Analytically tractable studies of traveling waves of activity in integrate-and-fire neural networks

In contrast to other large-scale network models for propagation of electrical activity in neural tissue that have no analytical solutions for their dynamics, we show that for a specific class of integrate and fire neural networks the acceleration depends quadratically on the instantaneous speed of the activity propagation. We use this property to analytically compute the network spike dynamics and to highlight the emergence of a natural time scale for the evolution of the traveling waves. These results allow us to examine other applications of this model such as the effect that a nonconductive gap of tissue has on further activity propagation. Furthermore we show that activity propagation also depends on local conditions for other more general connectivity functions, by converting the evolution equations for network dynamics into a low-dimensional system of ordinary differential equations. This approach greatly enhances our intuition into the mechanisms of the traveling waves evolution and significantly reduces the simulation time for this class of models.

Figure 1

(a) Theoretical results and numerical simulations for the dependence of wave acceleration on instantaneous speed, $a=a\left(c\right)$. The acceleration's quadratic dependence on speed (curve with merged blue, green, and red regions) is in perfect agreement with numerical simulations (dotted black line) on all areas except for the low speed regime where the agreement again becomes excellent with finer discretization of the spatial domain. The parameters used here are ${\tau }_1=4\text{ms}, {\tau }_2=30\text{ms}, \sigma =0.288\text{mm}, V_T=15\text{mV}, g_{\mathrm{syn}}=98.4\text{mV}$, yielding $c_1=0.0046\mathrm{m}/\mathrm{s}$, and $c_2=0.15\mathrm{m}/\mathrm{s}$. These parameters are in agreement with published data, and they are used as default values unless noted otherwise. (b) Theoretical results for acceleration vs speed for different excitability levels $g_{\mathrm{syn}}$. Depending on the overall excitability level, there are no traveling wave solutions (red line), one solution (green line), or two solutions (blue line). As excitability increases, $c_1$ and $c_2$ decrease or increase, respectively, provided that the overall excitability exceeds the critical value $g_{\mathrm{critical}}=0.0559V$ [see Eq. (13)].

Figure 2

Dependence of traveling wave solutions $c_1$ (red) and $c_2$ (blue) on neuron and network parameters. (a) Speed vs synaptic excitability $g_{\mathrm{syn}}$, with bifurcation occurring at $g_{\mathrm{critical}}=0.0559\mathrm{mV}$. (b) Speed vs $\sigma$, revealing linear correlations between propagating velocity and parameter $\sigma$. (c) Speed vs ${\tau }_1$, showing a decrease of $c_2$ as the neuron integration time, ${\tau }_1$, increases. (d) Speed vs ${\tau }_2$, indicating that fast solutions increase with the growth of decay time, ${\tau }_2$, in contrast to the slow solutions showing the opposite trend.

Figure 3

(a) Space vs firing times. Different initial conditions will determine if the transients evolve toward stable or transient propagation. When the speed (tangent) is less than $c_1$, propagation fails as expected (red line, failure when tangent becomes vertical). When the tangent is greater than $c_2$ (blue line), the traveling wave slows down and evolves asymptotically toward the constant speed traveling wave indicated by slope $c_2$. When the tangent is greater than $c_1$ but less than $c_2$ (green line), the traveling wave speeds up and evolves asymptotically toward a fast-stable solution at $c_2$. Results from numerical simulations, not shown here, are in perfect agreement with color lines shown in this graph. (b) Speed vs space. In agreement with results from part (a), propagation failure is achieved in a finite amount of space, while stability at $c_2$ is reached asymptotically.

Figure 4

Dependence of natural time scale ${\tau }_0$ on other network parameters. (a) ${\tau }_0$ vs $g_{\mathrm{syn}}$. As the excitability of the network ($g_{\mathrm{syn}}$) increases, it takes less time to achieve stability (at $c=0$ or at $c=c_2$). Here ${\tau }_0=0.12/g_{\mathrm{syn}}$, where $g_{\mathrm{syn}}>g_{\mathrm{critical}}=55.9\text{mV}$. (b) ${\tau }_0$ vs ${\tau }_1$. As the integration time ${\tau }_1$ increases it takes more time to reach the stable states. When ${\tau }_1$ becomes really small we obtain ${\tau }_0=0.4386{\tau }_1$. (c) ${\tau }_0$ vs ${\tau }_2$. Finally, when synaptic excitation lasts longer (at higher values of ${\tau }_2$), stable states are also reached faster. When ${\tau }_2$ becomes really large we obtain ${\tau }_0=1.7544$.

