the numerical delay is $[6.2\times {10}^{-6},0.0011], c_0$ [Eq. (16)] relative to the initial speed from simulations is $[2.5\times {10}^{-4},0.065]$, and $a_0$ [Eq. (17)] relative to the numerical acceleration is $[7.2\times {10}^{-4},0.35]$. For small values of $\epsilon$, our approximations from Eqs. (15, 16, 17) are accurate, but as $\epsilon$ increases, the system loses accuracy. (a) Delay computed from Eq. (15) and the delay from numerical simulations. (b) Speed computed from Eq. (16) and the speed from numerical simulations. (c) Acceleration computed from Eq. (17) and the acceleration from numerical simulations.

Figure 7

Wave speed oscillation as a function of both the synaptic coupling and the wave's state. In both panels the wave travels from $x=0$ to $x=12$. The blue line at the bottom of the graph represents how $g_{\mathrm{syn}}\left(x\right)$ oscillates as a function of space and the sinusoidal perturbation; $g_{\mathrm{syn}}$ is constant everywhere else. If $x>6$ and $x<7, g_{\mathrm{syn}}\left(x\right)=g[1+\epsilon sin\left(\frac{2\pi x}{\sigma }\right)]$; otherwise $g_{\mathrm{syn}}\left(x\right)=g$. The traveling wave speed fluctuates dramatically while the synaptic inhomogeneity perturbation is present. Because of the effect of the delayed parameters of the finite support kernel, this perturbation produces a peak of traveling wave speed after the perturbation is removed. Interestingly, the magnitude of the second peak is greater than the first peak. (a) The speed of the wave at $x=7$ affects the speed of the wave at $x=8$, although the perturbation is totally removed at $x>7$. The wave speed peaks twice: The increase in local excitability $g_{\mathrm{syn}}\left(x\right)$ causes the first peak. The delayed effect of wave speed, i.e., $t(x-\sigma )$ [Eq. (3) and $c(x-\sigma )$ [Eq. (4)], causes the second peak. This phenomenon exemplifies some of the complicated dynamics of the finite support connectivity kernel compared to the exponential. (b) The traveling wave speed at any given $x$ affects the instantaneous acceleration of the wave of subsequent $x+\sigma$. The instant acceleration oscillates transiently in a sinusoidal fashion while $g_{\mathrm{syn}}$ perturbation is present. The parameters are $g=15, \sigma =1, {\tau }_1=1, {\tau }_2=2, V_T=1, \delta =1\times {10}^{-3}$, and $\epsilon =0.0943$.

Figure 8

Neuronal traveling wave phenomena are all-or-none events: One small change in the control parameter separates activity propagation from propagation failure. In both panels the wave travels from $x=0$ to $x=12$. The blue line at the bottom of the graph represents how $g_{\mathrm{syn}}\left(x\right)$ oscillates as a function of space and the demyelination perturbation. At the perturbation, $g_{\mathrm{syn}}$ decreases sharply; the $g_{\mathrm{syn}}$ decay is determined as follows: If $x>6$ and $x<7, g_{\mathrm{syn}}\left(x\right)=g[1+10\epsilon sin\left(\frac{\pi x}{2\sigma }\right)]$; otherwise $g_{\mathrm{syn}}\left(x\right)=g$ The cyan and purple wave speeds demonstrate how a small increase in the parameter $\epsilon$ ($\pm 1\times {10}^{-6}$) determines whether the perturbation terminates the wave or the wave continues to propagate after removing the perturbation. The amplitude of the perturbation controls the breaking point of the traveling wave. Here two simulations with the same initial conditions and parameters but with one difference in $\epsilon$ ($\pm 1\times {10}^{-6}$) differ qualitatively. The cyan wave continues to propagate, at wave speed $c_{\mathrm{fast}}$; the purple wave fails to propagate. The parameters are $g=15, \sigma =1, {\tau }_1=1, {\tau }_2=2, V_T=1, {\epsilon }_1=-0.094289, {\epsilon }_2=-0.094288$, and $\delta =1\times {10}^{-3}$.

Figure 9

Dynamics of the traveling wave induced by a nonconducting gap. The simulation consists of a wave traveling at a constant speed $c_{\mathrm{fast}}$. Located at $x=6$ there is the nonconducting gap of “dead neurons,” which do not spike or synapse. The gap is relative to the synaptic space constant $\sigma$ and determined by the ratio $\alpha \text{:}\sigma$. Waves typically recover from smaller nonconducting gaps; the wave speed oscillates transiently but eventually settles back to a constant speed value, namely, $c_{\mathrm{fast}}$ ($\alpha <{\alpha }_{\mathrm{critical}}$, green curves). However, larger perturbations break traveling wave propagation ($\alpha \ge {\alpha }_{\mathrm{critical}}$, magenta curves). The traveling wave's speed is slower than $c_{\mathrm{fast}}$ right where synaptic coupling returns to the baseline; the speed after the break decreases monotonically as $\alpha$ grows and approaches ${\alpha }_{\mathrm{critical}}$. To compute ${\alpha }_{\mathrm{critical}}$, we uniformly sampled 100 points between 0 and 1 and then iterated between 0.8 and 0.86 to find ${\alpha }_{\mathrm{critical}}$ with four-point decimal accuracy. (a) The wave travels at a constant speed arriving at $x=6$ (black trace). The green and magenta traces show the speed of the wave after the nonconducting gap results from multiple simulations; green curves represent $\alpha <{\alpha }_{\mathrm{critical}}$ and magenta represent $\alpha >{\alpha }_{\mathrm{critical}}$. As the nonconducting gap becomes larger, the wave speed after the break decreases. Larger gaps induce propagation failure. (b) Speed after the gap for the simulations with $\alpha$ near ${\alpha }_{\mathrm{critical}}$. The green and magenta traces show a critical qualitative change in traveling wave evolution as a result of a small parameter change. (c) The speed after the gap is a monotonically decreasing function of the gap size.

Figure 10

Smallest gap lengths that induce propagation failure, $\sigma \alpha$. Networks with strong excitatory coupling allow for robust wave propagation. The data above show the critical value of $\alpha$ as a function of network global excitability. The trend demonstrates that with greater excitability, the network's ability to sustain propagating traveling waves is more resilient to nonconducting gaps (areas of dead tissue). Intuitively, this indicates that when overall network excitation is larger, more drastic reduction in the nonconducting gap is needed to produce propagation failure. Curve fit analysis of ${\alpha }_{\mathrm{critical}}$ as a function of $g_{\mathrm{syn}}$ demonstrated the power function was a good fit for the curve. Here $f\left(x\right)=a{x}^b+c$. The coefficients (with $95\%$ confidence bounds) are $a=-2.336 (-2.357,-2.315), b=-0.982 (-0.9904,-0.9736)$, and $c=1.005 (1.003,1.007)$ [sum of squares error (SSE) equal to 0.00284, $R^2=1$, adjusted $R^2=1$, root mean square error (RMSE) equal to 0.002918, and $p<0.1]$. Interestingly, the exponent is so close to $-1$ that other curves such as $f\left(x\right)=\frac{a}{x}+c$ are also a relative good fit (SSE equal to $0.00318, R^2=1$, adjusted $R^2=1$, RMSE equal to 0.00317, and $p<0.01$).