Figure 5

(a) Spatial scales for achieving stable states. We examine the distance needed to achieve stability depends on the initial speed $c_0$. For propagation failure, this is defined as the distance traveled until speed becomes 0 (red line). For the asymptotically stable states, this is defined as the distance needed to reach $c=\alpha c_2$, where $\alpha =0.99$ if $c_0<c_2$ (green line), and $\alpha =1.01$ if $c_0>c_2$ (blue line). (b) Temporal scales for achieving stable state. Similar graphs are shown for the time needed to achieve stable states. In the limit where $c_0\to \infty$ (black dotted line) stability is achieved in a finite amount of time ($t=9.1\text{ms}$).

Figure 6

Dependence of traveling-wave acceleration on parameters. (a) Acceleration vs $g_{\mathrm{syn}}$. The maximum value for acceleration (blue line) increases with excitability while surprisingly the minimum value (red line, achieved at $c=0$ when propagation fails) does not depend on $g_{\mathrm{syn}}$. (b) Acceleration vs synaptic footprint $\sigma$. Both maximum and minimum acceleration are linear in $\sigma$. (c) Acceleration vs integration time ${\tau }_1$. As the neural integration time becomes very large, both maximum and minimum acceleration decay toward zero values. (d) Acceleration vs decay of excitability parameter ${\tau }_2$. In contrast, only the minimum acceleration decays to zero as ${\tau }_2$ becomes large, as maximum acceleration saturates to a nonzero fixed value.

Figure 7

Activity propagation changes induced by a nonexcitable region of length $L$. (a) Speed after gap vs length of gap. Not surprisingly, larger nonexcitable regions decrease the speed of the traveling waves past the gap, and they could even lead to propagation failure at $c=0$. We plot this relationship for four initial conditions: $c=0.0386, c=0.0773, c=0.15$, and $c=0.3$. (b) Speed after gap vs speed before gap. Slow traveling waves are more affected by a constant length gap and may even fail, while fast ones only show a moderate loss in speed as they propagate further away. The red dotted line is the $y=x$, corresponding to zero speed loss, and it is included here for comparison with the other contour lines. Obviously, with the increase of gap length, the change rate of speed before and after gap increases. Length of the gaps considered here range from $L=0.05$ to $0.35$ mm, with eight uniformly spaced values considered here. (c) Minimum length of gap that causes propagation failure as a function of the speed before gap. We determine that propagation eventually fails when speed becomes less than $c_1$. Taking into account the speed before gap, the graph determines the minimum length of gap needed to reach propagation failure.

Figure 8

Voltage value at the edge of a forced constant speed traveling wave in a network with a polynomial time exponential coupling function. A more general coupling function (28) is applied to the system with parameters $a=1,b=1,{\tau }_1=4\times {10}^{-3}\text{s},{\tau }_2=3.0\times {10}^{-2}\text{s},\sigma =2.88\times {10}^{-4}\text{m},g_{\mathrm{syn}}=9.85\times {10}^{-2}\text{V},V_T=1.5\times {10}^{-2}\text{V}$. Two speed solutions exist when voltage reaches threshold ($V_T=1.5\times {10}^{-2}\text{V}$).

Figure 9

Successive derivatives of firing times. Numerical simulations (blue) are in excellent agreement with the the dynamics of ODE system, as illustrated by the first four derivatives of firing map [Eqs. (2, 3, 4), (31)–(32), red lines]: (a) $t^{\prime }$, (b) $t^{\prime \prime }$, (c) $t^{\prime \prime \prime }$, and (d) $t^{\prime \prime \prime \prime }$. Similar to the previous cases considered, an initial region is induced to spike to the left of $x>0$ region in order to provide the initial activity propagation. This is used to extract the initial conditions for the ODE system, such as the initial speed, $1/t^{\prime }\left(0\right)$ as well as the next two derivatives, $t^{\prime \prime }\left(0\right)$ and $t^{\prime \prime \prime }\left(0\right)$.